research article design of finite word length linear-phase...

Research ArticleDesign of Finite Word Length Linear-Phase FIR Filters inthe Logarithmic Number System Domain

Syed Asad Alam and Oscar Gustafsson

Division of Electronics Systems, Department of Electrical Engineering, Linkoping University, 581 83 Linkoping, Sweden

Correspondence should be addressed to Oscar Gustafsson; [email protected]

Received 4 November 2013; Accepted 20 February 2014; Published 9 April 2014

Academic Editor: Antonio G. M. Strollo

Copyright © 2014 S. A. Alam and O. Gustafsson. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Logarithmic number system (LNS) is an attractive alternative to realize finite-length impulse response filters because ofmultiplication in the linear domain being only addition in the logarithmic domain. In the literature, linear coefficients are directlyreplaced by the logarithmic equivalent. In this paper, an approach to directly optimize the finite word length coefficients in theLNS domain is proposed.This branch and bound algorithm is implemented based on LNS integers and several different branchingstrategies are proposed and evaluated. Optimal coefficients in the minimax sense are obtained and compared with the traditionalfinite word length representation in the linear domain as well as using rounding. Results show that the proposed method naturallyprovides smaller approximation error compared to rounding. Furthermore, they provide insights into finite word length propertiesof FIR filters coefficients in the LNS domain and show that LNS FIR filters typically provide a better approximation error comparedto a standard FIR filter.

1. Introduction

Finite-length impulse response (FIR) filters constitute a classof digital filters commonly used for their stability propertiesand the ability to obtain a linear phase response. The transferfunction of an𝑁th-order FIR filter is

𝐻(𝑧) =

𝑁

∑

𝑛=0

ℎ𝑛𝑧−𝑛, (1)

where ℎ𝑛are the impulse response coefficients.

The filter order and, therefore, the number of multi-plications and additions for a straightforward realizationgrows approximately inversely proportional to the transitionbandwidth of the magnitude response [1, 2]. As multiplica-tions traditionally have a larger area complexity and powerconsumption compared to additions, much work has focusedon reducing the number of multiplications in FIR filterrealizations by using sparse filters or frequency responsemasking filters [3–5]. Work has also been done to reduce thecomplexity of each multiplication, for example, by introduc-ing filter coefficients easily realizable using shifts, additions,

and subtractions, sometimes referred to as multiplierlessrealizations [6–8].

Furthermore, the representation of data and coefficientsaffects both the switching activity and implementation com-plexity which, in turn, affects power consumption as well.Commonly, a fixed-point two’s complement number rep-resentation is used to represent data in DSP systems, butother number representations have also been investigated asan efficient way of data representation for such systems [9–11]. Among them is the logarithmic number system (LNS)[12], which over the past few decades has been studiedas an alternative to fixed-point number systems. The mainmotivation for doing so is the inherent simplification of basicarithmetic operations as multiplication, division, roots, andpowers, due to its properties, which are reduced to addition,subtraction, multiplications, and divisions, respectively.Theyalso have interesting numerical properties as higher dynamicrange compared to fixed-point representations for a givennumber of bits [13] and better round-off noise perfor-mance than floating-point arithmetic for a given number ofbits [14–16].

Hindawi Publishing CorporationVLSI DesignVolume 2014, Article ID 217495, 14 pageshttp://dx.doi.org/10.1155/2014/217495

2 VLSI Design

With LNS reducing multiplication to an addition, itfinds application in low power signal processing systems,including digital filters [13, 17–23]. In [18, 19] LNS has alsobeen proposed to reduce power dissipation in hearing aiddevices [24], subband coding [25], and video processing [26].It has been shown that LNS requires a reduced data wordlength to achieve the same SNR when compared to fixed-point representations [21, 23], with a reported power savingof nearly 60% in some cases. However, no discussion aboutthe coefficient word length was included nor how to obtainfinite word length coefficients.

Most efforts towards utilizing LNS for digital havefocused on either implementing the nonlinear conversion toand fromLNS, selecting the logarithmbasis, or implementingthe LNS addition and subtraction efficiently [12, 23, 27–30].The finite word length filter design has to the best of theauthors’ knowledge not been considered but instead reliedon rounding the obtained coefficients to the nearest LNSnumber.

In this work, an integer linear programming (ILP)approach to design optimal finite word length linear-phaseFIR filters in the LNS domain is proposed. Here, insteadof optimizing filters in the linear domain and convertingthem into LNS with rounding in the LNS, we optimize thefilter directly in the LNS domain with finite word lengthconstraints in the LNS domain. By optimizing the filtersdirectly in the LNS domain it is also possible to compare therequiredword lengths to obtain further insights regarding theefficiency of LNS.

The rest of the paper is outlined as follows. In the next sec-tion LNS numbers are reviewed. In Section 3 ILP is reviewedand the issues involved by performing it in the LNS domainare discussed. This section also includes the formulationand the proposed variable selection and branching directionstrategies. Finally, Sections 4 and 5 present the results andconclusions, respectively.

2. The Logarithmic Number System (LNS)

The LNS takes advantage of the fact that multiplicationsbecome additions, however, at the cost of increased com-plexity to implement addition. The LNS representation of anumber𝑋 consists of a tripletX as follows [12]:

X = (𝑧𝑥, 𝑠𝑥, 𝑚𝑥) , (2)

where 𝑧𝑥is a one-bit flag to indicate if𝑋 is zero, 𝑠

𝑥is the sign

of𝑋, and𝑚𝑥= log𝑏|𝑋| is the base-𝑏 logarithm of the absolute

value of 𝑋 [21]. Representation capabilities, computationalcomplexity, and conversion to and fromLNS numbers greatlydepend on the choice of the base [10].

As stated before, the motivation behind using LNS is thesimplicity of implementing the multiplication of X and Ywhich is reduced to the computation of the tripletZ [21]:

Z = (𝑧𝑧, 𝑠𝑧, 𝑚𝑧) , (3)

where 𝑧𝑧= 𝑧𝑥or 𝑧𝑦; that is, the zero flag of the output, 𝑠

𝑧=

𝑠𝑥xor 𝑠𝑦; that is, the sign of the output and the output itself,

𝑚𝑧= 𝑚𝑥+ 𝑚𝑦.

The addition and subtraction are more complex and aregiven by (4)

add = max (𝑚𝑥, 𝑚𝑦) + log

𝑏(1 + 𝑏

−|𝑚𝑥−𝑚𝑦|) ,

sub = max (𝑚𝑥, 𝑚𝑦) + log

𝑏(1 − 𝑏

−|𝑚𝑥−𝑚𝑦|) .

(4)

In case of FIR filters, the filter coefficients are typicallybetween −1 and 1. Representing the coefficients in LNSmeans that all the exponents in the LNS domain will benegative. This means that the larger the magnitude in LNSdomain, the smaller it is in the linear domain. As the coeffi-cients become smaller, the LNS number magnitude becomeslarger and reaches themaximumnumber representable using𝑖 integer and 𝑓 fractional bits, given by ±2−(2

𝑖

−2−𝑓

). Importantto note is that since the LNS numbers will always be negativefor (most) FIR filter coefficients, there is no need to use a signbit to represent the exponent in this case.

Any number smaller than this is not representable. Thisphenomenon is shown on a linear scale in Figure 1 where twoscales are shown for one and two integer bits, respectively.As expected, the logarithmic numbers are unevenly spaced.However, there is a gap between the smallest nonzero numberand zero which is much larger than the distance between thesmallest and second smallest nonzero numbers. Any numberthat lies within this space is rounded to either zero or thesmallest number. Hence, the number of integer bits will havean impact on the smallest number that can be represented.

2.1. Finite Word Length Effects. It is well established thatcoefficient quantization results in a static deviation of thetransfer function, while data quantization results in round-off noise [1, 2]. In the linear domain the quantization happensafter the multiplications, as the fractional word length isincreased there. However, in the LNS domain, the fractionalword length is not increased after the multiplications, asthis corresponds to an addition. Instead, the quantizationoccurs in the addition, because addition, as shown in (4),requires look-up tables and cannot be represented exactly[15]. In the linear domain, the multiplier complexity growslinearly with the coefficient word length, while the addercomplexity is independent of the coefficient word length. Inthe LNS domain, the multiplier complexity is dependent onthe smallest of the data and coefficient word lengths, whilethe adder complexity is dependent on the largest of those.

3. Proposed Integer Linear ProgrammingDesign in the LNS Domain

3.1. Integer Linear Programming. Integer linear programming(ILP) is a class of linear programming where some or all thevariables are restricted to have integer values [31]. Generally,

VLSI Design 3

−2−(21−(2−3+2−2))

−2−(21−2−2)

−2−(21−2−3)

2−(21−(2−3+2−2))2−(2

1−2−2)

2−(21−2−3)

−0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.350.30.250.20.150.10.05

(a)

−2−(22−(2−3+2−2))

−2−(22−2−2)

−2−(22−2−3)

2−(22−(2−3+2−2))

2−(22−2−2)

2−(22−2−3)

−0.15 −0.1 −0.05 0 0.150.10.05

(b)

Figure 1: Distribution of smallest logarithmic values using base 2: (a) one integer bit and three fractional bits and (b) two integer bits andthree fractional bits.

h5

h4

h4

h3

h3h3h3

h2 h2

h1

h1

h1 h1h0 h0

LNS solution Better solutionexists

LNS solution Better solutionexists

h3 ≤h3 ≤

h2 ≥ ⌈h2⌉ h2 ≤ ⌊h2⌋h2 ≤ ⌊h2⌋h2 ≥ ⌈h2⌉

h3 ≥ ⌈h3⌉ ⌊h3⌋⌊h3⌋

h3 ≤ ⌊h3⌋

h3 ≥ ⌈h3⌉

h3 ≥ ⌈h3⌉h4 ≥ ⌈h4⌉

h4 ≤ ⌊h4⌋

h5 ≥ ⌈h5⌉h5 ≤ ⌊h5⌋

Figure 2: Example branch and bound tree for LNS FIR filter design.

the standard form to express a linear programming problemis given by

maximize 𝑐𝑇𝑥

subject to 𝐴𝑥 ≤ 𝑏𝑥 ≥ 0

𝑥, 𝑐 ∈ R𝑛, 𝑏 ∈ R𝑚, 𝐴 ∈ R𝑛×𝑚,

(5)

where 𝑛 is the number of variables and 𝑚 is the number ofconstraints.

However, some real world problems require the resultto be integers. This requirement is incorporated into theprogramming model by having an additional constraint 𝑥 ∈Z𝑛. There are two important algorithms for solving ILPs:branch and bound (BB) and cutting plane. (Naturally, for FIRfilters the “integers” are fixed-point values with a fractionalpart. However, to keep the standard nomenclature we selectto denote any fixed-point value with integer.)

In this work we use the branch and bound (BB) algorithmwhich is a general algorithm for finding optimal solutions tointeger or combinatorial problems. It consists of enumeratingall candidate solutions and discarding candidates by usingupper and lower bounds of the quantity being optimized.

ILP problems are nondeterministic polynomial (NP)hard. In order to solve them the integer constraint is relaxed

LNS integer pointsLinear integer points

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x2

Figure 3: Example solution space for ILP with LNS (⬦) and linear(×) integers with three fractional bits with valid region in gray andthe line representing the pareto front.

and the variables are allowed to take noninteger values.This, and subsequent, LP-relaxations provide a bound on thebest possible solution without the integer constraint. Onenoninteger variable is selected and branched on by formingtwo subproblems. One where the variable is restricted to beat least as large as the next higher integer value and onewhere it is restricted to be at most as large as the next lowerinteger value. This is illustrated in Figure 2, where the topnode represents the LP-relaxation, variable ℎ

5is selected to

branch on, and two subproblems are formed by adding theconstraint on the edges. This procedure continues until anyof the following happens.

(1) The subproblem is infeasible, as any possible furtherbranching will not change that.

(2) The subproblem results in a solution where all coeffi-cients are integers.

4 VLSI Design

Total word length (Q(3, y))10 20

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00

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Scheme 1Scheme 3Scheme 4

(d)

Figure 4: Number of visited nodes (solved subproblems) for specification 6: (a)𝑁 = 20, 𝑄(3, 𝑦), (b) 𝑁 = 20, 𝑄(4, 𝑦), (c) 𝑁 = 34, 𝑄(3, 𝑦),and (d)𝑁 = 34, 𝑄(4, 𝑦).

(3) The solution of the subproblem is worse than thebest obtained integer solution, and, hence, no furthersubproblems generated from that node can possiblyhave better values.

In any of these cases the algorithm returns to the previousnode and selects another subproblem to solve until allpossibilities are exhausted and the optimal integer solutionis obtained.

The nontrivial challenge in ILP is to know which variableto branch on and inwhich order.Wewill suggest and evaluatea number of branching schemes in later sections.

3.2. Linear Programming Design of FIR Filters. Linear-phaseFIR filters can be designed to be linear and are a goodcandidate for linear programming optimization.The work byRabiner [32] is one of the earliest papers using this technique

to solve digital FIR filter problems. However, this techniquehas its limitations when finite word length restriction isimposed [33].

In ILP, the FIR optimization problem is the same as inthe linear programming case. A general FIR optimizationproblem can be stated as the followingminimization problem[1]:

minimize 𝛿

subject to −𝛿 ≤ 𝐸 (𝜔𝑖𝑇) ≤ 𝛿, 𝑖 = 1, 2, . . . , 𝐾,

(6)

where 𝛿 is the approximation error, 𝐾 is the number offrequency points, and

𝐸 (𝜔𝑇) = 𝑊 (𝜔𝑇) [𝐻𝑅 (𝜔𝑇) − 𝐷 (𝜔𝑇)] , 𝜔𝑇 ∈ Ω, (7)

where Ω is the union of the passband and stopbandregions and is a dense set of frequency samples taken from

VLSI Design 5M

agni

tude

resp

onse

(dB)

Normalized frequency

20

0

0 10.2 0.4 0.6 0.8

−20

−40

−60

−80

−100

−120

−140

Q(4, 1)Q(4, 6)

Q(4, 10)

Q(4, 14)

Figure 5: Magnitude responses for different fractional word lengthof spec. 3,𝑁 = 34.

the passbands and stopbands, including the band edges.𝐷(𝜔𝑇) is the desired function to be approximated by𝐻

𝑅(𝜔𝑇).

For a standard low-pass filter,𝐷(𝜔𝑇) is given by [1]

𝐷 (𝜔𝑇) = {1, 𝜔𝑇 ∈ [0, 𝜔

𝑐𝑇]

0, 𝜔𝑇 ∈ [𝜔𝑠𝑇, 𝜋] .

(8)

The weighting function 𝑊(𝜔𝑇) specifies the cost of thedeviation from the desired function.This basicallymeans thatby using this weighting function, one can obtain differentdeviations in ripples in different frequency bands. The largerthe weighting function, the smaller the relative ripple. For astandard low-pass filter, the weighting function can be givenas [1]

𝑊(𝜔𝑇) ={

{

{

1, 𝜔𝑇 ∈ [0, 𝜔𝑐𝑇]

𝛿𝑐

𝛿𝑠

, 𝜔𝑇 ∈ [𝜔𝑠𝑇, 𝜋] .

(9)

𝐻𝑅(𝜔𝑇) is the real zero-phase frequency response, which

is related to the frequency response as [1]

𝐻(𝑒𝑗𝜔𝑇) = 𝑒𝑗Θ(𝜔𝑇)

𝐻𝑅 (𝜔𝑇) . (10)

Themagnitude response of𝐻(𝑒𝑗𝜔𝑇) is equal to themagni-tude of𝐻

𝑅(𝜔𝑇). However,𝐻

𝑅(𝜔𝑇) can take on negative val-

ues where |𝐻(𝑒𝑗𝜔𝑇)| is a nonnegative function. The resultingfilter will be optimized in the Chebyshev or minimax sense.Such filters are also called equiripple filters because all theripples will be of equal size, subject to the weighting function[1].

The linear relaxation obtainedwithout integer constraintswill be the same minimax solution obtained using otherFIR filter design techniques, such as the Remez exchangealgorithm. Therefore, in later results the linear relaxation is

used as a lower bound. Introducing integer constraints willconstrain the solution space and increase the approximationerror. Furthermore, the resulting filters may no longer beequiripple.

3.3. ILP Design of FIR Filters in the LNS Domain. In the LNSdomain, our proposed design method focuses on finding theLNS equivalent of the coefficient values and implementing abranch and bound tree based on LNS integer values. In thebranch and bound tree, the coefficients are, when picked asthe branching variable, constrained to be LNS integers.

The solution space for an LNS ILP problem is differentfrom that of an ILP problem using linear representation. Anexample of this is shown in Figure 3, clearly showing thedifference between the two spaces.

In the LNS branch and bound tree, after solving a node,some of the coefficients will be LNS integers and some not.The linear relaxation of a noninteger coefficient, if selected,would be rounded up and down.

The first proposed branch variable selection scheme,named as Scheme 1, is to scan all the noninteger coef-ficients and pick the noninteger variable which has thelargest absolute value. This may be efficient since this willintroduce the largest quantization error in the linear domain,and, hence, possibly reduce the convergence time. Anotherscheme, denoted as Scheme 2, instead selects the coefficient,after scanning all noninteger coefficients, which is farthestfrom being an integer in the LNS domain.This is again basedon the idea of introducing large quantization errors early.

For both schemes described above, both combinationsof branching direction are evaluated. This means that forboth schemes, the difference is calculated between the linearrelaxation of the selected variable and its upper and lowerbounds. In one implementation, the branching variable isconstrained to the bound which gives the least difference firstand then upon returning to that node constraining it to theother bound. In the other implementation, this selection isreversed.

A third scheme is proposed, denoted as Scheme 3, wherethe branching direction was limited to the bound with theleast difference. In this scheme, all noninteger coefficients arescanned and the minimum distance to each of its boundsfor each variable is calculated. The coefficient having themaximum of these minimum distances is selected as thebranching variable.

All the above schemes either picked up the floor orceiling value of the linear relaxation for the next branchwithout actually knowing whether that would produce alower approximation error. The final proposed approach,Scheme 4, uses the same variable selection scheme as Scheme3 but both subproblems are evaluated before the selection isdone on which path to continue along. The decision is basedon the most promising path, that is, the one with the smallestapproximation error.

All these branch variable selection schemes are for locat-ing the best variable to branch on. Apart from these schemes,the rest of the algorithm is the same for all cases. However, anerror threshold is allowed while testing each coefficient for

6 VLSI Design

0

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Total word length

LNS, Q(3, y)LNS, Q(4, y)

LNS, Q(5, y)Linear, Q(y)Linear relaxation

Appr

oxim

atio

n er

ror (

dB)

(c)

Figure 6: Effect of changing integer and fractional word length on approximation error: (a) spec. 3,𝑁 = 34, (b) spec. 6,𝑁 = 34, and (c) spec.6,𝑁 = 22.

being an LNS integer, which means that the coefficient neednot be exactly an LNS integer. Any number within a definedthreshold of the corresponding LNS integer is taken as anLNS integer. This is a standard procedure in ILP solving asnumerical errors in the computations may lead to nonintegervalues in the optimal solution, even though they in practiceshould be considered integers [31].

4. Results

For the results a number of low-pass filter specifications weredevised arbitrarily to cover both narrow-band andwide-bandfilters.The passband and stopband edges are shown in Table 1and the weighting function 𝑊(𝜔𝑇) is 1 for both passband

and stopband. Unless otherwise stated the logarithm base isselected as 𝑏 = 2. Regarding word length, 𝑄(𝑥, 𝑦) indicatesthe number of integer bits, 𝑥 and fractional bits, 𝑦 is used forthe LNS integer, and 𝑄(𝑦) denotes 𝑦 fractional bits for thelinear integers.

4.1. Comparison of Branching Schemes. A number of opti-mization runswere carried outwith different alternatives.Thepurpose was to see the efficiency of each scheme and see theimpact of word length on different filters.

Table 2 shows the number of nodes visited (subproblemssolved) for the first two schemes with different branch-ing directions. “L” denotes that the branching direction isdetermined on the closest integer, while “H” then denotes

VLSI Design 7

0

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LNS, Q(4, y)LNS (rounded), Q(4, y)

Linear relaxation

Appr

oxim

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n er

ror (

dB)

(c)

Figure 7: Approximation error as a function of coefficient word length for proposed method and direct rounding of the linear relaxation: (a)spec. 3,𝑁 = 34, (b) spec. 6,𝑁 = 34, and (c) spec. 6,𝑁 = 22.

Table 1: Filter specifications considered.

Filter spec. Spec. 1 Spec. 2 Spec. 3 Spec. 4 Spec. 5 Spec. 6𝜔𝑐𝑇, rad 0.2𝜋 0.2𝜋 0.2𝜋 0.2𝜋 0.1𝜋 0.75𝜋𝜔𝑠𝑇, rad 0.3𝜋 0.35𝜋 0.4𝜋 0.5𝜋 0.25𝜋 0.9𝜋

the integer with the largest difference from the relaxed value.The number of integer and fractional bits is four and six,respectively, that is,𝑄(4, 6). Table 2 establishes that branchingon the direction based on the least difference gives the bestresult. Also, Scheme 1 proves superior to Scheme 2.

Scheme 1 is further compared with Schemes 3 and 4 andthe results are shown in Table 3, again with𝑄(4, 6). This tableshows them to be comparable, apart from the case when thefilter length is 64.This shows that Scheme 1 does not convergequickly when filter length is increased. This behavior is

8 VLSI Design

20 30 40 50Filter order, N

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LNS, Q(4, 6)LNS, Q(3, 5)Linear, Q(10)

Linear, Q(8)Linear relaxation

Appr

oxim

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n er

ror (

dB)

(c)

Figure 8: Approximation error as a function of filter length: (a) spec. 5, (b) spec. 6, and (c) spec. 3.

further observed in numerous optimization runs. For variousinteger and fractional bits and filter lengths, Scheme 3 gives,on average, the best convergence time. Scheme 1 appears toperform better when given a narrow transition band andsmall filter length.

The effect of word length is further studied and resultsare shown in Figure 4. In Figures 4(a) and 4(b), the numberof nodes required to solve spec. 6, 𝑁 = 20, for three andfour integer bits, respectively, while the number of fractionalbits varying from 5 to 14 is shown. Similarly, Figures 4(c)and 4(d) show the same for spec. 6, 𝑁 = 34. When thefilter length is 20, Scheme 1 converges in the least amount oftime. Decreasing integer word length to three did not alterthe efficiency of this scheme. However, as the filter length was

increasedwhile keeping the passband and stopband edges thesame, one can see that decreasing the integer word lengthfrom four to three had a major impact on the convergencetime of Scheme 1. Further optimization runs show the sameeffect. Table 4 shows, for a few cases, the number of nodesneeded by each of these three schemes.

The figures and tables clearly show that if either (a) thetransition bandwidth is increased or (b) the filter length isincreased, Scheme 1 converges very slowly. This effect ismagnified when the integer word length is decreased. Thiscan be seen in the table in the case of spec. 4 with 𝑁 = 20,where, in relation to spec. 6,𝑁 = 20, the filter length is keptthe same but the transition bandwidth is increased.The sameeffect is seen for narrow band filters as well. Based on these

VLSI Design 9


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LNS, Q(4, 6)LNS, Q(3, 5)LNS, Q(4, 6) (rnd)

LNS, Q(3, 5) (rnd)Linear relaxation

Appr

oxim

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n er

ror (

dB)

(c)

Figure 9: Approximation error as a function of filter length for proposed method and direct rounding of the linear relaxation: (a) spec. 5, (b)spec. 6, and (c) spec. 3.

observations, one can conclude that either Scheme 3 or 4 ison average the best alternative to solve the given optimizationproblem.

4.2. Effect of Word Length. Next, the effect of changing theinteger and fractionalword length on themagnitude responseand approximation error was studied. Optimization runswere carried out for a number of specifications given inTable 1. For various optimization runs, either the integerwordlength was fixed and fractional word length varied or vice-versa. In Figure 5, the magnitude responses for filters withspecification 3 are shown. The integer word length has beenfixed at four while the fractional word length takes on valuesof one, six, ten, and fourteen.

To study the effect of changing integer and fractionalword length on the approximation error in more detailfor specification 3, the approximation error is plotted inFigure 6. In these figures, there are five curves, one showingthe optimal approximation error for continuous coefficients(linear relaxation) and one shows the approximation errorfor linear integers, while the other three curves are for LNSinteger word length three, four, and five, respectively, withtotal word length changing from four to 20 for each curve. Allthese figures give a detailed picture of the impact of changinginteger and fractional word length (the largest coefficientvalue, and, hence, the number of most significant bits neededfor the coefficients in the linear domain is also a functionof filter specification. For a narrow-band low-pass filter, one

10 VLSI Design

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−80

−100

−1200.2 0.4 0.6 0.8

LNS, Q(4, 6) LNS, Q(4, 6) (rounded)Linear, Q(10) Linear, Q(10) (rounded)

Figure 10: Magnitude response of finite word length LNS and linearcoefficients for spec. 3,𝑁 = 40.

Table 2: Comparison of first two variable selection schemes.

(a)

Scheme Branchingdirection

Number of nodesSpec. 1𝑁 = 64

Spec. 2𝑁 = 42

Spec. 3𝑁 = 34

1 L 30704 1410 948H 11113942 238320 20494

2 L 82336 5990 1088H 3887918 458974 12342

(b)

Scheme Branchingdirection

Number of nodesSpec. 4𝑁 = 20

Spec. 5𝑁 = 34

Spec. 6𝑁 = 26

1 L 378 1334 210H 4256 38650 13352

2 L 365 1940 356H 3710 37248 17190

would require less number of bits than awide-band filter.Thiscan be realized by considering Mth band (or Nyquist) filterswhere a filter with band with 𝜋/𝑀 rad will have a maximumcoefficient value of 1/𝑀. However, for simplicity, this aspectis ignored in the comparisons in this paper).

As shown in Figure 6, the LNS filters converge to opti-mum with both four and five integer bits. However, if theoptimal approximation error is small, approximately less than40 dB, filters with three integer bits also converge. Clearly,the number of fractional bits also has a large impact onthe resulting approximation error. It is also shown that thelinear domain finite word length filters are always beaten byan LNS filter.

Table 3: Comparison of variable selection schemes 1, 3, and 4 with𝑄(4, 6).

(a)

SchemeNumber of nodes

Spec. 1𝑁 = 64

Spec. 2𝑁 = 42

Spec. 3𝑁 = 34

1 30704 1410 9483 8986 4814 10104 10240 1618 1092

(b)

SchemeNumber of nodes

Spec. 4𝑁 = 20

Spec. 5𝑁 = 34

Spec. 6𝑁 = 26

1 378 1334 2103 366 1800 2904 164 3312 258

Table 4: Comparison of variable selection schemes 1, 3, and 4.

Spec. Filter length 𝑄(𝑥, 𝑦) Scheme Nodes

4 20 𝑄(4, 7)

1 2503 1784 96

4 20 𝑄(3, 7)

1 40568643 11504 84

6 20 𝑄(4, 10)

1 19483 21284 1982

6 20 𝑄(3, 10)

1 19483 21284 1982

3 34 𝑄(4, 3)

1 6903 9624 1442

3 34 𝑄(3, 3)

1 3054903 48804 4694

1 64 𝑄(4, 6)

1 307043 89864 10240

Results on comparing against plainly rounding the coeffi-cients are shown in Figure 7. As expected, results obtained byour proposed optimization method gives a smaller approx-imation error than obtained by rounding the real valuedcoefficients for the same specification. This means that itis possible to reduce the coefficient word length usingthe proposed technique and still meet the specification,hence, reducing the complexity of the arithmetic operatorsas discussed earlier. Also, by using the proposed method,

VLSI Design 11

0 10 20

−20

−40

−60

Total word length

Appr

oxim

atio

n er

ror (

dB)

(a)

−20

−40

−60


Appr

oxim

atio

n er

ror (

dB)

(b)

−20

−30

−40

−500 10 20

Total word length

1.95

2.00

2.05

2.10

Appr

oxim

atio

n er

ror (

dB)

(c)


−20

−30

−40

−50

1.95

2.00

2.05

2.10

Appr

oxim

atio

n er

ror (

dB)

(d)

Figure 11: Approximation error as a function of logarithm base. ((a), (b)) Spec. 3,𝑄(3, 𝑦),𝑄(4, 𝑦). ((c), (d)) Spec. 6,𝑄(3, 𝑦),𝑄(4, 𝑦).𝑁 = 34.

monotonic results are obtained; that is, increasing the wordlength always improves the performance.

To further compare the performance of LNS againstlinear integers, approximation error as a function of filterlength is plotted in Figure 8. The figure shows that for thesame number of total bits, LNS gives a better approximationerror. This improvement increases as the filter length isincreased. To strengthen the argument made earlier thatfilters optimized with word length constraints in the LNSdomain give better results than just rounding the linearrelaxation, Figure 9 compares the approximation error ofoptimized filters with that of direct rounding, as a function offilter length.The plot further establishes the advantage of ourproposed optimization over results obtained by rounding thereal valued coefficients. In this case, the optimization leadsto that a lower filter order can be reduced, and, hence, fewer

arithmetic operators. Again, the proposed design methodleads to monotonically decreasing approximation errors.

Finally, a comparison of the magnitude response ofspecification three for a filter length of 40 is shown inFigure 10. This further shows that LNS gives better resultsas compared with linear integers as well as illustrates thebenefit of actually optimizing to obtain finite word lengthcoefficients.

4.3. Changing the Base. Earlier works have indicated that theselection of logarithm base has an impact on the round-offnoise performance. Hence, several filters are designed withvarying bases. In addition, it was earlier shown that the num-ber of integer bits is limiting the obtainable approximationerror in some cases and changing the base can be seen to

12 VLSI Design

1 2 3 4 5

0

−60

−20

−40

Base, b

Appr

oxim

atio

n er

ror (

dB)

(a)

1 2 3 4 5Base, b

0

−60

−20

−40

Appr

oxim

atio

n er

ror (

dB)

(b)

1 2 3 4 5Base, b

0

−60

−20

−40

Appr

oxim

atio

n er

ror (

dB)

(c)

1 2 3 4 5Base, b

0

−60

−20

−40

Appr

oxim

atio

n er

ror (

dB)

(d)

Figure 12: Approximation error as a function of logarithmic base with different word length. ((a), (b)) Spec. 3,𝑄(3, 4),𝑄(4, 4). ((c), (d)) Spec.6 𝑄(3, 4), 𝑄(4, 3).𝑁 = 34.

give a similar effect as having a noninteger number of integerbits. For example, using three integer bits and using a baselarger than two will reduce the amplitude of the smallestrepresentable number (compare to Figure 1).

In Figure 11 the approximation error for specifications 3and 6 are shown using three and four integer bits. As seen,the above mentioned effect is clear; as for the three integerbit cases, the approximation error decreases when increasingthe base (and increases when decreasing the base). For fourinteger bits, the effect of the base is not so clear. However, forshort word lengths, it can be seen that there is some impactof the base.

To further illustrate the effect of selecting base the samefilters were redesigned with a total word length of seven bits,that is, 𝑄(3, 4) and 𝑄(4, 3), but with a larger range of basevalues. The results are shown in Figure 12. Here, the sametrend is clear and it is also evident that a base of at least twoshould be selected. However, it should be noted that the word

length is rather small and that the same clear effect may beseen for a longer word length.

5. Conclusion

In this paper, a method for designing finite word lengthlinear-phase FIR filters in the LNS domain was presented.Several branch variable selection and branching directionschemes were suggested and evaluated. The scheme wherethe branching variable was selected based on finding thelargest minimum distance from the closest integer variableand branching in that direction was found to be the beston average among those suggested. The resulting filters areoptimal in the minimax sense under finite word lengthconditions.

It was illustrated by examples that three or four integerbits were the best selection, with three being applicable forlarger approximation errors. As opposed to finite word length

VLSI Design 13

coefficients in the linear domain, the number of integerbits in the LNS domain determines the smallest nonzerocoefficient that can be represented. Using a different andlarger logarithm base than two can reduce the amplitude ofthe smallest representable number and slightly improve theresults, especially when using three integer bits.

An interesting topic for further studies is the relationbetween coefficient word length (determining the magnituderesponse) and the data word length (determining the round-off noise). While these are completely disconnected in thelinear scenario as well [1], the impact is different when LNSnumbers are considered. Assuming that the data word lengthis shorter than the coefficientword length, the least significantbits of each multiplication will be independent of the dataand vice-versa for the longer data word length.The impact ofthis on an architectural level may be interesting to investigatefurther.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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14 VLSI Design

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