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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 88 (2008) 2518– 2525

0165-16

doi:10.1

� Cor

Control

E-m

bregovi

journal homepage: www.elsevier.com/locate/sigpro

Design of low-delay nonuniform oversampled filterbanks

Bogdan Dumitrescu �, Robert Bregovic, Tapio Saramaki

Department of Signal Processing, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland

a r t i c l e i n f o

Article history:

Received 28 May 2007

Received in revised form

15 February 2008

Accepted 29 April 2008Available online 14 May 2008

Keywords:

Nonuniform filterbanks

Low-delay filterbanks

Semidefinite programming

Frequency sampling technique

84/$ - see front matter & 2008 Elsevier B.V

016/j.sigpro.2008.04.018

responding author. On leave from Depar

and Computers, ‘‘Politehnica’’ University of

ail addresses: bogdand@cs.tut.fi (B. Dumitre

c@cs.tut.fi (R. Bregovic), ts@cs.tut.fi (T. Sara

a b s t r a c t

Low-delay nonuniform oversampled filterbanks have a good applicability in real-time

audio applications. An appealing trade-off between complexity and frequency division is

obtained by using filterbanks having uniform sections generated by generalized DFT

modulation, with transition filters between the uniform sections. We propose two

methods for designing such filterbanks, with no restrictions on the widths of transition

filters. The first method is iterative and each iteration consists of convex optimization

problems. In the second, faster method, the transition filters are designed using a

frequency sampling technique. We present an example of design showing the good results

of the proposed methods.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

To mimic the human hearing system, we desire toanalyze audio signals in subbands whose widths followthe Bark scale [1]. This would imply the use of nonuniformfilterbanks (NUFB), with channels having distinct widths.With the current technology, it is not possible toefficiently implement, in real time applications, a filter-bank (FB) with such channels; its design would be also adifficult problem. Due to their fast implementation,modulated FBs are often used; for example, it has beenshown that various classes of oversampled nearly perfectreconstruction (NPR) uniform filterbanks (UFBs) performvery well in connection with typical speech enhancementapplications, like adaptive filtering, echo cancellation, etc.[2–5]. Oversampling allows the design of filters with goodstopband attenuation and of FBs with practically negli-gible aliasing. However, the uniform division of thefrequency scale is far from the psychoacoustic model. Itis assumed that NUFBs are better suited to the abovementioned applications. Therefore, the current research in

. All rights reserved.

tment of Automatic

Bucharest, Romania.

scu),

maki).

this area concentrates on searching for compromisesolutions between the complexity of implementationand design, and the departure from the Bark scaledivision. Furthermore, when designing FBs for audioapplications, it should be taken into account that theoverall FB delay is limited by standards in many cases.Therefore, the FB delay is an important design parameter.

Existing methods for NUFB design use one or moreUFBs whose filters are obtained by generalized DFT(GDFT) or cosine modulation of a few prototype filters.1

There are two main classes of NUFBs introduced so far.The first class contains FBs that are generated by mergingadjacent channels of one UFB [7–9]. The second classcontains FBs that are made of two or more UFBs, joinedwith transition filters (TFs) between two adjacent UFBs[10,11]. This paper proposes design methods for FBsbelonging to the second class, using GDFT modulation inthe UFB sections. The only method for this type of NUFBsappeared in [11]. With respect to it, we propose twoimprovements. The first is that we design low-delay

1 It should be pointed out that NUFBs can be also constructed by

using two-channel FBs in a tree structure, which can be useful in

mimicking the Bark scale [6]. Although these FBs are easy to design, the

overall delay tends to be large, due to the tree structure, and therefore

they are not considered here.

ARTICLE IN PRESS

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–2525 2519

(biorthogonal) NUFBs, while [11] deals with orthogonalFBs, whose delay is determined by the order of the longestfilter. The second is structural, i.e. the channels corre-sponding to the TFs can have in principle arbitrary widths.Also, our design methods are completely different fromthose proposed in [11].

In this paper, we present two methods for NUFBdesign, our main contribution being in the design oftransition filters. The prototypes of the uniform sectionsare designed using our previous work [12], which is also amodel for the first method: starting from an orthogonalNUFB, each of the analysis and synthesis banks isoptimized considering the other as fixed; the TFs areobtained by solving convex optimization problems, be-longing to semidefinite programming (SDP) or secondorder cone programming (SOCP). The second methodoptimizes the TFs using a frequency sampling technique(FST). Some of the results from this paper have beenpresented in the conference papers [13,14].

The paper is organized as follows. Section 2 describesthe structure and properties of the NUFBs under con-sideration. Section 3 contains our main contribution,the two methods for NUFB design. Section 4 presentsthe results of the two design methods for a NUFBspecification.

2. NUFB structure

An M-channel NUFB has the general block diagramshown in Fig. 1. The NUFB is oversampled if the down-sampling factors Rk, k ¼ 0 : M � 1, are chosen such that

XM�1

k¼0

1

Rk41. (1)

The NUFB structure studied here is built of S uniformsections, with TFs between each pair of consecutivesections. Section s contains ms filters, extracted from aUFB with Ms filters. Overall, we have the obvious relation

M ¼XS�1

s¼0

ms þ S� 1. (2)

For example, the idealized frequency responses shown inFig. 2 belong to a FB with M ¼ 9 channels, S ¼ 2 sections

H1(z)

↓R0H0(z)

↓R1

x[n]

HM-1(z) ↓RM-1

Proc

essi

ng U

nit

x0[n]

x1[n]

xM-1[n]

y0

y1

yM

Analysis Bank

Fig. 1. Nonuniform

and S� 1 ¼ 1 TFs; we have M0 ¼ 12, m0 ¼ 6, M1 ¼ 6,m1 ¼ 2. We use two notations for the transfer functions ofthe filters. As in Fig. 1, HkðzÞ, with k ¼ 0 : M � 1, is the filteron channel k of the FB. Also, by HðsÞk , with s ¼ 0 : S� 1,k ¼ 0 : ms � 1, we denote the filter on the k-th channel ofsection s. The TFs are denoted HðsÞt , s ¼ 1 : S� 1, thesubscript t being reserved for TFs. Similarly, we denoteRðsÞ ¼ Rk the down-sampling factor in the channel k,belonging to section s, and RðsÞt the down-sampling factorin the channel of the s-th TF. The channel width of a filterin section s is p=Ms. We denote Ds, s ¼ 1 : S� 1, the widthof the transition channel joining sections s� 1 and s. Tocover the whole spectrum of real-valued signals, thefollowing relation must hold:

pXS�1

s¼0

ms

MsþXS�1

s¼1

Ds ¼ p. (3)

The filters in section s are obtained by GDFT modulationfrom a single prototype, using the relation

hðsÞk ½n� ¼ hðsÞp ½n� ejpðkþasÞðn�D=2Þ=Ms ; k ¼ 0 : ms � 1, (4)

where D is the overall delay of the FB. The subscript p isreserved for prototype filters. The numbers as 2 R aredetermined by the position of the first filter of the section.For the first section we always have a0 ¼

12, since the

frequency response Hð0Þ0 ðejoÞ must be centered on p=ð2M0Þ

(the width of its channel is p=M0). For the second sectionin Fig. 2, the shift is a1 ¼ 4:5, since the frequency responseHð1Þ0 ðe

joÞ must be centered on p=2þ D1 þ p=ð2M1Þ ¼

4:5p=M1 (here, p=2 is the total width of the channels insection 0 and D1 ¼ p=6).

The orders of the prototype filters for section s aredenoted NðsÞh for the analysis bank, NðsÞf for the synthesisbank and NðsÞ if they are equal. The order of the s-thanalysis TF is NðsÞht , the order of the s-th synthesis TF is NðsÞf t ;we use the notation NðsÞt if the orders are equal.

The NUFB structure described above was proposed byPrincen [10] (using cosine modulation, i.e. real filters) andadapted by Cvetkovic and Johnston [11] to GDFT modula-tion. We note that all filters have complex coefficients andso, even if the input signal is real, the subband signals arecomplex-valued. However, obtaining real subband signalsis possible with an efficient modulation scheme describedin [15]. The implementation complexity of this NUFB

F1(z)

F0(z)

↑R1

↑R0

y[n]FM-1(z)↑RM-1

+

+

[n]

[n]

-1[n]

Synthesis Bank

filterbank.

ARTICLE IN PRESS

← S0 → ← S1 →t1

π0

H0(z) H1(z) H2(z) H3(z) H4(z) H5(z) H6(z) H 7(z) H 8(z)

)()0(0 zH )()0(1 zH )()0(2 zH )()0(3 zH )()0(4 zH )()0(5 zH )()1( zHt )()1(0 zH )()1(1 zH

π /M0 π /M1Δ12π /R4=2π /R(0)

Hk(z)

)()( zH sl

Fig. 2. Idealized frequency responses of the filters in the analysis bank.

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–25252520

structure can be approximated with that of S stand-aloneUFBs (one for each section), which can be implementedefficiently, due to the modulation, and of S� 1 directlyimplemented TFs. Hence, to meet practical complexityrequirements, the NUFBs with this structure should haveonly few uniform sections, typically two or three. In thenext section, we will present optimization methods forlow-delay NUFBs, with arbitrary bandwidth TFs.

3. Design methods

The input–output relation for the NUFB in Fig. 1 is

YðzÞ ¼ T0ðzÞXðzÞ þXM�1

k¼0

FkðzÞ

Rk

XRk�1

‘¼1

HkðzW‘RkÞXðzW‘

RkÞ, (5)

where WR ¼ e�j2p=R and

T0ðzÞ ¼XM�1

k¼0

1

RkHkðzÞFkðzÞ (6)

is the distortion transfer function and determines thedistortion caused by the overall system for the inputsignal. The terms of the double sum in (5) determine theeffect of the aliased components XðzW‘

RkÞ on the output

signal.If the filters HkðzÞ and FkðzÞ have a good attenuation

outside a band of width 2p=Rk, as suggested in Fig. 2, thealiasing terms FkðzÞHkðzW‘

RkÞ, ‘a0, from (5) are bounded

by a small constant for all frequencies. As argued in[11,12], it follows that the aliasing versions XðzW‘

RkÞ of the

input have an almost negligible contribution to the outputand so the input–output behavior of the FB can bedescribed by

YðzÞ � T0ðzÞXðzÞ, (7)

i.e. it depends practically only on the distortion transferfunction (6).

3.1. Design goals and strategy

An NUFB with the structure described in the previoussection is defined by 2S prototype filters and 2ðS� 1Þ TFs.In some particular cases, the filters of the synthesis bankare related to those of the analysis bank, and so the totalnumber of independent filters is smaller. A basic goal ofthe design is to obtain filters with good attenuation

outside a band of width 2p=Rk. This can be realized byminimizing their stopband energies or stopband ripples.This optimization is subject to NPR constraints. In view of(7), NPR can be obtained by imposing the condition

jT0ðejoÞ � e�jDojpdd; o 2 ½0;2p�, (8)

where dd is a preset bound. The condition (8) ensures thatthe output of the FB is (almost) a delayed version of theinput, i.e. y½n� � x½n� D�.

The design data are the orders of the prototype and TFs,the delay D and the tolerance dd. Since the simultaneousdesign of so many filters is difficult, we follow aprogressive approach, in which some filters are designedbefore the others. In particular, the prototypes of theuniform sections are designed independently and beforethe TFs.

For UFB design, we use the algorithms from [12], whichcan be classified as follows (their names are given hereonly for simpler reference).

�

ORTH_UFB: design a (near)orthogonal UFB, in whichthe synthesis filters are the reversed versions of theanalysis ones. The UFB is defined by a single prototypefilter. The delay is equal to the order of the filters. � PAIR_UFB: given the prototype filter defining the

analysis or synthesis bank, design the prototype filterdefining the other bank. The delay is arbitrary.

� ITER_UFB: design a complete (two-prototype) FB, with

arbitrary delay. The algorithm starts by designing anorthogonal FB, then alternately optimizes one of theprototypes, considering the other as fixed. A very smallnumber of iterations is usually sufficient.

The algorithms ORTH_UFB and PAIR_UFB optimize thestopband energy of the prototype filters, subject to the NPRconstraint (8). Each step of ITER_UFB performs the sameoptimization, so the overall algorithm minimizes the sumof the stopband energies of the two prototypes.

After designing the prototypes of the uniform sections,the TFs are designed one at a time. We denote ~T0ðzÞ thedistortion transfer function (6), where the transfer func-tions of the filters not yet designed are replaced by zero.When imposing the NPR constraint (8), we use ~T0ðe

joÞ

instead of T0ðejoÞ; thus, we take into account all the

information that is available at the moment of designinga certain TF. We design the S� 1 TFs from left to right

ARTICLE IN PRESS

Table 1Algorithm for designing orthogonal NUFBs

Algorithm ORTH_NUFBInput data: NUFB structure parameters and filter orders

bound dd

frequencies oðsÞb1, oðsÞb2, oðsÞe1, oðsÞe2, s ¼ 1 : S� 1

1. for s ¼ 0 : S� 1

design prototype filter of section s using ORTH_UFB2. for s ¼ 1 : S� 1

solve for GðsÞt ðejoÞ the SDP problem equivalent to (16)

compute TF HðsÞt ðejoÞ via spectral factorization

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–2525 2521

(from 1 to S� 1). This is the natural order, since therightmost channels have larger widths, and so the NPRconstraint is easier to satisfy; the narrower filters aredesigned first. More details are given below.

3.2. Design of orthogonal NUFBs

We start with the case of orthogonal FBs, where thesynthesis filters are

FkðzÞ ¼ z�DH�kðz�1Þ; k ¼ 0 : M � 1 (9)

the delay D being the order of the longest filter. (The namenear-orthogonal is probably more appropriate, as onlyperfect reconstruction FBs can be orthogonal.) We assumethat the prototype filters of the S uniform sections areavailable, as well as the first s� 1 TFs, and we want todesign the s-th TF, HðsÞt ðzÞ, of order NðsÞt . Let oðsÞc be thecenter of the channel corresponding to this TF. Thefrequency response of HðsÞt ðzÞ must be small outside aninterval ½oðsÞt1 ;o

ðsÞt2 �, defined by the down-sampling factor of

the channel by

oðsÞt1 ¼ oðsÞc � p=RðsÞt ; oðsÞt2 ¼ oðsÞc þ p=RðsÞt . (10)

We minimize the stopband energy of the TF onIs ¼ ½�p;o

ðsÞb1� [ ½o

ðsÞb2; p�, with oðsÞt1poðsÞb1, oðsÞt2XoðsÞb2; taking

the interval ½oðsÞb1;oðsÞb2� slightly narrower than ½oðsÞt1 ;o

ðsÞt2 �

allows an increase in the minimum attenuation of thefilter outside ½oðsÞt1 ;o

ðsÞt2 �, see [12] for a similar approach in

UFB design. The stopband energy has the expression

Es ¼1

2p

ZIs

jHðsÞt ðejoÞj2 do ¼ ðhðsÞt Þ

HUhðsÞt , (11)

where hðsÞt is the (complex) vector of the TF coefficientsand U is a positive definite Hermitian Toeplitz matrix withthe element on diagonal n defined by

fn ¼

1�oðsÞt2 � oðsÞt1

2p; if n ¼ 0;

j

2pnðe�joðsÞ

t1 � e�joðsÞt2 Þ; otherwise:

8>>><>>>:

(12)

Denoting GðsÞt ðejoÞ ¼ jHðsÞt ðe

joÞj2, the stopband energy canbe written as

Es ¼ f0gðsÞt ½0� þ 2ReXNðsÞt

n¼1

fngðsÞt ½n�

8<:

9=;9Ref/HgðsÞt g, (13)

where the vector gðsÞt contains the distinct coefficients ofGðsÞt ðe

joÞ.On account of (9), the distortion transfer function (6) is

T0ðejoÞ ¼ e�jDo

XM�1

k¼0

1

RkjHkðe

joÞj29e�jDoP0ðoÞ, (14)

where P0ðoÞ is real and positive. We denote ~P0ðoÞ afunction defined as PðoÞ from (14), but with the not yetdesigned filters (the TFs with indices from s to S� 1)replaced by zero. For a given frequency o, condition (8)becomes

1� ~P0ðoÞ � ddp1

RðsÞt

jHðsÞt ðejoÞj2p1� ~P0ðoÞ þ dd. (15)

We impose the NPR condition on an interval ½oðsÞe1;oðsÞe2�

respecting the conditions oðsÞe1ooðsÞt1 , oðsÞe24oðsÞt2 . Let G be agrid covering this interval. Using (13) and (15), theoptimization problem can be formulated as

mingðsÞt

Ref/HgðsÞt g

s.t. 1� ~P0ðoÞ � ddpGðsÞt ðejoÞ=RðsÞt p1� ~P0ðoÞ þ dd; 8o 2 G;

GðsÞt ðejoÞX0; 8o:

(16)

This is a convex optimization problem. The criterion andthe NPR constraints are linear in the coefficients ofGðsÞt ðe

joÞ. The difficulty of (16) comes from the positivityconstraint on GðsÞt ðe

joÞ. However, using the trace para-meterization of nonnegative trigonometric polynomials[16], this constraint is expressed as a linear matrixinequality (LMI) and (16) becomes an SDP problem.

The complete algorithm for designing orthogonalNUFBs is summarized in Table 1. We note that theconstraint (15) could be imposed exactly for all frequen-cies in ½oðsÞe1;o

ðsÞe2� (not only on a grid), using another

LMI [17]. However, due to the approximations inherentto our approach, this would complicate the problemunnecessarily.

3.3. Iterative design of low-delay NUFBs

We move now to the case of low-delay FBs. For thebeginning, we assume that the synthesis bank is given.Also, the uniform sections of the analysis bank and thefirst s� 1 TFs are known. We want to design HðsÞt ðzÞ, oforder NðsÞht .

We express the optimization objective and constraintsas functions of real variables, i.e. the real and imaginaryparts of the coefficient vector hðsÞt ¼ hðsÞtr þ jhðsÞti , gathered inthe vector ~h

ðsÞ

t ¼ ½ðhðsÞtr Þ

TðhðsÞti Þ

T�T. Denoting U ¼ Ur þ jUi, the

stopband energy (11) can be expressed as

Es ¼ ð~hðsÞ

t ÞT

Ur �Ui

Ui Ur

" #~hðsÞ

t 9ð~hðsÞ

t ÞTC ~h

ðsÞ

t , (17)

where the matrix C is symmetric and positive definite.The NPR condition (8) becomes

~T0ðejoÞ þ

1

RðsÞt

HðsÞt ðejoÞFðsÞt ðe

joÞ � e�jDo

����������pdd, (18)

where the only unknown is HðsÞt ðejoÞ. (Remind that ~T0ðe

joÞ

is computed using only the known filters.) Using only real

ARTICLE IN PRESS

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–25252522

variables, for a given frequency o, inequality (18) can bewritten in the second-order cone form

kAðoÞ~hðsÞ

t � bðoÞkpdd, (19)

where, denoting

wðoÞ ¼1

RðsÞt

FðsÞt ðejoÞ½1 e�jo . . . e�jNðsÞt o�, (20)

we have

AðoÞ ¼RefwðoÞg �ImfwðoÞg

ImfwðoÞg RefwðoÞg

" #,

bðoÞ ¼Ref ~T0ðe

joÞg � cos Do

Imf ~T0ðejoÞg þ sin Do

24

35. (21)

The minimization of the stopband energy subject toNPR constraints (imposed again on a grid G covering½oðsÞe1;o

ðsÞe2�) can now be cast as the following SOCP problem

min~hðsÞt ; a

a

s.t. kC1=2 ~hðsÞ

t kpa,

kAðoÞ~hðsÞ

t � bðoÞkpdd; o 2 G. (22)

To design the complete FB, we follow the same iterativeapproach used in [12] for UFBs. We start by designing anorthogonal NUFB, as described in Section 3.2, with filterorders smaller than the NUFB specifications. In the secondstage, we optimize the analysis bank considering thesynthesis bank as fixed, and then the synthesis bankconsidering fixed the analysis bank. This second stage isiterated a few times (a single iteration is often satisfac-tory) until no significant improvement is obtained. Whendesigning a bank (analysis or synthesis), first the proto-types of the uniform sections are designed independently,then the TFs are designed successively, one at a time.

The complete algorithm is described in Table 2.Although a convex optimization problem is solved in eachstep and the algorithm is convergent (since the sum ofstopband energies decreases in each step), the final resultis not necessarily a global optimum (moreover, we do notdefine a global optimization criterion). A nonnegligibleimpact on the result have the orders of the filters in the

Table 2Iterative algorithm for designing NUFBs

Algorithm ITER_NUFBInput data: NUFB structure parameters and filter orders

delay D

bound dd

frequencies oðsÞb1, oðsÞ

b2, oðsÞe1, oðsÞe2, s ¼ 1 : S� 1

orthogonal NUFB orders (for stage I)

Stage I: design orthogonal NUFB, using algorithm ORTH_NUFBStage II: iterate the following operations

II.A: design analysis bank keeping fixed the synthesis bank

1. for s ¼ 0 : S� 1

design prototype filter HðsÞp of section s using PAIR_UFB

2. for s ¼ 1 : S� 1

solve for HðsÞt ðejoÞ the SOCP problem (22)

II.B: design synthesis bank keeping fixed the analysis bank

same operations as in II.A, reversing filters roles

orthogonal design of stage I. A trial and error procedure isoften necessary. From our practice, we recommend valuesnear the orders of the final NUFB, but smaller than them;however, the order of the longest filters should be higherthan the delay D. To be able to meet the NPR constraints,we have noticed that the uniform sections that liebetween two TFs must have at least three filters; however,this is a constraint that is not restrictive in practice.

3.4. Frequency sampling technique for designing TFs

As we will report in Section 4, the iterative algorithmdescribed above is computationally costly, especially forthe design of the TFs of the orthogonal NUFB needed atinitialization. Hence, we consider a faster approach. Withthe specified delay and filter orders, we optimize theprototype filters of the uniform sections of both analysisand synthesis banks using ITER_UFB. We are left with theproblem of designing the TFs. To alleviate its difficulty, weconsider the particular structure with

HðsÞt ¼ FðsÞt ; s ¼ 1 : S� 1, (23)

i.e. the analysis and synthesis TFs on the same channel areidentical.

We assume that the first s� 1 TFs (out of the total S� 1TFs) are known and we want to design HðsÞt ðzÞ, of order NðsÞt .We adopt a frequency sampling technique (FST) [18,19]approach. The main idea of FST is to sample the frequencyresponse HðsÞt ðe

joÞ in NðsÞt þ 1 points, optimize a fewsamples (the others being given fixed values) and thengenerate the filter coefficients by an inverse DFT. In ourcase, we set to zero the samples that fall outside the band½oðsÞt1 ;o

ðsÞt2 � defined in (10). Denoting Do ¼ 2p=ðNðsÞt þ 1Þ the

grid step and oZ ¼ �pþ Do : Do : p, Z ¼ 0 : NðsÞt , thefrequencies of the sampling grid, the sampled frequencyresponse can be constructed as

HðsÞt ðejoZ Þ ¼

XðsÞt ðejoZ Þ if oZ 2 ½o

ðsÞt1 ;o

ðsÞt2 �;

0 otherwise;

((24)

where XðsÞt ðejoZ Þ are complex valued samples to be

determined. Fig. 3 illustrates definition (24). The numberof nonzero samples used in the optimization procedureis thus

NðsÞx ¼ bðNðsÞt þ 1ÞðoðsÞt2 � oðsÞt1 Þ=2pc, (25)

typically much smaller than NðsÞt . When a set of samples isavailable, the coefficients of the TF are obtained with an

Fig. 3. Characteristic frequencies and sampled frequency response for

designing a TF.

ARTICLE IN PRESS

Table 4NUFB channel parameters

ms Ms Width R N

Section 0 24 48 p=48 32 159

Transition – – p=16 10 149

Section 1 7 16 p=16 10 79

−20

0

20Prototype frequency response

)

AnalysisSynthesis

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–2525 2523

inverse DFT by

hðsÞt ½n� ¼1

NðsÞt þ 1

XNðsÞt

Z¼0

HðsÞt ðejoZ ÞejnoZ . (26)

Denoting X ðsÞt the vector of NðsÞx variables (frequencyresponse samples), and reminding that G is the gridcovering the interval ½oðsÞe1;o

ðsÞe2� on which the NPR condition

is imposed, we can state the optimization problem

minX ðsÞt

max WðoÞjHðsÞt ðXðsÞt ; e

joÞj; o 2 ½�p;oðsÞt1 � [ ½oðsÞt2 ; p�

s.t. ~T0ðejoÞ þ

1

RðsÞt

HðsÞt ðXðsÞt ; e

joÞ2� e�jDo

����������pdd; 8o 2 G

(27)

The notation HðsÞt ðXðsÞt ; e

joÞ stresses the fact that samples(24) determine the frequency response. This problem isnot convex, unlike the previous ones. Also, here weoptimize the stopband attenuation, not the stopbandenergy; this is due to the better results obtainedpractically with this criterion, for this setup. The weightWðoÞ allows different attenuations in different zones ofinterest; typically, WðoÞ is constant on each of theintervals ½�p;oðsÞt1 � and ½oðsÞt2 ;p�. The criterion is evaluatedon a frequency grid. The NPR constraint has the same formas in the previous design problems, taking (23) intoaccount.

For solving problem (27), we have used the Matlabfunction fminimax. Since the problem has many localminima, the initial values of the unknown samples (24)are important. We have used the initialization

XðsÞt ðejoZ Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðsÞt j1� j

~T0ðejoZ Þjj

qe�jDoZ=2; oZ 2 ½o

ðsÞt1 ;o

ðsÞt2 �,

(28)

which can be justified by the form of the constraint from(27). As another practical remark, we have noticed that itis recommended to start by choosing the order of the TF asthe higher order of the prototype filters of the adjacentuniform sections, i.e.

NðsÞt ¼ maxfNðs�1Þh ;Nðs�1Þ

f ;NðsÞh ;NðsÞf g. (29)

After a successful design, the order can be reduced as longas the obtained attenuations are satisfactory. Due to thelow number of variables NðsÞx in (25), we expect that thetime needed for solving (27) is considerably lower than forITER_NUFB. The algorithm described in this section issummarized in Table 3 and named FST_NUFB.

Table 3Frequency sampling technique algorithm for designing NUFBs

Algorithm FST_NUFBInput data: NUFB structure parameters and filter orders

delay D

bound dd

frequencies oðsÞe1, oðsÞe2, s ¼ 1 : S� 1

1. design prototypes of uniform sections using ITER_UFB2. for s ¼ 1 : S� 1

solve for XðsÞt ðejoZ Þ problem (27), using fminimax initialized with

(28)

compute coefficients of HðsÞt ðzÞ using (24) and (26)

4. Design example

To illustrate the algorithms presented in the previoussection, we design a NUFB with S ¼ 2 uniform sections.The parameters describing the NUFB structure are given inTable 4. Although the transition channel width is equal tothe width of the second section channels, we remind thatsuch a condition is not necessary for our algorithm. Thefilters orders are also given in Table 4; analysis andsynthesis filters on the same channel have the samelengths. The NPR error from (8) is dd ¼ 0:01. The delay ofthe FB is D ¼ 128 samples; for a sampling rate of 16 kHz,this corresponds to a delay of 8 ms. We report the resultsfor our algorithms as follows

ITER_NUFB. The SDP problems (16) and (22) have beensolved using the library SeDuMi [20]. The frequencyresponses of the two uniform sectionsprototypes, TF andcomplete analysis bank, designed using a single iterationof ITER_NUFB, are shown in Figs. 4–7. The dashed verticallines show the extent of the baseband, of width 2p=Rk. The(worst) stopband attenuations As of the filters are given inTable 5; these attenuations are obtained typically at theedges of the stopband. In other zones of the stopband, theattenuations go to over 75 dB in section 0, 90 dB in section1 and 100 dB for the TFs. The down-sampling factor of thetransition channel can be raised to 12, in which case As

becomes 65 dB, i.e. still a convenient value. A seconditeration slightly improves the filters, both in terms ofstopband energy and attenuation. For example, the

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−100

−80

−60

−40

Normalized frequency (ω/π)

Mag

nitu

de (d

B

Fig. 4. Frequency responses of the prototype filters for section 0.

ARTICLE IN PRESS

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−120

−100

−80

−60

−40

−20

0

20Prototype frequency response

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

AnalysisSynthesis

Fig. 5. Frequency responses of the prototype filters for section 1.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−140

−120

−100

−80

−60

−40

−20

0

Transition filter frequency response

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

AnalysisSynthesis

Fig. 6. Frequency responses of transition filters.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−120

−100

−80

−60

−40

−20

0Analysis bank

Normalized frequency

Mag

nitu

de (d

B)

Fig. 7. Frequency response of all analysis filters.

Table 5Stopband attenuations (in dB) obtained with ITER_NUFB

Analysis As Synthesis As

Section 0 44.6 43.6

Transition 84.8 83.8

Section 1 74.7 74.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−43

−42

−41

−40

−39

−38

−37

−36

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

PSD of reconstruction error

Fig. 8. Power spectral density of the reconstruction error, for the NUFB

designed with ITER_NUFB.

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–25252524

attenuation of the analysis TF is 84.7 dB (almost un-changed with respect to the first iteration), but theattenuation of the synthesis filter grows to 88 dB. Theorders of the orthogonal NUFB in the initialization stageare 145 and 71 for the prototypes of the uniform sectionsand 119 for the TF. Small modifications (few units) of theseorders still give good designs.

For checking the NPR quality of the overall design, wepass a white noise x½n� through the above NUFB. Thepower spectral density (PSD) of the reconstruction errory½n� � x½n� D� is shown in Fig. 8. The PSD is slightly largerthan dd ¼ �40 dB, due to the aliasing neglected in the NPRcondition (8). However, it proves that the approximationsused in the method are sound.

The design time for the NUFB with the abovespecifications is about 5 min, on a computer with AMDSempron 3000þ processor (at 1.6 GHz). The design of theTFs takes more than 4 min (mostly for the orthogonalNUFB TF) and thus is the most expensive part of thealgorithm. The time for an iteration is only about 10 s.

FST_NUFB. The prototype filters defining the uniformsections are the same as for ITER_NUFB. The frequencyresponse of the TF (remind that the analysis and synthesisTFs are identical) is shown in Fig. 9. The (worst)attenuation is 60 dB on ½�p;oð1Þt1 � and 90 dB on ½oð1Þt2 ; p�.(The weights in (27) have been chosen such that this ratiois obtained.) The PSD of the reconstruction error is similarto that from Fig. 8, but flatter (the maximum value isbelow�38 dB). The design time for the TF is about 50 s, i.e.

ARTICLE IN PRESS

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−140

−120

−100

−80

−60

−40

−20

0

Transition filter frequency response

Normalized frequency (ω/π)

Mag

nitu

de (d

B)

Fig. 9. Frequency response of transition filter designed with FST_NUFB.

B. Dumitrescu et al. / Signal Processing 88 (2008) 2518–2525 2525

about five times smaller than for the TF designed byITER_NUFB.

5. Conclusions

We have presented two methods for designing NUFBsmade of uniform sections obtained via GDFT modulationand joined with TFs. The main contribution of the paperresides in the design algorithms for TFs. The first isiterative and a convex optimization problem is solvedeach iteration. The second is based on a FST and has alower complexity. Both algorithms give good results, asshown by our experimental results.

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