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Int. J. Communications, Network and System Sciences, 2010, 3, 934-942 doi:10.4236/ijcns.2010.312127 Published Online December 2010 (http://www.SciRP.org/journal/ijcns) Copyright © 2010 SciRes. IJCNS Design of NPR-Type Cosine Modulated Filterbank Using Combinational Window Functions Ram Kumar Soni1, 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 E-mail: {soniram04, alokjain6}@rediffmail.com, rajiv.saxena@jiet.ac.in Received September 8, 2010; revised October 10, 2010; accepted November 18, 2010 Abstract This paper presents the design of near perfect reconstruction (NPR) cosine modulated filterbank (CMFB). The prototype filter is designed by the combinational window functions. These window functions provide high side-lobe-fall-off-rate (SLFOR) with better far-end attenuation which suppresses the undesired inter- ferences occur in the filterbank. A linear optimization is used to minimize the error parameters. Design ex- amples have been included to illustrate the effectiveness of the proposed technique over the earlier reported work. Keywords: Combinational Window Functions, SLFOR, Far-End Attenuation, Stopband Energy 1. Introduction Cosine modulated filterbank are used in a wide range of applications, from data compression of speech, audio and video signals to data transmission [1-6]. They are special subclass of the general M-channel filterbank, which has some very attractive features, first, it is a highly selective and discrimination system. Second, all of the analysis Hz(z) and synthesis Fz(z) filters of filterbank can be si- multaneously generated by the cosine modulation of sin- gle linear phase FIR prototype filter as shown in Fig- ure 1. Thirdly, the error occurs at the reconstructed out- put in NPR types system can easily optimized by using suitable optimization technique [2,7]. The CMFB has been studied by many prominent authors in PR and NPR conditions and finds that the NPR type is simple, more realizable and computationally efficient [3-9]. Also the performance of the system can be further improved with high stopband attenuation and narrow transition band- width of the prototype filter [8]. In speech and audio communication the stopband at- tenuation (s) of 40-60dB provides the signals of ade- quate quality if the crosstalk suppression is efficient [10,11]. It mainly depends on the different parameters of prototype filter, i.e., stopband energy (Es), far-end at- tenuation (f) and SLFOR [2,10]. A window function with minimum stopband energy, better far-end attenua- tion and high SLFOR is the most suitable in such appli- cations. Combinational window functions provide all these characteristics. These are designed by combining a data window and a lag window function in a linear man- ner [2,10]. Figure 1. M-band cosine modulated filterbank. R. K. SONI ET AL. 935 In this proposed work 8-, 16-, 32-band NPR CMFB is designed. The prototype FIR filter is formulated using Parzen-Cos6 (PC6) and Papoulis-Cos4 (PC4) combina- tional window functions [10,12]. The main contribution of this paper is summarized below: 1) The prototype filter is designed using high SLFOR combinational window functions. 2) The NPR CMFB is designed using cosine modula- tion. 3) The reconstructed output is not exact replica of the input. A linear optimization is applied in order to ap- proximate the power complementary property. 4) Examples are included to evaluate the performance of the proposed technique in terms of error parameters, far-end attenuation and stopband energy. 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 [2]: pn wnhn (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. wn is the used window function. PC6 and PC4 combinational window functions are used to design prototype filter. These window functions are character- ized for high SLFOR and far-end attenuation [10]. The value of SLFOR for PC4 and PC6 window functions are –24 dB/octave to –30 dB/octave, –24 dB/octave to –42 dB/ octave, respectively [10,12]. Whereas this figure for Kaiser window is only –6 dB/octave [2]. The expressions for these window functions in time domain are given below [10,12]. 2.1. Parzen-Cos6(n /N) (PC6) Combinational Window The expression for Parzen (lag window, l(n)) and Cos6(n /N) (data window, d(n)) combinational window with 6 as window shape parameter is given as [10,12] 666 6 6 6 1, 2 0, 2 where 03.7 and PC N lndn n wn N n 2 63 12412 ,4 21 2,42 nn N n NN ln nN n N N 6 6cos, 2 n dn n N N (3) FIR filter design relationships are given by the fol- lowing equations [10]: 1) Relationship between window shape parameter ( 6) and desired stopband attenuation (s): 2 6 s s ab c (4) where, 8.15414,0.236709, 0.00218617 for 30.3251.25 s ab c 21.3669,0.605789, 0.00434808 for 51.2568 s ab c 2) Relationship between normalized window width parameter (D) and (s): 2 s s Da bc (5) where, 1.82892,0.0275481, 0.00157699 for 30.3243.60 s ab c 1.67702,0.0450205,0 for 43.6049.44 s ab c 85.4738,3.41969,0.035784 for 49.4457.48 s ab c 8.60006, 0.477004,0.00355655 for 57.4868.69 s abc 2.2. Papoulis-Cos4 (n /N) (PC4) Combinational Window The combinational window of Papoulis (lag window, l(n)) and Cos4(n /N) (data window, d(n)) with 4 as combina- tional factor is given by [12]: 444 4 4 4 1, 2 0, 2 where 08.235 PC N lndn n wn N n (6) Copyright © 2010 SciRes. IJCNS 936 R. K. SONI ET AL. 4 12 2 sin1 2cos,2 nnn ln n NNN N 4 4cos, 2 nN dn n N FIR filter design relationships for PC4 window are given by the following equation [12]. 1) Relationship between window shape parameter ( 4) and desired stopband attenuation (s): 23 4 for 26.1961.08 4 s ss s ab cde s 4 (7) where, a = –69.058755, b = 8.409918, c = –0.321364, d = 0.005044; e = –0.000028. 2) Relationship between normalized window width parameter (D) and (s): 23 , for 26.1961.08 ss ss s Da bcde (8) where, a = 8.728537, b = –0.412899, c = –0.000713, d = 0.000355; e = –0.000004. The order (N) of the filter designed by using above mentioned window functions can be estimated by the following formula: 1 D N (9) where, D is the window width parameter and is the normalized transition width = 2 sp with s and p are the stopband and passband frequencies, respectively. x represents the smallest integer greater than or equal to x. 3. Cosine Modulation Cosine modulation is one of the efficient techniques which provide minimum computational cost in the de- sign of filterbank. All filters of synthesis and analysis sections are obtained by cosine modulation of a linear phase lowpass prototype filter [2,13]. 2cos 211 22 4 2cos 211 224 for0 1,and0 k k k k N hn pnkn M N fn pnkn M kMnN (10) where, M is the number of bands and and k hn k f n are the impulse responses of the analysis and synthesis sections, respectively. 4. Optimization Technique In NPR system, the PR condition is relaxed by allowing small amount of errors. There are three types of errors occur at the reconstructed output, as follows amplitude (Epp), phase and aliasing (Ea) [2,10]. The measures of these error parameters are given by the following equa- tions: 0for j Pe M (11) 21 2 0 1, where j o MjkM j o k Te Te Pe (12) is the overall distortion transfer function. The accuracy of first approximation gives aliasing and the accuracy of the second one gives the reconstruction error. The phase error is eliminated by linear phase prototype filter. However, other two distortion parameters can be mini- mized by applying suitable optimization technique. Much of work have been done in this field [10,11,14,15]. Initially, Johnston [15] developed a nonlinear optimiza- tion technique, later on many prominent authors such as Creusere et al. [11] and Lin et al. [14] and Jain et al. [10] have simplified it and developed single variable linear optimization techniques with different objective func- tions. In the proposed work, a linear gradient optimiza- tion technique is used with objective function given in Equation (13). The cutoff frequency of prototype filter is varied to obtain the minimum value of the objective function. 2 2 max 1 for 0/ jM j Pe Pe M (13) Initially, input parameters, i.e., sampling rate, number of band, passband and stopband frequencies, passband ripple and stopband attenuation of prototype filter are specified and determine the cutoff frequency, transition band and filter length. Initialize, different optimization pointers like step size, search direction, flag and initial (p_err) as well as expected minimum possible values (e_err) of objective function. Inside the optimization loop, design the prototype lowpass filter and determine the bandpass filters for analysis and synthesis sections using cosine modulation. In optimization routine cutoff frequency is gradually changed as per the search direc- tion and calculates the corresponding value of the objec- tive function. Algorithm halts when it attains the mini- mum value of the objective function. The flowchart of optimization is given in appendix section and simulated on MATLAB 7.0. Copyright © 2010 SciRes. IJCNS R. K. SONI ET AL. Copyright © 2010 SciRes. IJCNS 937 5. Design Examples The performance evaluation of the proposed technique has been made with the help of design examples. It is examined in terms of stopband attenuation, reconstruc- tion error, aliasing error, computational complexity, group delay and stopband energy. The effectiveness of the proposed work is compared with the earlier reported work for the same input parameters. Example 1: An eight-band CMFB has been designed using PC6 and PC4 window functions. Same specifica- tions are taken as given by Kha et al. [16], i.e., stopband attenuation (Δs) = 35.8 dB and stopband frequency 0.12 s , respectively. The frequency response of prototype FIR filter, eight-band CMFB and the magni- tude responses of reconstruction and aliasing errors for PC6 and PC4 window functions are shown in Figures 2 (a)-(d) and 3(a)-(d), respectively. The obtained values of reconstruction error, aliasing error, far-end attenuation, stopband energy are given in Table 1. Example 2: In this example, sixteen-band CMFB is Table 1. Performance comparison with Kha et al. [16]. Work Window functionM Δs (dB)ωs N Epp E a E s Δf 8 35.8 0.12π 40 5.50 × 10-3 2.47 × 10-3 1.17 × 10-2 50 16 45 0.059π102 5.95 × 10-3 3.89 × 10-4 1.46 × 10-4 70 Kha et al. [16] Kaiser 32 102 0.031π466 9.12 × 10-4 2.38 × 10-7 1.04 × 10-6 120 PC4 8 35.8 0.12π 47 5.79 × 10-3 7.42 × 10-4 1.05 × 10-2 110 PC6 8 35.8 0.12π 49 2.60 × 10-3 6.25 × 10-3 8.60 × 10-3 105 PC4 16 45 0.059π151 4.04 × 10-4 3.20 × 10-4 3.05 × 10-5 137 PC6 16 45 0.059π123 7.25 × 10-4 4.85 × 10-5 1.41 × 10-5 140 Proposed PC6 32 61 0.031π370 8.93 × 10-4 6.77 × 10-4 2.06 × 10-7 160 (a) (b) (c) (d) Figure 2. Magnitude responses of 8-band CMFB using PC-6 window at Δs = 35.8 dB. (a) Prototype lowpass filter; (b) Filter- bank; (c) Plot of reconstruction error; (d) Plot of aliasing error. 938 R. K. SONI ET AL. (a) (b) (c) (d) Figure 3. Magnitude responses of 8-band CMFB using PC-4 window at Δs = 35.8 dB. (a) Prototype lowpass filter; (b) Filter- bank; (c) Plot of reconstruction error; (d) Plot of aliasing error. (a) (b) (c) (d) Figure 4. Magnitude responses of 16-band CMFB using PC6 window at Δs = 45 dB. (a) Prototype lowpass filter; (b) Filter- ank; (c) Plot of reconstruction error; (d) Plot of aliasing error. b Copyright © 2010 SciRes. IJCNS R. K. SONI ET AL. Copyright © 2010 SciRes. IJCNS 939 designed using same specifications as given in [16]. The stopband attenuation and stopband frequency are taken as (Δs) = 45dB, ( s ) = 0.0590 , respectively. The mag- nitude responses of prototype FIR filter, 16-band CMFB, reconstruction error and aliasing error for both window functions are shown in Figure 4 and Figure 5, respec- tively. The optimized value of performance parameters are given in Table 1. Example 3: A thirty two-band CMFB has been de- signed using PC6 window function with the same stop- band frequency as taken by Kha et al. [16]. Since, the stopband attenuation for PC6 window functions is re- stricted up to 68.69 dB, therefore, 61 dB stopband attenua- tion is taken for the design. The frequency responses are shown in Figures 6(a)-(d), respectively. The obtained values of different parameters are given in Table 1. 6. Discussion The comparative performance of proposed work with the reported publication is given in Table 1. In case of 8-band CMFB the obtained values of reconstruction error for PC4 and PC6 window functions is 5.79 × 10-3; 2.60 × 10-4, respectively, which is smaller than the reported value of Kha et al. [16]. Similarly, the aliasing error in reported and proposed work is 2.47 × 10-3 and 7.42 × 10-4, 6.25 × 10-3, respectively. Apart from lower values of error parameters the proposed prototype filter using combinational window functions are providing reduction in stopband energy and better far-end attenuation in con- trast to Kha et al. Similarly, in case of 16-band CMFB the value of error parameters are much smaller than the Kha et al. [16] with smaller value of stopband energy and more far-end attenuation. In 32-band CMFB the proposed technique provides smaller value of reconstruction error than the existing journal with less stopband energy and better far-end attenuation. 7. Conclusions An efficient design for M-band NPR CMFBs has been proposed using combinational window functions. Simu- lation studies shows that the developed algorithm provides smaller values of error parameters than the previous pub- lication. High values of SLFOR and far-end attenuation (a) (b) (c) (d) Figure 5. Magnitude responses of 16-band CMFB using PC4 window at Δs = 45 dB. (a) Prototype lowpass filter; (b) Filter- bank; (c) Plot of reconstruction error; (d) Plot of aliasing error. 940 R. K. SONI ET AL. (a) (b) (c) (d) Figure 6. Magnitude responses of 32-band CMFB using PC-6 window at Δs = 61 dB. (a) Prototype lowpass filter; (b) Filter- bank; (c) Plot of reconstruction error; (d) Plot of aliasing error. provide significant reduction in stopband energy. These filterbank can be used in real time applications such as echo cancellation and cross-talk suppression. 8. References [1] L. Lin and B. Farhang-Boroujeny, “Cosine-Modulated Multitone for very High Speed Digital Subscribe Lines,” EURASIP Journal on Applied Signal Processing, Vol. 2006, 2006, p. (19329)79. [2] P. P. Vaidyanathan, “Multirate Systems and Filter Banks,” Prentice-Hall, Englewood Cliffs, 1993. [3] W. S. Lu, T. Saramäki and R. Bregovic, “Design of Prac- tically Perfect-Reconstruction Cosine-Modulated Filter Banks: A Second-Order Cone Programming Approach,” IEEE Transactions on Circuits and System-I, Vol. 51, No. 3, May 2004, pp. 552-563. [4] M. B. Furtado, P. S. R. Diniz and S. L. Netto, “Numeri- cally Efficient Optimal Design of Cosine Modulated Fil- ter Banks with Peak-Constrained Least-Squares Behav- ior,” IEEE Transactions on Circuits and System-I, Vol. 52, No. 3, May 2005, pp. 597-608. [5] M. B. Furtado, P. S. R. Diniz, S. L. Netto, and T. Saramäki, “On the Design of High-Complexity Cosine-Modulated Transmultiplexers Based on the Frequency-Response Masking Approach,” IEEE Transactions on Circuits and System-I, Vol. 52, No. 11, November 2005, pp. 2413- 2426. [6] M. Blanco-Velasco, F. Cruz-Roldán, E. Moreno-Martínez, J. I. Godino and K. E. Barner, “Embedded Filter Bank- Based Algorithm for ECG Compression,” Signal Proc- essing, Vol. 88, No. 6, June 2008, pp. 1402-1412. [7] F. Cruz-Roldán and M. Monteagudo, “Efficient Imple- mentation of Nearly-Perfect Reconstruction Cosine-Mo- dulated Filterbanks,” IEEE Transactions on Signal Proc- essing, Vol. 52, No. 9, September 2004, pp. 2661-2664. [8] T. Q. Nguyen, “Near-Perfect-Reconstruction Pseudo QMF Banks,” IEEE Transactions on Signal Processing, Vol. 42, No. 1, January 1994, pp. 65-76. [9] C. K. Goh and Y. C. Lim, “Design of Two Channel Per- fect Reconstruction Linear Phase FIR Filterbanks,” IEEE Transactions on Circuits and System-II: Analog and Digital Signal Processing, Vol. 45, August 1998. [10] A. Jain, R. Saxena and S. C. Saxena, “An Improved and Simplified Design of Cosine Modulated Pseudo-QMF Filterbanks,” Digital Signal Processing, Vol. 16, No. 3, May 2006, pp. 225-232. [11] C. D. Creusere and S. K. Mitra, “A Simple Method for Designing High-Quality Prototype Filters for M-Band Copyright © 2010 SciRes. IJCNS R. K. SONI ET AL. 941 Pseudo QMF Banks,” IEEE Transactions on Signal Pro- cessing, Vol. 43, No. 4, April 1995, pp. 1005-1007. [12] S. N. Sharma, R. Saxena and S. C. Saxena, “Design of FIR Filters Using Variable Window Families: A Com- parative Study,” Journal of Indian Institute of Science, Vol. 84, October 2005, pp. 155-161. [13] S. K. Mitra, “Digital Signal Processing—A Computer Based Approach,” Tata McGraw-Hill Publishing Co. Ltd., New Delhi, 1998. [14] Y. Lin and P. P. Vaidyanathan, “A Kaiser Window Ap- proach for the Design of Prototype Filters of Cosine Modulated Filterbanks,” IEEE Signal Processing Letters, Vol. 5, No. 6, June 1998, pp. 132-134. [15] J. D. Johnston, “A Filter Family Designed for Use in Quadrature Mirror Filter Banks,” Proceedings of IEEE International Conference on Acoustics, Speech and Sig- nal Processing, Denver, 9-11 April 1980, pp. 291-294. [16] H. H. Kha, H. D. Tuan and T. Q. Nguyen, “Efficient De- sign of Cosine-Modulated Filterbanks via Convex Opti- mization,” IEEE Transactions on Signal Processing, Vol. 57, No. 3, 2009, pp. 966-976. Copyright © 2010 SciRes. IJCNS 942 R. K. SONI ET AL. Appendix Specify sampling frequency, num- ber of bands, stopband attenuation, and passband ripple Initialize passband and stopband frequency. Calculate the cutoff fre- quency, transition band, filter length and window coefficients of selected window function. Initialize step size (step), search direction (dir), flag, minimum value of reconstruction error (e_err) and initial value of reconstruction error (p_err) Design prototype filter and obtain filters for synthesis and analysis sections using Cosine modulation scheme. Calculated reconstruction error and absolute value of objective function (obj) ( = + dir × step) c c p_err = obj Display optimized value of reconstruc- tion error No No No Is obj ≤ e_err Is obj = p_err Is obj > p_err End No Yes step = step/2 dir = –dir Yes Flag = 1 (comes out from loop) Yes Copyright © 2010 SciRes. IJCNS |