Journal of Signal and Information Processing, 20 10 , 1, 35 -43
doi:10.4236/jsip.2010.11004 Published Online November 2010 (http://www.SciRP.org/journal/jsip)
35
Design of M-Band NPR Cosine-Modulated
Filterbank Using IFIR Technique
Ram Kumar So ni1, Alok Jain2, Rajiv Saxena3
1Department of Electronics & Communication Engineering, Samrat Ashok Technological Institute (Polytechnic), Vidisha, India,
2Department of Electronics & Instrumentation Engineering, Samrat Ashok Technological Institute, Vidisha, India, 3Department of
Electronics & Communication Engineering, Jaypee University of Engineering & Technology, Raghogarh, Guna, India.
Received September 28th, 2010, revised November 1st, 2010; accepted November 3rd, 2010.
ABSTRACT
This paper presents a design of M-band near perfect reconstructed (NPR) cosine-modulated filterbank (CMFB). The
prototype filter is formulated as an interpolated finite impulse response (IFIR) filter. Suitable stretched factor is used to
provide considerable reduction in computation cost as well as minimum value of error parameters. Further minimiza-
tion in errors has been achieved by applying a linear gradient optimization technique. The IFIR approach provides
reduction in stopband energy as side-lobe-fall-off-rate (SLFOR) in magnitude response of the prototype filter is im-
proved. Design examples have been included to illustrate the effectiveness of the proposed technique over the existing
work.
Keywords: CMFB, IFIR, NPR, Window
1. Introduction
Cosine modulated filterbank find wide applications in
many areas of signal processing such as data compres-
sion of speech, audio and video signals, denoising, fea-
ture detection and extraction and adaptive signal proc-
essing [1-5]. In this approach all the analysis filters

k
H
z and synthesis filters

k
F
z are designed by the
cosine modulation of prototype lowpass filter as shown
in Figure 1. As a result, the design of filterbank becomes
simple, more realizable and the computation cost reduces
to the cost of one prototype filter plus modulation over-
head. This design technique has been extensively studied
by many prominent authors [3-7]. The use of finite im-
pulse response (FIR) filter is oftenly preferred over the
infinite impulse response (IIR) filter in the design of
prototype lowpass filter. Such type of filter has some
desirable features like linear-phase, inherent-stability,
negligible-quantization noise, etc. [4]. However, the
computational requirements of a FIR filter are usually
greater than an IIR filter. This is essentially true in
CMFB design for high stopband attenuation and narrow
transition band width. The high computational require-
ment of FIR filter can be reduce by exploiting the redun-
dancy in filter coefficients. Neuvo et al. [8] have devel-
oped a technique for FIR filter design in which the re-
quired specifications are obtained by the cascade of FIR
sections. The overall structure is known as interpolated
finite impulse response (IFIR) filter. This approach re-
quires less multipliers than the conventional single stage
FIR implementation.
In the proposed work, an NPR type cosine modulated
filterbank is designed. The required prototype filter is
formulated by IFIR technique. The comparison has been
made with earlier reported work of Furado et al. [9] and
Kha et al. [10]. In first one the frequency response
masking (FRM) approach has been used for the design of
prototype filter of a filterbank. Similarly, in second case
prototype filter is designed using standard FIR approach
and a convex optimization technique is used to minimize
the associated errors of filterbank. The contribution of
this paper is summarized below:
1) The prototype filter is designed using IFIR ap-
proach to provide reduction in computation complexity at
high stopband attenuation and narrow transition band
width.
2) Most popular variable Kaiser window function is
used to design the prototype filter.

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
36
Figure 1. M-band cosine modulated filterbank.
3) NPR type cosine modulation approach is used for
the design of filterbank. It avoids the computation of
large matrix sets of perfect reconstruction (PR) condition
and thereby further reduces the computational burden
during the implementation.
4) The reconstructed output of NPR system is not ex-
act replica of input due to reconstruction and aliasing
error. A linear gradient optimization is applied in which
the cutoff frequency is changed in order to approximate
the power complementary property.
5) Examples are included to evaluate the performance
of the proposed technique in terms of error parameters,
computational complexity, group delay and stopband
energy.
This paper is organized as follow: The IFIR filter with
their performance parameters are described in Section 2.
This section is also included the design procedure of
prototype lowpass filter using proposed IFIR. Section 3
covered the description of cosine modulation approach
for filterbank design and parameters related with compu-
tational complexity. The algorithm of linear gradient
optimization technique with required objective function
is provided in Section 4. Section 5 included design ex-
amples of eight, sixteen and thirty two band NPR CMFB.
The performance of proposed work with earlier reported
publications are discussed in Section 6. Finally conclud-
ing remarks are made in Section 7.
2. Design of Prototype Filter
The impulse response coefficients of a causal Nth-order
linear phase FIR filter p[n] using window technique is
given by [4]:






pnw nhn (1)
where, h[n] is the impulse response of the ideal lowpass
filter and is expressed as:





c
sin 0.5
0.5
nN
hn nN



 (2)
where,


c

is the cutoff frequency of the ideal low-
pass filter and


wn is the Kaiser window function. For
the given value of passband

p

and stopband


s

frequencies the filter order is calculated by [4]:

7.95
14.36
As
N


 (3)
where,


2
sp

    is the normalized transi-
tion bandwidth, As is the stopband attenuation and


2
csp
  .
In the design of narrowband FIR lowpass filter the fil-
ter order is very high consequently the required number
of adders and multipliers are also high. Therefore, in pro-
posed work an IFIR technique is used for the design of
prototype lowpass filter. This technique is computation-
ally more efficient than standard FIR filter design tech-
nique. In IFIR technique the final prototype lowpass fil-
ter can be obtained by Eq. (4) [11]




L
PzGz Iz (4)
where,


L
Gz is the up-sampled version of model filter


Gzwith the stretching factor of L and I(z) is the inter-
polator filter. In the proposed work, both filters are de-
signed by using Kaiser window function. Input parame-
ters like stopband attenuation, passband ripple are taken
same as desire in final prototype IFIR filter. Only the
passband and stopband frequencies are different. For the
model filter the values are relaxed by factor of L, i.e. ,

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
37

p
L

and

s
L

, respectively. Similarly, for the in-
terpolator filter the value of passband and stopband fre-
quencies are

p

and

2
s
L

 , respectively.
The reduction in computation complexity provided by
IFIR filter is very much depends upon the stretch factor.
There is an optimum value of stretch factor L given in (5),
which provides upto 90% reduction in computation at the
cost of other performance parameters [11]. Therefore, the
selection of L is depends upon the applications, for which
the IFIR filter is designed. The optimum value of L is
given as [11].

2
2
opt
ps sp
L

 



 


(5)
The computational complexity is stated in terms of
number of multipliers needed for implementation. The
number of multiplier in case of linear phase FIR structure,
polyphase structure and IFIR structure can be calculated
as below [4,12]
Multiplier in linear phase FIR structure = 2N


(6)
Multiplier in polyphase structure =2
N



(7)
Multiplier in IFIR structure =


22
N
Ni
m





(8)
Similarly,
Adders in linear phase FIR structure = N


(9)
Adders in polyphase structure = 2N


(10)
Adders in IFIR structure = i
N
m
N


(11)
where, N, Nm and Ni are the filter order for standard FIR
filter, model filter and interpolator, respectively. Since,
the filter order of model filter and interpolator are less
because first one is stretched version of desired prototype
filter and second one has wide band and large transition
width. Therefore, the multiplier (multi) needed in IFIR
filter is much less than standard FIR filter and achieved
computational reduction is given as:
()
%Computational reduction
F
IR IFIR
FIR
multi multi
multi

 (12)
A FIR filter exhibits property of linear phase. There-
fore, its phase response is constant and derivatives of
phase with respect to frequency called the group delay
(
g

) is given by [12]:

2
g
dN
d

  (13)
Hence, in linear phase FIR filter the groups delay de-
pends upon filter order. The proposed technique provides
improvement in SLFOR, consequently reduction in
stopband energy (Es). For a lowpass prototype filter
stopband energy is given as [13]
2
()
s
s
EHd




 (14)
3. Cosine Modulation
Cosine modulation is one of the efficient technique
which provides minimum computational cost in filter-
bank design. In this technique all the filters of synthesis
and analysis sections are obtained by cosine modulation
of a linear phase lowpass prototype filter [4,12].
 
 
2cos 211
22 4
2)cos 211
224
k
k
k
k
N
hn pnkn
M
N
fn pnkn
M




























for 0 1, and 0kM nN

  (15)
Here, M is the number of bands in filterbank and


k
hnand


k
f
n are the impulse responses of the fil-
ters of analysis and synthesis sections, respectively. The
relation between the reconstructed output Y(z) and input
X(z) is expressible in the z-domain as
 

12
0
1
M- -j πlM
l
l
Yz TzXzTzXze


 (16)
Here, T0(z) is the overall distortion transfer function
and determines the distortion caused by the overall sys-
tem for the unaliased component X(z) of the input signal
and
 

12
0
M- jπ lM
k
lk
k
Tz FzHze


 (17)
for l = 1,2,,M-1 is called the aliased transfer func-
tion and determine how well the aliased components


2 jlM
Xze
 of the input signal are attenuated. To can-
cel aliasing and achieve PR, it is required that [4]


0,0, d
Tz cznc


(18)
where, ndis positive integer and


0,
l
Tz

for l = 1,2,3,, M -1 (19)
For PR the criteria for the prototype filter are very
strict. Fortunately, in many practical applications the PR
property can be slightly relaxed, resulting in NPR [4].
4. Optimization Technique
In NPR system, the PR conditions are certainly relaxed

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
38
by allowing small amount of error. There are three types
of errors occurs at the reconstructed output, viz., ampli-
tude, phase and aliasing [4,5]. The stopband attenuation
plays an important role in analysis of these parameters.
High stopband attenuation results smaller aliasing error
but larger reconstruction error. The peak to peak recon-
struction and the peak aliasing errors are define in (19-20)
[10]
   
00
max min
jj
pp
EMTeMTe


 (19)

max j
a
EEe


 (20)
where,
 
12
12
1
M
jj
l
l
EeT e







In NPR system, the measures of these error parameters
are approximately and are given by the following equa-
tions:

0for /
j
Pe M

  (21)




0
2
/
0
1,
21
where
0
j
jkM
j
Te
M
Te Pe
k








(22)
The accuracy of the first approximation gives aliasing
and accuracy of the second gives the reconstruction error.
The phase error can be eliminating completely by using
linear phase prototype filter. Other two distortion pa-
rameters can be minimized by applying suitable optimi-
zation technique. Much of work have been done in this
field [5,14-16]. Initially, Johnston [16] developed a
nonlinear optimization technique, later on many promi-
nent authors such as Creusere et al. [14] and Lin et al.
[15] and others [5] have simplified it and developed sin-
gle variable linear optimization techniques with different
objective functions. In the proposed work, a linear gra-
dient optimization technique is used with objective func-
tion given in (23). The cutoff frequency of the model
filter is selected as variable parameter to optimize the
objective function.


2 2
max( 1
jM
j
Pe Pe




 (23)
for 0≤ ω < π / M
Initially, input parameters, i.e., sampling rate, number
of bands, stretch factor, passband and stopband frequen-
cies, passband ripple and stopband attenuation of proto-
type filter are specified. Based on these input parameters
the passband and stopband frequencies of model filter
and interpolator filter are calculated. Determine the cut-
off frequency, transition band and filter length of model
filter as well as interpolator filter. Initialize, different
optimization pointers like step size, search direction, flag
and initial as well as expected minimum possible values
of objective function. Inside the optimization loop, de-
sign the model filter, interpolated filter, up-sampled
model filter and finally, obtained the prototype lowpass
filter using Equation (4). Determine the other bandpass
filters of analysis and synthesis sections using cosine
modulation. In optimization routine cutoff frequency of
model filter is gradually change as per the search direc-
tion and calculate the corresponding value of objective
function. Algorithm comes out from the loop as it at-
tained the minimum value of objective function and ob-
tained the optimized value of reconstruction error.
The flowchart of the optimization algorithm is given in
appendix and implemented on MATLAB 7.0 on Pentium
IV processor.
5. Design Examples
In this section, the performance evaluation of the pro-
posed technique has been done with the help of design
examples. The performance of the filterbank is examined
in terms of stopband attenuation, reconstruction error,
aliasing error, computational complexity, group delay
and stopband energy in used prototype filter. The effec-
tiveness of proposed work is compared with the earlier
reported work for same input parameters.
Example 1: An eight-band cosine modulated filter-
bank has been designed for same specifications as given
by Kha et al. [10] with stopband attenuation (As) =
35.8dB, stopband frequency

s

= 0.12

. Figure
2(a), shows the frequency response of the prototype filter
designed by standard FIR technique. In this proposed
work, the prototype filter is designed using IFIR tech-
nique. For the given design parameters of prototype filter,
the required model, up-sampled model and interpolator
filters are obtained for stretched factor 2L. The stop-
band attenuation and passband ripple for these filters are
same as desire in prototype filter, however the stopband
frequency is different. For model and interpolator filters
these values are


s
L

= 0.24

and

2/
s
L

 =
0.88

, respectively. The obtained magnitude responses
of these filters are shown in Figure 2(b). The magnitude
response of the resultant IFIR filter is shown in Figure
2(c). Filter coefficients are optimized by the proposed
gradient optimization algorithm (given in Appendix) to
reduce the reconstruction error. The optimized eight-
band CMFB is shown in Figure 2(d). The obtained val-
ues of reconstruction error, aliasing error, group delay,
stopband energy and the required number of multipliers
and adders in case of Kha et al. [10] and in proposed
IFIR technique are given in Table-1. The magnitude re-
sponses of reconstruction and aliasing errors are shown

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
39
in Figures 3(a) and 3(b), respectively.
Example 2: In this example, sixteen-band cosine
modulated filterbank is designed using same specifica-
tions as given in [10]. The stopband attenuation and
stopband frequency are taken as same, i.e., (As) = 45dB,

s

= 0.0590

. The stopband attenuation for model
and interpolator are same as in prototype filter, i.e., 45dB.
The stopband frequency for these filters are


s
L

=
0.118

and

2
s
L

 = 0.941

, respectively. The
magnitude response of prototype IFIR filter is shown in
Figure 4(a) and the magnitude responses of sixteen band
CMFB is shown in Figure 4(b). The optimized value of
reconstruction error, aliasing error and reduction in
computation cost, stopband energy and group delay are
given in Table 1. The reconstruction error and aliasing
error are shown in Figure 4(c) and 4(d), respectively.
Example 3: A thirty two-band cosine modulated fil-
terbank has been designed for same specifications as
given by Kha et al.[10] with stopband attenuation (As) =
100 dB, stopband frequency

s

=0.03125

. The
same stopband attenuation, i.e., 100dB is taken for model
filter as well as for interpolator. The stopband frequency
for these filters are

s
L

= 0.0625

and


2
s
L

 = 0.968

, respectively. The magnitude
response of prototype IFIR filter is shown in Figure 5(a)
and the magnitude responses of first and last four bands
of thirty two-band CMFB is shown in Figure 5(b). The
reconstruction error and aliasing error are shown in Fig-
ure 5(c) and 5(d), respectively. The optimized value of
reconstruction error, aliasing error and reduction in
computation cost and group delay are given in Table 1.
Example 4: In this example, the performance of pro-
posed technique is compared with the reported work of
Furado et al. [9]. Same input parameters as reported are
taken for comparison purposes. The obtained error pa-
rameters from the proposed technique are given in Table
2. The plot of reconstruction error and aliasing error are
given in Figure 6.
Figure 2. (a) FIR lowpass filter, (b) (–––) Up-sampled model,(– – – –)model filter,(--------) Interpolator filter, (c) IFIR proto-
type lowpass filter, (d) Eight-band CMF bank.
Figure 3. Magnitude response of (a) Reconstruction error, (b) Aliasing error for eight-band filterbank.

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
40
Figure 4. Magnitude response of (a) IFIR prototype filter, (b) Sixteen-bands CMF bank(c) Reconstruction error, (d) Aliasing
error.
Table 1. Performance comparison.
As Filter order
Arithmetic
elements
%Computa-
tional
saving
Work Approach
dB
M ωs
N NmNiEpp Ea E
s Ad-
ders
Mult-
liers
g

FIR 35.8 8 0.12π 40 - -5.50e-32.47e-31.18e-240 20 20 -
FIR 45.0 160.059π 102 - -5.95e-3 3.89e-4 9.00e-4102 51 51 -
FIR 100 320.031π 466 - -9.12e-4 2.38e-7 9.9e-10466 233 233
35.8 8 40 - - 20 20
45.0 16 102 - - 51 51
Kha et
al.[10]
Polyphase
100 32 466 - - 233 233
35.8 8 0.12π - 20065.46e-31.41e-31.0e-226 13 13 35
45.8 16 0.059π - 46062.1e-32.62e-41.0e-352 26 26 49
Pro-
posed IFIR
100 320.031π - 267153.3e-31.80e-78.8e-10282 141 141 39

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
41
Figure 5. Magnitude response of (a) IFIR prototype filter, (b) Thirty two-band CMF bank (c) Reconstruction error, (d)
aliasing error.
Table 2. Performance comparison with Furado et al. [9].
work Approach As M ωs N Nm N
i E pp E a
Furado et al. [9] FRM 60 8 0.035π639- - - 2.1e-4
Proposed IFIR 60 8 0.035π- 12208 9.1e-3 0.92e-5
00.1 0.2 0.3 0.4 0.50.5
-5
0
5x 10-3
Magnitude
Normal ized Frequ ency
00.1 0.2 0.3 0.4 0.50.5
0
0.2
0.4
0.6
0.8
11x 10
-5
Magnitude
Normalized Frequency
(a) (b)
Figure 6. Plot of magnitude response of (a) reconstruction error (b) aliasing error.
6. Discussion
In last two decades, lot of cost effective techniques have
been developed for the design of the filterbank. IFIR
approach is one of the techniques, which provides sig-
nificant reduction in computational cost. In this proposed
work, the reduction in computation is achieved by using
stretch factor of two and better results are obtained.

Design of M-Band NPR Cosine-Modulated Filterbank Using IFIR Technique
42
In the first example, 8-bands CMFB is designed. The
values of resulting reconstruction and aliasing error in
Kha et al. [10] and proposed work are 5.508 × 10-3;
2.477 × 10-3 and 5.46 × 10-3; 1.4 × 10-3 respectively.
Hence, the proposed method offers much better per-
formance in terms of reconstruction and aliasing errors.
Apart from lower values of error parameters the pro-
posed method provides reduction in stopband energy,
less group delay and 35% reduction in computational
cost in contrast to Kha et al.
Similarly, in second example, 16-band CMFB is de-
signed. It is quite clear from Table 1 that the value of
error parameters in the proposed work is much smaller
than the Kha et al. [10] with 49% reduction in computa-
tional cost. In third example, 32-band CMFB is designed
with the same input parameters as of Kha et al. [10]. As
it is clear from the results given in Table 1 that the pro-
posed technique provides smaller value of aliasing error
then reported work with reduction in computational cost
in terms of multipliers.
The fourth example is quoted for the comparison with
the work of Furado et al. [9]. In this approach the proto-
type filter is designed using FRM technique and the re-
quired number of filter order is very large, therefore
computational complexity is also high. However, the
proposed IFIR technique provides smaller value of alias-
ing error using much lower value of filter order.
The results in Table 1 indicate that the proposed
method provides better performance in terms of compu-
tational complexity then the equivalent polyphase im-
plementation. It is also clear from the magnitude re-
sponse of prototype filter that there is improvement in
SLFOR, which provides reduction in stopband energy of
prototype filter. The proposed method is not only suitable
for lower range of stopband attenuation, but also appli-
cable at higher value of stopband attenuations.
7. Conclusion
Efficient design for M-band NPR CMFBs has been pre-
sented. IFIR technique is used to design the prototype
filter, which provides significant reduction in computa-
tion cost as well as improvement in SLFOR of the proto-
type filter. The obtained values of reconstruction and
aliasing errors are much smaller than the error values in
previous publications for wide range of stopband at-
tenuation. Therefore, the proposed method is useful in
variety of applications such as audio and image compres-
sion. The improvement in SLFOR can be utilized for
better suppression of cross-talk and echo cancellation.
The improvement in proposed approach is possible by
using multistage IFIR prototype filter.
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Appendix