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EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 1
MAHALAKSHMI ENGINEERING COLLEGE
TIRUCHIRAPALLI – 621213
QUESTION BANK
DEPARTMENT: ECE SEMESTER: V
SUBJECT CODE / Name: EC2301 – DIGITAL COMMUNICATION
UNIT III
BASEBAND CODING TECHNIQUES
PART -A (2 Marks)
1. Mention is the properties of cyclic codes [AUC NOV/DEC 2011]
Linearity property
The sum of any two code word is also a valid code word
Cyclic property
Every cyclic shift of a valid code vector produces another valid code vector
2. Define hamming distance. [AUC APR/MAY 2011]
The hamming distance between two code vectors is equal to the number of elements in
which they differ. For example, let the two code words be,
X = (101) and Y= (110)
These two code words differ in second and third bits. Therefore the hamming distance
between X and Y is two.
3. What is meant by transparency with respect to line codes [AUC APR/MAY 2011]
The line code is said to be transparent if the synchronization between the transmitter
and receiver is maintained for any type of input data sequence.
4. Define hamming distance and calculate its value for two code words 11100 and
11011
The hamming distance between two code vectors is equal to the number of
elements in which they differ. For example, let the two code words be,
X = (11100) and Y= (11011)
D= 2 These two code words differ in second and third bits. Therefore the hamming
distance between X and Y is two.[AUC APR/MAY 2010]
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 2
5. Draw the NRZ and RZ code for the digital data 10110001 [AUC APR/MAY 2010]
6. Draw the RZ bipolar line code format for the information {10110}[AUC NOV/DEC
2011]
(Similarly can done for the given data)
7. What is convolution code? How is it different from block codes? [AUC APR/MAY
2012]
Fixed number of input bits is stored in the shift register & they are combined with the
help of mod 2 adders. This operation is equivalent to binary convolution coding.
8. What is Manchester code? Draw the Manchester format for the data stream
10110? [AUC APR/MAY 2012] (Similarly can done for the given data)
In Manchester code each bit of data is signified by at least one transition. Manchester
encoding is therefore considered to be self-clocking, which means that accurate clock
recovery from a data stream is possible. In addition, the DC component of the encoded
signal is zero. Although transitions allow the signal to be self-clocking, it carries
significant overhead as there is a need for essentially twice the bandwidth of a simple
NRZ or NRZI encoding
9. State any four desirable properties of line code[AUC NOV/DEC 2012]
The PAM signal should have adequate timing content,
The PAM signal should immune to channel noise and interference
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 3
The PAM signal should allow error detection and error correction
The PAM signal should be transparent to digital data being transmitted
10. Find the hamming distance 101010 and 010101.If the minimum hamming distance
of a (n,k) linear block code is 3, what is its minimum hamming weight? [AUC
NOV/DEC 2012]
d(x1,x2)=x1 exor x2
=111111
d(x1,x2)=6
Dmin=3 then Wmin=dmin=3
11. What is meant by syndrome of linear block code?
The non zero output of the produce YHT is called syndrome & it is used to detect errors
in y. Syndrome is denoted by S & given as,
S=YHT
12. What does meant by RS coding?
These are non binary BCH codes. The encoder for RS code operates on multiple bits
simultaneously. The (n, k) RS code takes the groups of m- bit symbols of incoming
binary data stream. It takes such „k‟ number of symbols in one block. Then the encoder
acts (n – k) redundant symbols to form the code word of „n‟ symbols
RS code has:
Block Length : n=2m-1 symbols
Message size: K symbols
Parity check size: n-k= 2t symbols
Minimum distance: dmin=2t+a symbols
13. What is convolutional code? Explain the fundamental difference between block
codes and convolutional codes.
Block codes takes‟k‟ number of bits simultaneously form „n‟-bit code vector. This code
vector is also called block. Convolutional code takes one message bits at a time and
generates two or more encoded bits. Thus convolutional codes generate a string of
encoded bits for input message string.
14. Define constraint length in convolutional code?
Constraint length is the number of shift over which the single message bit can
influence the encoder output. It is expressed in terms of message bits.
15. Define free distance and coding gain.
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 4
Free distance is the minimum distance between code vectors. It is also equal to
minimum weight of the code vectors.
Coding gain is used as a basis of comparison for different coding methods. To
achieve the same bit error rate the coding gain is defined as,
A= (Eb/No) encoded
(Eb/No) coded
For convolutional coding, the coding gain is given as,
A = rdf /2
Here „r‟ is the code rate
And „df is the free distance.
16. What are the advantages of convolutional codes?
Advantages:
1. The decoding delay is small in convolutional codes since they operate o
smaller blocks of data.
2. The storage hardware required by convolutional decoder is less since the
block sizes are smaller.
Disadvantages:
1. Convolutional codes are difficult to analyze since their analysis is complex.
2. Convolutional codes are not developed much as compared to block codes.
18. Compare between code tree and trellis diagram?
19. Write the futures of BCH Codes?
BCH codes are most extensive and powerful error correcting cyclic codes. The
decoding of BCH codes is comparatively simpler.The decoding schemes of BCH codes
can be implemented on digital computer. Because of software implementation of
decoding schemes they are quite flexible compared to hardware implementation of
other schemes.
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 5
20. What is Golay codes?
Golay code is the (23,12) cyclic code whose generating polynomial is,
G(p) = p11+p9+p7+p6+p5+p+1
This code has minimum distance of dmin = 7. This code can correct upto 3 errors. But
Golay code cannot be generalized to other combinations of n and k.
21. What is meant by systematic and non-systematic codes?
In a Systematic block code, message bits appear first and then check bits. In the non-
systematic code, message and check bits cannot be identified in the code vector.
22. What is meant by linear code?
A code is linear if modulo-2 sum of any two code vectors produces another code vector.
This means any code vector can be expressed as linear combination of other code
vectors.
23. What are the error detection and correction capabilities of hamming codes?
The minimum distance (dmin) of hamming codes is „3‟. Hence it can be used to detect
double errors or correct single errors. Hamming codes are basically linear block codes
with dmin =3.
24. What is meant by cyclic codes?
Cyclic codes are the subclasses of linear block codes. They have the property that a
cyclic shift of one codeword produces another code word.
PART -B(16 Marks)
1. i)Consider a single error correction (7,4) linear code and the corresponding decoding
table(10)
2. Find the (7,4) linear systematic block code word corresponding to 1101.Assume a
suitable generator matrix. [AUC APR/MAY 2011]
Let
n=7 k=4
q=n-k=3
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 6
code vector G=[Ik: P]
Check matrix C=MP
C1 = m1+m2+m3
C2= m2+m3+m4
C3= m1+m2+m4
C=[010]
Complete code word can be calculated X={M:C}={1 1 0 0 0 1 0}
The parity matrix H=[pT :I] =[I: pT] =
Minimum weight W(X)=3
3. ii)Briefly describes the concept of error free communication (6) [AUC NOV/DEC 2011]
Redundancy can e added as a block code or as in colvolutional codes. Addition
on (n-k) bits of redundancy in k message bits reduces transmission rate by a
factor of k/n
Because of redundancy there are 2n possible messages and only 2k are
required. This increase the distance among the message vector and hence error
possibility is reduced
the channel capacity gives an upper limit on transmission rate for error free
transmission. The channel capacity depends upon signal power and channel
noise of the system.
Shannon established that error free transmission is possible with the help of
proper channel coding technique and selecting a transmission rate less than
channel capacity
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 7
4. i) Determine the generator polynomial g(X) FOR A (7,4) cyclic code and fine the code
vector for the following data vector 1010, 1111 and 1000 (8)
n=7 k=4
q=n-k=3
to obtain the generator polynomial
(p7+1) =(p+1)(p3+p2+1)(p3+p+1)
Let G(p)= (p3+p+1)
To obtain the generator matrix in systematic form
To determine the code vector
1. code vector for M=1010
X=MG
2. code vector for M=1111
3. code vector for M=1000
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 8
5. ii)List and explain the properties of line codes [AUC NOV/DEC 2011]
DC Component:
Eliminating the dc energy from the single power spectrum enables the transmitter to be
ac coupled. Magnetic recording system or system using transformer coupling are less
sensitive to low frequency signal components. Low frequency component may lost, if
the presence of dc or near dc spectral component is significant in the code itself.
Self synchronization
Any digital communication system requires bit synchronization. Coherent detector
requires carrier synchronization.
For example Manchester code has a transition at the middle of every bit interval
irrespective of whether a 1 or 0 is being sent This guaranteed transmitter provide a
clocking signal at the bit level.
Error detection
Some codes such as duo binary provide the means of detecting data error without
introducing additional error detection bits into the data sequence.
Band width compression:
Some codes such as multilevel codes increase the efficiency of the bandwidth
utilization by allowing a reduction in required bandwidth for a given data rate, thus more
information transmitted per unit band width.
DIFFERENTIAL ENCODING
This technique is useful because it allow the polarity of differentially encoded waveform
to be inverted without affecting the data detection. In communication system where
waveform to be inverted having great advantage
NOISE IMMUNITY
For same transmitted energy some codes produces lesser bit detection error than
other in the presence of noise. For ex. The NRZ waveforms have better noise
performance than the RZ type.
SPECTRAL COMPATABILITY WITH CHANNEL:
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 9
On aspect of spectrum matching is dc coupling. Also transmission bandwidth of the
code musts is sufficient small compared to channel bandwidth so that ISI is not
problem.
TRANSPARENCY
A line doe should be so designed that the receiver does not go out of synchronization
for any line sequence of data symbol. A code is not transparent if for some sequence of
symbol, the clock is lost.
6. Assume a (2,1) convolutional coder with constraint length 6.Draw the tree diagram,
state diagram and trellis diagram for the assumed coder [AUC APR/MAY 2011]
Design block code for a message block of size eight that can correct for single
errors Briefly discuss on various error control codes and explain in detail with one
example for convolution code. (12)
N=2, K=1 AND K=6(CONSTRAIN LENGHT)
M=K/n=6/2=3, snce constrain length k=n*M
3 storage element in shift register
N=2 two output bits
One set k=1 of shift register having 3 storage element the convolutional code structure is easy
to draw from its parameters. First draw m boxes representing the m memory register. Then
draw n modulo-2 adders to represent the n output bits. Now connect the memory registers to
the adders using the generator polynomial
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 10
Convolutional codes k = number of bits shifted into the encoder at one time
k=1 is usually used!!
n = number of encoder output bits corresponding to the k0020information bits
r = k/n = code rate
K = constraint length, encoder memory Each encoded bit is a function of the present input bits and their past ones. Generator Sequence
Convolutional Codes An Example – (rate=1/2 with K=2)
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 11
Trellis Diagram Representation
Encoding Process
Viterbi Decoding Algorithm
Maximum Likelihood (ML) decoding rule
Viterbi Decoding Algorithm An efficient search algorithm
Performing ML decoding rule.
Reducing the computational complexity.
Basic concept Generate the code trellis at the decoder
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 12
The decoder penetrates through the code trellis level by level in search for the transmitted code sequence
At each level of the trellis, the decoder computes and compares the metrics of all the partial paths entering a node
The decoder stores the partial path with the larger metric and eliminates all the other partial paths. The stored partial path is called the survivor.
Viterbi Decoding Process
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 15
7. Derive the power spectra of polar codes and on-off codes. Discuss their
characteristics (16) Derive the expression for power spectral density of unipolar
NRZ line code. Hence discuss its characteristics [AUC NOV/DEC 2012]
Line coding: Line coding refers to the process of representing the bit stream (1s and
0s) in the form of voltage or current variations optimally tuned for the specific properties
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 16
of the physical channel being used. The selection of a proper line code can help in so
many ways: One possibility is to aid in clock recovery at the receiver. A clock signal is
recovered by observing transitions in the received bit sequence, and if enough
transitions exist, a good recovery of the clock is guaranteed, and the signal is said to be
self-clocking.
Another advantage is to get rid of DC shifts. The DC component in a line code is called
the bias or the DC coefficient. Unfortunately, most long-distance communication
channels cannot transport a DC component. This is why most line codes try to eliminate
the DC component before being transmitted on the channel. Such codes are called DC
balanced, zero-DC, zero-bias, or DC equalized. Some common types of line encoding
in common-use nowadays are unipolar, polar, bipolar, Manchester, MLT-3 and
Duobinary encoding. These codes are explained here: 1. Unipolar (Unipolar NRZ and
Unipolar RZ):
Unipolar is the simplest line coding scheme possible. It has the advantage of being
compatible with TTL logic. Unipolar coding uses a positive rectangular pulse p(t) to
represent binary 1, and the absence of a pulse (i.e., zero voltage) to represent a binary
0. Two possibilities for the pulse p(t) exist3: Non-Return-to-Zero (NRZ) rectangular
pulse and Return-to-Zero (RZ) rectangular pulse. The difference between Unipolar NRZ
and Unipolar RZ codes is that the rectangular pulse in NRZ stays at a positive value
(e.g., +5V) for the full duration of the logic 1 bit, while the pule in RZ drops from +5V to
0V in the middle of the bit time. A drawback of unipolar (RZ and NRZ) is that its average
value is not zero, which means it creates a significant DC-component at the receiver
(see the impulse at zero frequency in the corresponding power spectral density (PSD)
of this line code
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 17
The disadvantage of unipolar RZ compared to unipolar NRZ is that each rectangular pulse in
RZ is only half the length of NRZ pulse. This means that unipolar RZ requires twice the
bandwidth of the NRZ code. Polar (Polar NRZ and Polar RZ):
In Polar NRZ line coding binary 1s are represented by a pulse p(t) and binary 0s are
represented by the negative of this pulse -p(t) (e.g., -5V). Polar (NRZ and RZ) signals .Using
the assumption that in a regular bit stream a logic 0 is just as likely as a logic 1,polar signals
(whether RZ or NRZ) have the advantage that the resulting Dc component is very close to
zero.
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 18
that polar signals have more power than unipolar signals, and hence have better SNR at the
receiver. Actually, polar NRZ signals have more power compared to polar RZ signals. The
drawback of polar NRZ, however, is that it lacks clock information especially when a long
sequence of 0‟s or 1‟s is transmitted.
Non-Return-to-Zero, Inverted (NRZI): NRZI is a variant of Polar NRZ. In NRZI there are two
possible pulses, p(t) and –p(t). A transition from one pulse to the other happens if the bit being
transmitted is logic 1, and no transition happens if the bit being transmitted is a logic 0.
This is the code used on compact discs (CD), USB ports, and on fiber-based Fast Ethernet at
100-Mbit/s .
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 19
Manchester encoding: In Manchester code each bit of data is signified by at least one
transition. Manchester encoding is therefore considered to be self-clocking, which means that
accurate clock recovery from a data stream is possible. In addition, the DC component of the
encoded signal is zero. Although transitions allow the signal to be self-clocking, it carries
significant overhead as there is a need for essentially twice the bandwidth of a simple NRZ or
NRZI encoding
ii)Derive the power spectral density of polar signaling and explain[AUC APR/MAY
2012]
POWER SPECTRA OF LINE CODE
Unipolar most of signal power is centered on origin and there is waste of power due to DC
component that is present.
Polar format most of signal power is centered on origin and they are simple to implement.
• Bipolar format does not have DC component and does not demand more bandwidth, but
power requirement is double than other formats.
• Manchester format does not have DC component but provides proper clocking.
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 20
[AUC APR/MAY 2011]
8. For(6,3) systematic linear block code the codeword comprises I1,I2,I3,P1,P2,P3
where the 3 parity bits P1,P2,P3 are formed from the information bits as follows:
P1= I1 EX-OR I2
P2= I1 EX-OR I3
P3= I2 EX-OR I3
1. Find parity check matrix, 2.generator matrix,3.all possible codewords,4.
Minimum weight 5. Minimum distance, the error correcting and detecting
capability of the code [AUC APR/MAY 2010]
The parity check bits, the sub matrix and information bits are related as
Minmimum weight and minimum distance
D min =3
Minmimum weight =3
Error detecting and correcting capability
Dmin>= s+1
3>= s+1
S<=2
Two error will be detected
Dmin>= 2t+1
3>=2t+1
T<=1
Ony one error will be corrected
9. Explain how encoding is done by convolution codes with a suitable example [AUC
APR/MAY 2010]
10. Design a convolution coder of constraint length and rate efficiency ½.Draw its tree
diagram and trellis diagram. [AUC NOV/DEC 2012]
Convolutional codes are widely used as channel codes in practical communication systems for
error correction. *The encoded bits depend on the current k input bits and a few past input bits.
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 21
The main decoding strategy for convolutional codes is based on the widely used Viterbi
algorithm. Convolutional codes are commonly described using two parameters: the code rate
and the constraint length. The code rate, k/n, is expressed as a ratio of the number of bits into
the convolutional encoder (k) to the number of channel symbols output by the convolutional
encoder (n) in a given encoder cycle.
The constraint length parameter, K, denotes the "length" of the convolutional encoder, i.e. how
many k-bit stages are available to feed the combinatorial logic that produces the output
symbols. Closely related to K is the parameter m, which can be thought of as the memory
length of the encoder. A simple convolutional encoder is shown below(fig 3.1). The information
bits are fed in small groups of k-bits at a time to a shift register. The output encoded bits are
obtained by modulo-2 addition (EXCLUSIVE-OR operation) of the input information bits and the
contents of the shift registers which are a few previous information bits
.
The operation of a convolutional encoder can be explained in several but equivalent ways such
as, by a) state diagram representation. b) tree diagram representation. c) trellis
diagram representation.
a) State Diagram Representation: A convolutional encoder may be defined as a finite
state machine. Contents of the rightmost (K-1) shift register stages define the states of
the encoder. So, the encoder in has four states. The transition of an encoder from one
state to another, as caused by input bits, is depicted in the state diagram. A new input
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 22
bit causes a transition from one state to another
State diagram representation for the encoder
b) Tree Diagram Representation: The tree diagram representation shows all possible
information and encoded sequences for the convolutional encoder. The encoded bits
are labeled on the branches of the tree. Given an input sequence, the encoded
sequence can be directly read from the tree.
c) Representing convolutional codes compactly: code trellis and state diagram:
Inspecting state diagram: Structural properties of convolutional codes:
• Each new block of k input bits causes a transition into new state
• Hence there are 2k branches leaving each state
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 23
• Assuming encoder zero initial state, encoded word for any input of k bits can thus be
obtained. For instance, below for u=(1 1 1 0 1), encoded word v=(1 1, 1 0, 0 1, 0 1, 1 1, 1 0, 1
1, 1 1) is produced
Encoder state diagram for (n,k,L)=(2,1,2) code - note that the number of states is 2L+1
= 8 Distance for some convolution code
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 24
Trellis Diagram Representation:
The trellis diagram of a convolutional code is obtained from its state diagram. All state
transitions at each time step are explicitly shown in the diagram to retain the time
dimension, as is present in the corresponding tree diagram. Usually, supporting
descriptions on state transitions, corresponding input and output bits etc. are labelled in
the trellis diagram. It is interesting to note that the trellis diagram, which describes the
operation of the encoder, is very convenient for describing the behaviour of the
corresponding decoder, especially when the famous Viterbi Algorithm (VA) is followed.
trellis diagram for the encoder
11. Explain the error detecting and correcting capabilities of linear block code.
BLOCK CODE A block code is linear if any linear combination of two code words is also a codeword. In the
binary case this requires that if ci and c j are code words then ci ⊕c j is also a code word,
where ⊕ denotes component-wise modulo-2 addition.)
HAMMING DISTANCE
The hamming distance between two code vectors is equal to the number of elements in which
they differ. For example, let the two code words be, X = (101) and Y= (110)
These two code words differ in second and third bits. Therefore the hamming distance between
X and Y is two
The Hamming distance between two code words ci and c j is the number of components at
which the two code words differ, and is denoted by
d(ci , c j ).†
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 25
The Hamming weight, or simply the weight of a code word ci is the number of nonzero
components of the code word and is denoted by w(ci ).
The minimum distance of a code is the minimum Hamming distance between any two different
code words; i.e.,
The minimum weight of a code is the minimum of the weights of the code words except the all-
zero code word.
Hamming Codes. Hamming codes are a class of linear block codes with n = 2m − 1, k = 2m −
m − 1 and dmin =3, for some integer m ≥ 2.this minimum distance, these codes are capable of
providing error-correction capabilities for single errors. The parity check matrix for these codes
has a very simple structure. It consists of all binary sequences of length m except the all zero
sequence. The rate of these codes is given by
A code can detect any combination of d or fewer errors if
E.g., (4,1) repetition code: detect up to 3 errors.
A code can correct any combination of c or fewer errors
if E.g., (4,1) repetition code: correct 1 error.
A code can simultaneously detect d errors and correct c < d errors if
E.g., (4,1) repetition code: correct 1 error and detect up to 2 errors.
Definition: A block code is linear if any linear combination of two code words is also a
codeword. In the binary case, it is equivalent to the fact that the sum of any two code words is
also a codeword where summation is defined by component wise modulo-2 addition linear
codes.)
Definition: An (n; k) binary linear block code is a binary linear block code with 2
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 26
k code words of length n.
ii)Consider a (7,4) linear block code whose parity check matrix is given by
1. Find the generator matrix 2. How many error thios code can be detected? 3 How
many error can this code be correct? 4. Draw circuit for encoder and syndrome
computation [AUC APR/MAY 2012]
To generator matrix
G= [Ik:P]
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 27
To determine the code word:
Modulo 2 addition
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 28
Minimum distance between code vector
D min =3
Error detection and correction capabilities
Dmin>=3
Dmin>=s+1
3>= s+1
S<=2
Two error will be detected
Dmin>=2t+1
3>= 2t+1
t<=1
Thus one error will be corrected.
12. Explain the transform domain approach analysis of convolution code
let the sequence
denotes the impulse response of the adder
let the second sequence
let in coming message be m0, m1,m2,m3……. The encoder generated where output
sequence x1 and x2.
Here Mi-1 =0 for all i>i. similarly the sequence x2 is given as
Explain the coding and decoding process of block codes (16) (AUC NOV/DEC 2011)
Let m0, m1...mk-1 constitute a block of k message. C0, c1, c2...cn-1 be code word.
Let b0, b1, b2....b n-k-1 denotes (n-k) parity bit of k messages
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 29
The(n-k) parity bit are linear sum of k message
And the coefficient as follows
The code vector is given as follows
The common message vector as follows
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 30
Where Ik is the k by k identity matrix
Let H denotes (n-k) by n matrix
Syndrome
Generator matrix G is used in encoding operation at the transmitter is parity check matrix used
in decoding operation at the receiver. Let r denoted the 1 by n received vector, C code vector
The vector e is called error pattern or error pattern
Syndrome is define as
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 31
1. What are cyclic codes. Explain the merits and demerits Give the properties of cyclic
codes. (AU- May/June 2010)
Cyclic codes forms as subclass of linear block code.
Properties of cyclic code
1. Linearity property.
The sum of two code word in the code is also a code word.
2. Cyclic property
A cyclic shift of a code word in the code is also a code word.
GENERATOR POLYNOMIAL
Encoding procedure
Encoder of cyclic codes
EC2301 –DIGITAL COMMUNICATION V Sem EEE – R.Vanitha Asst.Prof./ECE Page 32
SYNDROME CALCULATOR
EXPLAIN THE PROPERTY OF SYNDROME DECODING
Let c be the original codeword sent and r = c + e be the received codeword, where e is the error vector. ei = {n1, if an error has occured in the ith position0, otherwise Decoding of r A 1 × (n − k) matrix called error syndrome matrix is calculated
Properties of Syndrome:
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