Cosine Modulated Non-uniform Filter Banks

Traditional designs for non-uniform filter bank (NUFB) are usually complex; involve complicated nonlinear optimization with a large number of parameters and lack of linear phase (LP) property. In this paper, we describe a simple design method for multirate near perfect reconstruction (NPR) cosine modulated filter banks with non-uniform frequency spacing and linear phase property that involves optimization of only single parameter. It is derived from the uniform cosine modulated filter bank (CMFB) by merging some relevant band pass filters. The design procedure and the structure of the uniform CMFB are mostly preserved in the non-uniform implementation. Compared to other design methods our method provides very good design and converges very rapidly but the method is applicable, only if the upper band edge frequency of each non-uniform filter is an integral multiple of the bandwidth of the corresponding band. The design examples are presented to show the superiority of the proposed method over existing one.


Introduction
Multirate filter bank find wide applications in many areas of digital signal processing such as sub-band coding, transmultiplexer, image, video and audio compression, adaptive signal processing [1-3].On the basis of timefrequency resolution, filter bank can be classified in two categories, i.e., uniform and non-uniform filter bank.Uniform filter bank provides fixed and uniform time frequency decomposition [1].However in some applications like audio analysis and coding, broadband array signal processing non-uniform and variable time-frequency resolution may lead to better performance and reduced arithmetic complexity, which is provided by non-uniform filter bank [NUFB] [4][5][6].Therefore efficient structure and design procedures for NUFB are highly desirable.Over the years, a number of design methods have been proposed by different authors [6][7][8][9][10].Among these, only few of them possess linear phase (LP) property.The tree structure method [1] is an easy way to design LP-NUFB via cascading uniform filter bank.However, the limitation of decimation factors and the long system delay are two major drawbacks of using this method.Most of the available approaches [6][7][8][9][10] for NUFBs, use standard constrained or unconstrained optimization techniques to obtain the design, which tend to be computationally expensive, when high order filters are used.In wideband audio signal analysis and coding, filter banks with high stop band attenuation greater than 100 dB is required.Moreover, it is difficult to design NUFBs with high stop band attenuation and LP property.In [11], a simple design method for NUFBs was proposed.It is based on the design of a uniform cosine modulated filter bank and is applicable only to non-uniform integer-decimated filter banks.Moreover, it still involves complicated nonlinear optimization with large number of parameters.Recently, Zing et al. [12] proposed interpolated FIR prototype filter to design the NUFB.
In this work, a simple design approach for linear phase NUFB is presented.The approach is based on uniform CMFBs] as shown in Figure 1.The constituent NUFB as shown in Figure 2 is obtained by merging some relevant uniform filters in the associated uniform CMFB.The design procedure is therefore reduced to the design of the prototype filter in the associated uniform CMFB.With this approach NUFBs with high stop band attenuation up to 110 dB can be easily designed.A single variable optimization is used to obtain minimum value of amplitude (E max ) and aliasing (E a ) distortions [1].

Uniform
Cosine modulation is a cost effective technique for M--band filter bank [1].In this approach all the filters of analysis and synthesis section are obtained by cosine modulation of single linear phase prototype low pass filter which normally has linear phase and a finite length impulse response as shown in Figure 1.Let H(z) be the transfer function of the prototype filter.It is given as: The impulse responses of filters of analysis and synthesis sections are obtained from the closed form expressions as given by [1]: The required prototype filter is designed by window technique using Kaiser Window function.

Non-uniform
In the case of non-uniform NPR filter banks, the concept of cosine modulating low pass filters is applied [11,13].After designing the required uniform CMFB, the corresponding NUFB is obtained by merging the relevant band pass filters of analysis and synthesis section of the uniform filter bank as described below [13].
We define   i H z , 0,1, 2, , 1 i  M , to be the filters obtained by merging the adjacent analysis filters, i.e.,   ) in a uniform M-channel CMFB.More specifically, where n i+1 = n i + l i .We define  M in a similar manner for the synthesis filters , form a new set of analysis and synthesis filters in the Mchannel non-uniform CMFB.Note that , and 0 1 2 Figure 2 shows the resulting overall structure of the M -channel non-uniform CMFB where , amounts to the decimation ratio for the ith channel.

Optimization Technique
In NPR, perfect reconstruction condition is relaxed by allowing small amount of distortion.Three types of distortions occur at the reconstructed output, i.e, amplitude (E max ), phase and aliasing (E a ) [1].The aliasing and phase distortion can be eliminated by careful design of the linear phase FIR filter.However, amplitude distortion can not be eliminated completely but can be minimized by applying optimization technique [14].Initially; Johnston [15] developed a nonlinear optimization technique.Later on many prominent authors such as Creusere et al. [16], Lin et al. [17], Jain et al. [18], have simplified it using linear optimization technique with objective function as given below: In this work same objective function in modified form is used for the design of non-uniform filter bank, as given below vaidynathan [1]: where, M is number of channels in non-uniform filter bank and is the frequency responses of the filters of the non-uniform section.Initially, input parameters, i.e., sampling rate, number of band, pass band and stop band frequencies, pass band ripple and stop nd attenuation of prototype filter are specified.Cutoff ba -channel nonuniform filter bank. M freque itio eter-ncy, trans n band and filter length is than d mined.Initialize, different optimization pointers like step size, search direction, flag and initial (perror) as well as expected minimum possible values (terror) of the objective function.Inside the optimization loop, design the prototype low pass filter and determine the band pass filters for analysis and synthesis sections using cosine modulation.Obtain the desired NUFB using merging of relevant band pas filters.In optimization routine cutoff frequency of the prototype filter is gradually changed as per the search direction and calculates the corresponding value of the objective function.Algorithm halts when it attains the minimum value of the objective function.The flowchart of optimization Figure 3 is given below and Specify stop band attenuation (As), number of bands (M) Initialize: passband (ωp), stopband freq (ωs), terror, perror, step, dir, and flag Calculate cutoff frequency (ωc).Filter order (N) and design the prototype filter.Obtained filters of analysis section using cosine modulation.Obtain NUFB using merging filters.

Design Examples
In this section 3-channel an signed and the performance of proposed technique is compared with the earlier reported work [5,11,12,19].
In this example a 3-channel NUFB with decimatio ctor (4, 4, 2) has been designed using same specifications as given in Xie et al. [5] and Li et al. [11].The design specifications of the filter are: stop band attenuation 100 Th nd e uencie e ba edg freq s are . The magnitude responses of protooptimized value of amplitude distortion is shown in Figures 4-6.The obtained value of maximum amplitude distortion is E max = 2.99 × 10 -3 .
This example is quoted to compare the performa type filter, nce with recent work of Zing et al. [12].In the work of [12], 5-channel NUFB with integer decimation factors (4,4,8,8,4)  1 and N = 39, respectively.Therefore, the obtained 3 i overall filter length of IFIR prototype filter becomes N = (L.N m + N i ) = 163 [14].Here, L is the stretch factor.In this example, 5-band NUFB is designed with the following specifications as in [12]: The band edge frequencies are and

Discussion
Two design exam demonstrate effectiveness of the design.A three and five channel symmetric non-uniform filter banks were designed and the amplitude characteristics of analysis filters are shown in Figure 5 and qu ncy range is clearly divided into three and five non-e uniform bands.These filter banks have integer decimation factors.The filter lengths of analysis FIR filters are 63 and 85.For both the designs the Kaiser windowed LPF were used as initial filters for minimization of the performance function.And, as a tool for optimization, the linear iterative algorithm was utilized.Comparisons with Tree-Structure NUFBs show that the Tree-Structure can be either PR or NPR depending on the FBs used in the design.On the other hand proposed method can only design NPR FBs.The advantage of our method is that it can be used to design a feasible or non feasible partition NUFB with good performance.Since it is derived from uniform CMFBs by cosine modulating a prototype filter, its implementation also consists of one prototype filter and a discrete cosine transform (DCT).Since the number of parameter is reduced, the speed of convergence is faster, and filter bank with high attenuation can be designed.It is clear from

6
A simple and comput is presented.In traditional design approaches, it is difficult to design the NUFB at high stop band attenuation above 100 dB.The proposed work eliminated this constraint by exploiting the design process of cosine modulation and obtained NUFB with a feasible partition property.The performance comparison of proposed with previously reported work shows that the resulting overall distortion and aliasing errors are smaller than the previous reported work.In addition, this method has lower system delay compared with the LP NPR NUFBs by the indirect method.This method is suitable particularly for large number of channels where high order filters with unequal pass bands have to be designed with small distortion and aliasing.Such filter banks are needed in a wide variety of applications like speech coding and speech enhancement.

Figure 3 .
Figure 3. Flow chart of optimization algorithm.

7 - 9 .
The obtained prototype filt s t ngth 163, the stop band attenuation A s = 110 dB.The magnitude responses of prototype filter, filter bank and distortion parameters are shown in Fig- ures The obtained resulting distortion parameter is maximum amplitude distortion E max = 0.0065 dB., n = 0. n = 4, n = 5, n 6, n = 8; ples for the NUFB are presented to er ha he le

Table 1 and
Table 2 that for same decimation factors the proposed work provided better results for peak amplitude distortions (E max ).