arj september 2012 vol 103 no 3 · 2016-08-25 · 108 south african institute of electrical...

Vol.103(3) September 2012 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 105

VOL 103 No 3 September 2012

SAIEE Africa Research Journal

SAIEE AFRICA RESEARCH JOURNAL EDITORIAL STAFF ...................... IFC

A Simulation and Graph Theoretical Analysis of Certain Properties of Spectral Null Codebooks

by K. Ouahada and H. C. Ferreira .............................................................106

Error Performance of Concatenated Super-Orthogonal Space-Time-Frequency Trellis Coded MIMO-OFDM System

by I. B. Oluwafemi and S. H. Mneney ......................................................... 116

Fault Diagnosis of Generation IV Nuclear HTGR Components using the Enthalpy-Entropy Graph Approach

by C.P. du Rand and G. van Schoor ............................................................127

Simulation Study of the Performance of the Viterbi Decoding Algorithm for Certain M-Level Line Codes

by Khmaies Ouahada ..................................................................................134

Vol.103(3) September 2012SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS106

A SIMULATION AND GRAPH THEORETICAL ANALYSIS OFCERTAIN PROPERTIES OF SPECTRAL NULL CODEBOOKS

K. Ouahada and H. C. Ferreira ∗

∗ Department of Electrical and Electronic Engineering Science, University of Johannesburg, SouthAfrica E-mail: {kouahada, hcferreira}@uj.ac.za

Abstract: The spectral shaping technique and the design of codes providing nulls at rationalsub-multiples of the symbol frequency, as the case with spectral null (SN) codes, have enhanced digitalsignaling over communication channels as digital mass recorders and metallic cables. The study of thespecial structure of these codes helps in investigating and analyzing certain of their properties whichhave been proved and emphasized from a mathematical perspective using graph theory. The cardinalityof spectral null codebooks reflects the rate of spectral null codes and therefore the amount of transmittedinformation data. The rate of these codes can also play a role in their error correction capability. Thepaper presents in different ways the special structure of spectral null codebooks and analyze better theirproperties. A possible link between these codes and other error correcting codes as the case of LowDensity Parity Check (LDPC) is presented and discussed in this paper.

Key words: Spectral shaping, spectral null codes, error correcting codes.

1. INTRODUCTION

The design of a code having power spectral density(PSD) zero at its DC-component, called DC-freecodes [1, 2], becomes a necessity for AC coupling ofthe signal to the medium. DC-balanced codes havefound widespread applications in digital transmissionand recording systems [3]– [5]. DC balance can beachieved by using an appropriate transmission code orby balancing each transmitted symbol. Any drift in thetransmitted signal from the center baseline level, due toan uncontrolled running digital sum (RDS) or the effectsof an AC coupling, will create a DC component, which isknown as baseline wander [6], or create an intersymbolinterference, which is caused by the AC coupling atvarious points in the communication channel [4]. Insome applications low-frequency channel noise, such asa fingerprint on an optical disk [7] or impulse noise dueto dial pulses in a subscriber loop plant, can be filteredout by sending the encoded data through a high pass filter.To minimize the effect of this filtering on the symbolshape of the coded sequence, the encoded data stream musthave very little or no DC or low-frequency component.Also magnetic recording systems often require that thechannel sequences have a spectral null at zero frequency.This technique is called the spectral shaping techniqueor the design of nulls at certain specific frequencies in aspectrum.

Spectral null codes are codes with simultaneous nulls atthe rational submultiples of the symbol frequency andhave great importance in certain applications like in thecase of transmission systems employing pilot tones forsynchronization and that of track-following servos indigital recording [8, 9].

The paper is organized as follows. In Section 2 wepresent two different design techniques of spectral null

codebooks. Section 3 emphasizes better the relationshipin the calculation of the cardinality of the codebook and itscorresponding spectral null equation. Section 4 derives andpresents proofs of certain properties of spectral null codes.A link and approach between spectral null codebooks andLDPC codes is presented in Section 5. We conclude withan analysis of these properties in Section 6.

2. SPECTRAL NULL CODES DESIGN

In this section we present two different techniques fordesigning spectral null codes based on the calculationof the power spectral density function and the binaryrepresentation of permutation sequences.

2.1 Using Gorog Construction

Gorog [10] was first to simplify and formulate the wayof calculating the values of the frequencies for spectralnull codes. To calculate the value of the frequencies atthe corresponding nulls at the rational submultiples of thesymbol frequency fc for block codes, he considered thevector y = (y1, . . . ,yM), yi ∈ {−1,+1}, to be an element ofa set S, which is called the codebook of codewords withelements in {−1,+1}. For the sake of simplification andgood presentation, we represent−1 with a 0. Applying theFourier transform to those codewords we get [10]:

Y =M

∑i=1

yie− jiw,−π≤ w≤ π. (1)

The power spectral density function denoted by H(w) ofthe concatenated sequence when transmitted serially [11]is defined as:

H(w) =1

CSM

M−1

∑i=0

∣∣Y i (w)∣∣2

, (2)


where Y i (w) is the Fourier transform of the i-th elementof S and CS is the cardinality of S. Having nulls atcertain frequencies is the same as having the powerspectral density function H(w) equal to zero at thosefrequencies [7].

A sequence of length M having a null at the frequency f =ω/2π = 1/N, with N an integer, means that it is sufficientto satisfy |Y (2π/N)| = 0. For purposes of simplificationwe choose the codeword length M as an integer multiple ofN, where f = r/N represents the spectral nulls at rationalsubmultiple r/N. The parameter N could be chosen eitherprime or not prime and divides M [7], i.e.

M = Nz. (3)

We denote the vector amplitudes by the summation:

Ai =z−1

∑r=0

yi+rN , i = 1,2,3, . . . ,N. (4)

In the case where N is a prime number [12], we have tosatisfy [13],

A1 = A2 = · · ·= AN , (5)

where Ai is the same as in (4). As an example, if N = 3 andM = 6, the following relationship must hold,

A1 = A2 = A3,y1 + y4 = y2 + y5 = y3 + y6.

Definition 1 A spectral null binary block code of length

M is any subset Cb(M,N)⊆ {0,1}M of all binary M-tuplesof length M and have spectral nulls at the rationalsubmultiples of the symbol frequency 1/N.

For codewords of length M consisting of N interleavedsubwords of length z, the cardinality of the codebookCb(M,N) for the case of N considered as a prime numberis presented by the following formula [14],

|Cb(M,N)|=M/N

∑i=0

(M/N

i

)N

, (6)

where

(M/N

i

)denotes the combinatorial coefficient

(M/N)!i!(M/N−i)! .

Example 1 The spectral null codebook for N = 2 and z =2 is:

Cb(4,2) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 0 0 00 0 1 10 1 1 01 0 0 11 1 0 01 1 1 1

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

.

The cardinality of this codebook Cb(4,2) is clearly equalto 6, which could be verified from (6). The spectrum isshown in Fig. 1, where we can see the null appearing at thefrequency 1/2 since N = 2.

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6

Normalized Frequency

P.S.

D.

Figure 1: Power spectral density of codebook N = 2; M = 4.

In the case where N is not prime we have to suppose thatN = cd, where c and d are integer factors of N. Theequation, which leads to nulls, is

Au = Au+vc,

u = 0,1,2, . . . ,c−1,

v = 1,2, . . . ,d−1,

N = cd,

(7)

where Au is the same as in (4). The complete spectral nullcodebook for a given N is the union of the solutions to (7)for each possible pair of factors. For example, if N = 12, itcan be written as the following products: 2×6, 6×2, 3×4and 4×3 [15].

Example 2 If we take N = 4 and M = 8, we have thefollowing relationships:

A1 = A3, y1 + y4 = y3 + y5,

A2 = A4, y2 + y6 = y4 + y8.(8)

We expect that the null will appear at the frequencies 1/4and 3/4 of the normalized frequency since N = 4. Thespectrum is shown in Fig. 2.

The corresponding spectral null codebook is:

Cb(8,4)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

00000000,00000101,00001010,00001111,00010100,00011110,00101000,00101101,00111100,01000001,01001011,01010000,01010101,01011010,01011111,01101001,01111000,01111101,10000010,10000111,10010110,10100000,10100101,10101010,10101111,10110100,10111110,11000011,11010010,11010111,11100001,11101011,11110000,11110101,11111010,11111111

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

.

2.2 Using Permutation Sequences

We consider permutation sequences written in the passiveform, such as 12 . . .M, where each of the symbols arewritten as a binary sequence of length M, with the

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6


P. S

. D.

Figure 2: Power spectral density of codebook N = 4; M = 8.


symbol value indicating where a 1 is to appear and zeroseverywhere else (similar to pulse position modulation).For example, if we take M = 3, we have

1 → 1 0 0,2 → 0 1 0,3 → 0 0 1.

(9)

The permutation sequences for M = 3 are thus changed tothe binary form as follows:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1 2 31 3 22 1 32 3 13 1 23 2 1

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭→

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1 0 0 0 1 0 0 0 11 0 0 0 0 1 0 1 00 1 0 1 0 0 0 0 10 1 0 0 0 1 1 0 00 0 1 1 0 0 0 1 00 0 1 0 1 0 1 0 0

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

Therefore, each of the M! permutation sequences can beconverted to binary sequences of length M2.

An alternative representation is that of (0,1)-matrices,where only a single 1 is allowed in every column and everyrow. For example, the permutation sequence 231 will be⎡

⎣0 0 11 0 00 1 0

⎤⎦ . (10)

The binary sequence representation of the permutationsequence 231 is then constructed by concatenating thecolumns to form 010001100. The matrix in (10) has onlyone single 1 in each row and each column.

We denote by Pω(M2) the binary permutation code thatcontains all the binary sequences of length M2 as a resultof the conversion of the permutation sequences of length Mto binary sequences. The value of ω represents the weightof the binary sequences in each row and each column. Forthe case of ω = 1, as in the matrix presented in (10), thecardinality of the code P1(M2) is |P1(M2)|= M!.

For the case of ω = 2, the (0,1)-matrix can be constructedfrom two ω = 1 (0,1)-matrices by XOR-ing them, as shownbelow⎡

⎣1 0 00 1 00 0 1

⎤⎦⊕

⎡⎣0 1 0

0 0 11 0 0

⎤⎦ =

⎡⎣1 1 0

0 1 11 0 1

⎤⎦ , (11)

or equivalently 100010001⊕001100010 = 101110011 forthe binary sequences.

In general, we will use Pω(M2) to denote the codecontaining all the possible binary sequences that areobtained from (0,1)-matrices with ω 1s in each row andeach column.

It is clear that for P1(3) and P1(4), we have spectral nullcodes with nulls at frequency multiples of 1/3 and 1/4respectively, as depicted in Fig. 3 and 4, in addition to itnot being DC-free.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

Normalised Frequency

P.S

.D.

Figure 3: Power spectral density of P1(3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

Normalised Frequency

P.S.

D.

Figure 4: Power spectral density of P1(4)

3. COMPUTATION OF THE SPECTRUM

We present in this section a few examples of designed spec-tral null codebooks where we compute their cardinalitiesbased on their spectral null equations defined in (5) and (7).

The value of N, can be prime or non prime. In thefollowing section we limit our work only on the case ofN prime since the other one case be derived similarly.

In the case of N prime, we substitute (4) into (5), and weget:

M/N︷︸︸︷y1 + · · ·+ y1+(M−N) =y2 + · · ·+ y2+(M−N)

= · · ·=yN + · · ·+ yM

(12)

It is clear from (12) that the codeword of length M consistsof N groupings of subwords of length z = M/N. We canrewrite (4) as follow:

Ai = ∑m

ym, i = 1,2, . . . ,N, (13)

where m∈{i, i+N, i+2N, . . . , i+(M/N−1)N}, with 1≤i≤ N.

It is also clear from (12) that the value Ai is the sum of“M/N” binary elements, which could be presented in alimited form as follow:

Ai ∈ {−M/N,−M/N +2, . . . ,M/N−2,M/N} (14)

A Matlab c© program, based on an exhaustive search,was used to calculate all possible binary codewordscorresponding to different combinations of Ai as presentedin (14). A few results of our Matlab exhaustive search willbe presented later in Table 1.


0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6


P. S

. D.

Figure 5: Power spectral density of codebook N = 2,M = 6.

To satisfy (12), we need to have the same sum value of theaddition of the M/N elements in all different groupingsAi. Thus the number of the binary sequences or binarycodewords, which satisfy (12) is the number of codewordsin the codebook Cb(M,N) of the spectral null code.

Following are few examples of Cb(M,N) codebooks withtheir power spectral densities graphs for different values ofM and N.

Example 3 For M = 6 and N = 2, we have

3︷︸︸︷y1 + y3 + y5 = y2 + y4 + y6︸︷︷︸

2

. (15)

The cardinality of the codebook Cb(6,2) is the result ofa number of combinations that satisfy (15). The value ofeach grouping Ai could be −3, +3, −1 or +1 since weare dealing with binary sequences. We can see from (15)that there is one combination of six bits, A1 = A2 = −3,when all the elements in the groupings are equal to −1and another combination, A1 = A2 = +3 when all theelements in the groupings are equal to +1. There is anothercombination which yields A1 = A2 = −1 and another onewhich is A1 = A2 = +1. The last two combinations arein fact a result of a permutation of the three elementsin each grouping, Thus the number of combinations isequal to 32 = 9. Finally the total number of combinationsis 1 + 1 + 32 + 32 = 20, which is in fact equal to thecardinality of the codebook Cb(6,2).

The spectral shaping codebook for N = 2 and z = 3 is:

Cb(6,2) =

⎧⎪⎨⎪⎩

000000,000011,000110,001001,001100,001111,010010,011000,011011,011110,100001,100100,100111,101101,110000,110011,110110,111001,111100,111111

⎫⎪⎬⎪⎭ .

We can see that the total number of codewords in thecodebook Cb(6,2) found by our computer search is thesame found by our combinatorial analysis. The spectrumis shown in Fig. 5. Since N = 2, we expect that the null willappear at the frequency 1/2 of the normalized frequency.This is confirmed in Fig. 5.

Example 4 For M = 6 and N = 3, we have

2︷︸︸︷y1 + y4 = y2 + y5 = y3 + y6︸︷︷︸

3

. (16)

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6


P. S

. D.

Figure 6: Power spectral density of codebook N = 3, M = 6.

Using a similar approach for the codebook Cb(6,3), wenote from (16) that the value of each grouping Ai couldbe −2, +2 or 0 since the elements in each groupingare binary bits. We can see from (15) that there is onecombination of six bits, A1 = A2 = A3 = −2, when allthe elements in the groupings are equal to −1 and anothercombination such that A1 = A2 = A3 = +2, when all theelements in the groupings are equal to +1. There is anothercombination which yields A1 = A2 = A3 = 0. The lastcombination is in fact a result of a permutation of thetwo elements in each groupings, so the total number ofcombinations is 2 + 2 + 2 = 6. Taking into considerationthe permutation of the three groupings A1, A2 and A3,which still satisfy the relationship A1 = A2 = A3 = 0, wefind that the number of combinations is 2. Finally, thetotal number of combinations is 1+1+2+2+2+2 = 10,which is the cardinality of the codebook Cb(6,3). Thespectral shaping codebook for N = 3 and z = 2 is:

Cb(6,3) ={

000000,000111,001110,010101,011100,100011,101010,110001,111000,111111

}.

The total number of codewords in the codebook Cb(6,2)found by our computer search is the same found by ourcombinatorial analysis. The spectrum is shown in Fig. 6.Since N = 3, we expect that the nulls will appear at thefrequencies 1/3, 2/3 of the normalized frequency. This isconfirmed in Fig. 6.

Table 1 summarizes few results of the values ofcardinalities and their corresponding values of N andz. It is important to mention that the cardinality playsa role in leading to have an idea about the code ratewhich might be helpful in the improvement of the errorcorrection capability of the code. The cardinality also canbe increased by satisfying the spectral null equation andhaving more codewords in the spectral null codebook.

4. PROPERTIES OF SPECTRAL NULL CODES

4.1 Complementary Symmetry of Codewords

From a simple observation from the design of spectralnull codes, we can see that their codebooks are usuallyhalf-complement symmetrically. We discuss this propertyfor N prime only. The case of N not prime is similar.

Proposition 1 For any spectral null codebook Cb ={∀yi ∈ {−1,+1}/A1 = · · ·= AN}, there exists a subsetC′b, where C′b is a subset of Cb.


Table 1: Cardinalities for Codeword length M = Nz and

spectral null at f = 1/N with N prime

M N z Cardinality Spectral Null

Frequencies

4 2 2 6 1/2

6 2 3 20 1/2

8 2 4 70 1/2

10 2 5 252 1/2

12 2 6 924 1/2

14 2 7 3432 1/2

16 2 8 12870 1/2

18 2 9 48620 1/2

20 2 10 184756 1/2

6 3 2 10 1/3, 2/3

9 3 3 56 1/3, 2/3

12 3 4 346 1/3, 2/3

15 3 5 2252 1/3, 2/3

18 3 6 15184 1/3, 2/3

10 5 2 34 1/5, 2/5, 3/5, 4/5

15 5 3 488 1/5, 2/5, 3/5, 4/5

20 5 4 9826 1/5, 2/5, 3/5, 4/5

25 5 5 206252 1/5, 2/5, 3/5, 4/5

PROOF For all y = (y1,y2, . . . ,yM) ∈ C′b we have:

A1 = · · ·= AN ⇔ y1 + · · ·+ y1+zN =· · ·=yn + · · ·+ yn+zN

⇔ y1 + · · ·+ y1+zN =· · ·=yn + · · ·+ yn+zN

⇔ y1 + · · ·+ y1+zN =· · ·=yn + · · ·+ yn+zN

⇔ A1 = · · ·= AN ,wa

therefore for all y ∈C′b we have all y ∈C′b and thus C′b is asubset of Cb.

4.2 Repetition of Codewords

As defined previously, N represents the number ofgroupings and z represents the number of elements ineach grouping. Satisfying (5), in the case of N prime asexample, means having the same value of the sum in eachgrouping. The value of z can be reduced or increased byeither eliminating or adding certain number of elementsequally in each grouping. The power spectral density isnot effected by the variations of the value of z, since thenulls are always a multiple of 1/N, where N stays thesame. In this section we show that for any value of N wehave codebooks, that are included in other codebooks withlonger codewords.

From previous sections it is clear that the variables of anycodeword y element of the set Cb, satisfies the spectralnull equation of the corresponding codebook Cb. Similarlywith sub-sets, if any codebook C′b ⊂Cb, the codewords ofthe codebook C′b satisfy the spectral null equation of the

codebook Cb. We can prove this in a detailed way in thefollowing proposition.

Proposition 2 For two different spectral null codebooksCb and Cα

b , with the same value of N and different valuesof z, where zα = z+α, α≥ 1, we have y ∈ Cb ⇒ y ∈ Cα

b .

As we know

Ai =z−1

∑λ=0

yi+λN , i = 1,2, . . . ,N,

we consider M = Nz the length of the codewords of thecodebook Cb and Mα = Nzα the length of the codewordsof the codebook Cα

b .

PROOF In this case we have:

A1 = A2 = · · ·= AN

In the case where Mα = Nzα, with zα = z + α, α ≥ 1,which means we have more elements in each grouping, thecodeword length can be written as follows:

Mα = Nzα

Mα = N(z+α)= Nz+Nα= M +Nα.

(17)

For all yα ∈ Cαb and all y ∈ Cb, we have length (yα) =

length (y) + Nα as shown below,

∀yα ∈ Cαb ⇒ yα = (y1,y2, . . . ,yM,yM+1, . . . ,yM+N,

wwwwww . . . ,yM+αN) ,∀y ∈ Cb ⇒ y = (y1,y2, . . . ,yM) .

It is clear from (17), that any spectral null codebookwith codewords of length Mα is different to any otherspectral null codebook with codewords of length M, onlywith an extra number of bits which is equal to αN. Asis known for any codebook with longer codewords, wehave higher cardinality. This will let us predict that thespectral null codebook for the codewords of length M canbe found in the codebook with codewords of length Mα.The addition or the reduction of the number of elementswithin a grouping could be achieved whether we use zerosor ones.

Taking into consideration (4), for ∀y ∈ Cb, (5) could bewritten as:

y1 + y1+N + · · ·+ y1+(z−1)N = y2 + y2+N+

wa · · ·+ y2+(z−1)N

=...

= yN + y2N+wa · · ·+ yzN

(18)

We can extend (18), by adding αN elements from thecodeword y, which can be 0 or 1. We can then show the


idea in (19) by using the canceled variables, such that

y1 + · · ·+��y1+(z−1)N+N +a

· · ·+��y1+(z−1)N+αN = y2 + · · ·+ y2+(z−1)N+��y2+(z−1)N+N + · · ·+��y2+(z−1)N+αN

=...

= yN + · · ·+ yN+(z−1)N+��yN+(z−1)N+N + · · ·+��yN+(z−1)N+αN

(19)

The addition of yi, of the same value as shown beforeregarding the elements in each grouping, to all theequations will not change the sum of the equations. Wehave then the following relation,{ ∀y ∈ Cb

A1 = A2 = · · ·= AN⇒

{ ∀yα ∈ Cαb

Aα1 = Aα

2 = · · ·= AαN

(20)

The equations in (20) show that all the elements of thecodebook Cb are also elements of the codebook Cα

b . Wedenote by Aα

i , the same value of the grouping Ai but for thevalues of zα. The equations in (20) can be proven from theopposite direction, which means from the elements of Aα

ito the elements of Ai and this just by deducting elements.

Example 5 The following example shows the codebookCb is within the codebook Cα

b .

Consider N = 2 and z = 2 for Cb = Cb(4,2) and z1 = 3for C1

b =Cb(6,2). This means that in this example we have

α = 1, so M1 = M+2 as shown in the following codebook:

Cb

N bits

0 0 0 0 0 0⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

C1b

0 0 0 0 1 10 0 0 1 1 00 0 1 0 0 10 0 1 1 0 00 0 1 1 1 1

0 1 0 0 1 0⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

C′b

0 1 1 0 0 00 1 1 0 1 10 1 1 1 1 01 0 0 0 0 11 0 0 1 0 01 0 0 1 1 11 0 1 1 0 1

1 1 0 0 0 0⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

C1b

1 1 0 0 1 11 1 0 1 1 01 1 1 0 0 11 1 1 1 0 01 1 1 1 1 1

(21)

1

2

3

4

1

2

3

4

1

2

3

4G

G1

G2

y1 = y2 = y3 = y4

y1 + y3 = y2 + y4

M = 4

Figure 7: Equation representation for Graph M = 4

This shows clearly the difference between the codewordsof length 6 for C1

b and 4 for Cb as it has been explained

previously. It also shows that Cb ⊂ C1b as it was defined

previously and thus the codewords from Cb appear aselements of the codebook C1

b .

4.3 Concept of Graph Theory

In this section we present and emphasize certain propertiesof spectral null codebooks from graph theoreticalperspective. The concept of subsets and subgraphs [16]–[17] are studied. We link between the indices of thevariables in a spectral null equation and the permutationsequences formed from these indices.

As an example if we take the case of M = 4 with N = 2,we have the spectral null equation:

A1 = A2 → y1 + y3 = y2 + y4 (22)

The corresponding permutation sequences to the variablesin (22) is (1)(3)(2)(4). These permutation symbols canbe presented graphically by just being lying on a circle,which it is called a state. The state design follows theorder of appearance of the indices in (22). The symbolsare connected in respect of the addition property of theircorresponding variables in (22) as depicted in Fig. 7.

The elimination of states from any graph correspondingto the index-permutation symbols is in fact the sameas the elimination of the corresponding variables fromthe spectral null equation (5). The elimination of thevariables is performed in such a way that the spectral nullequation is always satisfied. This leads to the basic idea ofeliminating an equivalent number of variables equal to N asa total number from different groupings in the spectral nullequation. This is true when we eliminate only one variablefrom each grouping. In the case when we eliminate t


1

2

3

4

5

6

1

2

3

4

1

2

3

4

G6G4

M = 6 M = 4

Figure 8: Subgraph design from M = 6 to M = 4 with N = 2

variables with 1 < t < z from each grouping, we have atotal number of eliminated variables of t×N.

Example 6 We construct a spectral null code for the caseof M = 6, with N = 2 and z = 3, which is represented bythe codebook Cb(6,2) in (21) and which is designed fromthe spectral null equation (23). The corresponding graph isG6 in Fig. 8.

From the spectral null equation (23), we can eliminate thevariables y5 and y6 using the addition property and thiswill lead to the equation (24), which is the spectral nullequation for the case of M = 4 with N = 2.

N=2︷︸︸︷z=3︷︸︸︷

y1 + y3 + y5 =z=3︷︸︸︷

y2 + y4 + y6(23)

The obtained codebook is denoted by Cb(4,2). Fig. 8depicts the elimination of the states from a graph theoryperspective. The elimination of the states “5” and “6”results in the elimination of the links between them andthe other states.

N=2︷︸︸︷z=2︷︸︸︷

y1 + y3 =z=2︷︸︸︷

y2 + y4(24)

It is clear that in the codebook presented in (21), wehave Cb(4,2) ⊂ Cb(6,2), in terms of the existence ofthe elements from the codebook Cb(4,2) in the codebookCb(6,2), which is the same as for the subgraphs where wehave G4 ⊂ G6.

4.4 Frequency Spectra of Spectral Null Codes

From the designed spectral null codes Cb, we can observethat each codebook has balanced codewords within it.These balanced codewords form DC-free subsets of thedesigned spectral null codes denoted by CB

b . Anotherproperty that can be observed from the designed spectralnull codebook is that they have codewords with a sequencewhere half of it, is a complement of the other half orwith another word like a mirror of the other half. We callthis class of codes the complementary symmetrical codes,which are subsets of the spectral null codes and denoted byCS

b .

Definition 2 A balanced code, denoted by CBb has all its

codewords with an even length where the number of onesand zeros are equal.

Definition 3 A complementary symmetrical code, de-noted by CS

b has all its codewords with an even length insuch a way that its first half is the conjugate of its secondhalf.

From Table 2, we can see that for the same length of thecodeword and certain specific values of N and z, we alwayshave CB

b ⊂Cb and CSb ⊆CB

b .

Taking into consideration the definitions, we havesummarized our results in Table 2 where it can be seenthat we have a few important properties to be derived fromthese results:

1. For any prime value of z, we cannot design asymmetric codebook except for the special case ofz = 2.

2. For any not prime value of z, we can design a balancedcode and we can produce a symmetric codebook witha predictable cardinality equal to

∣∣CSb

∣∣ = 2n/2.

3. For the values of z, which are not prime, we canhave nulls at the Nyquist frequency for the followingconditions:

(a) if z = 2 and N not prime with N ≥ 2 we can getnulls at the Nyquist frequency,

(b) if z≥ 4 and ∀N we can always have nulls at theNyquist frequency.

4. In the case of symmetric codes, we can alwayspredict the values of the nulls and their correspondingfrequencies as shown in the following equation:

fM = 2(i−1)/M, i = 1, . . . ,M/2.

5. SPECTRAL NULL CODES APPROACH:LOW-DENSITY PARITY-CHECK CODES

Ouahada et al [18] have shown that for any permutationsequences of length N, the binary representation ofthese permutation symbols, where the bit 1 representsthe symbols at its corresponding position, e.g. 123 →100010001, is a subset of spectral null codes with N = zand codewords length of M = N2 and cardinality of N!.The obtained codebook is a N!×N2 matrix, denoted by Mand the number of 1s in each row is equal to N.

The LDPC matrices, denoted by H, were first introducedby Gallagar [19], who defined them as (n, j,k) matriceswith n columns that have j ones in each, and k ones ineach row, and zeros elsewhere.

The number of 1s in each row in the obtained codebookis equal to N with a rate of pr = N/N2 and the number of1s in each column is equal to (N−1)!, which represents arate of pc = (N−1)!/N!. We can see that pr = pc = 1/N,


Table 2: Frequency Spectra and Cardinalities of Spectral Null Codes

M N z |Cb| Nulls |CBb | Nulls |CS

b | Nulls

4 2 2 6 1/2 4 0, 1/2, 1 4 = 22 0, 1/2, 1

6 2 3 20 1/2 — — — —

8 2 4 70 1/2 36 0, 1/2, 1 16 = 24 0, 1/4, 1/2, 3/4, 1

10 2 5 252 1/2 — — — —

12 2 6 924 1/2 400 0, 1/2, 1 64 = 26 0, 1/6, 1/3,1/2, 2/3, 5/6, 1

14 2 7 3432 1/2 — — — —

8 4 2 36 1/4, 3/4 18 0, 1/4, 3/4, 1 16 = 24 0, 1/4, 1/2, 3/4, 1

12 4 3 400 1/4, 3/4 164 0, 1/4, 3/4, 1 — —

16 4 4 4900 1/4, 3/4 1810 0, 1/4, 3/4, 1 256 = 28 0, 1/8, 1/4, 3/8,5/8, 3/4,

7/8, 1

12 6 2 250 1/6, 5/6 90 0, 1/6, 5/6, 1 64 = 26 0, 1/6, 1/3,1/2, 2/3, 5/6, 1

which means that the rates are very low at very large valuesof N.

We can define two numbers that describe a low-densityparity-check matrix with a dimension of n×m; wr for thenumber of 1s in each row and wc for the columns. To havea low-density parity-check matrix we need to satisfy twoconditions wc � n and wr � m.

Proposition 3 The matrix H = M T , is a regular LDPCmatrix, for N ≥ 4.

PROOF The matrix M is a N!×N2 matrix. So H is a N2×N! matrix, with n = N!, k = (N − 1)! and j = N, whichmeans that H is regular [20].

It is clear that the Gallagar condition is satisfied, wherethe number of rows is N2 = n j/k. We can also see thateach submatrix of N×N!, has a single 1 in each one of itscolumns.

For example for N = 4, with

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

111111000000000000000000000000111111000000000000000000000000111111000000000000000000000000111111000000110000110000110000110000000000001100001100001100001100000000000011000011000011000011000000000000001010001010001010001010000000100001100001100001100001000000010100010100010100010100000000000000000101000101000101000101000000010010010010010010010010000000101000101000101000101000000000

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

we can see that

n = N! = 24k = (N−1)! = 6j = N = 4

⎫⎬⎭⇒ 24×4

6= 42 = N2.

It is important to notice that for N = 3, we have H = M .For example, we obtain for N = 3 the following,

H =

⎡⎢⎢⎢⎢⎢⎣

1 0 0 0 1 0 0 0 11 0 0 0 0 1 0 1 00 1 0 1 0 0 0 0 10 1 0 0 0 1 1 0 00 0 1 1 0 0 0 1 00 0 1 0 1 0 1 0 0

⎤⎥⎥⎥⎥⎥⎦ ,

where H is our low-density parity-check matrix with thedimension of N!×N2.

The example N = 3 is just used to show how ourbinary representation of permutation codes is a low-densityparity-check matrix. In reality we should have N verylarge.

Our low-density parity-check matrix is regular. As can beseen in the case of N = 3, where wr and wc are constant.

The regularity is also clear when we form the Tanner graphdepicted in Fig. 9, where we have the same number ofincoming edges for every v nodes and also for all the cnodes.

For all codewords v, we have

v ·HT = 0.

Any LDPC code is encoded via generator matrix G. For agiven information vector u, the corresponding codeword vis encoded via

v = u ·G,

H = [H1|H2], where H1 and H2 have dimensions (n−k)× k and (n− k)× (n− k), respectively. H2 should benon-singular. In the case where H2 is singular, we have


f0 f1 f2 f3 f4 f5

c0 c1 c2 c3 c4 c5 c6 c7 c8

c nodes

v nodes

Figure 9: Tanner graph

to eliminate some rows and columns to get a non-singularmatrix

G = [I|HT1 H−T

2 ].

As example N = 3, we have

H =

⎡⎢⎢⎢⎢⎢⎣

100010001100001010010100001010001100001100010001010100

⎤⎥⎥⎥⎥⎥⎦⇒

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

v1 + v5 + v9 = 0v1 + v6 + v8 = 0v2 + v4 + v9 = 0v2 + v6 + v7 = 0v3 + v4 + v8 = 0v3 + v5 + v7 = 0

with

H1 =

⎡⎢⎢⎢⎢⎢⎣

1 0 01 0 00 1 00 1 00 0 10 0 1

⎤⎥⎥⎥⎥⎥⎦ , H2 =

⎡⎢⎢⎢⎢⎢⎣

0 1 0 0 0 10 0 1 0 1 01 0 0 0 0 10 0 1 1 0 01 0 0 0 1 00 1 0 1 0 0

⎤⎥⎥⎥⎥⎥⎦ .

In the case where N is very large we have N!�N2 and thiswill cause some problems to get the previous conditions ofH satisfied.

We denote by Hμ, where μ is the number of concatenatedLDPC matrices, the generalized form of the constructionof our low-density parity check matrix from our binaryrepresentation,

Hμ =

μ︷︸︸︷[H|H| . . . |H] . (25)

Putting H in serial concatenation μ times can increase theweight wr. We can see that H is always a regular matrixwith a dimension equal to N!× (μN2).

For example if N = 4, we have H with wc = 6 and wr = 4.We choose μ = 4, and we get a H4 with wc = 6 and wr = 16.

It is important to notice that the concatenated constructionmight causes the dependency in the columns of the matrixHμ. Thus some columns could be eliminated and thematrix might become a singular matrix. To solve thisproblem we can permute randomly the columns of eachH. We denote by Hpϕ the matrix H when we permute its

columns ϕ times. Thus (25) will be presented as follows:

Hμ =

μ︷︸︸︷[Hpϕ |Hpϕ | . . . |Hpϕ ], 1≤ ϕ≤ N2.

It is important to mention that the values of wr and wccan be further increased by satisfying the spectral nullequation, which leads to the increase of ones in the LDPCmatrix. Therefore the code rate will be increased. FromFig. 9 we can also see that the girth of the code is higherthan four, which means that we have good error correctioncodes.

6. CONCLUSION

In this paper, with certain observations of the structureof spectral null codes, we could have derived importantproperties that can be useful in the field of digitalcommunications. The paper does not present constructionsof any type of codes but just analysis of existent propertiesof spectral null codes.

The relationship between the spectral null equations, thegenerated nulls and the cardinalities of spectral null codeswere investigated. The importance of the cardinality ofthe codebook and the corresponding rate of the code andalso the error correction capability are emphasized andclarified.

The properties and the approaches that we have presentedusing the binary structure of the codebooks and thegraph theory approach could help in similar research indiscovering more properties that can be used in importantapplications telecommunications and data recording tohelp improve the quality of the transmitted dateinformation.

Certain spectral null codes properties can lead to certainerror correcting codes for certain channels as the examplein [21, 22] or the improvement in the structures of certainspectral null codebooks for better design of Low DensityParity Check codes.

REFERENCES

[1] R. Karabed and P. H. Siegel, “Matched spectral-nullcodes for partial-response channels,“ IEEE Trans. onInfo. Theory, vol. 37, no. 3, May. 1991, pp. 818–855.

[2] K. A. S. Immink, “Spectrum shaping with binaryDC2-constrained channel codes,“ Philips Journal ofResearch, vol. 40, May. 1985, pp. 40–53.

[3] K. W. Cattermole, “Principles of digital line coding,“International Journal on Electronics, vol. 55, Jul.1983, pp. 3–33.

[4] K. A. Immink, Coding Techniques for DigitalRecorders, Prentice Hall International, UK., 1991.


[5] K. A. S. Immink and P. H. Siegel and J. K. Wolf,“Codes for Digital Recorders,“ IEEE Trans. on Info.Theory, vol. 44, no. 6, Oct. 1998, pp. 2260–2299.

[6] E. Eleftheriou and R. D. Cideciyan, “On codes sat-isfying M-th order running digital sum constraints,“IEEE Trans. on Info. Theory, vol. 37, no. 5, Sept.1991, pp. 1295-1313.

[7] K. A. Immink, Codes for Mass Data Storage Systems,Chapter 12, Shannon Foundation Publishers, TheNetherlands, 1999.

[8] N. Hansen, “A head-positioning system using buriedservos,“ IEEE Transactions on Magnetics, vol. 17,no. 6, Nov. 1981, pp. 2735–2738.

[9] M. Haynes, “Magnetic recording techniques forburied servos,“ IEEE Transactions on Magnetics, vol.17, no. 6, Nov. 1981, pp. 2730-2734.

[10] E. Gorog, “Redundant Alphabets with DesirableFrequency Spectrum Properties,“ IBM J. Res.Develop., vol. 12, pp. 234–241, May 1968.

[11] G. L. Pierobon, “Codes for zero spectral density atzero frequency,“ IEEE Trans. on Info. Theory, vol.30, Mar. 1984, pp. 435–439.

[12] T. Estermann, Introduction to Modern Prime NumberTheory, Cambridge Tracts in Mathematics andMathematical Physics., 1961.

[13] B. H. Marcus and P. H. Siegel, “On codes withspectral nulls at rational submultiples of the symbolfrequency,“ IEEE Trans. on Info. Theory, vol. 33, no.4, Jul. 1987, pp. 557–568.

[14] K. A. S. Immink, “Spectral null codes,“ IEEETransactions on Magnetics, vol. 26, no. 2, Mar. 1990,pp. 1130–113.

[15] L. K. Hua, Introduction to Number Theory, NewYork: Springer-Verlag, 1982.

[16] R. J. Wilson, Graph theory and Combinatorics.England: Pitman Advanced Publishing Program.,1979.

[17] J. L. Gross and J. Yellen, Graph theory and itsApplications. USA: Chapman and Hall/CRC., 2006.

[18] K. Ouahada and T. G. Swart and H. C. Ferreira andL. Cheng, “Binary permutation sequences as subsetsof Levenshtein codes, spectral null codes, run-lengthlimited codes and constant weight codes,“ Designs,Codes and Cryptography, vol. 48, no. 2, Aug. 2008,pp. 141–154.

[19] R. G. Gallagar, “Low-density parity-check codes,“IRE Transactions on Information Theory, vol. 8, no.1, Jan. 1962, pp. 21–28.

[20] S. Lin and D. J. Costello Jr., Error Control Coding:Fundamentals and Applications, Prentice Hall Inc.,1983.

[21] L. Cheng, H. C. Ferreira and K. Ouahada, “k-BitGrouping Moment Balancing Templates for SpectralShaping Codes,” in Proceedings of the IEEEInformation Theory Workshop, Porto, Portugal, May5-9, 2008, pp. 426–430.

[22] L. Cheng, H. C. Ferreira and K. Ouahada, “Momentbalancing templates for spectral null codes,” IEEETransactions on Information Theory, vol. 56, no. 8,pp. 3749–3753, Aug. 2010.


ERROR PERFORMANCE OF CONCATENATED SUPER-ORTHOGONAL SPACE-TIME-FREQUENCY TRELLIS CODED MIMO-OFDM SYSTEM I. B. Oluwafemi* and S. H. Mneney* *School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Durban, South Africa. E-mail: [email protected]; [email protected] Abstract: In this paper, we investigate the performance of serially concatenated convolutional code with super-orthogonal space-time trellis code (SOSTTC) in orthogonal frequency division multiplexing (OFDM) over frequency selective fading channels. We consider both recursive systematic convolutional code (RSC) and non-recursive convolutional code (NRC) as the outer code, and 16-state QPSK SOSTTC as the inner code. Employing these, two concatenated schemes consisting of single convolutional outer code and two serially concatenated convolutional outer codes are proposed. We evaluate the performance of the concatenated schemes by means of computer simulations with maximum a posteriori (MAP) algorithm based iterative decoding. Simulation results indicate that the performance of the proposed concatenated schemes improved significantly when compared with schemes without concatenation under the same channel condition. Keywords: Coding gain, convolutional code, frequency selective fading channels, iterative decoding, orthogonal frequency division multiplexing, super-orthogonal space-time trellis code.

1. INTRODUCTION

Space-time coding (STC) has been shown to be an effective method of increasing the capacity of wireless communication channels by combining the benefits of diversity transmission and error correction coding to combat impairments of wireless channels [1-4]. Super-orthogonal space-time trellis code (SOSTTC) is the recently proposed space-time code (STC) that combine set partitioning based on the coding gain distance and a super set of orthogonal space-time block code in a systematic way, to provide full diversity and improved coding gain over the earlier proposed space-time trellis code constructions [5-8]. Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier modulation scheme used to combat frequency selective fading channels [9]. Although OFDM eliminates the inter-symbol interference (ISI) problem caused by the multipath effect, it does not eliminate errors caused by channel fading and additive white Gaussian noise (AWGN) in wireless channels [10]. Antenna diversity, through space-time coding, is one of the adopted techniques being used to improve the performance of OFDM systems in the presence of channel fading and AWGN impairment. Due to its high bandwidth efficiency and suitability for high data-rate wireless applications, OFDM was chosen as a modulation scheme for the physical layer in several new wireless standards such as digital audio and digital video broadcasting (DAB, DVB) in Europe, the three broadband wireless local area networks (WLAN), European HIPERLAN/2, American IEEE 802.11a and Japanese MMAC [11].

Space-time coded OFDM system was first introduced in [12] where OFDM technique was employed to transform a frequency selective fading channel into many flat fading channels. The initial work of [12] led to many design considerations for space-time coded OFDM system in order to improve its performance [13-17]. It is known that STCs are designed to maximize the diversity gain for a given number of transmit antennas and that the coding gain of STC is low. Increasing the number of states of STC will lead to an increase in the achievable coding gain but the decoding complexity also increases exponentially [10]. Concatenated coding schemes with sub-optimum, yet powerful iterative decoding, have been shown to guarantee improved error performance while the complexity of the decoders is kept comparable to single coding schemes [18, 19]. Several concatenated schemes with constituent codes of STC and convolutional codes were proposed in [20-27], with reported improved coding gain over their un-concatenated counterparts. In order to improve the coding gain of SOSTTC in frequency selective fading channels, we hereby propose two concatenated schemes consisting of convolutional codes and SOSTTC for OFDM systems. The first involves a serially concatenated convolutional outer code with SOSTTC inner code while the second scheme involves two outer serially concatenated convolutional codes with SOSTTC inner code. The systems have the advantage of achieving diversity gain by exploiting available diversity resources of the frequency selective fading channel. Also, by iterative information exchange, the concatenation schemes achieve additional decoding gain without bandwidth expansion. It is well known that the theoretical evaluation of the exact performance of such concatenated schemes using iterative decoding in

[email protected]; [email protected]


frequency selective fading channel is a very difficult task, and hence we used computer simulations to evaluate the performance of the proposed systems. In particular, we considered rate ½ and rate 2/3 outer convolutional codes for both recursive and non recursive outer codes. As pointed out in [5-8], SOSTTC has a large number of parallel transitions in their trellis which limit their error performance in frequency selective fading channels. To avoid such parallel transitions, at least a 16-state code for QPSK is required [28, 29]. This is the reason for the choice of the 16-state QPSK SOSTTC as the inner code for the proposed concatenated schemes. The rest of the paper is organized as follows. In section 2, we describe the system model which includes a brief description of the channel model, the transmitter and the receiver structure. Section 3 describes the outer and the inner code used in this paper while the error performance of the proposed schemes is evaluated by computer simulation in section 4. Finally, section 5 concludes this paper.

2. SYSTEM MODEL 2.1 Channel model We considered a MIMO-OFDM system consisting of two transmit antenna and Mr receive antennas. Each transmit antenna employs an OFDM modulator with K subcarriers. We assume no spatial correlation exists between the antennas and that the receiver has perfect knowledge of the channel while the channel is unknown to the transmitter. The channel impulse response (CIR) between the transmit antenna p and receive antenna q with L independent delay paths on each OFDM symbol and an arbitrary power delay profile can be expressed as [30]

)()()(

1

0,, l

L

lqpqp tlth , (1)

where l represents the lth path delay and qp, )(l are

the fading coefficients at delay l . Note that each

qp, )(l is a complex Gaussian random variable with

zero mean and variance 2

2l on each dimension. For

normalization purposes we assumed that 10

2L

l l in

each of the transmit –receive links. The channel frequency response (CFR), that is the fading coefficient for the kth subcarrier between transmit antenna p and receive antenna q with a proper cyclic prefix and a perfect sampling time, is given by

1

0

2,, )()(

L

l

njqpqp

lfelkH (2)

where f is the inter-subcarrier spacing, sl Tl is the

lth path delay and f

s KT 1 is the sampling interval of

the OFDM system. A space-frequency codeword for two transmit antennas transmitted at the tth OFDM symbol period can be represented by 2

21 )()( KtttSF CkckcC , where

)(kctp is the complex data transmitted by the pth transmit

antenna at the kth subcarrier for, k = 0, …, K-1. Moreover, t

SFC satisfies the power constraint

KEF

tSF

2C . A STC codeword has an additional time

dimension added to the above space frequency codeword, and can be represented as 221 Kt

SFtSFSTF CCCC .

At the receiver, after matched filtering, removal of the cyclic prefix and application of fast Fourier transform (FFT), the signal at the kth subcarrier and antenna q is given by

)()()()( ,

2

1

kNkHkctr tq

tqp

n

i

ti

tq

T

, (3)

where q = 1, …, Mr, and )(kN tq is a circularly symmetric

Gaussian noise term, with zero-mean and variance N0 at tth symbol period. 2.2 Encoder structure Convolutional code with super-orthogonal space-time trellis code (CC-SOSTTC-OFDM): We consider a serially concatenated Multi-Input Multi-Output (MIMO) OFDM communication system that employs nT = 2 antennas at the transmitter, and nR = 1 antenna at the receiver. The transmitting block diagram of the concatenated scheme is shown in Figure1. The encoder consists of an outer convolutional code concatenated with an inner SOSTTC code. In this system, a block of N independent data bits is encoded by the convolutional outer encoder and the output block of coded bits are interleaved by using a random bit interleaver ( ). The interleaved sequence are then passed to the SOSTTC encoder to generate a stream of QPSK symbols. Each of the symbols from the SOSTTC encoder is converted to a parallel output, and an inverse fast Fourier transform (IFFT) is performed on each of the parallel symbols. At the end, cyclic prefix (CP) is added to each of the transformed symbols before transmission from each of the transmit antennas.


Double convolutional code with super orthogonal space-time trellis code (CC-CC-SOSTTC-OFDM): The encoder structure of the double concatenated scheme is shown by Figure 2. Two outer serially concatenated convolutional codes are concatenated with an inner SOSTTC in a bid to improve the overall coding gain of the system. The encoding process is similar to that described above, except for the addition of an extra convolutional outer encoder. In this system, a block of N independent data bits is encoded by the first convolutional outer encoder and then interleaved using a random bit interleaver ( 1 ). The output stream from the interleaver is then encoded by the second convolutional outer code and thereafter interleaved by the second interleaver ( 2 ). The permuted sequence is thereafter SOSTTC encoded. The remaining process of encoding follows the description given above. In the two systems, both the recursive systematic (RSC) and non recursive convolutional (NRC) codes were considered and each of the encoders was terminated using appropriate tail bits.

2.3 Decoder structure

In this section, a description of the iterative decoding of the two concatenated schemes is given. The employed decoders operate on bit streams using the Soft–Input Soft-Output (SISO) algorithm [31]. Extrinsic information is exchanged between the component decoders using the soft estimates of their Log Likelihood Ratio (LLR) with the presence of the feedback loop. CC-SOSTTC-OFDM decoder: The decoder structure of the CC-SOSTTC-OFDM is shown in Figure 3. As shown in the figure, the inserted CP is first removed and thereafter fast Fourier transform (FFT) is performed on each of the symbols. The parallel outputs obtained from this transformed symbol are then converted to serial streams for computation of the coded intrinsic LLR of the SOSTTC SISO module.

Figure 1: Encoder block diagram of CC-SOSTTC-OFDM

system.

1 2

Figure 2: Encoder block diagram of double serially concatenated SOSTTC-OFDM (CC-CC-SOSTTC-

OFDM) system.

Given that the received symbol from subcarrier k is

),()()()( kwkHkxkr (4)

the coded intrinsic LLR for the SOSTTC SISO ( , )stC I is computed as [26]

0

Pr[ | ( )]( , ) = log

Pr[ | ( )]st

x r kC I

x r k, (5)

where x0 is a reference symbol. By dropping the subcarrier index k for simplicity, we can express (5) by

2

1 102

2

1 12 2

121),(

R TR T n

q

n

ppq

n

q

n

ppqst xHrxHrIC , (6)

where 2 is the variance of the independent complex Gaussian noise variable. The SOSTTC SISO takes ),( ICst and the a priori information from the CC-SISO (initially set to zero) and compute the extrinsic information given by

ˆ( , ) = ( , ) - ( , ).st st stU O U O U I (7)

This extrinsic information is de-interleaved ( )1 and fed to the CC-SISO to become it’s a priori information

),( ICcc . The a priori information is then used to compute the extrinsic information for the convolutional code SISO (CC-SISO). The extrinsic information for the CC-SISO is given by

).,(),(),(~ ICOCOC cccccc (8)


The extrinsic LLR in (8) is then interleaved to become the a priori information ),( IU st for the SOSTTC SISO for the next iteration. During the first iteration, we set

),( IU st to zero, as no a priori information is available at the SOSTTC-SISO. We assumed that the source symbols transmitted are equally likely. Therefore, the input LLR

),( IUcc to the CC-SISO is permanently set to zero. We iterated the process several times. On the final iteration, a decision is taken on the extrinsic information ),( OCcc to obtain the estimate of the original transmitted bit stream. CC-CC-SOSTTC-OFDM decoder: The decoder structure of the CC-CC-SOSTTC-OFDM is shown in Figure 4. During the first iteration, the LLRs ),( IUst and ),( 2 IU are set to zero as no a priori information is available. Since we assumed that the source symbols transmitted are equally likely, the input LRR ),( 1 IU to the C1-SISO is permanently set to zero. The coded intrinsic LLR for the SOSTTC SISO ),( ICst is computed using (6). The SOSTTC SISO takes ),( ICst and the a priori information from the C2-SISO (initially set to zero) for the computation of the extrinsic information given by

ˆ( , ) = ( , ) - ( , ).st st stU O U O U I (9)

The extrinsic LLR ),(~ OU st is then de-interleaved

through 12 to become the input LLR ),( 2 IC of the

SISO decoder for the second outer convolutional code. The C2-SISO takes the input LLR ),( 2 IC and the a priori information from the C1-SISO to compute the extrinsic information for the C2-SISO of the coded and the un-coded values. The un-coded extrinsic information for the C2-SISO decoder is given by

).,(),(),(~222 IUOUOU (10)

The extrinsic information from (10) is then de-interleaved by 1

1 to become the intrinsic LLR information for the C1-SISO decoder ),( 1 IC . The intrinsic information is then used by the C1 SISO to compute the extrinsic information for the C1 SISO as

1 1 1ˆ( , ) = ( , ) - ( , ).C O C O C I (11)

The C1-SISO output LLR is thereafter passed through the interleaver 1 to obtain the a priori information for the C2-SISO. The coded LLR output obtained from the C2-SISO given by

),(),(),(~222 ICOCOC . (12)

is also passed through the interleaver 2 to obtain the a priori information ),( IU st for SOSTTC- SISO for the next iteration. During the final iteration, decision is taken on ),( 1 OU from the C1-SISO output to obtain the estimate of the original transmitted symbol.

3. COMPONENT CODES

3.1 Inner code The SOSTTC code with the transmission matrix given by (13) is considered as the inner codes.

12

2121 ),,(

xexxex

xxC j

j

, (13)

where for M-PSK signal constellation, the signals x1 and x2 which are selected by input bits can be represented by

,2M

lj

e where l = 0,1,…, M-1 and which is the rotation angle can take on the values = 2 l’/M, where l’ = 0,1,…, M-1. As noted in [28, 29] and in [32], SOSTTC have parallel transitions that limit its error performance in a frequency selective fading channel. To exploit the multipath diversity of frequency selective fading channels, at least 16-state SOSTTC is needed for QPSK constellation. We considered 16-state QPSK SOSTTC with the trellis diagram shown by Figure 5 [33]. The SOSTTC was designed using the set partitioning principle of [5] and design rules outlined in [7, 8]. For comparison, we also considered a 16-state QPPK STTC presented in [1] as inner code. 3.2 Outer code Convolutional codes are considered as the outer codes. We considered both recursive systematic and non recursive codes. The generator matrices of the outer codes considered in our investigation are given in Table 1.

4. RESULTS

In this section, we provide the simulation results illustrating the performance of the proposed concatenated schemes for multiple-antenna OFDM systems. The performance is presented in terms of frame error rate (FER) versus the received SNR for QPSK constellation. To properly model a frequency selective fading channel, we considered the typical urban six-path (TU6) COST 207 model reported in [34]. The OFDM modulator utilizes 64 subcarriers with a total system bandwidth of 800 kHz and FFT duration of 100 s.


1

OccU ,IccU ,

IccC ,

IUst ,

uICst, OU st ,

OUst ,~

OCcc,~OCcc,

Figure 3: Decoder block diagram of CC-SOSTTC-OFDM.

12

21

11 u

),( Icst

),( Iust

),( Oust ),(~ Oust ),( 2 Ic ),( 2 Oc

),( 2 Iu

),(~2 Ou

),( 2 Ou ),(~2 Ou

),(~1 Oc

),( 1 Ic

),( 1 Iu

),( 1 Oc

),( 1 Ou

Figure 4: Decoder block diagram of CC-CC-SOSTTC-OFDM.

The system’s subcarrier spacing is 12.5 kHz with symbol duration of 80 s while the guard band interval is s.The channel is assumed to be static during one OFDM symbol duration and the receiver is assumed to have the full knowledge of the channel (perfect channel estimation). We also assume perfect time and frequency synchronization between the transmitter and the receiver. 4.1 CC-SOSTTC-OFDM Figure 6 shows the FER simulation results for the first concatenated scheme for various numbers of iterations. The amount of improvement resulting from iteration is seen to become smaller as the number of iteration increases. This is especially evident after the 4th iteration. In Figure 7 the FER performance is shown for normalized Doppler frequencies fdt of 0.05, 0.02, 0.01 and 0.005, corresponding to mobile speed of 150 m/s, 60 m/s, 30 m/s and 15 m/s respectively. Rate ½, 4-state recursive convolutional code is considered in this investigation. As it can be observed, the scheme’s FER performance improves as the mobile speed decreases. This is due to the fact that the channels are more correlated at high normalized Doppler frequencies. In other words, channel correlation may cause long burst errors during deep fades, which degrades the FER performance. Figure 8 illustrates the effect of the number of the states of the outer codes on the performance of the proposed

schemes. Here, the systems have 2 transmit antennas and 1 receive antenna. We evaluate the system using the 4-state, 8- state and 16-state rate ½ non recursive convolutional codes as the outer code. As can be observed from Figure 8, the system with 4-state exhibits the best performance. This shows that increasing the number of the states of the outer code results in degradation of the system’s performance. This is a typical phenomenon in iterative decoding [20, 35]. 4.2 CC-CC-SOSTTC-OFDM

For the CC-CC-SOSTTC-OFDM scheme, we considered the rate 2/3 outer codes for the case of both recursive and non recursive code. Figure 9 shows the FER variation with the number of decoding iterations. The performance improvement produced by each iteration is observed to converge with increase in the number of iterations. Figure 10 also shows the effect of the mobile speed on the FER performance of the proposed scheme for various normalized Doppler frequencies. In Figure 11 we show the FER results for four various combinations of the outer channel codes. From the FER performance curve, it is observed that having RSC as the middle code achieves the best performance irrespective of having either RSC or NRC as the outer code.


Figure 5: Trellis diagram of 16 states SOSTTC at rate 2 bits/s/Hz [33].

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

1st iter2nd iter

3rd iter

4th iter

5th iter6th iter

Figure 6: FER performance of CC-SOSTTC-OFDM for

various numbers of iterations.

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

fdt=0.05

fdt=0.02

fdt=0.01

fdt=0.005

Figure 7: FER performance of the CC-SOSTTC-OFDM

with different normalized Doppler frequency.


Table1: Generating matrices for the outer convolutional codes.

No Code

Description G(D)

1 4-state rate ½ NRC

[1 + D + D2, 1 + D2 ]

2 4-state rate

½ RSC 2

2

11,1

DDD

3 8-state rate

½ NRC 3232 1,1 DDDDD

4 16-state

rate½ NRC 4432 1,1 DDDDD

5 4-state rate

2/3 NRC DDDD1,1,1

1,,1

6 4-state rate

2/3 RSC 2

2

2

11,1,0

110,1

DDD

DDD

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

16 states CC-SOSTTC-OFDM

8 states CC-SOSTTC-OFDM4 states CC-SOSTTC-OFDM

Figure 8: FER performance of CC-SOSTTC-OFDM for different states of outer code.

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

1st iter

2nd iter

3rd iter4th iter

5th iter

6th iter

Figure 9: FER performance of CC-CC-SOSTTC-OFDM for various number of iteration.

4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

fdt=0.05

fdt=0.02

fdt=0.01

fdt=0.005

Figure 10: FER performance of CC-CC-SOSTTC-OFDM over various normalized Doppler frequencies.

For the purpose of performance comparison we show the FER of 16-state SOSTTC-OFDM, CC-STTC-OFDM, CC-SOSTTC-OFDM, CC-CC-STTC-OFDM, and the CC-CC-SOSTTC-OFDM schemes in Figure 12. The CC-SOSTTC-OFDM is seen to outperform the SOSTTC-OFDM by about 5 dB at FER of 10-3 and also outperform the CC-STTC-OFDM by about 2.5 dB at the same FER. The CC-CC-SOSTTC-OFDM outperforms the SOSTTC-OFDM by about 7 dB at FER of 10-3 and also outperforms the CC-CC-STTC-OFDM with about 1.4 dB at the same FER. The CC-CC-SOTTC-OFDM scheme also outperforms the CC-SOSTTC-OFDM counterpart by about 2 dB at FER 10-3 albeit with loss in bandwidth efficiency.


4 5 6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

NRC-NRC-SOSTTC-OFDM

RSC-NRC-SOSTTC-OFDMNRC-RSC-SOSTTC-OFDM

RSC-RSC-SOSTTC-OFDM

Figure 11: FER performance of CC-CC-SOSTTC-OFDM for different outer code combinations for 2 transmits and

1 receive antennas.

4 6 8 10 12 14 16 18 2010

-4

10-3

10-2

10-1

100

SNR[dB]

FE

R

SOSTTC-OFDM

CC-STTC-OFDM

CC-SOSTTC-OFDMCC-CC-STTC-OFDM

CC-CC-SOSTTC-OFDM

Figure 12: FER comparisons of SOSTTC-OFDM, CC-STTC-OFDM, CC-SOSTTC-OFDM and CC-CC-

SOSTTC-OFDM for 2 transmit and 1 receive antenna.

5. COMPARATIVE DECODING COMPLEXITY AND BANDWIDTH EFFICIENCY

In this section, the relative estimated complexity and bandwidth efficiency of the proposed concatenated

schemes in MIMO-OFDM systems are presented. The approach used in [36] is herewith adopted in analyzing the complexity of the various proposed schemes. As stated in [36], the complexity of the channel decoders depends directly on the number of trellis transition per information data bits. This will be used as the basis for our comparison. For SOSTTC, the number of trellis leaving each state is equivalent to 2BPS, where BPS is the number of transmitted bits per modulation symbols. For QPSK SOSTTC, four bits are used for every modulation symbol. The approximate complexity of the SOSTTC - OFDM system decoder using Viterbi algorithm can therefore be calculated by

Comp{SOSTTC} = BPS

statesofNoBPS2 (14)

For the concatenated schemes, we applied the Log-MAP

decoding algorithm for iterative decoding. Since the Log-MAP algorithm has to perform forward as well as backward recursion and soft output calculation, the number of trellis in Log-MAP decoding algorithm is assumed to be three times higher than that of a conventional Viterbi algorithm. For the rate 1/2 4-state convolutional code (CC) decoder, the complexity is approximated as

iterationsofnoK 322=K)}1,Comp{CC(2, 1 , (15)

where K is the constraint length of the convolutional code. For the rate 2/3 4-state CC decoder, the complexity is estimated as

.323=K)}2,Comp{CC(3, 1 iterationsofnoK (16)

For SOSTTC with six iterative decoding, the complexity is estimated as

.263=Citer}Comp{SOSTT

BPSstatesofnoBPS

(17)

During each OFDM frame, 256 information bits are encoded to generate coded QPSK sequence which is OFDM modulated on 64 subcarriers. Considering the tail bits of the encoders and the guard interval of the OFDM modulation, the bandwidth efficiency of the concatenated scheme is given by

.

2

ratecodealloverbitstailbitofno

bitsofnobandguardK

K (18)


Table 2: Estimated decoder complexity and bandwidth of the proposed schemes.

S/N SCHEME Decoding Algorithm CC STC

states Complexity Bandwidth Efficiency ( b/s/Hz)

SNR at FER of 10-2

1 SOSTTC-OFDM VA 16 states

64 1.575 16.8 dB

2 CC-STTC-OFDM Log-MAP rate 2/3 16 states 792 1.04 13 dB

3 CC-CC-STTC-OFDM Log-MAP rate 2/3 16 states 1008 0.68 10 dB

4 CC-SOSTTC-OFDM

Log-MAP rate 2/3 16 states 1368 1.04 11.6 dB

5 CC-CC-SOSTTC-OFDM

Log-MAP rate 2/3 16 states 1584 0.68 9 dB

Applying equations (14) to (18) and considering six iterations for all the concatenated schemes, we summarize the estimated complexity and bandwidth efficiency for the proposed schemes in OFDM systems in Table 2. From Table 2, the CC-CC-SOSTTC-OFDM have the highest decoding complexity but with the best FER performance.

6. CONCLUSIONS

In this paper, investigation of concatenated SOSTTC over time varying frequency selective fading channel for MIMO-OFDM systems was carried out. In the investigation, two concatenated schemes were proposed for OFDM systems. We compared the error performance of the schemes with the use of QPSK STTC inner code combination. The system performance was observed to depend on the time selectivity of the channels, exhibiting high performance over slow varying channel and poor performance over fast varying channels. Results showed performance degradation when outer code with higher number of states and iterative decoding was used. The concatenated SOSTTC-OFDM systems have the advantage of achieving high diversity gain by exploiting available diversity resources of frequency selective fading channels and high coding gain was provided by serial concatenation coding scheme. The decoding complexity of the proposed schemes was also evaluated and compared with the reference systems.

7. ACKNOWLEDGMENT

The authors are indebted to Center for Engineering Postgraduate Study (CEPS), University of KwaZulu-Natal for the support made available for this study.

8. REFERENCES [1] V. Tarokh, N. Seshadri and A. R. Calderbank:

“Space-time codes for high data rate wireless communication; Performance criterion and code construction”, IEEE Transactions on Information Theory, Vol. 44 No.2, pp. 744–765, 1998.

[2] V. Tarokh, A. Naguib and N. Seshadri: “Space-time

codes for high data rate wireless communication: Performance criteria in the presence of channel estimation errors, mobility, and multiple paths”, IEEE Transactions on Information Theory, Vol. 47 No. 2, pp. 199-207, 1999.

[3] S. M. Alamouti: “A simple transmitter diversity

scheme for wireless communication”, IEEE Journal on Selected Areas of Communication, Vol. 16 No. 8, pp. 1451-1458, 1998.

[4] V. Taroch, H. Jafarkhani and A.R. Calderbank:

“Space-time block codes from orthogonal designs”, IEEE Transactions on Information Theory, Vol. 45 No. 5, pp. 1456-1467, 1999.

[5] H. Jafarkhani and N. Sashadri: “Super-orthogonal

space-time trellis codes”, IEEE Transactions on Information Theory, Vol. 49 No. 4, pp. 937-950, 2003.


[6] S. Siwamogsatham and P. Fitz: “Improved high–rate space-time codes via concatenation of expanded orthogonal block code and M-TCM”, Proceedings: IEEE International Conference on Communication, New York, NY, USA, pp. 636-640, 28 Apr 2002 - 02 May 2002.

[7] M. Bale, et al: “Computer design of super-orthogonal

space-time trellis code”, IEEE Transactions on Wireless Communication, Vol. 6 No 2, pp. 463-467, 2007.

[8] E. R. Hartling and H. Jafarkhani: “Design rules for

extended super-orthogonal space-time trellis codes”, Proceedings: 21st Canadian Conference on Electrical and Computer Engineering CCECE, Ontario, pp. 621-626, 2008.

[9] L. Cimini Jr: “Analysis and simulation of a digital

mobile channel using orthogonal frequency division multiplexing”, IEEE Transactions on Communication, Vol.33 No.7, pp. 665-675, Jul 1985.

[10] P. Patcharamaneepakorn, R. M. A. P. Rajatheva and

A. Ahmed: “Performance analysis of serial concatenated space-time code in OFDM over Frequency selective channels”, Proceedings: IEEE Vehicular Technology Conference, pp. 746- 750, 2003.

[11] T. Djordje, J. Markku and L. Matti: “Space-time

turbo coded modulation: design and applications”, Eurasip Journal on Applied Signal Processing, Vol. 3, pp. 236-248, 2002.

[12] D. Agrawal, V. Tarokh, A. Naguib and N. Seshadri:

“Space-time coded OFDM for high data-rate wireless communication over wideband channels”, Proceedings: IEEE Vehicular Technology Conference, pp. 2232-2236, 1998.

[13] B. Lu and X. Wang: “A space-time trellis code

design method OFDM systems”. Wireless Personal Communications, Vol. 24, pp. 403-418, 2003.

[14] Y. Hong and Z. Y. Dong: “Performance analysis of

space-time trellis coded OFDM systems”, International Journal of applied Mathematics and Computer Science, Vol.2 No.2, pp. 59-65, 2006.

[15] B. Lu and X. Wang: “Space-time code design in

OFDM systems”, Proceedings: IEEE Conference on Global Communications, San Francisco, CA, USApp.1000-1004, 27 Nov 2000 - 01 Dec 2000.

[16] Z. Liu, Y. Xin and G. B. Giannakis: “Space-time

frequency coded OFDM over frequency selective fading channels”, IEEE Transactions on Signal Processing, Vol. 50 No.10, pp. 2465-2476, 2002.

[17] J. Yue and J. D. Gidson: “Performance of OFDM systems with space-time coding”, Proceedings: IEEE Wireless Communications and networking Conference, pp. 280-284, 2002.

[18] S. Benedetto and G. Montorsi: “Serial concatenation

of interleaved codes: analytic performance bounds”, Proceedings: IEEE Global Communications Conference, pp. 106-110, 1996.

[19] S. Benedetto and G. Montorsi: “Serial concatenation

of interleaved codes: performance analysis, design and iterative decoding”, IEEE Transactions on Information Theory, pp. 909-926, 1998.

[20] I. Altunbas and A. Yongacoglu: “Error performance

of serially concatenated space-time coding”, Journal of Communication Networks (JCN), 5(2), pp. 135-140, 2003.

[21] H. J. Su and E. Geraniotis: “Space-time turbo codes

with full antenna diversity”, IEEE Transactions on Communication, Vol. 49 No.1, pp. 47-57, 2000. Jan 2001.

[22] W. Firmanto, Z. Chen, Vucetic, and B.J. Yuan:

“Space-time turbo trellis coded modulation for wireless data communications” EURASIP Journal on Applied Signal Processing, Vol. 5, pp. 459-470, 2002.

[23] X. Lin and R. S. Blum: “Improved space-time codes

using serial concatenation” IEEE Communications Letters, Vol. 4 No. 7, pp. 221-223, July 2000.

[24] I. Altunbas: “Performance of serially concatenated

coding scheme for MIMO systems” International Journal Electronic Communication, Vol. 6, No.1, pp. 1-9, 2007.

[25] K. Bon-Jin, C. Jong Moon and K. Changeon:

“Analysis of serial and hybrid concatenated space-time codes applying iterative decoding” International Journal Electronics and Communication, Vol. 58 No. 6, pp. 420-423, October 2004.

[26] C. Dongzhe and A. M. Haimovich: “Performance of

parallel concatenated space-time codes”, IEEE Communications Letters, Vol. 5 No. 6, pp. 236-238, 2001.

[27] J. N. Pillai and S. H. Mneney: “Turbo decoding of

super-orthogonal space-time trellis codes in fading channels”, Wireless personal communications, Vol. 37, pp. 371-385, 2006.


[28] K. Aksoy and U. Aygolu: “Super–orthogonal space-time frequency trellis coded OFDM”, IET communications, Vol. 1 No. 3, pp. 317-324, 2007.

[29] O. Sokoya and B. T. Maharaj: “Performance of

space-time coded orthogonal frequency division multiplexing scheme with delay spreads”, Proceedings: IEEE MELECON, pp. 198-203. 2008.

[30] J. G. Proakis and M. Salehi: Digital

Communications, Mc Graw- Hill. New York, NY, USA, 5th edition, 2008.

[31] S. Benedetto, D. Divsalar, G .Montorsi and F.

Pollara: “A Soft-Input Soft-Output APP module for iterative decoding of concatenated codes”, IEEE Communications Letters, Vol. 1 No.1, pp. 22-24. 1997.

[32] J. Flores, J. Sanchez and H. Jafarkhani: “Quasi-

orthogonal space-time frequency trellis codes for two transmit antenna”, IEEE Transactions on wireless communications, Vol. 9 No. 7, pp. 2125-2129, July 2010.

[33] I. B. Oluwafemi and S. H. Mneney: “Performance of extended super orthogonal space-time trellis coded OFDM systems”, Proceedings: Southern African Telecommunications Networks and Applications Conference (SATNAC), Spier Estate, Stellenbosch, South Africa, pp. 35-40, 5th - 8th September 2010.

[34] Digital Land Mobile Radio Communications.

Commission of the European Communities, Luxembourg, Belgium, COST 207, 1989.

[35] V. Gulati and K. R. Narayanan: “Concatenated codes

for fading channel based on recursive space-time trellis codes”, IEEE Transactions on Wireless Communications, Vol. 2 No. 1, pp. 118-128, Jan. 2003.

[36] L. Hanzo, T. H. Liew and B. L. Yeap: Turbo coding,

turbo equalization and space-time coding for transmission over fading channel. John Willey and Son, England, pp. 411-414, 2002.


FAULT DIAGNOSIS OF GENERATION IV NUCLEAR HTGR COMPONENTS USING THE ENTHALPY-ENTROPY GRAPH APPROACH C. P. du Rand* and G. van Schoor** * School of Electrical, Electronic, and Computer Engineering, North-West University, Potchefstroom Campus, Private Bag X6001, Potchefstroom, 2520, South Africa E-mail: [email protected] ** Unit for Energy Systems, North-West University, Potchefstroom Campus, Private Bag X6001, Potchefstroom, 2520, South Africa E-mail: [email protected] Abstract: Fault diagnosis (FD) is an important component in modern nuclear power plant (NPP) supervision to improve safety, reliability, and availability. In this regard, a significant amount of experience has been gained in FD of generation II and III water-cooled nuclear energy systems through active research. However, new energy conversion methodologies as well as advances in reactor and component technology support the study of different FD methods in modern NPPs. This paper presents the application of the enthalpy-entropy (h-s) graph for FD of generation IV nuclear high temperature gas-cooled reactor (HTGR) components. The h-s graph is adapted for fault signature generation by comparing actual operating plant graphs with reference models. Multiple input feature sets (patterns) are generated for the fault classification algorithm based on the error, area, and direction of the fault residuals. The effectiveness of the FD method is demonstrated by classifying 24 non-critical single faults in the main power system of the Pebble Bed Modular Reactor (PBMR) during normal steady state operation as well as load following of the plant. Reference and fault data are calculated for the thermo-hydraulic network by means of a simulation model in Flownex® Nuclear. The results show that the proposed FD method produces different uncorrelated fault signatures for all the examined fault conditions. Keywords: Fault diagnosis, enthalpy-entropy (h-s), high temperature gas-cooled reactor (HTGR), Pebble Bed Modular Reactor (PBMR), nuclear power plant (NPP).

1. INTRODUCTION

High levels of safety, reliability, and availability are challenging technology goals in the development of the next-generation nuclear power plants (NPPs) [1]. Accurate fault diagnosis (FD) plays an important role in these safety-critical systems providing crucial information regarding component health [2]. This work considers the fourth generation nuclear high temperature gas-cooled reactor (HTGR) energy system, with thermo-hydraulic models such as the Gas Turbine-Modular High Temperature Reactor (GT-MHR), and the Pebble Bed Modular Reactor (PBMR). FD in NPPs comprises the fault isolation and identification tasks, and can be classified into methods that either utilise model-based or process history based (data-driven) techniques [2] - [4]. Model-based methods require an explicit mathematical system model and knowledge about the domain as well as the relationships between the different patterns of fault evolution. In practice, model-based methods are difficult to implement for systems with complex nonlinear dynamics such as NPPs, and are therefore mostly limited to linear applications or linear model approximations [2], [5]. Data-driven methods on the other hand, develop models from historical input-output process measurements, and require high quality (calibrated) and volume training data [2], [6]. These techniques are extensively studied for

application in current water-cooled NPPs owing to the availability of system data and knowledge [2], [7 - 11]. Ma et al. [2] presents a complete review of current model-based and data-driven FD methods for monitoring different subsystems in NPPs. Transients in NPPs can be initiated by component failure or malfunction [2], [11]. During a transient period, plant measurements develop unique patterns that differ from normal operating conditions. To date, artificial neural networks (ANNs) are the mostly studied algorithm for pattern classification in NPPs [2], [11]. However, practical applications in NPPs are still limited [2]. The main disadvantages of ANNs are their “black-box” structure (they do not offer justification or reasoning for their output), inflexibility regarding network adjustments (retraining needed), and the high volume of training samples required. Given these challenges, this research adapts the enthalpy-entropy (h-s) graph originally employed for analysis of a process’s thermodynamic cycle to facilitate FD of new HTGR components. The application of this method inherently provides a transparent classification model that can be physically interpreted (i.e. heat transfer and work of different thermo-hydraulic sub-processes) while supervising only a minimum number of system variables. Furthermore, the proposed approach does not require large quantities of historical plant data for model development or retraining of the entire network for changes in reference parameters.

[email protected]

[email protected]


FD via the h-s graph approach is accomplished by comparing the actual operating h-s graph with a reference graph model. Changes in the shape of the supervised graph are therefore associated with specific component malfunctions. Representative fault signatures (patterns) are generated via multiple ensembles named the error and area error methods. Classifiers with multiple input feature sets obtained via different feature extraction methods normally exhibit complementary classification behaviour [12]. The application of the proposed FD approach is demonstrated by classifying 24 non-critical component faults in the PBMR during normal steady state operation as well as load following of the plant.

2. THE FAULT DIAGNOSIS APPROACH

2.1 Description of the HTGR The HTGR under investigation, the PBMR, is a fourth generation nuclear energy system concept that employs a single-shaft, closed-loop direct Brayton thermodynamic cycle with helium gas as the primary coolant. This energy conversion methodology offers several advantages over traditional water nuclear reactor systems concerning operation, safety, and economics [13]. The HTGR design parameters allow for a maximum pressure of 9 MPa, reactor inlet and outlet temperatures of 500 °C and 900 °C respectively, and a 165 MWe output for a 400 MWt input. Figure 1 depicts the power cycle and the interconnection of the different main power system (MPS) components. The figure shows the gas flow path through the MPS and the eight thermo-hydraulic sub-processes. The power output of the MPS is manipulated by regulating the gas inventory via the pressure differential between the MPS and the inventory control system (ICS). By using only the ICS, maximum cycle efficiency is achieved at all power levels for normal power operation [14].

Figure 1: Schematic layout of the MPS.

2.2 Analysis of the HTGR thermodynamic cycle Normal irreversibilities in the thermodynamic cycle, such as fluid friction, pressure losses, heat leakage, and non-isentropic compression and expansion losses, are best described using an h-s graph. The properties h and s facilitate a “non-black-box” fault classification structure that is physically interpretable. The property h describes the system’s internal energy as well as the energy that is required to start fluid flow. The change in h for helium gas is defined as [15]

( )2 1 2 1ph h h c T TΔ = − = − (1) with cp and T the constant pressure specific heat (temperature independent) and temperature respectively. The property s signifies the amount of internal energy that is not converted into work, and is applied to calculate the theoretical limits of energy conversion. The change in specific s is given by [16]

2 22 1

1 1

ln lnpT P

s s s c RT P

Δ = − = − (2)

where T, R, and P denote the temperature, gas constant, and pressure respectively. Normal system irreversibility alters the characteristics of the ideal cycle and accordingly, changes the shape of the h-s graph. Therefore, if the normal system irreversibility is known (practical cycle), discrepancies can be identified in the actual operating cycle. Plant transients cause the actual cycle graph to deviate from the reference graph model, thus producing residuals (errors) for h and s. The residuals are then used to generate different fault signatures (patterns) that correspond to the specific component malfunction. Figure 2 illustrates the proposed FD approach graphically. The node numbers in the figure correspond to the different sub-processes in the power cycle. The primary power control mechanism of the HTGR under investigation produces a reference h-s graph model that remains relatively constant over the range of normal power operation. Therefore, different operating points are described with only one reference model. Figure 3 shows the practical cycle of the PBMR on a T-P and h-s graph for minimum and maximum continuous rating (MCR). 2.3 The error enthalpy-entropy method The first FD method utilises the residual errors between the reference and actual operating h-s graphs to generate fault signatures

( )( ) ( )

( )1, 2, ,ref actual

ref

i x ifs i i n

i

xx

−= = (3)


Figure 2: Fault diagnosis using the h-s graph.

with x denoting either h or s respectively, xref (i) the reference node value, xactual (i) the actual node value, and i the node number (Figure 2). The actual and reference s are calculated independently by assigning reference values in (2) to T1 and P1. However, (3) produces uncorrelated fault signatures for dynamic fault conditions. Accordingly, the h and s signatures are normalised to obtain graphic patterns that are independent from the fault magnitude. The normalised signatures are described by

( ) ( )( )1

1, 2, ,maxnorm n

i

fs ifs i i n

fs i=

= = (4)

with max |fs (i)| the maximum value for all the nodes.

Figure 3: T-P and h-s graphs of the thermodynamic cycle.

Figure 4 presents an example of the normalised h and s signatures of a turbine pressure ratio fault for various fault magnitudes (percentage of maximum). The figure shows that the signatures are highly correlated with regard to fault direction and magnitude. Therefore, dynamic fault symptoms are described with one normalised h and s reference signature. This characteristic reduces the fault database to a minimum. 2.4 The area error enthalpy-entropy method The second FD method utilises the area and direction of the residual shift between the reference and the actual operating h-s graphs to generate signatures. On each graph, two consecutive nodes (i) and (i + 1) are employed to describe the residual shift by means of a 2-dimensional boundary area. Figure 5 shows a graphical illustration of this signature generation method. The distance between two graph nodes is given by [17]

( ) ( ) ( ) ( ) ( )2 2- -j l j lr d h k h m s k s m= + (5)

where j, l signify the reference a or fault b graphs, and k, m denote nodes i or (i + 1). Equation (5) is computed for nodes that are on the same or different graphs by setting j = l or k = m respectively. The residual area is then described according to the following fault graph variations: 1. the area is an irregular quadrilateral if the fault

directions of nodes (i) and (i + 1) are the same (e.g. area 1 in Figure 5);

2. the area is triangular if only one fault node shifts (e.g. area 4 in Figure 5); and

3. the total area is given by the sum of two triangular areas if the fault directions of nodes (i) and (i + 1) differ (e.g. area 3 in Figure 5).

For the first variation, the area is determined by dividing the bounded segment into two triangular areas via a diagonal [18]. In the third variation, an intermediate value is defined at the graphs’ crossover point to establish two individual triangular areas.

Figure 4: Normalised h and s error signatures for a decrease in turbine pressure ratio fault.

0 2000 4000 6000 8000 100000

200

400

600

800

1000

1200

Pressure (kPa)

Tem

pera

ture

(ºC

)

MPS at 100% MCRMPS at 40% MCR

0 2 4 6 8 100

1000

2000

3000

4000

5000

Entropy [kJ/(kg.K)]

Ent

halp

y [k

J/kg

]

MPS at 100% MCRMPS at 40% MCR

1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Nor

mal

ised

erro

r

Node number

h for 1 %h for 5 %h for 10 %s for 1 %s for 5 %s for 10 %

h

s

6

1 3

24 5 7

8

s h

Practical thermodynamic cycle (no faults) Ideal theoretical thermodynamic cycle Actual thermodynamic cycle (faults)


Figure 5: Description of the residual area errors between

the reference and actual h-s graphs. In the first method, different h and s signatures can be derived since (3) is calculated independently for each property. However, the residual areas derived via the area error method jointly summarise the shift in h and s. Therefore, to generate independent h and s signatures, separate fault directions are defined for h and s at fault node (i). Next, the h and s fault directions are multiplied with the corresponding residual area. The h and s area error fault signatures fsarea (i) are computed using

( ) ( )( )( )( ) ( )1

.q

area p p p pfs i s s x s y s z dir i= − − −

(6) with sp the semiperimeter of each triangular area, x, y, and z the triangle side lengths, q the number of triangular areas, and dir (i) the h or s fault direction at node (i) (i.e. positive or negative). Similar to (4), (6) is normalised to obtain signatures that are independent from the fault magnitude. Figure 6 shows the normalised h and s area error signatures for the turbine pressure ratio fault. If compared to Figure 4, it can be seen that different uncorrelated h and s signatures describe the component malfunction. The results therefore demonstrate that different fault feature sets can be extracted for the classification procedure via the h-s error and area error methods respectively.

3. APPLICATION OF THE FAULT DIAGNOSIS APPROACH

3.1 Modelling the HTGR The MPS shown in Figure 1 is modelled using Flownex®

Nuclear [19]. The main advantage of using Flownex® is that transient simulations are supported as well as advanced nuclear reactor models. Figure 7 shows the thermo-hydraulic model as a component network. The operating point of the model is manipulated by controlling the reactor outlet temperature (ROT), the

pressure in the high pressure manifold (via the ICS), and the bypass valves. The system parameters for normal power operation at 100 % MCR are: ROT 900 °C, manifold pressure 9 MPa, compressor bypass valves fully closed, and the turbine-compressor shaft speed 6000 r/min (generator synchronised to power grid). Injection and extraction of helium gas are limited by a maximum ramp rate during MPS power output control. The T and P signals are monitored at the inlet nodes of the eight sub-processes to compute (1) and (2). 3.2 Categorisation of the HTGR fault classes Three classes of component faults are identified via simplified cycle analyses of a three-shaft HTGR [20, 21]. The parameters and range of variation also attempt to describe many system uncertainties, originating from design, manufacturing, measuring, control, etc. The fault classes comprise variations in: 1. leak flows of the primary working fluid; 2. pressure losses in the primary circuit; and 3. component effectiveness or efficiency. Leakage flows between system nodes are emulated using a resistive element with discharge coefficient (represents a throttling process) [19]. The secondary loss coefficients (k) of the inlet pipe models are changed to vary the pressure losses around the primary circuit. The heat transfer area between the hot and cold streams is decreased to reduce the effectiveness of the heat exchangers. The performance characteristics of the turbo machinery are varied by scaling the pressure ratio and efficiency as functions of corrected mass flow rate for various speed curves [19]. This study considers 24 non-critical single faults in the HTGR MPS that mainly influence the thermodynamic performance of the plant. The resulting fault transients are characterised by incipient time behaviour (small fault perturbation) and will therefore not initiate a PCU trip or reactor SCRAM (emergency shutdown of reactor). Table 1 presents a summary of the 24 single faults.

Figure 6: Normalised h and s area error signatures for a decrease in turbine pressure ratio fault.

1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Nor

mal

ised

are

a er

ror

Node number

h for 1 %s for 1 %

1a

1b

2a2b

3a

3b

4a 4b

area 1 area 3

area 2

s

h

area 4

8a

8b

5b 5a

3im

reference graph a fault graph b


Figure 7: The Flownex® simulation model of the MPS.

Table 1: Summary of the single faults in the MPS

Description Direction Fault parameter 1. Pre-cooler heat transfer area Decrease a (m2) 2. Intercooler heat transfer area Decrease a (m2) 3. Recuperator heat transfer area Decrease a (m2) 4. Recuperator HP to LP leakage Increase a (m2) 5. LPC pressure ratio Increase scaling factor 6. LPC pressure ratio Decrease scaling factor 7. LPC efficiency Decrease scaling factor 8. HPC pressure ratio Increase scaling factor 9. HPC pressure ratio Decrease scaling factor 10. HPC efficiency Decrease scaling factor 11. Turbine pressure ratio Increase scaling factor 12. Turbine pressure ratio Decrease scaling factor 13. Turbine efficiency Decrease scaling factor 14. HP manifold leakage to outlet Increase a (m2) 15. HP manifold leakage to inlet Increase a (m2) 16. LPC inlet pipe losses Increase k 17. Intercooler inlet pipe losses Increase k 18. HPC inlet pipe losses Increase k 19. Recuperator HP inlet pipe losses Increase k 20. Reactor inlet pipe losses Increase k 21. Turbine inlet pipe losses Increase k 22. Recuperator LP inlet pipe losses Increase k 23. Pre-cooler inlet pipe losses Increase k 24. Reactor core bypass leakage Increase a (m2)

3.3 Fault signature generation Normal steady-state operation of the MPS at 100 % MCR is utilised to model the reference fault-free system. The fault transients are modelled individually as abrupt faults with a 1 % bias value. A small fault perturbation is chosen to demonstrate early FD of incipient faults. Since

this work is mainly concerned with fault classification, it is assumed that the fault detection system recognises the onset of a component malfunction. As an example, Figure 8 depicts the signatures of different single faults. The figure shows that the correlation between the signatures is small for both signature generation methods regarding the nodes’ residual magnitude and direction. FD is important during load following of the plant to classify probable dynamic fault conditions. In this example, the following control characteristics are applied to manipulate the steady state operating point: 1. helium is injected and removed at a rate of 7.5 kg/s

from the MPS via the ICS; 2. ROT is controlled at 900 °C via the control rods; and 3. GBPC valves are partially opened and closed. Variations of the steady state reference graph model are utilised to facilitate GBPC valve operation. The results suggest that although GBPC valve operation changes the shape of the reference graph, the altered models remain relatively constant over the range of normal power operation. The fault transients evolve by way of an incipient time pattern (drift) representing a dynamic fault condition. A maximum fault magnitude of 2 % is reached at t = 4000 s. In practice, the examined fault symptoms will develop slowly over time. However, the fault evolution rate is increased to prevent lengthy simulations. Figure 9 shows the signatures of fault 18 during load following of the MPS. The figure shows that the correlation between the signatures is good for both signature generation methods. The results show that static


Figure 8: Normalised error and area error fault signatures for different fault classes.

signatures can describe dynamic fault conditions during normal power operation of the MPS, therefore reducing the size of the fault database.

4. CONCLUSION

This paper presented the h-s graph approach for FD of generation IV nuclear HTGR components. The h-s graph is implemented for fault isolation and identification, and aims to generate fault signatures (patterns) that are representative of NPP component malfunctions. The application of the proposed approach not only provides a transparent classification model (outputs can be interpreted), but also a flexible structure for model adjustment (e.g. retraining of entire model not required for changes in reference fault conditions). FD is realised by comparing the actual operating cycle h-s graph with a reference graph model that includes normal system irreversibilities. Deviations from the practical cycle graph are associated with specific component malfunctions and used to generate different fault feature sets based on the error, area, and direction of the residual. An inventory power control mechanism produces a practical reference graph model that remains relatively constant over the range of normal power operation, which greatly simplifies the computational complexity of the fault diagnostic system. Accordingly, one reference model as well as single fault signatures facilitate FD of various component faults in different sub-processes of the MPS.

Steady state as well as dynamic Flownex® simulations indicate that all the examined faults can be described with uncorrelated signatures for variations of the normal process. The results show that the proposed approach successfully extracts different fault characteristics via the h-s graph, which can provide a robust FD solution in generation IV HTGRs. Continued research will be focused on applications in different thermo-hydraulic systems.

5. REFERENCES [1] U.S. DOE Nuclear Energy Research Advisory

Committee (NERAC) and the Generation IV International Forum (GIF): A Technology Roadmap for Generation IV Nuclear Energy Systems, Report no. GIF-002-00, December 2002.

[2] J. Ma and J. Jiang: “Applications of fault detection and diagnosis methods in nuclear power plants: A review”, Progress in Nuclear Energy, Vol. 53 No. 3, pp. 255-266, April 2011.

[3] V. Venkatasubramanian, R. Rengaswamy, K. Yin and S.N. Kavuri: “A review of process fault detection and diagnosis - Parts I, II, III”, Computers and Chemical Engineering, Vol. 27 No. 3, pp. 293-346, March 2003.

[4] M.D. Shah: “Fault detection and diagnosis in nuclear power plant - A brief introduction”, IEEE Conference on Engineering - Current Trends in Technology (NUiCONE), India, pp. 1-5, December 2011.

Figure 9: Normalised error and area error fault signatures of fault 18 during variations of the normal process.

1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Nor

mal

ised

erro

r

Node number

h for f4h for f8h for f13h for f20s for f4s for f8s for f13s for f20

1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Nor

mal

ised

are

a er

ror

Node number

h for f4h for f8h for f13h for f20s for f4s for f8s for f13s for f20

1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Node number

Nor

mal

ised

erro

r

h for injection at t = 2100 sh for steady state at t = 2500 sh for extraction at t = 3000 sh for valve operation at t = 3650 ss for injection at t = 2100 ss for steady state at t = 2500 ss for extraction at t = 3000 ss for valve operation at t = 3650 s

1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

Node number

Nor

mal

ised

are

a er

ror

h for injection at t = 2100 sh for steady state at t = 2500 sh for extraction at t = 3000 sh for valve operation at t = 3650 ss for injection at t = 2100 ss for steady state at t = 2500 ss for extraction at t = 3000 ss for valve operation at t = 3650 s


[5] A. Evsukoff and S. Gentil: “Recurrent neuro-fuzzy system for fault detection and isolation in nuclear reactors”, Advanced Engineering Informatics, Vol. 19 No. 1, pp. 55-66, January 2005.

[6] E. Zio, F. Di Maio and M. Stasi: “A data-driven approach for predicting failure scenarios in nuclear systems”, Annals of Nuclear Energy, Vol. 37 No. 4, pp. 482-491, April 2010.

[7] B. Lu and B.R. Upadhyaya: “Monitoring and fault diagnosis of the steam generator system of a nuclear power plant using data-driven modelling and residual space analysis”, Annals of Nuclear Energy, Vol. 32 No. 9, pp. 897-912, June 2005.

[8] C.M. Rocco and E. Zio: “A support vector machine integrated system for the classification of operation anomalies in nuclear components and systems”, Reliability Engineering and System Safety, Vol. 92 No. 5, pp. 593-600, May 2007.

[9] B.R. Upadhyaya and K. Zhao: “Adaptive fuzzy inference causal graph approach to fault detection and isolation of field devices in nuclear power plants”, Progress in Nuclear Energy, Vol. 46 No. 3-4, pp. 226-240, August 2005.

[10] E. Zio and G. Gola: “Neuro-fuzzy pattern classification for faults diagnosis in nuclear components”, Annals of Nuclear Energy, Vol. 33 No. 5, pp. 415-426, March 2006.

[11] T.V. Santosh, G. Vinod, R.K. Saraf, A.K. Ghosh, and H.S. Kushwaha: “Application of artificial neural networks to nuclear power plant transient diagnosis”, Reliability Engineering and System Safety, Vol. 92 No. 10, pp. 1468-1472, October 2007.

[12] Y. Lei, Z. He, Y. Zi and Q. Hu: “Fault diagnosis of rotating machinery based on multiple ANFIS combination with GAs”, Mechanical Systems and Signal Processing, Vol. 21 No. 5, pp. 2280–2294, July 2007.

[13] P.G. Rousseau and G.P. Greyvenstein: “Changing the face of nuclear power via the innovative Pebble Bed Modular Reactor”, Proceedings: Power Generation World, South Africa, pp. 1-22, March 2004.

[14] C. Nieuwoudt: Demonstration plants operations summary, PBMR (Pty) Ltd., Revision 4, South Africa, November 2002.

[15] A. Bejan and A.D. Kraus: Heat transfer handbook, John Wiley & Sons, Inc., New Jersey, p. 111, June 2003.

[16] A. Bejan: Entropy generation minimization, CRC Press, Florida, pp. 1-16, 205-231, October 1995.

[17] L.F. Costa and R.M. Cesar: Shape analysis and classification: Theory and practice, CRC Press, second edition, Boca Raton Florida, April 2009.

[18] C.B. Clapham: Arithmetic for engineers, Clapham Press, 2007.

[19] Flownex®: General User & Library user manuals, Version SE Nuclear, M-Tech Industrial, South Africa, 2009.

[20] P.G. Rousseau, G.P. Greyvenstein, B.W. Botha and C.G. Du Toit: “Sensitivity analysis of the PBMR gas

cooled nuclear reactor cycle with the aid of a simplified simulation model”, IFAC Conference on Technology Transfer in Developing Countries, South Africa, pp. 1-5, July 2000.

[21] E. Van Der Linde: “Evaluating the PBMR main power system turbo machines with the aid of sensitivity and Monte Carlo analyses”, Proceedings: 2nd International Topical Meeting on High Temperature Reactor Technology, China, Paper D09, pp. 1-11, September 2004.


SIMULATION STUDY OF THE PERFORMANCE OF THE VITERBIDECODING ALGORITHM FOR CERTAIN M-LEVEL LINE CODES

Khmaies Ouahada ∗

∗ Department of Electrical and Electronic Engineering Science, University of Johannesburg, SouthAfrica. Email: [email protected].

Abstract: In this paper we study the performance of different classes of M-level line codes under theViterbi decoding algorithm. Some of the presented M-level line codes inherited the state machinestructure by using the technique of distance mappings which preserve the properties of binaryconvolutional codes. Other M-level line codes were enforced to have the state machine structure tomake use of the Viterbi decoding algorithm. The technique of spectral shaping was combined withdistance mappings to generate spectral null distance mappings (SNDM) M-level line codes.The 2-dB gain between soft and hard decisions decoding for the different classes of M-level line codesis investigated. The standard technique for assessing the stability and the accuracy of any decodingalgorithm, which is the error propagation is used to analyze the stability and the accuracy of the Viterbidecoding algorithm of the M-level line codes.The obtained results have shown advantages and outperformance of SNDM codes compared to the restof line codes presented in this paper.

Key words: Viterbi decoder, Soft/Hard decision, Error propagation, Line codes

1. INTRODUCTION

In literature, many authors’ works contributed towards thedevelopment of the design of multi-level line codes andthe improvements of their error-correction capabilities [1–4]. For certain applications, researchers have shown thatM-level line codes may be preferable to binary codesfor high speed digital transmission as in the case of theoptical fiber channel [5–8]. The additional signal levels orsymbols in a pulse amplitude modulated signal sequencecan be used to reduce the symbol rate and hence thebandwidth of the coded signal [9]. The lower switchingrate required can also be used to obtain higher data-transferrates in an optical local area network (LAN) system wherethe transmission rate is limited by complementary metaloxide semiconductor (CMOS) technology [10].

It is still common practice to use combinational logicdecoders for M-level line codes. As these codes areconsidered to be non-linear codes, we make use ofthe technique of distance mappings [12–14] to mappermutation sequences to the outputs of convolutionalcodes that can give our new M-level line codes the trellisstructure and thus make use of the Viterbi decodingalgorithm.

The spectral shaping technique used in this paper isto create nulls at certain specific frequencies includingthe lowest ones, which can give our new designedM-level line codes another advantage to overcome somecommunications problems like channels not transmittingzero frequency components.

This paper is organized as follows. Section 2 introducesbriefly the techniques of spectral shaping and distancemappings and presents a few examples of algorithms forthe design of the related class of M-level line codes.

Section 3 investigates the implementation of the Viterbidecoding to our designed codes and to a range of differentpublished M-level line codes. Simulation results for thebit error rate (BER) performance of these M-level linecodes in soft and hard decision to verify the 2-dB gainare also presented. The Viterbi decoding error propagationfor M-level line codes for the assessment of the stabilityand the accuracy of our Viterbi decoding algorithm isinvestigated in Section 4. Finally a conclusion is presentedin Section 5 to compare between the obtained resultsand present the advantages and disadvantages of codesinheriting the convolutional codes structures.

2. DESIGN OF M-LEVEL LINE CODES

The techniques of spectral shaping and distance mappingsare combined and implemented to permutation sequencesto generate our new designed M-level line codes [15]. Asline codes are usually DC-free codes, we make use of thespectral shaping technique to shape the spectrum of ourM-level line codes to suit certain applications. The useof distance mappings technique is actually for the purposeof having codes with better error correction capabilityinheriting the trellis structure from the base codes whichare the convolutional codes. This makes the use of theViterbi decoding algorithm possible.

2.1 Spectral Shaping M-level Line Codes

Shaping the spectrum of any sequence whether it is binaryor non-binary to create nulls at certain frequencies is thesame as forcing the power spectral density (PSD) functionto zero at those corresponding frequencies [16]. Thespectral shaping technique is usually applied to basebanddata stream, which is represented by the vector y =(y1,y2, . . . ,yM). We make use in this paper of spectral


null equations to create nulls at rational submultiples of thesymbol frequency. The reason of the use of non binary orpermutation sequences is to be able to generate multilevelpulse amplitude modulated signals. The design of DC-freeM-level line codes is also considered in our work.

For a codeword of length, M, there exists an integermultiple of k, where

M = ks.

The frequency of value f = r/k represents the spectralnulls at rational sub multiples r/k, with r as an integer. Togenerate nulls at those frequencies, we have to satisfy [17]

A1 = A2 = · · ·= Ak, (1)

If all the codewords in a codebook satisfy these equations,the codebook will exhibit nulls at the required frequencies.This is true for binary and non binary sequences despiteshaping non-binary sequences is much complicated thanthe binary ones since more constraints should be takeninto considerations. Hence, we have to choose a suitablepermutation sequences with suitable pulse amplitudechannel symbols to satisfy the spectral shaping equationin (1).

Each permutation symbol (PS) is mapped to a channelsymbol (CS), which represents the level of the signal. Ingeneral, for odd values of M, the symbol mapping is

PS: 0 1 · · · M−12 · · · M−2 M−1

↓ ↓ ↓ ↓ ↓CS:−M−1

2 −M−32 · · · 0 · · · +M−3

2 +M−12

and for even values of M, the symbol mapping is

PS: 0 1 · · · M−22

M2 · · · M−2 M−1

↓ ↓ ↓ ↓ ↓ ↓CS:−M

2 −M−22 · · · −1 +1 · · · +M−2

2 +M2

As an example, if we allocate the channel symbols of−3 −1 +3 +1 to the permutation sequence 0132, then we willguaranty nulls at frequencies 0, 1/2 and 1 since M = 4 andtherefore k = 2.

2.2 Decoding Algorithm for M-level Line Codes

As was mentioned in the introduction, M-level line codesusually use combinational logic decoders. To get benefitof the Viterbi decoder, we make use of the distancemappings technique to present our new designed M-levelline codes in a state machine [18] form. The techniqueis simply mapping the outputs of a convolutional code toother codewords from a code with lesser error-correctioncapabilities. In our case we use the spectrally shapedpermutation sequences. This mapping will allow us toobtain suitably well shaped output code sequences andbetter decoded by using the Viterbi algorithm [13, 19, 20].

Our new codes are also called as M-level line trellis codesin view of their trellis structure that is inherited from thebase codes, the convolutional codes.

The technique of mapping was introduced by Ferreira et alin their papers [12] and [13], where they have shown howthe output binary n-tuple code symbols from an R = m/nconvolutional code can be mapped to non-binary M-tuplepermutation code symbols, thereby creating a permutationtrellis code.

Ferreira et al have introduced two types of matrices D =[dij] and E = [eij], which are respectively related to theHamming distances between the codewords of the basecode, the convolutional code and the mapped code, theM-level line code.

As an example, we take the mapping of the set of binary2-tuple code symbols, {00,01,10,11} to a set of 4-tuplesspectral null codewords, {0123,0231,3102,3201}.For this mapping we have

D =

⎡⎢⎣

0 1 1 21 0 2 11 2 0 12 1 1 0

⎤⎥⎦ and E =

⎡⎢⎣

0 2 2 42 0 4 22 4 0 24 2 2 0

⎤⎥⎦ .

It is clear that ei j ≥ di j + 1, ∀i �= j, and this guarantees anincrease in the distance of the resulting code.

In general, if ei j ≥ di j + δ, δ ∈ {1,2, . . .}, ∀i �= j we callsuch mappings distance-increasing mappings (DIMs). Inthe case where ei j ≥ di j, ∀i �= j and equality achievedat least once, we have distance-conserving mappings(DCMs). Finally, if ei j ≥ di j +δ, δ∈ {−1,−2, . . .}, ∀i �= j,we have distance-reducing mappings (DRMs).

Since we make use of spectral null technique combinedwith distance mapping technique, we denote our newdesigned M-level line codes as SNDM-line codes.

2.3 Examples of Designed Codes

To design our M-level line codes with well shapedspectrum, we need to start with a permutation sequencethat will lead to the construction of our multilevelcodebook. By using the property of commutativityfor addition between the variables in (1) to permutethe channel symbols of these variables and to keep thespectral null property satisfied, we can generate ourspectrally shaped M-level line codewords or sequences.The swapping of our permutation symbols needs aspecial construction algorithm that will help in avoidingrepetitive sequences which cause loss on the calculationof distances between the generated codewords of our codeand therefore the error correction capability will be less.Here we make use of the cube graph construction [21],which has proven to be optimum.

As was explained before, in the spectral null equation andfor sequences of length M = ks, we have k groupings with


0 0.2 0.4 0.6 0.8 102468

10121416


P. S

. D.

Figure 1: PSD for M = 8 and k = 4

s symbols in each grouping. Hence, our mappings willconsist of several smaller mappings. For each Ai we havea mapping of length s that is used to permute the yi in thegrouping. For all the Ai as a grouping we have anothermapping of length k that is used to permute the Ai. Thefollowing example illustrates this.

Example 1 For M = 8, with k = 4 and s = 2, we have thefollowing spectral null equation:

s=2︷︸︸︷y1 + y5 =

s=2︷︸︸︷y2 + y6 =

s=2︷︸︸︷y3 + y7 =

s=2︷︸︸︷y4 + y8︸︷︷︸︸︷︷︸︸︷︷︸︸︷︷︸

A1 = A2 = A3 = A4︸︷︷︸k=4

(2)

Let swap(ya,yb) denote the swapping of symbols in thevariables ya and yb. The following sequences can then beobtained from the original SN sequence. All swaps arepresented in (3).

The swaps in (3), can be written in an algorithm formto show in details all steps in the generation of our newdesigned M-level line codes. The inputs xi represent theoutputs of the convolutional codes.

Input: (x1,x2,x3,x4,x5,x6,x7,x8)Output: (y1,y2,y3,y4,y5,y6,y7,y8)(y1,y2,y3,y4,y5,y6,y7,y8)← (0,1,2,3,7,6,5,4)beginif x1 = 1 then swap(y1,y5)if x2 = 1 then swap(y2,y6)if x3 = 1 then swap(y3,y7)if x4 = 1 then swap(y4,y8)if x5 = 1 then swap(y1,y2)(y5,y6)if x6 = 1 then swap(y3,y4)(y7,y8)if x7 = 1 then swap(y1,y3)(y5,y7)if x8 = 1 then swap(y2,y4)(y6,y8)

end.

�

We can see from the algorithm that the permutationsequence that we start our swapping with is 01237654 andthis to make sure that the corresponding channel symbolsfor this sequence satisfies the spectral null equation (2).The designed M-level line code will generate spectral nullsat frequencies 0, 1/4, 1/2, 3/4 and 1 as depicted in Fig. 1.

0 0.2 0.4 0.6 0.8 102468

101214


P. S

. D.

Figure 2: PSD for M = 8 and k = 2

Example 2 For the case of M = 8, with k = 2 and s = 4,we have the following spectral null equation:

s=4︷︸︸︷y1 + y3 + y5 + y7 =

s=4︷︸︸︷y2 + y4 + y6 + y8︸︷︷︸︸︷︷︸

A1 = A2︸︷︷︸k=2

(4)

The corresponding algorithm to generate our M-level linecode is as follows:

Input: (x1,x2,x3,x4,x5,x6,x7,x8,x9)Output: (y1,y2,y3,y4,y5,y6,y7,y8)(y1,y2,y3,y4,y5,y6,y7,y8)← (0,1,3,2,4,5,7,5)beginif x1 = 1 then swap(y1,y3)if x2 = 1 then swap(y5,y7)if x3 = 1 then swap(y1,y5)if x4 = 1 then swap(y3,y7)if x5 = 1 then swap(y2,y4)if x6 = 1 then swap(y6,y8)if x7 = 1 then swap(y2,y6)if x8 = 1 then swap(y4,y8)if x9 = 1 then swap(y1,y2)(y3,y4) · · ·

· · ·(y5,y6)(y7,y8)end.

When the corresponding channel symbols for thissequence satisfies the spectral null equation (4), thedesigned M-level line code will generate spectral nulls atfrequencies 0, 1/2, and 1 as depicted in Fig. 2. �

In general, we have to conduct the swaps based on thek-cube construction algorithm [21] to respect the distancebetween the indices of the permutation sequences yi.And the general algorithm for our codes mapping issummarized as follows:

1. Comparison between n ( convolutional outputscodewords length) and M (permutation sequencelength).

(a) n > M: Reducing mappings

(b) n < M: Increasing mappings

(c) n = M: Conserving mappings


A1︷︸︸︷y1 + y5 =

A2︷︸︸︷y2 + y6 =

A3︷︸︸︷y3 + y6 =

A4︷︸︸︷y4 + y8

SN sequence −7 + +7 =−5 + +5 =−5 + +5 =−5 + +5

swap(y1,y5) +7 + −7 =−5 + +5 =−5 + +5 =−5 + +5

swap(y2,y6) −7 + +7 = +5 + −5 =−5 + +5 =−5 + +5

swap(y1,y5) +7 + −7 =−5 + +5 =−5 + +5 =−5 + +5

swap(y2,y6) −7 + +7 = +5 + −5 =−5 + +5 =−5 + +5

swap(y1,y2)(y5,y6) −5 + +5 =−7 + +7 =−5 + +5 =−5 + +5

swap(y1,y2)(y5,y6) −5 + +5 =−7 + +7 =−5 + +5 =−5 + +5

swap(y1,y2)(y5,y6) −5 + +5 =−7 + +7 =−5 + +5 =−5 + +5

swap(y1,y2)(y5,y6) −5 + +5 =−7 + +7 =−5 + +5 =−5 + +5

(3)

2. Mappings

• xi, 1≤ n

• y j, 1≤ j ≤ 2�log2M�

• for i = 1 : n if xi = 1 then swap (y j,y j+1) end.

3. VITERBI DECODING M-LEVEL LINE CODES

In the literature, it was shown that with soft decisionsViterbi decoding, we have an improvement of 2 dB gainover hard decisions [22].

Following are a few examples of published M-level linecodes including our designed codes. We run our simulationfor soft and hard decisions for each of these M-level linecodes and see if all of them have achieved the 2 dB gaindifference between soft and hard decisions.

All M-level line codes investigated in this section arerepresented in a state machine form and this for the sakeof using the Viterbi decoding algorithm.

The values of bit error rate (BER) corresponding todifferent values of signal to noise ration (SNR) for themapped code are less than those of the base code. Besidesthat we have to emphasize the fact that mapped codeusing the Euclidean distance is slightly outperforming themapped code using the Hamming distance. This is almostrelated to the fact that the Euclidean distance is used torefine the distance metrics and thus a soft decision couldbe used.

3.1 Three-Level Line Codes: Ternary Line Codes

Ternary line codes [24, 25] are often used on channelssuch as PCM metallic cable systems with transformerdecoupling and repeaters. As the first world countrieshave started moving to high speed DSL technologies, usingother modulation techniques, the developing countrieshowever, the already installed digital subscriber loops,utilizing line codes such as AMI or HDB3, will still haveto function for many years to come.

High Density Bipolar n (HDBn): This class of ternary linecodes, used in European countries, has a maximum number

01/0

10/−

10/+00/−

HDBn

xx/- xx/0

xx/+xx/001/0 00/0

00/0

01/0

10/−00/−

01/0

10/+

11/−

11/+

11/−

Figure 3: State Machine of HDBn.

of consecutive zeros to be limited to n. HDBn codes areconsidered to be good codes for better synchronization ofreceiver and transmitter and with low frequency cut-offpoint provided in power spectral density function. Fig. 3shows the general form of the state machine of this classof codes.

If we take the case of n = 3, the digital data in HDB3encoding is represented in almost identical fashion toAMI except for allowances made to accommodate certainviolation as will be explained later.

The patterns of HDB3 codes are described as there isno changes in voltage for a sequence of 0s is solved bychanging any incidence of four consecutive ‘0’ bits into

0 2 4 6 8 10 12 14 16 18 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR (dB)

BER

Hard decisionSoft decision

Figure 4: HDB3: Viterbi decoding soft and hard decisions.


00/0

xx/+xx/-

xx/0 xx/-

xx/+

01/0

10/+

11/+

10/−11/−

xx/0xx/+

xx/-

xx/- xx/+

01/0

00/0

Figure 5: State Machine of BnZS.

a stream containing 000V, where the polarity of the Vbit is the same as the previous non-0 voltage (oppositeto a ‘1’ bit, which causes a V signal with an alternatevoltage according to the previous one). But a new problemarises - because the polarity of the non-zero bits is thesame, a non-zero DC level is formed. This is overcomeby changing the polarity of the V bit to the opposite ofthe previous V bit. This changes the bit stream to B00V,where the polarity of the B bit is the same as the polarityof the V bit. The change “fools” the receiver into thinkinga received B bit is a ‘1’ bit, but when it receives the V bit(with the same polarity), it understands the B and the V bitsas a ‘0’. In HDB3, the maximum number of consecutivezeros allowed in the substituted string is 3.

Using the same simulation setup as previously done withAMI, we found that the difference between Soft and Harddecisions decoding is near the 2 dB gain at the BER = 10−6

as depicted in Fig. 4.

Binary n Zeros Substitution (BnZS): The encoder forBnZS codes uses the 0VB0VB filling pattern. In Fig. 5,the convention of using a bold transition arrow labeledXX/output has been introduced to indicate that transitionsand outputs for all four input combinations are the same.Otherwise, the input of the encoder is arranged in the sameway as the output of detection, where the most significantbit represents the delayed data and the least significant bitrepresents the all-zeros flag.

A simple modification to the output code converts thefilling sequence to B0VB0V, or indeed any desired fillingpattern. Care must be taken only to ensure that the firstsigned pulse of the filling sequence is indeed a B or V pulseas required.

Using the VBVB filling pattern, the B4ZS line codeconsists of 18 states arranged symmetrically around ahorizontal center line.

Every transition from a state in the upper-half, followinga data ‘1’, has its destination in the lower half, and viceversa. This feature corresponds to adherence to the bipolaralternation rule. The pair of states at the left-hand side ofthe state diagram is occupied whenever the data contains along string of consecutive data 1s. They can be consideredto be the remnants of the parent bipolar encoder, with itsdata ‘0’ self-loops replaced by the remaining 16 states.

0 2 4 6 8 10 12 14 16 18 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR (dB)

BER

Hard decisionSoft decision

Figure 6: B4ZS: Viterbi decoding soft and hard decisions.

One of the pair of states at the right-hand side of thediagram is occupied whenever a data ‘1’ is followedby 3 consecutive data 0s. Exiting from them on adata ‘0’ corresponds to commencing the filling sequence.Considering just the upper state of the pair, it is enteredonly by an arc associated with the previous output +,and on data ‘0’, it begins producing the output sequence+−−+, that is VBVB.

Using the same simulation setup, Fig. 6 shows the 2 dBgain between hard and soft decisions Viterbi decoding.

SNDM-3Binary 6Ternary Line Code: (SNDM-3B6T): Aswas explained earlier, this class of codes is actuallythe combination of two techniques, which are thespectral shaping and distance mappings techniques. TheSNDM-3B6T code is a ternary line code where we map3 binaries to six ternaries. Our base code which is theconvolutional code has a code rate of R = 3/4 and aconstraint length of K = 3. The four bit outputs will bemapped onto six permutation symbols which are in factrepetitive symbols for the sake to drop the pulse amplitudemodulated levels to three as it will be depicted in thefollowing algorithm.

We consider the case of M = 6 with k = 2 and s = 3. Thechannel symbols must satisfy

s=3︷︸︸︷y1 + y3 + y5 =

s=3︷︸︸︷y2 + y4 + y6.︸︷︷︸︸︷︷︸

A1 = A2︸︷︷︸k=2

(5)

We can see from (5) that we can assign two input bits toswap yi in each equation, which means that we need fourinput bits in total to swap all symbols. Since we havechosen by purpose that all symbols are the same in eachequation, then we do not need to swap A1 and A2 andtherefore no need for extra input bits for swapping. Thusthe convolutional base code may have a rate of R = 3/4.


110/++–00

00 10

01 11

101/–++–00

000/–++00–100/0–+0–+

000/–0+–0+

010/

++

–00–

110/

0+–

–+0

000/

00+

––+

100/

––+

00+ 01

0/+0–

–0+

110/

0––0

++

001/

–0+–0

+

101/

0–+0–

+

011/0+–0+

–

010/00– – ++110/+– – 00+

011/++–00–

111/0+– –+0

100/0++– –0

001/00+– –+

101/– –+00+

000/0++0–

–

111/++––00

011/+0– –0+

111/0– –0++

011/0++0– –

100/–++–00

010/0+–0+–

001/–++

00–

101/0++

––0

111/+–

–00+

011/00––

++

Figure 7: State machine of SNDM-3B6T.

0 2 4 6 8 10 12 1410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

SNR (dB)

BER

Soft decisionHard decision

Figure 8: SNDM-3B6T: Viterbi decoding soft and hard

decisions.

The mapping algorithm for the SNDM-3B6T code isdescribed below.

Input: (x1,x2,x3,x4)Output: (y1,y2,y3,y4,y5,y6)(y1,y2,y3,y4,y5,y6)← (0,1,2,0,1,2)beginif x1 = 1 then swap(y1,y3)if x2 = 1 then swap(y1,y5)if x3 = 1 then swap(y2,y4)if x4 = 1 then swap(y2,y6)

end.

The state machine of the resultant SNDM-3B6T code, ispresented in Fig. 7.

The BER performance of this code is depicted in Fig. 8.We can see clearly the 2 dB gain between soft and harddecisions at BER = 10−6.

0 2 4 6 8 10 12 14 16 18 20 2210-7

10-6

10-5

10-4

10-3

10-2

10-1

1

SNR (dB)

BER


Figure 9: SNDM-7B8Q: Viterbi decoding soft and hard

decisions.

3.2 Four-Level Line Codes: Quaternary Line Codes

Quaternary line codes, as the 2B1Q line code, have beenused in transmission and also been used in modem ISDNcircuits. Following are a few examples of quaternary linecodes.

SNDM-7Binary 8Quaternary (SNDM-7B8Q): We considerthe example of permutation sequences of length M = 8 forthe case of k = 2 and s = 4. The symbols in each groupingAi are equal or repeating and the resultant permutationsequence is an eight symbol sequence with four channelsymbols, which will be used to generate a 4-Level linecode. As explained previously, we need 8 bits from theconvolutional code’s output to be able to swap all the yisymbols. Since the two groupings A1 and A2 are equal,then there is no need for input bits to swap them. Thereforethe convolutional base code may have a rate of 7/8. Themapping algorithm for the SNDM-7B8Q code is describedbelow.

Input: (x1,x2,x3,x4,x5,x6,x7,x8)Output: (y1,y2,y3,y4,y5,y6,y7,y8)(y1,y2,y3,y4,y5,y6,y7,y8)← (0,0,1,1,2,2,3,3)beginif x1 = 1 then swap(y1,y3)if x2 = 1 then swap(y3,y7)if x3 = 1 then swap(y1,y5)if x4 = 1 then swap(y3,y7)if x5 = 1 then swap(y2,y4)if x6 = 1 then swap(y6,y8)if x7 = 1 then swap(y2,y6)if x8 = 1 then swap(y4,y8)

end.

The BER performance of this code is depicted in Fig. 9.We can see clearly the 2 dB gain between the soft and harddecisions.


Table 1: Encoder for 2B1QI line code

Inputs

00 01 10 11

State Outputs Next State Outputs Next State Outputs Next State Outputs Next State

A 0 B -1 C +1 D +2 E

B -2 A -1 C +1 D +2 E

C -2 A 0 B +1 D +2 E

D -2 A -1 C 0 B +2 E

E -2 A -1 C +1 D 0 B

0 2 4 6 8 10 12 14 16 18 20 22 2410-6

10-5

10-4

10-3

10-2

10-1

1

SNR (dB)

BER


Figure 10: 2B1QI: Viterbi decoding soft and hard decisions.

3.3 Five-Level Line Codes

2 Binary 1 Quaternary Inverse Line Code: (2B1QI): Thiscode, as its encoder is presented in Table 1, is consideredto be similar to the well known 2B1Q quaternary linecode [9], except that this line code is with 5-levels andwhich give him a favorably built-in properties for clockextraction.


SNDM-4Binary 6Quaternary Inverse (SNDM-4B6QI): Wetake the case of M = 6 with k = 2 and s = 3. The channelsymbols must satisfy (5). By repeating only one symbol inboth A1 and A2, we can make the number of symbols equalto five. The corresponding five channel-level symbols cangenerate a 5-Level line code. To chose the convolutionalbase code rate, we can see that we need 4 bits to swapall the symbols yi and just 1 bit to swap A1 and A2 sincek = 2. In total, we need 5 bits from the outputs of thecorresponding convolutional base code, which means thatwe need a convolutional code with a rate of R = 4/5.

The new designed code belongs to the quaternary linecodes. This code is called a 4Binary 6Quaternary Inverse

0 2 4 6 8 10 12 14 16 1810-7

10-6

10-5

10-4

10-3

10-2

10-1

1

SNR (dB)

BER


Figure 11: SNDM-4B6QI: Viterbi decoding soft and hard

decisions.

and denoted by SNDM-4B6QI. The mapping algorithm forthe SNDM-4B6QI code is described below.

Input: (x1,x2,x3,x4,x5)Output: (y1,y2,y3,y4,y5,y6)(y1,y2,y3,y4,y5,y6)← (4,0,2,4,1,3)beginif x1 = 1 then swap(y1,y3)if x2 = 1 then swap(y1,y5)if x3 = 1 then swap(y2,y4)if x4 = 1 then swap(y2,y6)if x5 = 1 then swap(y1,y2)(y3,y4)(y5,y6)

end.


3.4 Six Level Line Codes

4B2H Line Code: This new line code [9] with encoderpresented in Table 2, generates 6-level from permutationsequences. Different to the published results, we makeuse of the Viterbi decoding algorithm since a state machinepresentation was given to this code. The BER performanceof this code is depicted in Fig. 12. We can see clearly the2 dB gain between the soft and hard decisions.


Table 2: Encoder for 4B2H Line Code

State

A B C D E

Input Outputs Next State Outputs Next State Outputs Next State Outputs Next State Outputs Next State

0000 -1+1 A -1+1 B -1+1 C -1+1 D -1+1 E

0001 +1-1 A +1-1 B +1-1 C +1-1 D +1-1 E

0010 +3-3 A +3-3 B +3-3 C +3-3 D -5-3 A

0011 +3+5 E -3+3 B -3+3 C -3+3 D -3+3 E

0100 +5+3 E -3+1 A -3+1 B -3+1 C -3+1 D

0101 +3+3 D +3+3 E -1-3 A -1-3 B -1-3 C

0110 +1+5 D +1+5 E +1-5 A +1-5 B +1-5 C

0111 +5+1 D +5+1 E -3+5 D -3+5 E -3-5 A

1000 +1+3 C +1+3 D +1+3 E -5+3 C -5+3 D

1001 +3+1 C +3+1 D +3+1 E +3-5 C -5-1 B

1010 -1+5 C -1+5 D -1+5 E -3-1 B -3-1 C

1011 +5-1 C +5-1 D +5-1 E -5+1 B -5+1 C

1100 +1+1 B +1-3 A +1-3 B +1-3 C +1-3 D

1101 +3-1 B +3-1 C +3-1 D +3-1 E -3-3 B

1110 -1+3 B -1+3 C -1+3 D -1+3 E -1-1 D

1111 +5-3 B +5-3 C +5-3 D -1-5 A -1-5 B

0 2 4 6 8 10 12 14 16 18 20 22 2410-6

10-5

10-4

10-3

10-2

10-1

1

SNR (dB)

BER


Figure 12: 4B2H: Viterbi decoding soft and hard decisions.

3.5 High-Multilevel Line Codes

For the generation of higher levels codes, we make use ofour previous techniques the spectral shaping and distancemappings in order to generate pulse amplitude modulatedline codes with m binary inputs and M symbol outputs orM channel levels, we denote them by SNDM-mBML. Westudy the case of M = 12 to explain our technique.

Example 3 For M = 12, with k = 3 and s = 4, we have

A1 =A2 =A3 =

s=4︷︸︸︷y1 + y4 + y7 + y10

y2 + y5 + y8 + y11

y3 + y3 + y6 + y12

(6)

In this case where all symbols are not repeated, we need 4

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30


P. S

. D.

Figure 13: PSD of SNDM-13B12L Line code

bits to swap all symbols in each grouping. This will leadto 12 bits since we have three groupings. On the other sidewe need 2 bits to swap the three groupings as it is basedon the cube construction. Therefore we need in total 14bits to swap all the channel symbols. The correspondingconvolutional base code rate should be convolutional codewith a rate of R = 13/14. The corresponding mappingalgorithm for this high level line codes is presented below.The designed code has nulls at the frequencies 0, 1/3, 2/3and 1 as depicted in Fig. 13.

Input: (x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14)Output: (y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12)(y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12)← (0,1,2,5,4,3,6,7,8,11,10,9)beginif x1 = 1 then swap(y1,y4)if x2 = 1 then swap(y7,y10)if x3 = 1 then swap(y1,y7)if x4 = 1 then swap(y4,y10)if x5 = 1 then swap(y2,y5)if x6 = 1 then swap(y8,y11)if x7 = 1 then swap(y2,y8)


Table 3: Some high-level line codes

M k s Frequency Nulls Base Code Rate Multilevel line code

8 4 2 0,1/4,1/2,3/4,1 7/8 SNDM-7B8L

8 2 4 0,1/2,1 8/9 SNDM-8B8L

12 4 3 0,1/4,1/2,3/4,1 11/12 SNDM-11B12L

12 3 4 0,1/3,2/3, 1 13/14 SNDM-13B12L

12 2 6 0,1/2,1 14/15 SNDM-14B12L

15 5 3 0,1/5,2/5,3/5,4/5,1 14/15 SNDM-14B15L

15 3 5 0,1/3,2/3, 1 16/17 SNDM-16B15L

16 4 4 0,1/4,1/2,3/4,1 19/20 SNDM-19B16L

16 8 2 0,1/8,1/4,3/8,1/2,5/8,3/4,7/8,1 19/20 SNDM-19B16L

16 2 8 0,1/2,1 24/25 SNDM-24B16L

18 2 9 0,1/2,1 26/27 SNDM-26B18L

18 9 2 0,1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9,1 20/21 SNDM-20B18L

18 3 6 0,1/3,2/3, 1 22/23 SNDM-22B18L

18 6 3 0,1/6,1/3,1/2,2/3,5/6,1 18/19 SNDM-18B18L

20 2 10 0,1/2,1 30/31 SNDM-30B20L

20 10 2 0,1/10,1/5,3/10,2/5, 1/2, 3/5,7/10,4/5,9/10,1 22/23 SNDM-22B20L

20 4 5 0,1/4,1/2,3/4,1 23/24 SNDM-23B20L

20 5 4 0,1/5,2/5,3/5,4/5,1 24/25 SNDM-24B20L

if x8 = 1 then swap(y5,y11)if x9 = 1 then swap(y3,y6)if x10 = 1 then swap(y9,y12)if x11 = 1 then swap(y3,y9)if x12 = 1 then swap(y6,y12)if x13 = 1 then swap(y1,y2)(y4,y5) · · ·

· · ·(y7,y8)(y10,y11)if x14 = 1 then swap(y1,y3)(y4,y6) · · ·

· · ·(y7,y9)(y10,y12)end.

In view of the large number of states for the base codes,we will not be able to present their state machines in viewof the space restriction in this paper. �

Table 3 presents a few examples of high-level line codesthat we are able to design, taking into consideration therates of the corresponding base codes.

4. VITERBI DECODING ERROR PROPAGATION

Viterbi algorithm is based on the calculation ofthe distances between the received and the expectedtransmitted information data in each branch of thetrellis diagram designed from the state machine of theconvolutional base code. Previous published work [18] hasshown the modeling of line codes for the sake of havingthe state machine presentation, which will be used for theViterbi algorithm.

A simulation experiment was conducted to analyze andprove that the coding gain is predominantly determined bythe error propagation when Viterbi decoding fails. Theexperiment [28] is simply based on the generation of

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of errors

Prob

abili

ty o

f err

ors

AMIHDB1HDB2HDB3CHDB3B4ZSB6ZS

Figure 14: Error propagation of certain ternary line codes, due to

a random single isolated channel error.

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Number of errors

Prob

abili

ty o

f err

ors

SNDM-1B4QSNDM-2B4QSNDM-4B6QISNDM-7B8Q1B1Q3B2Q4B2H

Figure 15: Error propagation of certain quaternary line codes,

due to a random single isolated channel error.


Table 4: Probability of i errors

Multilevel Number of propagation errors iLine Codes

0 1 2 3 4 5 6 7

AMI 0.29 0.02 0.69 0 0 0 0 0

HDB1 0.54 0.03 0.25 0.14 0.04 0 0 0

HDB2 0.49 0.01 0.28 0.08 0.09 0.05 0 0

HDB3 0.22 0.06 0.49 0.09 0.08 0.03 0.02 0.01

CHDB3 0.38 0.01 0.48 0.06 0.02 0.05 0 0

B4ZS 0.34 0 0.66 0 0 0 0 0

2B2T 0.17 0.11 0.02 0.16 0.54 0 0 0

2B2TA 0.16 0.07 0.58 0.19 0 0 0 0

SNDM-3B6T 1 0 0 0 0 0 0 0

SNDM-5B9T 1 0 0 0 0 0 0 0

1B1Q 0.62 0 0.38 0 0 0 0 0

3B2Q 0.05 0.2 0.16 0.16 0.25 0.08 0.1 0

4B2H 0.02 0.2 0.13 0.06 0.15 0.16 0.2 0.08

SNDM-1B4Q 1 0 0 0 0 0 0 0

SNDM-2B4Q 1 0 0 0 0 0 0 0

SNDM-4B6QI 1 0 0 0 0 0 0 0

SNDM-7B8Q 1 0 0 0 0 0 0 0

Table 5: Number of levels vs error propagation

Number of Codes Multilevel Gain using 3-bit Expected number of Maximum number of

Levels Line Codes quantization propagated errors propagated errors

AMI 1.90 1.40 3

Three B4ZS 2.05 1.32 2

B6ZS 2.00 1.34 2

HDB1 1.85 1.40 5

HDB2 1.55 1.42 6

HDB3 1.45 1.48 6

CHDB3 1.35 1.66 7

2B2T 1.70 1.8 3

2B2TA 1.50 2.79 4

SNDM-3B6T 2.10 0 0

SNDM-5B9T 2.00 0 0

1B1Q 2.00 0.76 2

2B1Q 1.40 3.17 6

Four 3B2Q 1.50 3.06 6

SNDM-2B4Q 2.00 0 0

SNDM-4B6Q 2.00 0 0

SNDM-7B8Q 2.00 0 0

Five 2B1QI 1.30 3.2 6

SNDM-4B6QI 2.1 0 0

Six 4B2H 1.1 3.8 7


widely separated single random errors between the levelsof the code’s symbols, and the observation of the numberof errors propagated.

Table 4 shows that our designed codes have zeroerror propagation and this is expected in view of theconvolutional codes properties that our codes are generatedfrom. We can see also these results from another anglefrom Fig. 14 and 15.

Table 5 shows the variation of the number of errorpropagation and the values of the expected number ofpropagated error in terms of the number of the code’s level.

So from Tables 4 and 5, we can summarize that the increaseof the number of levels in line codes and the increase ofthe complexity of the pattern within the same class of linecodes are the major factors in the increase of the errorpropagation of the Viterbi decoding. This is always nottrue if the codes are designed from the convolutional codesas in the case of our new designed M-level line codes.

It is clear from the previous results that the expectednumber of error propagation increases for the followingreasons:

• When the complexity of the pattern increases, whichis true even within the same class of line codes, thevalue of the expected error propagation increases.As an example, the error propagation for HDBn linecodes, the expected number of error propagation ishigher than the one for BnZS line codes. Similar casewhen we compare 2B2T line codes and 2B2TA linecodes.

• When the number of levels increases, even within thesame number of states, the expected number of errorpropagation increases. As an example, if we take thecase of 2B1QI line code, which has five levels, we cansee that the expected number of error propagation ishigher than the one for 4B2H line codes since it hassix levels. Both line codes have similar number ofstates which is five.

It is important to notice that the number of states is not animportant criteria in the increase of the expected numberof error propagation. This is clear with the following twoexamples. In the 2B1QI and 4B2H line codes which havesimilar number of states, we can see that they differ withthe number of expected number of error propagation. Inthe case of 1B1Q and 3B2Q line codes, we can see that1B1Q code has less expected number of error propagationthan 3B2Q code despite it has more number of states.

5. CONCLUSION

We have combined two techniques, spectral shaping anddistance mappings to design M-level line codes. This classof codes inherited the same state structure and distance

properties as convolutional codes. Thus the use of theViterbi algorithm as a decoding solution is possible.

The possible 2-dB gain is investigate and the resultsverified some theoretical assumption that made SNDMcode outperforming other M-level line codes.

The obtained results have shown that the error propagationViterbi decoding increases with the complexity of thepattern of the code as the case between the HDBn codesand BnZS line codes even both classes belong to the sameclass of M-level line codes which is the ternary line codes.As the number of levels, M, increase the complexity ofcoding increase as well and therefore the error propagationof the decoding algorithm increase. This problem isreduced and simplified with the design SNDM line codes.

High rate M-level line codes were also achieved with wellshaped spectrum. Table 3 presents their correspondingspectral nulls frequencies. It is clear from Table 3 and fromall previously presented PSD figures, that all our designedM-level line codes are also DC-free codes [29], which willhelp to overcome some communications problem like zerofrequency components.

REFERENCES

[1] H. Imai and S. Hirakawa, “A new multilevel codingmethod using error-correcting codes”, in IEEETransactions on Information Theory, vol. 23, no. 3,pp. 371–377, May. 1977.

[2] U. Wachsmann and R. F. H. Fisher and J. B.Huber,“Multilevel codes: Theoretical concepts andpractical design rules”, in IEEE Transactions onInformation Theory, vol. 45, no. 5, pp. 1361–1391,Jul. 1999.

[3] Y. Kofman and E. Zehavi and S.Shamai,“Performance analysis of a multilevelcoded modulation system”, in IEEE Transactions onCommunications, vol. 42, no. 234, pp. 299–312, Feb.1994.

[4] J. Wu and D. J. Costello Jr.,“New Multilevel CodesOver GF(q)”, in IEEE Transactions on InformationTheory, vol. 38, no. 3, pp. 933–939, May. 1992.

[5] A. R. Calderbank and M. A. Herro and V. Telang,“AMultilevel Approach to the Design of DC-freeLine Codes”, in IEEE Transactions on InformationTheory, vol. 35, no. 3, pp. 579–583, May. 1989.

[6] R. M. Brooks and A. Jessop,“Line coding foroptical fibre systems”, in International Journal ofElectronics, vol. 55, no. 1, pp. 81–120, 1983.

[7] S. D. Personick,“Optical Fiber Transmission Sys-tems”,New York: Plenum, 1981.

[8] C. Matrakidis and J. J. O’Reilly, “A Block decodableline code for high speed optical communication”,in Proceedings of the International Symposium on


Information Theory, pp. 221, Ulm, Germany, June29–Jul 4, 1997.

[9] L. Botha, H. C. Ferreira and I. Broere, “Multilevelsequences and line codes,” in IEE Proc. I Commun.Speech Vis., 1993, vol. 140, no. 3, pp. 255–261.

[10] J. K. Pollard, “Multilevel data communication overoptical fiber,” in IEE Proc. I Commun. Speech Vis.,1991, vol. 138, no. 3, pp. 162–168.

[11] N. J. Lynch-Aird,“Review and Analytical compar-ison of Recursive and Nonrecursive EqualizationTechniques for PAM Transmission Systems”, inInternational Journal on Electronics,vol. 55, no. 1,pp. 81–120, 1983.

[12] H. C. Ferreira and A. J. H. Vinck, “Interferencecancellation with permutation trellis codes,” in Proc.IEEE Veh. Technol. Conf. Fall 2000, Boston, MA,Sep. 2000, pp. 2401–2407.

[13] H. C. Ferreira, A. J. H. Vinck, T. G. Swart and I.de Beer, “Permutation trellis codes,” IEEE Trans.Commun., vol. 53, no. 11, pp. 1782–1789, Nov. 2005.

[14] T. G. Swart and H. C. Ferreira, “A generalizedupper bound and a multilevel construction fordistance-preserving mappings,” IEEE Trans. Inf.Theory, vol. 52, no. 8, pp. 3685–3695, Aug. 2006.

[15] K. Ouahada, T. G. Swart and H. C. Ferreira,“Permutation sequences and coded PAM signals withspectral nulls at rational submultiples of the sym-bols,” accepted for publication in Communicationsand Cryptography, Springer, 2011.

[16] E. Gorog, “Alphabets with desirable frequencyspectrum properties,” IBM J. Res. Develop., vol. 12,pp. 234–241, May 1968.

[17] K. A. S. Immink, Codes for mass data storagesystems, Shannon Foundation Publishers, TheNetherlands, 1999.

[18] D.B. Keogh, “Finite-State Machine Descriptions ofFilled Bipolar Codes,” A.T.R. vol. 18, no. 2, pp. 3-12,June 1984.

[19] T. G. Swart, I. de Beer, H. C. Ferreira andA. J. H. Vinck, “Simulation results for permutationtrellis codes using M-ary FSK,” in Proc. Int. Symp.on Power Line Commun. and its Applications,Vancouver, BC, Canada, Apr. 6–8, 2005, pp.317–321.

[20] G. David Forney, JR., “The Viterbi Algorithm,”Proceedings of the IEEE, vol. 61, no. 3, pp. 268–278,Mar. 1973.

[21] K. Ouahada and H. C. Ferreira, “k-Cube constructionmappings from binary vectors to permutationsequences,” in proceedings of IEEE InternationalSymposium on Information Theory, Seoul, SouthKorea, June 28July 3, 2009.

[22] K. Ouahada and H. C. Ferreira, “Simulation study ofthe performance of ternary line codes under Viterbidecoding,” in IEE Proc-Commun., vol. 151, no. 5, pp.409–414, 2004.

[23] A. Viterbi and J. Omura, “Principles of Digital Com-munication and Coding, McGraw-Hill KogakushaLTD, Tokyo Japan, 1979.

[24] Buchner J. B. “Ternary Line Codes,” PhilipsTelecommunications Review, vol. 34, no. 2, pp.72–86, June 1976.

[25] H. C. Ferreira, J. F. Hope, and A. L. Nel, “On TernaryError Correcting Line Codes,” IEEE Trans. Commun.vol. 37 no. 5, pp. 510–515, May 1989.

[26] A. Croisier, “Introduction to Pseudo-ternary Trans-mission Codes,“ IBM J. Res. Develop. , vol. 14, pp.354–367, Jul. 1970.

[27] V. I. Johannes, “Comments onCompatible-High-Density Bipolar Codes: AnUnrestricted transmission Plan for PCM Carriers,”IEEE Trans. Commun. Tech., vol. COM-20, no. 1,pp. 78–79, Feb. 1972.

[28] N. Q. Duc, “Line coding techniques for basebanddigital transmission,” in Australian Telecommunica-tions Research, vol. 9, no. 1, pp. 351–357, 1977.

[29] G. L. Pierobon, “Codes for zero spectral density atzero frequency,” IEEE Trans. Inf. Theory, vol. 30, pp.435–439, Mar. 1984.


Notes

arj september 2012 vol 103 no 3 · 2016-08-25 · 108 south african institute of electrical...

Documents