[66]

CHAPTER 4 SYNTHESIS OF COSINE MODULATED

NPR TYPE QMF BANKS

VERSAMPLED filter banks offer more design freedom and better noise immunity

than critically sampled filter banks. Due to the increased computational complexity caused

by oversampling, oversampled filter banks allowing an efficient implementation, such as CMFBs. CMFB

which is a special sub-class of the general M-channel filter bank. These filter banks are extremely used in

different areas of the digital signal processing such as equalization of wireless communication channel,

sub-band coding, bio-signal processing, spectral analysis, adaptive signal processing, denoising, feature

detection and extraction. In CMFB the analysis and synthesis filters are cosine modulated version of low

pass prototype filter [15].

4.1 COSINE MODULATION

Two-channel QMF bank finds limited applications as it divides the incoming signal into two sub-

band signals. The M band extension of two channel QMF filter bank consists of M parallel band pass

filters followed by M-fold down sampler called analysis filter bank section. All the filters of the bank

have same bandwidth and uniformly spaced centre frequencies. The analysis section decomposes the

spectrum of the input signal into a set of M-band signals each having a bandwidth of π/M. Since the

bandwidth of each sub-band signal is reduced by a factor of M, therefore it is possible to down sample

each by M without violating the Nyquist criteria.

If the decimation factor M is the same as the number of bands, it is called critically sampled filter

bank. Conversely, at the receiving end the filters F0(z) to FM-1(z) have basically the same characteristics as

the analysis filters H0(z) to HM-1(z), as shown in Figure-4.2.2 and Figure 4.2.1 respectively. This filter

section is called synthesis section. It up-samples the received sub-band signals by the same factor i.e., M,

and reconstructed the output with the help of synthesis filters. If, the reconstructed output is exact replica

of input, the filter bank is said to be perfect reconstruction type. In such type of filter bank the aliasing is

completely cancelled out and the distortion in transfer function is pure delay, thus the system is free from

aliasing, amplitude and phase distortion. Cosine modulation is one of the efficient techniques which

provides minimum computational cost in the design of filter banks [12 and 33].

O

[67]

Figure-4.2.1: M-Channel analysis filter bank

Figure-4.2.2: M- Channel synthesis filter

4.2 COSINE MODULATED MULTIBAND FILTER BANK

The cosine modulated analysis and synthesis filter banks are very cost effective and consists of

realizable aspect than that of a general filter bank system. There are different methods available in design

of cosine modulated filter bank system. The first method involves the design of two prototype filters and

therefore the implementation cost for the filter bank becomes the cost of two prototype filters along with

cosine and sine modulation. The second method requires only one prototype filter. Its reconstruction

distortion has linear phase and approximately flat magnitude response, but modulated analysis/synthesis

filters do not have linear phase, which is an important requirement in sub-band coding of images. In third

method, single prototype filter is needed and modulated filters have linear phase, but the resulting

distortion has peak or null at 0 and π frequencies [12 and 74].

F0 (z)

F1 (z)

M

M .

.

.

X1 (z)

FM-1 (z) M

.

.

XM-1 (z)

H0 (z)

H1 (z)

M

M X1 (z)

XM-1 (z)]

X0 (z)

HM-1 (z) M

.

.

X (z)

푋 (푧) X0 (z)

.

.

.

[68]

4.3 M-CHANNEL POLYPHASE COSINE MODULATED QMF BANKS

4.3.1 Model

In a two-channel QMF bank, by parallel application of a low-pass filter H0(z) and a high-pass

filter H1(z), the input signal is divided into two overlapping sub-bands decimated by 2:1. For the

reconstruction, decimated signals are interpolated and filtered by a similar set of filters F0(z) and F1(z)

before being recombined to produce reconstructed output. In the absence of channel and quantization

noise, QMF banks design requirements allow an alias free reconstruction of the original input signal. But,

perfect reconstruction of the original signal is not possible in the presence of aliasing distortion, phase

distortion and amplitude distortion. Aliasing distortion is removed by using suitable design of the

synthesis filters, whereas phase distortion is eliminated with the use of the linear phase FIR filters [75, 80

and 106].The amplitude distortion can only be minimized by using computer-aided techniques or

equalized by cascading with a filter. The generalized structure of M-band filter bank is shown in Figure-

4.2.3.

Figure-4.2.3: M-band filter bank

4.3.2 Design Procedure of Window Based Cosine Modulated QMF

The elementary block in realization of such applications is CMFBs, in which analysis and

synthesis filter banks are obtained by cosine modulated versions of low pass prototype filters [34 and 98].

Secondly, all the analysis Hz(z) and synthesis Fz(z) filters of filter bank can be simultaneously generated

by the cosine modulation of single linear phase FIR prototype filter. Thirdly, the error occurs at the

reconstructed output in NPR types system can easily be optimized by using suitable optimization

technique. The NPR type CMFB is simple, efficient realizable and computationally efficient [83 and

H0 (z) M

H1 (z) M

HM-1 (z) M

X (z)

M F0 (z)

M F1 (z)

M FM-1 (z)

Y(z)

[69]

131]. The performance of the system can be further improved with high stop-band attenuation and narrow

transition bandwidth of the prototype filter [21].

Therefore, it is required to optimize the coefficients of the prototype filter only during the design

phase. This significantly reduces the complexity and computational overheads. In speech and audio

communication, the stop-band attenuation (ATT) in the range of 40-60 dB provides the signals of

adequate quality if the crosstalk suppression is efficient [21 and 135]. It mainly depends on the different

parameters of prototype filter, i.e., stop-band energy (Es), far-end attenuation and SLFOR [21]. A window

function with minimum stop-band energy, better far-end attenuation and high SLFOR is the most suitable

window in such applications. All filters of synthesis and analysis sections are obtained by cosine

modulation of a linear phase low-pass prototype filter response p(n) [2 and 21].

ℎ (푛) = 2푝(푛)푐표푠 (2푘 + 1)휋

2푀푛 −

푁2

+ (−1)휋4

,

푓 (푛) = 2푝(푛)푐표푠 (2푘 + 1)휋

2푀푛 −

푁2

− (−1)휋4

,

for 0 ≤ k ≤ M-1, and 0 ≤ n ≤ N (4.3.2.1)

where, M is the number of bands and hk(n) and fk(n) are the impulse responses of the analysis and

synthesis sections, respectively.

In NPR system, the perfect reconstruction condition is relaxed by allowing small amount of

errors. There are three types of errors occur at the reconstructed output and these are amplitude (Epp),

phase and aliasing (Ea) [2 and 59]. The measures of these error parameters are given by the following

equations:

푃 푒 = 0,

for |푤| > (휋 푀⁄ ) (4.3.2.2)

푇 푒 = 1.

(4.3.2.3)

[70]

where, overall distortion is given as -

푇 푒 = 푃 푒 ( ⁄ ) .

(4.3.2.4)

The accuracy of first approximation generates aliasing and the accuracy of the second one gives

the reconstruction error whereas the phase error is eliminated by linear phase prototype filter. However,

other two distortion parameters (phase and aliasing) can be minimized by applying suitable optimization

technique. In the proposed work, a linear gradient based optimization technique is used with objective

function, as given in (4.3.2.5). Here, the cut-off frequency of prototype filter is varied to obtain the

minimum value of the objective function.

휑 = 푚푎푥 푃 푒 + 푃 푒 ( ⁄ ) − 1 ,

for 0 ≤ w ≤ 휋 푀⁄ (4.3.2.5)

The desired steps required to be undertaken for designing the prototype filter using the proposed

method and bi-section optimization algorithm is shown below-

(i). Specify sampling frequency, number of bands, stop-band attenuation, and pass-band ripple.

(ii). Initialize pass-band and stop-band frequency. Calculate the cut-off frequency, transition-band,

filter length and window coefficients of selected window function. Thereafter, initialize step size

(step), search direction (dir), flag, minimum value of reconstruction error (e_err) and initial value

of reconstruction error (k_err).

(iii). Design of prototype filter and filters for synthesis and analysis sections using cosine modulation

scheme.

(iv). Calculation of reconstruction error and absolute value of objective function (obj).

(v). Check whether obj > k_err. If yes, step=step/2, dir = - dir. If no, follow the next step.

(vi). Check obj ≤ e_err. If yes, flag comes out of loop and optimized value of reconstruction error. If

no, follow the next step.

(vii). Check obj = k_err. If yes, flags comes out of loop and optimized value of reconstruction error.

If yes, compute the new cut off frequency wc*= wc +dir *step and follow the step (iv).Program

halts when k_err = obj.

[71]

4.3.3 Design of Cosine Modulated NPR QMF Bank

The prototype filter has been designed by using high SLFOR combinational window functions.

The NPR type CMFB is designed by using cosine modulation. The reconstructed output is not exact

replica of the input [86]. A linear optimization is applied in order to approximate the power

complementary property. The performance of the proposed technique in terms of error parameters, far-

end attenuation and stop-band energy is included. In NPR case, the reconstructed signal has two major

distortions, viz., amplitude distortion and aliasing. In such case, aliasing is canceled approximately and

the distortion becomes delay only. The input-output relationship for cosine modulated filter banks are as

below:

푌(푧) = 푇 (푧)푋(푧) + 푇 (푧)푋 푧푊 ,

(4.3.3.1)

where, 푊 = 푒 .⁄

To(z) is called the distortion transfer function which reflects the distortion caused by overall system for

un-alias component of X(z) of the input signal is given as-

푇 (푧) =1푀

퐹 (푧)퐻 (푧).

Similarly, Tl(z) represents aliasing transfer function which determines the extent of attenuation of the

aliased components X (zWlM) of the input signal which is defined as-

푇 (푧) =1푀

퐹 (푧)퐻 푧푊 ,

for 0 ≤ l ≤ M-1 (4.3.3.2)

The measure for the un-aliased distortion is the peak-to- peak amplitude distortion Epp which is given as-

퐸 = 푚푎푥 ∈[ , ] 푇 푒 −푚푖푛 ∈[ , ] 푇 푒

(4.3.3.3)

[72]

Subsequently, peak aliasing error Ea is defined as-

퐸 푒 = |푇 (푒 )|

(4.3.3.4)

Therefore, prototype filters with stop-band attenuations in the range 40–60 dB in nonadjacent

bands can provide an output signal of adequate quality e.g., for speech signal 40 dB is sufficient [34, 54

and 96].

4.3.4 Design examples

In this sub-section, three examples are illustrated for QMFBs.

Example 1: An 8-band CMFB has been designed using proposed window function. The specifications

are taken same as assumed by Kha et al. [54], i.e., stop-band attenuation (ATT) = 35.8 dB and stop-band

frequency (ws) = 0.12 π. The frequency response of prototype FIR filter, 8-band CMFB and the

magnitude responses of reconstruction and aliasing errors for proposed window function are shown in

Figures- 4.3.1 (a)-(d).

[73]

(a)

(b)

(c)

[74]

(d)

Figure- 4.3. 1: Magnitude responses of 8-band CMFB using proposed window at ATT = 35.8 dB. (a)

Prototype low pass filter; (b) Filter bank; (c) Plot of reconstruction error; (d) Plot of aliasing error.

The obtained values of reconstruction error, aliasing error, far-end attenuation, stop-band energy

are summarized in Table-4.1. Apart from lower values of error parameters obtained, the proposed

prototype filter using variable window functions are providing reduction in stop-band energy and better

far-end attenuation in contrast to Kha et al. [54]. Similarly, in the case of 16-band CMFB, the values of

error parameters are much smaller than the Kha et al. [54] with smaller value of stop-band energy and

more far-end attenuation. Also, in 32-band CMFB also, the proposed window provides smaller value of

reconstruction error than the existing design [54] with less stop-band energy and better far-end

attenuation.

Example 2: In this example, 16-band CMFB is designed using same specifications as assumed in Kha et

al. [54]. The stop-band attenuation (ATT) = 45dB and stop-band frequency (ws) = 0.059 π are considered.

The magnitude responses of prototype FIR filter 16-band CMFB, reconstruction error and aliasing error

for proposed window function are plotted and results obtained are shown in Figure- 4.3. 2 (a) - (d). The

optimized value of performance parameters are given in Table-4.1.

[75]

(a)

(b)

[76]

(c)

(d)

Figure- 4.3.2: Magnitude responses of 16-band CMFB using proposed window at ATT = 45 dB. (a)

Prototype low pass filter; (b) Filter bank; (c) Plot of reconstruction error; (d) Plot of aliasing error.

[77]

Example 3: A 32-band CMFB has been designed using proposed window function with the same stop-

band frequency as taken by Kha et al. [54]. Since, the stop-band attenuation for proposed window

functions is restricted up to 74 dB, therefore, 61 dB stop-band attenuation is taken for the design. The

frequency responses are shown in Figures- 4.3.3 (a)-(d), respectively.

(a)

[78]

(b)

(c)

[79]

(d)

Figure- 4.3.3: Magnitude responses of 32-band CMFB using proposed window at ATT = 61 dB. (a)

Prototype low pass filter; (b) Filter bank; (c) Plot of reconstruction error; (d) Plot of aliasing error.

The obtained values of different parameters are given in Table-4.1.

4.5 ANALYSIS OF RESULTS AND COMPARISON

In this chapter, the design of NPR type two channel QMF and 8-, 16-, and 32- bands filter banks

using CMFBs approach has been successfully proposed and analyzed. The required low-pass FIR

prototype filter is obtained by using Kaiser window, DC window, PC4, PC6 and proposed combinational

window functions. Bi-section and LM optimization techniques are applied to optimize the filter

coefficients and obtained minimum value of reconstruction and aliasing errors. Some other performance

parameters such as far-end attenuation, stop-band energy have also been calculated and a comparison is

established between different window functions based on these parameters. In this study, some examples

have also been included to analyze the performance of proposed window function in the design of

CMFBs with bisection algorithm. In example 1, an eight-band CMFB has been designed using proposed

window function with specifications as given by Kha et al. [54]. The obtained values of peak

reconstruction error (7.32×10-4), aliasing error (2.3×10-3), far-end attenuation (125 dB) and stop-band

energy (4.52×10-3) are reported and found to be better proposition than existing windows.

[80]

Apart from lower values of error parameters the proposed prototype filter using variable window

functions are providing reduction in stop-band energy and better far-end attenuation as compared to Kha

et al. [54].Similarly, in the case of 16-band CMFB the value of error parameters such as peak

reconstruction error (1.35×10-6), aliasing error (5.61×10-4), far-end attenuation (150 dB) and stop-band

energy (2.32×10-6) are obtained with proposed window based design which are much smaller than the

Kha et al. [54] with smaller value of stop-band energy and more far-end attenuation. Finally, in 32-band

CMFB the proposed window provides peak reconstruction error (1.45×10-4), aliasing error (5.92×10-5),

far-end attenuation (171 dB) and stop-band energy (3.81×10-9) which is better than the reported designs.

Table- 4.1: Performance Comparison of Cosine Modulated QMF using bi-section algorithm

S.

No.

Window

function

M ATT

(dB)

ws N PRE

(Epp)

Aliasing

error

(Ea)

Stop band

energy

(Es)

Far-end

attenuati

on

(dB)

1 Kaiser

8

35.8 0.12π 40 5.50×10-3 2.47×10-3 1.17×10-3 50

2 PC4 35.8 0.12π 47 5.79×10-3

7.42×10-4

1.05×10-2

110

3 PC6 35.8 0.12π 49 2.60×10-3

6.25×10-3

8.60×10-3

105

4 Proposed

35.8 0.12π 43 7.32×10-4 2.3×10-3 4.52×10-3 125

5 Kaiser

16

45 0.059π 102 5.95×10-3 3.89×10-4 1.46×10-4 70

6 PC4 45 0.059π 151 4.04×10-4

3.20×10-4

3.05×10-5

137

7 PC6 45 0.059π 123 7.25×10-4

4.85×10-5

1.41×10-5

140

8 Proposed

45 0.059π 99 1.35×10-6

5.61×10-4 2.32×10-6

152

9 Kaiser

32

61 0.031π 466 9.12×10-3 2.38×10-7 1.04×10-6 120

10 PC6 61 0.031π 377 8.93×10-4

6.77×10-4

2.06×10-6

160

11 Proposed

61 0.031π 243 1.45×10-4

5.92×10-5 3.81×10-9

171

[81]

In a similar pattern, the performance comparison of CMFBs using LM algorithm have also been

made with different window function as used in QMF bank. The proposed window gives better results

with this assumption in terms of peak reconstruction and aliasing errors with same specifications as used

in bi-section algorithm.

Table-4.2: Performance Comparison of Cosine Modulated QMF using LM Algorithm

S. No. Window

function

M ATT(dB) ws Filter order

(N)

PRE

(Epp)

Aliasing

error(Ea)

1 Kaiser

8

35.8 0.12π 43 0.9915 0.61×10-4

2 PC4 35.8 0.12π 47 0.93

0.856×10-4

3 PC6 35.8 0.12π 49 0.9755

0.762×10-4

4 Proposed 35.8 0.12π 49 0.9617 0.73×10-4

5 Kaiser

16

45 0.059π 89 1.45×10-4 7.86×10-6

6 PC4

45 0.059π 159 2.128×10-4 1.102×10-6

7 PC6

45 0.059π 129 2.24×10-4 8.73×10-6

8 Proposed

45 0.059π 99 5.28×10-5 9.07×10-6

9 Kaiser

32

61 0.031π 239 0.7282 6.03×10-5

10 PC4

61 0.031π 393 0.7761 1.734×10-4

11 PC6 61 0.031π 389 0.4786 1.76×10-4

12 Proposed 61 0.031π 243 0.7272 5.307×10-5

[82]

An efficient design for M-band NPR type CMFBs has been proposed using variable window

functions. The simulation results show that the optimization algorithm effectively reduced the peak

reconstruction error with comparable aliasing error. High values of SLFOR and far-end attenuation

provide significant reduction in stop-band energy. These CMFBs can be used in real time applications

such as echo cancellation and cross-talk suppression. Based on these two tables, it can be concluded that

that bi-section optimization algorithm provides better results as compared to LM algorithm in design of

the multichannel NPR type CMFBs.

*********************************************************