On an Iterative Method to Design Oversampled GDFT

Filterbanks ∗

Bogdan Dumitrescu†, Robert Bregovic, Tapio SaramakiInstitute of Signal Processing

Tampere University of TechnologyP.O.Box 553, 33101 Tampere, FINLAND

e-mail: bogdand,bregovic,ts@cs.tut.fi

ABSTRACTWe have recently proposed an algorithm for the simpli-fied design of generalized DFT modulated filterbanks.The algorithm produces very good designs, but its ex-ecution time becomes unacceptably long for prototypefilters lengths greater than 200. The main cause is theinitialization step, in which an orthogonal filterbank isfound. In the present paper, we study two alternativeinitialization methods. The most successful is based onthe design of a reduced length filterbank from which, byinterpolation, a good initialization of nominal length canbe found. Examples illustrate the quality and efficiencyof this approach.

1 INTRODUCTION

Subband processing has attracted considerable atten-tion in the past decade, and several studies [4, 5, 9, 8, 1]have shown that good performances can be achievedwith nearly perfect reconstruction (NPR) oversampledfilterbanks. To reduce the implementation cost, mod-ulated uniform filterbanks are commonly employed.Complex modulation is especially interesting, since itallows aliasing elimination for any (sufficiently high)down-sampling factor.

In the M -channel filterbank (FB) from Figure 1, theanalysis filters Hk(z) and synthesis filters Fk(z) are ob-tained via generalized DFT (GDFT) modulation fromtwo prototype filters, H(z) and F (z), respectively. Theimpulse responses of the filters are

hk[n] = h[n]ejπ(2k+1)(n−D/2)/M ,fk[n] = f [n]ejπ(2k+1)(n−D/2)/M ,

(1)

where h[n], f [n] are the impulse responses of the proto-types and D is the overall delay of the filterbank. Thefrequency responses of the filters are like in Figure 2b. Ifthe input signal is real, then only M/2 filters are neces-sary in each bank. We assume that the prototype filtersare FIR, of order Nh and Nf . In an NPR FB, in the

∗ Work supported by Nokia Research Center, Tampere, Fin-land.

†On leave from the Department of Automatic Control andComputers, ”Politehnica” University of Bucharest, Romania.

absence of subband processing, the output y[n] is ap-proximately equal to the delayed copy x[n − D] of theinput.

Real-time processing imposes bounds on the delay ofthe filterbank. As orthogonal1 GDFT FBs have a delayequal to prototypes order, we are interested in biorthog-onal FBs, for which the delay is arbitrary. Certainly, thecase of interest is when D ≤ Nh, Nf . Design algorithmsfor biorthogonal FBs are proposed in [4, 3, 1, 2]. Thealgorithm we have proposed in [2] differs from the oth-ers by offering almost complete control on the PR error.Although it has an iterative nature, as in [1], our algo-rithm uses a completely different initialization (and alsothe optimization criterion is different).

In this paper we aim to refine the work from [2]. Asinitially proposed, our algorithm is relatively fast fororders below 100, but becomes unacceptably slow fororders greater than 200. The main burden is the initial-ization step, where an orthogonal FB is designed with atechnique similar to that from [10]. So, the main subjectof this paper is to study fast initialization techniques. InSection 2, we review the basic algorithm from [2]. Sec-tion 3 presents two alternative initialization methods,based on interpolation of lower order prototypes and onthe window method [6], respectively. Section 4 is ded-icated to experiments, from which we derive the bestapproach to the design of high order prototypes.

2 ITERATIVE DESIGN OF GDFT FBs

The input-output relation for the FB from Figure 1 is

Y (z) = T0(z)X(z) +R−1∑

`=1

T`(z)X(ze−j2π`/R), (2)

where

T0(z) =1R

M−1∑

k=0

Hk(z)Fk(z) (3)

1Following the accepted definitions, only perfect reconstructionfilterbanks can be orthogonal. In a loose sense, we name hereorthogonal an NPR filterbank in which Nh = Nf and fk[n] =h∗k[Nh−n]. Such FBs are defined by a single prototype filter. Allother NPR filterbanks, usually defined by two prototypes (one foreach of the analysis and synthesis banks), are called biorthogonal.

1

F1(z) H1(z)

F0(z)

↑R

↑R ↓R H0(z)

↓R

y[n]

x[n]

HM-1(z) FM-1(z) ↑R ↓R

+

Pro

cess

ing

Un

it

+

Analysis Bank Synthesis Bank

x0[n]

x1[n]

xM-1[n]

y0[n]

y1[n]

yM-1[n]

Figure 1: M -channel filterbank.

π/M

H

2π/M

H0 H1 H2

ω 4π/M π

ω

−π 6π/M

M = 8 ,, k =1/2=

π −π

H3 H4 H5 H6 H7

(a)

(b)

π/R

Figure 2: Magnitude response of an eight-channelGDFT filterbank: a) prototype filter, b) analysis filters.

is the distortion transfer function, and

T`(z) =1R

M−1∑

k=0

Hk(ze−j2π`/R)Fk(z), (4)

for ` = 1 : R− 1, are called aliasing transfer functions.The algorithm from [2] is based on two conditions that

ensure the NPR property:

• The distortion transfer function (3) is approxi-mately equal to a pure delay, i.e.

|T0(ejω)− e−jDω| ≤ δd, ∀ω ∈ [0, 2π], (5)

where δd is a preset tolerance.

• The prototype filters, H(z) and F (z), have a mag-nitude response that is very small outside the base-band [0, π/R], as suggested in Figure 2a. Thus, theterms Hk(ze−j2π`/R)Fk(z) from (4) are small andthe aliasing transfer functions have little influenceon the output of the filterbank.

Let Eh and Ef be the stopband energies of the analy-sis and synthesis prototypes, respectively. The stopbandedge is

ωs = (1 + ρ)π/M, (6)

with ρ slightly smaller than M/R − 1 (such that ωs ≤π/R). The algorithm from [2] has the following form:

1. Design the prototype H(z) of an orthogonal FB oforder N0 by minimizing Eh subject to (5).

FB order 104 208 312 416Initialization time 16 95 500 -

Iteration time 8 30 80 160

Table 1: Times (in seconds) necessary for initializationand one iteration for the algorithm from [2].

2. Given H(z), design F (z) of order Nf by minimizingEf subject to (5).

3. Given F (z), design H(z) of order Nh by minimizingEh subject to (5).

The second and third step can be iterated and thestopband energies Eh and Ef decrease. However, inmany cases the improvement is marginal and so it isenough to perform only once steps 2 and 3; moreover,the average stopband energy

E = (Eh + Ef )/2 (7)

has low sensitivity with respect to the value of the initialorder N0 (the only parameter of the method which is nota genuine FB specification).

The optimization problems are convex: semidefiniteprogramming (SDP) in step 1 and second-order coneprogramming (SOCP) in steps 2 and 3. The complex-ity of the SDP problem grows faster with the order anddominates the computation when the order is in the hun-dreds. In Table 1, we present the average times neces-sary for initialization (step 1) and one iteration (steps2 and 3) of the algorithm, for orders Nh = Nf used inthe examples from the experimental section. All timesare measured on a PC with Pentium IV at 1GHz. It isclear that the initialization becomes too costly for orderssituated between 100 and 200 and even prohibitive (orimpossible) for larger orders. This is the main reasonfor searching fast initialization techniques.

3 INITIALIZATION TECHNIQUES

The algorithm presented in the previous section is con-vergent, but it does not solve a convex optimizationproblem, although each step is based on convex opti-mization. So, the solution may depend strongly on theinitialization, moreover if only one or few iterations areperformed. Our experience from [2] shows that the ini-tialization with an orthogonal FB is good and robust, inthe sense that similar values of the average stopband en-ergy (7) are obtained for a relatively large range of initialorders N0. However, as shown in Table 1, this initial-ization may be too complex. Our aim is to explore thepossibilities of producing initializations of similar qual-ity with significantly less computational effort.

We propose two distinct initialization techniques:

• The window method [6] is used for designing ap-proximately orthogonal GDFT FBs. The method

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is fast, as only one parameter is optimized, namelythe cutoff frequency of a Kaiser window. However,the PR error is not directly controlled and the stop-band edge is not guaranteed; moreover, there is nooptimization of the stopband energy.

• The initial analysis prototype is obtained by in-terpolation from the prototype of a shorter lengthNPR GDFT filterbank. As very good quality smallorder FBs can be designed easily with the algo-rithm from [2], this is a very appealing method.We mention that interpolation was proposed in [7]for designing orthogonal M -channels FBs startingfrom two-channels FBs. However, the PR error ofthe resulting FBs are out of control, as well as thestopband properties of the filters. So, we suggestits use mainly for initialization.

A short description of the interpolation method fol-lows. Let us assume that an M -channel GDFT filter-bank is available. Its delay is D, the down-samplingfactor R and the prototype filters are H(z) and F (z),of orders Nh and Nf , respectively. We want to designan NPR GDFT FB with M channels, delay D, down-sampling factor R and prototype filters orders Nh andNf , such that

M = mM,

D = mD,

R = mR,

Nh = mNh,

Nf = mNf ,

(8)

where m is a positive integer. The interpolation methodconsists of obtaining the impulse responses of H(z) andF (z) by interpolating the impulse responses of H(zm)and F (zm) (and then scaling properly). The Matlabfunction interp can be used to this purpose.

If the M -channel FB would be PR and interpolationfilter ideal, then the interpolated FB would be also PR.For an NPR FB and with finite length interpolation fil-ter, the interpolated FB is NPR, with a PR error (5)that is usually relatively small, but out of control. So,such a filterbank needs further refinement. Thus, wepropose to use only the analysis prototype of the inter-polated FB as initialization; this is the step 1 of thealgorithm presented in the previous section.

4 EXPERIMENTAL RESULTS

We compare here the three initialization methods dis-cussed above: orthogonal, window method and interpo-lation; the first one was proposed in [2], while the othertwo are the fast methods suggested in this paper. Asthe design by interpolation imposes constraints on theFB specification, we chose the smallest FB data as fol-lows: M = 8 channels, delay D = 20, down-samplingfactor R = 3 and prototype filters orders Nh = Nf = 26.All the other specifications are obtained by multiplying

FB order 104 208 312 416Orthogonal 6.66 3.52 2.39 -

Window 6.91 3.58 2.43 1.82Interpolation 6.69 3.55 2.39 1.80

Table 2: Average stopband energies (×10−10) obtainedwith the three initialization methods.

N0 170 180 190 200 210Orthogonal 3.65 3.58 3.52 3.57 4.43

Window 11.2 7.09 4.40 3.58 5.10

Table 3: Average stopband energies (×10−10), for dif-ferent initialization orders (Nh = 208).

these numbers with an integer m, which takes the val-ues 4, 8, 12 or 16. The PR error bound is δd = 0.001(i.e. −60dB) in all cases, while the factor from (6) isρ = 1.63 (while M/R− 1 = 1.667). The orthogonal andthe window method use an initialization order N0; whennot specifying this order we will actually report the bestresult among those obtained with several values of N0.For all initialization methods, we report the results ob-tained after 10 iterations, if not otherwise specified; thisnumber of iterations is sufficient for convergence in al-most all cases.

In Table 2, we give the best average stopband energy(7) obtained for the three initializations and differentorders of the prototype filters. It is clear that the fastinitializations are able to provide results similar to thosegiven by the original algorithm. However, we have to in-vestigate the robustness of the fast methods. The pur-pose is to establish an approach that can always give agood FB with just few runs of the method.

Window method results. Although the windowmethod can give very good initializations, it is sensitiveto the initial order N0. An illustrative example is givenin Tabel 3, which contains average stopband energies ofFBs with Nh = 208, designed with the window and or-thogonal methods, for different initialization orders N0.While the range of values N0 for which the orthogonalinitialization gives almost the same (best) result is verybroad, the same range is rather narrow for the windowmethod. Moreover, the convergence speed is smaller forthe window method. We conclude that, for large ordersof the FB, the window method can be used with goodresults, but several initializations with different N0 arenecessary to guarantee the quality of the FB.

Interpolation method results. In Table 4 we presentthe average stopband energy of FBs with Nh = 208(and M = 64, D = 160, R = 24), obtained with theinterpolation initialization, for different FBs of smallerorder (these FBs are designed as in [2], with an orthog-onal initialization of order N0). Expectedly, a greatermultiplication factor m implies a poorer stopband en-

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ergy after the interpolation; the difference between theFBs interpolated with m = 8 and m = 2 is very visi-ble; although the iterative process is able to provide thesame performance in both cases (although when m = 8the initialization is important), the convergence is slowfor large m. On the contrary, when m = 2, the firstiteration already gives a stopband attenuation near theoptimum and so, practically, further iterations are notneeded.

Also, the values of the actual distortion transfer func-tion error δd from (5), obtained for the interpolated pro-totypes, are displayed in Table 4. The degradation ofthe actual δd is relatively small, from 0.001 for the initialFB to at most 0.00144 for the interpolated FBs. Some-what surprisingly, the value of the multiplication factorm appears not to determine the amount of degradation.

Similar phenomena happen for FBs with Nh = 416(and M = 128, D = 320, R = 48), as it can be seenfrom Table 5.

Examples of design. Figure 3 contains the frequencyresponses of the prototype filters of a small order FB,with Nh = 26 (and M = 8, D = 20, R = 3), designedstarting from an orthogonal initialization with N0 = 27.Since only the analysis prototype is used for the initial-ization of the larger order FB, it is advisable to designa FB whose analysis prototype has better attenuationthan the synthesis one.

After interpolation with a factor m = 8, a FB withNh = 208 (and M = 64, D = 160, R = 24) is obtained;The frequency responses of its prototypes are given inFigure 4. We remind that the average stopband energyof the prototypes is E = 1.1 · 10−8 (this value can beseen in Table 4); in particular, the stopband energy ofthe analysis prototype is Eh = 3 · 10−9.

After 10 iterations of the optimization algorithm, weobtain the prototypes shown in Figure 5. The averagestopband energy is E = 3.57 · 10−10, as it can be seenin Table 4 (and Eh = 3.09 · 10−10, Ef = 4.05 · 10−10).The improvement with respect to the interpolated pro-totypes is more then tenfold.

Design recommendations. Based on the experimentspresented above (and others not reported), we can givesome simple and general rules on the selection of aninitialization procedure. As before, we assume that thethe initial specifications are such that (8) holds for atleast an integer m.

1. If the order Nh is about 100 or smaller, use directlythe standard algorithm from [2], i.e. the orthogonalinitialization.

2. If the order is larger, but below, say, 500, then de-sign first an FB of order Nh = Nh/m as near aspossible from 100, using the standard algorithm.Interpolate (with factor m) the prototypes of thisFB. Using the interpolated prototypes as initial-ization, perform a single iteration of the standardalgorithm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

20Prototype frequency response

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

AnalysisSynthesis

Figure 3: Frequency responses of ”short” prototypes(with Nh = 26).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

20Prototype frequency response

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

AnalysisSynthesis

Figure 4: Frequency responses of interpolated proto-types (Nh = 208).

3. For orders even larger, design as above an FB oforder Nh/m as near as possible from (but less than)500. Interpolate (with factor m) its prototypes toobtain the desired FB.

Comments. The effective computer time needed forsuch a design may be less than 10 minutes in the worstcase. The distortion transfer function error (5) is notguaranteed when rule 3 is applied, but for such largeorders optimization is clearly difficult. Finally, if thespecification is such that relation (8) is not satisfied forany convenient m, then initialization with the windowmethod can be successfully used.

5 CONCLUSION

We have investigated two fast methods for the initial-ization of an algorithm for the design of oversampledGDFT modulated filterbanks in [2]. We have found thatthe interpolation method is the most appropriate andhave suggested the initialization strategy described atthe end of Section 4.

4

Nh 26 104N0 24 25 26 27 90 96 100m 8 2

δd(×10−3) after interpolation 1.44 1.23 1.11 1.16 1.29 1.28 1.28E after interpolation 151 248 175 110 6.72 6.61 6.64E after 1st iteration 31.9 15.7 7.56 4.88 3.65 3.58 3.61E after 10 iterations 7.54 4.51 3.55 3.57 3.59 3.53 3.56

Table 4: Actual distortion transfer function errors and average stopband energies (×10−10), for different interpolationinitializations (Nh = 208).

Nh 26 104 208N0 26 27 96 190m 16 4 2

δd(×10−3) after interpolation 1.14 1.19 1.35 1.28E after interpolation 101 63.6 2.77 3.97E after 1st iteration 4.08 2.64 1.85 1.84E after 10 iterations 1.82 1.82 1.80 1.83

Table 5: Same as in Table 4, for Nh = 416.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−140

−120

−100

−80

−60

−40

−20

0

20Prototype frequency response

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

AnalysisSynthesis

Figure 5: Frequency responses of optimized prototypes(Nh = 208).

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