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Design of prototype filter of DFT filter banks withlow implementation complexity

Zhi Guo Feng and Ka Fai Cedric Yiu

Abstract—In this paper, we consider the design of FIR pro-totype filter of DFT filter banks, where the coefficients areexpressed as sums of signed powers-of-two terms. The infiniteprecision solution can be obtained for given amplitude distortionor aliasing power. However, the finite precision solution may notbe satisfied. This problem is formulated as a constrained opti-mization problem and can be transformed into an unconstrainedinteger programming problem. An efficient algorithm based on adiscrete filled function is developed for solving the finite precisionsolution.

I. PROBLEM FORMULATION

Digital filters with coefficients expressed as sums of signedpowers-of-two (SPT) have been widely studied due to theirease in implementation [1], [2]. It can be applied in multiratefilter banks, which have found important applications in audioand video signal processing [3]. We assume that the sameprototype filter is applied for both analysis and synthesis. Atypical prototype FIR filter can be defined by

H(ω) =

L−1∑

k=0

h(k)e−ikω , (1.1)

where L is the filter length. Then, the aliasing power and theamplitude distortion can be defined as

AP =1

2π

∫ π

−π

D−1∑

l=1

|Al(ω)|2dω, (1.2)

andAD =

1

2πD

∫ π

−π

(1 − |A0(ω)|)2dω, (1.3)

where D is the decimator factor, K is the number of subbands,and

Al(ω) =1

D

K−1∑

k=0

H(ω +2πk

K+

2πl

D)H(−ω −

2πk

K). (1.4)

Ignoring the linear phase term, (1.1) is expressed as

H(ω) = hᵀC(ω), (1.5)

where h = (h(0), · · · , h(m))ᵀ, and m is defined by m =(L− 1)/2 if L is odd and m = (L− 2)/2 if L is even. C(ω)is an appropriate cosine function vector.

Zhi Guo Feng is with College of Mathematics and Computer Science,Chongqing Normal University, Chongqing, P.R.C. and Department of AppliedMathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon,Hong Kong. Email: z.feng.scholar@gmail.com

Ka Fai Cedric Yiu is with Department of Applied Mathematics, The HongKong Polytechnic University, Hung Hom, Kowloon, Hong Kong. Email:macyiu@polyu.edu.hk

For each i, the coefficient h(i) is expressed by

h(i) =

bi∑

k=1

sk,i2−k, i = 0, . . . , m, (1.6)

where bi is the wordlength and sk,i ∈ {−1, 0, 1}. To reducethe hardware complexity in real applications, the coefficientsneed to satisfy the constraint

bi∑

k=1

m∑

i=0

|sk,i| ≤ N, (1.7)

where N is the total allowable number of the SPT terms used.The performance is to choose the coefficient h, which

satisfies the constraint (1.7), such that

E(h) = maxω∈F=P∪S

|D(ω) − H(ω)|, (1.8)

is minimized, where P and S are the passband and thestopband regions. The target response is D(ω) = 1 whenω ∈ P and D(ω) = 0 when ω ∈ S.

From [4], the cutoff frequency ωp and stopband frequencyωs can be considered as parameters and obtained by control-ling the amplitude distortion (1.2) and aliasing effect (1.3)when Remez exchange algorithm is applied. Then, the infiniteprecision solution can be obtained. However, we need todecide the finite precision solution in real applications.

II. METHOD

A. Problem Transformation

Introduce an integer vector x = (x(0), . . . , x(m))ᵀ, wherex(i) = 2bih(i). Then, x ∈ X , where X is

{1− 2b0 , . . . , 2b0 − 1} × · · · × {1 − 2bm , . . . , 2bm − 1}.

When the wordlength is taken as b-bit, the number of SPTterms for a coefficient x can be defined by

Px(x, b) = mins

{

b−1∑

k=0

|sk| : x =

b−1∑

k=0

sk2k

}

. (2.1)

Then, the constraint (1.4) is equivalent tom∑

i=0

Px(x(i), b) ≤ N. (2.2)

The penalty function for (2.2) is given by

g(x) = max

{

0, Q

(

m∑

i=0

Px(x(i), b) − N

)}

, (2.3)

where Q is a sufficiently large positive real number.

The 13th IEEE International Symposium on Consumer Electronics (ISCE2009)

978-1-4244-2976-9/09/$25.00 ©2009 IEEE 30

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By adding the penalized term, we can transform the probleminto an unconstrained integer programming problem as

Problem 1. Find an x ∈ X such that

E1(x) = E(h(x)) + g(x). (2.4)

is minimized.

Remark 1. The computation of Px(x, b) can be implementedby recursive functions. Detail is given in [2].

B. Algorithm

To solve Problem 1, we will start the search from agood initial point first. For this, we consider the respectivecontinuous optimization problem of Problem 1, where h isnot restricted in discrete value. This problem can be solvedby the sequential quadratic programming (SQP). Suppose thatthe infinite precision solution is h and Let x denotes thecorresponding integer vector. We choose the initial point x0

as the nearest point in X to x.With an initial point x0, we can find the local minimizer.

The definition of local minimizer is given by

Definition 1. A point x∗ is called a local minimizer of E1

over X if E1(x∗) ≤ E1(x), ∀x ∈ X ∩ N (x∗), where

N (x) = {x, x ± ei : i = 0, . . . , m},

where ei is the i-th unit vector (the Rm+1 matrix with the

i-th component equal to one and all other components equalto zero).

Based on Definition 1, we present a discrete steepest descentalgorithm to search for a local minimizer below as

Algorithm 1.1. Start from an initial point x

∗ = x0.2. For each point x ∈ X ∩ N (x∗) \ {x∗}, compute the

function value E1(x). Suppose x′ is such that E1(x

′)is the minimum. If E1(x

′) ≥ E1(x∗), then x

∗ is a localminimizer of E1 and stop, else goto Step 3.

3. Set x∗ = x

′. Goto Step 2.

After a local minimizer is found with Algorithm 1, we usea discrete filled function method together with Algorithm 1 tojump out and search for a better solution. The discrete filledfunction is based on the one constructed in [5] as

Fµ,ρ(x; x∗) = µ[E1(x)−E1(x∗)]2 − ρ||x − x

∗||2,

if E1(x) ≥ E1(x∗). (2.5)

When ρ > 0, 0 < µ < ρ/M (M is a sufficiently large realnumber), (2.5) is called a discrete filled function.

It is not necessary to define the function Fµ,ρ when E1(x) <E1(x

∗). For in this case, we can terminate the filled functionsearch and use Algorithm 1 again to search for the localminimizer by using x as the initial point. And the localminimizer obtained is better than x

∗. Hence, we can repeatthe local minimizer search and filled function search until asufficiently large searching steps is met. Then we stop andreturn the optimal solution.

III. SIMULATION RESULTS

The computation of the algorithm was performed in Matlab,where the coefficients are set as µ = 10−5, ρ = 1, Q = 10.

We consider that there are 16 subbands and the decimatorfactor is D = 16. The filter length is taken as L = 16. If theamplitude distortion must be less than -20dB and the aliasingpower is treated as cost function, the optimal infinite precisionsolution of the prototype filter can be obtained by [4] withωp = 0, ωs = 0.1002 and error E∗ = −35.8360dB. Whenthe wordlength is fixed by b = 10, we can obtain the truncatedsolution with 46 SPT terms and the error is −34.8733dB.Then, we fix N = 42 and apply our method. The error of theoptimized solution is −35.2366dB, better than the truncatedsolution. These results can be seen in Figure 1.

0 0.1 0.2 0.3 0.4 0.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency

Gai

n(dB

)

Infinite precision solutionTruncated solutionOptimized solution

Fig. 1. Comparison of the designed prototype filters.

ACKNOWLEDGMENT

The research is supported in part by RGC Grant PolyU.7191/06E and Research Committee of the Hong Kong Poly-technic University.

REFERENCES

[1] W. Lu, “Design of 2-d fir filters with power-of-two coefficients: asemidefinite programming relaxation approach,” Circuits and Systems,2001. ISCAS 2001. The 2001 IEEE International Symposium on, vol. 2,2001.

[2] Z. Feng and K. Teo, “A discrete filled function method for the designof fir filters with signed-powers-of-two coefficients,” IEEE Trans. SignalProcessing, vol. 56, no. 1, pp. 134–139, 2008.

[3] P. Vaidyanathan, Multirate Systems and Filter Banks, 1993.[4] K. Yiu, N. Grbic, S. Nordholm, and K. Teo, “Multicriteria design of

oversampled uniform dft filter banks,” IEEE Signal Processing Letters,vol. 11, no. 6, pp. 541–544, 2004.

[5] C. Ng, L. Zhang, D. Li, and W. Tian, “Discrete filled function methodfor discrete global optimization,” Computational Optimization and Appli-cations, vol. 31, pp. 87–115, 2005.

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