Digital Signal Processing 23 (2013) 412–417

Contents lists available at SciVerse ScienceDirect

Digital Signal Processing

www.elsevier.com/locate/dsp

Design and performance analysis of adjustable window functions based cosinemodulated filter banks

Ashutosh Datar a,∗, Alok Jain a, Pramod Chandra Sharma b

a Samrat Ashok Technological Institute (Engineering College), Vidisha (M.P.) 464001, Indiab S.D. Bansal College of Technology, Umaria, Indore (M.P.) 452001, India

a r t i c l e i n f o a b s t r a c t

Article history:Available online 27 July 2012

Keywords:Cosine-modulated filter banksKaiser windowSaramäki windowRoark’s transitional windowUltraspherical windowNear perfect reconstruction

In this paper, design and comparative analysis of adjustable window functions based cosine modulatedfilter banks are analyzed. Four adjustable windows, viz., Kaiser window, Saramäki window, ultrasphericalwindows and Roark’s transitional window are used to design prototype filters. Reconstruction error,which is used as an objective function, is minimized by optimizing the cutoff frequency of designedprototype filters. The gradient based iterative optimization algorithm is used. These optimized filters arelater cosine modulated to obtain filter banks. The performances of filter banks are compared on the basisof reconstruction error and aliasing error.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

The multirate filter banks find variety of applications in sub-band coding, transmultiplexing, image and video or audio com-pression, spectral estimation, biosignal processing, and adaptivesignal processing. The elementary block in realization of suchapplications is cosine modulated filter banks (CMFBs), in whichanalysis and synthesis filter banks are obtained by cosine mod-ulated versions of lowpass prototype filter [1–3], Fig. 1. Thisscheme is popular for its ease of designing high-selective andhigh-discrimination systems. Also, since analysis and synthesis fil-ter banks are generated by the lowpass prototype filter, the entiredesign of the filter bank is reduced to the design of the prototypefilter. Therefore, during the design phase, it is required to optimizethe coefficients of the prototype filter only. This significantly re-duces the complexities and computational overheads.

Suppose the prototype filter P (e jω) is a low pass linear phase.The quality of reconstructed signal depends on how closely P (e jω)

satisfies the following two conditions:∣∣P(e jω)∣∣ = 0 for |ω| > π/M (1)

and∣∣T (e jω)∣∣ = 1 for 0 < ω < π/M

where T(e jω) =

2M−1∑k=0

∣∣P(e j(ω−kπ/M)

)∣∣2(2)

where M is the number of channels.

* Corresponding author.E-mail address: addatar@yahoo.com (A. Datar).

1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.dsp.2012.07.007

If (1) is satisfied exactly, there is no aliasing between nonad-jacent bands, while if (2) is satisfied exactly, amplitude distortionis completely eliminated in the combined analysis/synthesis filterbank system. Aliasing between the adjacent bands is eliminated byselecting appropriate phase factor in the modulation [3]. Unfortu-nately, it is not possible to design a finite length filter that exactlysatisfies the constraints of (1) and (2). Hence the filter bank thatprovides approximate or near-perfect reconstruction (NPR) is de-signed that approximately satisfies the constraints laid down in (1)and (2). Different objective functions have been optimized usinglinear optimization [4,5] as well as nonlinear optimization tech-niques [6,7].

2. Filter bank design

The analysis and synthesis filter banks are based on cosinemodulation of P (e jω). The prototype filter with desired character-istics can be easily designed by window technique.

2.1. Window technique

The impulse response of the ideal low pass filter with cutofffrequency ωc is given as

hid(n) = sinωcn

πn, −∞ < n < ∞ (3)

hid(n) extends from −∞ to +∞, is not absolutely summableand, therefore, unrealizable [8]. Hence, shifted impulse responseof hid(n) will be

hid(n) = sin(ωc(n − 0.5N)), n ∈ Z (4)

π(n − 0.5N)

A. Datar et al. / Digital Signal Processing 23 (2013) 412–417 413

Fig. 1. M-channel maximally decimated filter bank.

For making a causal filter, direct truncation of infinite-durationimpulse response of a filter results in large passband and stop-band ripples near transition band. These undesired effects are wellknown Gibbs’ phenomenon. However these effects can be signifi-cantly reduced by appropriate choice of smoothing function w(n).Hence, a filter p(n) of order N is of the form [9]:

p(n) = hid(n)w(n) (5)

where w(n) are the time domain weighting functions or windowfunctions. Window function is of limited duration in time domain,which approximates band limited function in frequency domain.Window functions are broadly categorized as fixed and adjustablewindows. In fixed window, the window length N governs main-lobe width. Adjustable window has two or more independent pa-rameters that control the window’s frequency response character-istics.

2.2. Adjustable window functions

2.2.1. Kaiser windowKaiser window [10,11] achieves close approximations to the

discrete prolate functions, which have a maximum energy concen-tration in the main lobe relative to that of the side lobes.

The window function w(n) for Kaiser window is given as [8]:

w(n) ={

I0{β√

1−(n/N)2}I0(β)

; 0 � n � N0; otherwise

(6)

where I0(.) is the zeroth-order modified Bessel function, which canbe computed as:

I0(x) = 1 +∞∑

k=1

((0.5x)k

k!)2

(7)

Parameter β for desired As and filter order N , an appropriate cho-sen transition bandwidth �ω, can be estimated as:

β =⎧⎨⎩

0.1102(As − 8.7); As > 50

0.5842(As − 21)0.4 + 0.07886(As − 21); 21 � As � 50

0; As < 21(8)

N ≈ As − 7.95

14.36�ω/2π(9)

2.2.2. Saramäki windowThis window [12,13] is also close approximation to discrete pro-

late functions. Compared to Kaiser window, it has the advantageof having analytical expression in both the frequency and timedomains. Further, no power series expressions are needed in eval-uating the window functions. The FIR filters obtained are slightly

better than those obtained using Kaiser window. The window func-tion w(n) for the window is given as:

w(n) = v0(n) + 2N∑

k=1

vk(n) (10)

and

N = As − 8.15

14.36(ωs − ωp)/π(11)

vk(n) can be calculated according to the following recursion rela-tions:

v0(n) ={

1; n = 00; otherwise

(12a)

v1(n) ={

γ − 1; n = 0γ /2; |n| = 10; otherwise

(12b)

and

vk(n) ={2(γ − 1)vk−1(n) − vk−2(n) + γ [vk−1(n − 1)

+ vk−1(n + 1)]; |n| � k0; otherwise

(12c)

Parameter γ can be computed as

γ = 1 + cos 2π2N+1

1 + cos 2βπ2N+1

(13)

with

β =

⎧⎪⎨⎪⎩

0.000121(As − 21)2 + 0.0224(As − 21)

+ 1; 21 � As � 650.033As + 0.062; 65 < As � 1100.0342As − 0.064; As > 110

(14)

2.2.3. Ultraspherical windowThese windows are based on orthogonal polynomial known as

Gegenbauer or ultraspherical polynomial. These polynomials havea close relationship with Jacobi polynomial and Chebyshev poly-nomial. The window has three control parameters with additionalcapability of generating variety of sidelobe patterns.

The coefficients of ultraspherical windows can be generated as:

w(nT )

= 1

N

[Cα

N−1(x0) +(N−1)/2∑

i=1

CαN−1

(x0 cos

iπ

N

)cos

(2nπ i

N

)]

(15)

where CαN (x) is the ultraspherical polynomial of degree N defined

by recursion relationship

414 A. Datar et al. / Digital Signal Processing 23 (2013) 412–417

CαN (x) = 1

N

{2(N + α − 1)xCα

N−1(x) + (N + 2α − 2)CαN−2(x)

}(16)

with Cα0 = 1; Cα

1 = 2αx; Cα2 = −α + 2α(1 + α)x2. w(nT ) can be

obtained by inverse FFT of CαN (x).

[14,15] provide the efficient formulation for computation ofwindow coefficients and allied parameters.

2.2.4. Roark’s transitional windowTransitional windows function w(n) [16,17] has an additional

degree of freedom compared to Kaiser window and Saramäki win-dow. B-splines are used to replace the sharp transition edges. Thewindow function for principally-flat (PF) designs are expressed as:

w[n] ={( sin[πn/(N+1)]

πn/(N+1)

)ρ; for −N � n � N

0; otherwise(17)

The desired smoothness relation is

α = ρ

(ωc/π)(N + 1)(18)

Impulse response of prototype filter h[n] is given as

h[n] = ωc

π

sin(nωc)

nωc

[sin[πn/(N + 1)][πn/(N + 1)]

]ρ

(19)

where

ρ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0; As < 21 dB13

1+(126/As)1.6 − 0.7; 21 � As � 120 dB

0.51+[(As−120)/20]1.6 + 0.063(As − 120)

+ 5.06; As � 120 dB

(20)

N =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

π�ω

( 24.31+(149/As)1.6 − 0.085

) − 1; 21 � As < 120 dBπ

�ω (−0.00075(200.3 − As)2 + 14.74) − 1;

120 � As < 150 dBπ

�ω (0.00001087(As + 245.6)2 − 3.1) − 1;As � 150 dB

(21)

2.3. Cosine-modulated filter banks

The impulse responses of the analysis filter bank hk(n) and syn-thesis filter bank fk(n) are the cosine-modulated versions of theprototype filter p(n) [3,9].{

hk(n) = 2p(n) cos((2k + 1) π

2M

(n − N−1

2

) + θk)

fk(n) = 2p(n) cos((2k + 1) π

2M

(n − N−1

2

) − θk)

0 � n � N − 1, 0 � k � M − 1 (22)

where θk = (−1)k( π4 ) is the phase factor for cancellation of all sig-

nificant aliasing terms [3].

3. Optimization algorithm

3.1. Objective function

The cutoff frequency ωc of low pass prototype filter is taken asthe variable parameter, and is optimized to obtain the minimumvalue of reconstruction error, chosen as objective function φ.

φ = maxω

∣∣∣∣P(e jω)∣∣2 + ∣∣P

(e j(ω−π/M)

)∣∣2 − 1∣∣

for 0 �ω � π/M (23)

The algorithm evaluates ωc such that the 3-dB cutoff frequency ofoptimized prototype filter is located approximately at ω = π/2M .An iterative linear optimization algorithm (Fig. 2) developed in [18,19], is modified and is used to minimize the objective function φ.

Fig. 2. Flow chart of optimization algorithm.

3.2. Algorithm

A gradient based linear optimization algorithm is used to ad-just the cutoff frequency. Filter design parameters and optimizationcontrol parameters like step size (step), target error (t-error), direc-tion (dir) and previous error (prev-error) are initialized. Prototypefilter is designed using windowing technique. With each iteration,ωc of p(n) and reconstruction error (error) is computed, which isalso the objective function. If the error increases in comparison toprevious iteration (prev-error), step size (step) is halved and thesearch direction (dir) is reversed. This step size and direction isused to re-compute ωc for new prototype filter. The optimizationprocess is halted, when the difference in error between the nthand (n − 1)th iteration is within a specified tolerance (t-error).

A. Datar et al. / Digital Signal Processing 23 (2013) 412–417 415

4. Distortions in filter banks

In the near-perfect reconstruction (NPR) case, the reconstructedsignal has two major distortions, viz., amplitude distortion andaliasing. In such a case, aliasing is canceled approximately and thedistortion is a delay only approximately [5]. The input–output re-lationship for cosine-modulated filter banks is given by [9,20]

Y (z) = T0(z)X(z) +M−1∑l=1

Tl(z)X(zW l

M

)(24)

where W M = e− j2π/M and

T0(z) = 1

M

M−1∑k=0

Fk(z)Hk(z) (25)

is called the distortion transfer function, determining the distortioncaused by the overall system for un-aliased component X(z) of theinput signal, and

Tl(z) = 1

M

M−1∑k=0

Fk(z)Hk(zW l

M

)(26)

for 1 � l � M − 1 is the aliasing transfer function. It determineshow well the aliased components X(zW l) of the input signal areattenuated.

Also, filter banks are characterized by the error in the ampli-tude response [21]

Em(ω) = 1 − ∣∣T0(e jω)∣∣ for ω ∈ [0,π ] (27)

A measure for the un-aliased distortion is the peak-to-peak am-plitude distortion E pp [22].

E pp = maxω∈[0,π ]

{∣∣T0(e jω)∣∣} − min

ω∈[0,π ]{∣∣T0

(e jω)∣∣} (28)

Also measure of total aliasing distortion, peak aliasing error Ea isgiven as

Ea = maxω∈[0,π ]

{∣∣Talias(e jω)∣∣} (29)

where Talias(e jω) is the total aliasing distortion and given as

∣∣Talias(e jω)∣∣ =

√√√√M−1∑l=1

∣∣Tl(e jω

)∣∣2(30)

5. Results

Four adjustable windows based optimized low pass FIR pro-totype filters are used to design 32-channel cosine modulatedfilter bank. Stopband attenuation As is kept at −100 dB, stop-band frequency is ωs = (1 + ε)π/(2M) and passband frequencyis ωp = π/(2M). Here, ε is the roll-off factor which influences theoverlap of subbands in filter banks. Roll-off factor ε and ρ in [21]are related in terms of equal �ω. For various values of ε, prototypefilters are optimized and corresponding filter banks are developed.In each case, reconstruction error Em and maximum aliasing errorEa are computed and plotted (Fig. 3 and Fig. 4, respectively). Theresults are compared with results obtained in [21] for the samedesign parameter with M = 32 and As = −100 dB (Table 1).

For all four windows under study, prototype filters (with equalN and As = −100 dB) are designed and optimized using the algo-rithm. From these prototype filters, the 32-channel cosine modu-lated filter banks are generated (Fig. 5) and their parameters arecompared (Table 2).

Fig. 3. Variation of reconstruction error Em with roll-off factor.

Fig. 4. Variation of aliasing error Ea with roll-off factor.

6. Conclusion

In this work, performances of four adjustable windows basedcosine-modulated filter banks are analyzed. Optimization algo-rithm effectively optimized the prototype filters in each case andlocated cutoff frequencies at π/2M approximately. The algorithmalso significantly reduced Em and Ea .

It is observed that, for the same value of ε, ultraspherical win-dow dominated other windows in terms of lowest Em with lowervalue of N (Table 1). On the other hand, Roark’s transitional win-dow displayed superior performance with lowest Ea , but the win-dow has larger value of N , which is a drawback. Kaiser windowand Saramäki window showed almost identical performance. Thesetwo windows offered a reasonable performance in terms of Em , Ea

and N .It is also noted that the proposed method shows better recon-

struction as compared to earlier reported work [21] for the sameN and As (Table 1). Moreover, Ea is comparable.

Also, for the same value of N , ultraspherical window providedlowest value of Em , while Roark’s transitional window offered low-est Ea (Table 2).

Acknowledgment

Authors wish to acknowledge the anonymous reviewer for giv-ing constructive suggestions, which has helped in improving thequality of the manuscript.

416 A. Datar et al. / Digital Signal Processing 23 (2013) 412–417

Table 1Comparison of reconstruction error Em and max. aliasing error Ea (M = 32 and As = −100 dB).

Window Proposed method Bergen and Antoniou [21]

ε N Em Ea ρ N Em Ea

Kaiser window 1.70 483 2.48E−03 5.46E−07 1.00 483 3.13E−03 3.73E−071.79 460 2.52E−03 1.44E−07 1.05 460 3.23E−03 9.78E−081.88 437 2.49E−03 2.78E−07 1.10 437 3.40E−03 1.84E−072.05 401 2.48E−03 4.28E−07 1.20 401 3.85E−03 2.38E−07

Saramäki window 1.70 483 2.54E−03 5.40E−07 1.00 482 3.90E−03 3.86E−071.79 459 2.57E−03 1.29E−07 1.05 459 3.53E−03 9.13E−081.88 439 2.56E−03 2.53E−07 1.10 438 3.90E−03 1.85E−072.05 403 2.54E−03 3.23E−07 1.20 402 3.68E−03 2.29E−07

Ultraspherical window 1.70 468 2.18E−03 9.39E−07 1.00 468 3.85E−03 6.63E−071.79 445 2.26E−03 2.68E−07 1.05 446 3.80E−03 1.58E−071.88 424 2.18E−03 4.19E−07 1.10 425 4.41E−03 2.63E−072.05 388 2.14E−03 1.05E−06 1.20 390 4.20E−03 4.56E−07

Roark’s transitional window 1.70 625 4.29E−03 3.52E−09 – – – –1.79 593 4.36E−03 1.25E−08 – – – –1.88 565 4.30E−03 9.03E−09 – – – –2.05 518 4.31E−03 1.54E−07 – – – –

Fig. 5. Optimized analysis filter bank frequency responses (at ε = 1.5, As =−100 dB) using (a) Kaiser window, (b) Saramäki window, (c) Roark’s transitionalwindow and (d) ultraspherical window. Only first 8 bands are shown.

Table 2Comparison of reconstruction error Em and max. aliasing error Ea at equal N(M = 32 and As = −100 dB).

Window ε N Em Ea

Kaiser window 1.59

518

2.85E−03 3.58E−07Saramäki window 1.59 2.90E−03 3.43E−07Ultraspherical window 1.54 2.58E−03 7.92E−07Roark’s transitional window 2.05 4.31E−03 1.54E−07

Kaiser window 1.452

565

2.86E−03 1.25E−07Saramäki window 1.452 2.91E−03 1.23E−07Ultraspherical window 1.41 2.61E−03 1.22E−07Roark’s transitional window 1.88 4.30E−03 9.03E−09

Kaiser window 1.386

593

2.86E−03 3.09E−07Saramäki window 1.3878 2.91E−03 2.80E−07Ultraspherical window 1.342 2.58E−03 6.15E−07Roark’s transitional window 1.79 4.36E−03 1.25E−08

Kaiser window 1.315

625

2.83E−03 2.73E−07Saramäki window 1.316 2.87E−03 3.47E−07Ultraspherical window 1.273 2.54E−03 1.01E−06Roark’s transitional window 1.7 4.29E−03 3.52E−09

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Ashutosh Datar was born in Vidisha, India in1967. He has received his B.E. (Electronics & Instru-mentation) degree from Samrat Ashok TechnologicalInstitute, Vidisha (M.P.) India, and M.Tech. (BiomedicalEngineering) degree from the Institute of Technology,Banaras Hindu University (BHU), Varanasi, in 1989and 1998, respectively. He obtained his Doctoral de-gree from Rajiv Gandhi Proudyogiki Vishwavidyalaya(State Technological University of Madhya Pradesh),

Bhopal, India in 2012.He is presently working as associate professor in the Department

of Biomedical Engineering, Samrat Ashok Technological Institute, Vidisha(M.P.) India.

Dr. Datar is a life member of IE(I), ISTE, BMESI, Instrument Societyof India. His current research interests include digital signal processing,

multirate signal processing, filterbanks, bio-medical signal processing andmedical image processing.

Alok Jain was born in Vidisha, India in 1966.He received his B.E. (Electronics & Instrumentation)degree from Samrat Ashok Technological Institute,Vidisha, M.Tech. (Computer Science and Technology)from IIT Roorkee (University of Roorkee), in 1988 and1992, respectively. He obtained his Ph.D. degree fromThapar University (erstwhile Thapar Institute of Engi-neering and Technology), Patiala, India, in 2006.

He is presently serving as a Professor and Headin the Department of Electronics & Instrumentation Engineering, SamratAshok Technological Institute, Vidisha, India. He has published more than50 papers in journals and conference proceedings of international repute.He authored two monographs related to filterbank and transmultiplexerand two books on power electronics. He co-chaired the session in Int.Conf. SCI 2004 held at Orlando, USA.

Dr. Jain is a life member of IE(I), IETE, ISTE, BMESI, Instrument Societyof India and was the member of IET, UK for more than 10 years. His cur-rent research interests include digital signal processing, multirate signalprocessing, filterbanks, and their applications.

P.C. Sharma obtained BE and ME degrees from theUniversity of Indore (India) in 1969 and 1972 respec-tively. He earned Ph.D. from IIT Kanpur in 1982. In1972, he joined SGS Institute of Technology & Science,Indore as Lecturer in Electrical & Electronics Engineer-ing. He served this institute as Head of Electronics &Communication Department and later as Director un-til 2000.

During 1989–1990, he has been with the Depart-ment of Electrical & Computer Engineering, University of Colorado at Boul-der, USA as visiting scientist. He was with the Faculty of Electrical & Com-puter Engineering, Universiti Sains Malaysia, Ipoh, in 1995–96 and 1997–98.He served as Professor with Multi Media University, Kuala Lumpur during2000–2001 and with Universiti Teknologi Petronos, Ipoh, Malaysia during2002–2005. He is currently Executive Director, Bansal Group of Institutes,Indore (India).

For outstanding contribution to research, he has been awarded Statelevel Pt. Lajja Shankar Jha Award (1986), Anna University Award (1987) – theNational Award by ISTE New Delhi for the best teacher in engineering, andfor the best research paper in international journal, he was awarded Dr. RadhaKrishnan Award in 1992. Dr. Sharma has authored a book on transmissionlines and filters. He has to his credit over 150 research papers.

Dr. Sharma is senior member IEEE, Fellow ETE (India), and Life memberISTE.