Design of NPR DFT-Modulated Filter Banks Using Modified Newton s Method

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1 ISSN Vol.02,Issue.12, September-2013, Pages: Design of NPR DFT-Modulated Filter Banks Using Modified Newton s Method M.RAJESH 1 PG Scholar, Dept of EIE, Kakatiya Institute of Technology and Science, Warangal, AP-INDIA, rajeshmitukula@gmail.com. M.SREELATHA 2 Professor, Dept of EIE, Kakatiya Institute of Technology and Science, Warangal, AP-INDIA. mslata65@yahoo.com. Abstract: Multirate filter banks have been thoroughly investigated and widely applied in digital signal processing. As a special class of multirate filter banks, modulated filter banks are popular, due to their easy design and fast implementation. In this paper, an efficient algorithm is proposed to design nearly-perfect reconstruction (NPR) DFT-modulated filter banks. First, the perfect-reconstruction (PR) condition of the oversampled DFT-modulated filter banks in the frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF) in the time domain. Second, the design problem is formulated as an unconstrained optimization problem that involves PR condition and stopband energy of the PF. With the gradient vector of the objective function, an efficient iterative algorithm using modified Newton s method will be used to update the linear matrix equations at each iteration and the convergence is will be compared. Keywords: DFT-Modulated Filter Banks, Iterative Updating Program, Perfect Reconstruction (PR), Modified Newton s Method, Prototype Filter. I. INTRODUCTION Multirate filter banks are widely applied in digital signal processing [1 6, 8, 9, 11 15]. In the multirate filter banks, modulated filter banks are popular; because of their easy design and fast implementation [1 4, 6, 9, 12 15].In generally two main types of modulated filter banks are Cosine-modulated filter banks (CMFBs) and DFTmodulated filter banks. DFT-modulated filter banks possess a unique merit to separate positive and negative frequency components into different sub bands compared with CMFBs, a useful property in complex signal processing [7] and two-dimensional directional filter banks [8].Perfect reconstruction filter banks (PR FBs) have important applications in speech, audio, image and array processing. The discrete Fourier transform (DFT) filter bank is well-known for its computational efficiency in realizing channels with equal bandwidth. There are several methods to design NPR DFT-modulated filter banks. In [13], the NPR DFT-modulated filter banks are designed by the semi-definite program (SDP), and the designed prototype filters (PFs). However, the aliasing distortion of filter banks in [13] is controlled by the stopband characteristics of the PF. The reconstruction error of the obtained filter bank is restricted by the attainable stopband attenuation of the PFs. In [2], the bi-iterative quadratic program (BI-QP) algorithm is presented to design doubleprototype DFT-modulated filter banks, where the transfer function distortion is predefined in design, and the PFs are optimized to minimize the aliasing distortions. Therefore, the reconstruction error is also limited due to the unbalance between transfer function distortion and aliasing distortion in [2]. In [9], the bi-iterative secondorder cone program (BI-SOCP) algorithm is proposed to design double-prototype DFT-modulated filter banks, in which the frequency selectivity of the PFs are given and the PFs are iteratively optimized by minimizing the transfer function distortion and aliasing distortions. Compared with that in [2, 13], the BI-SOCP algorithms can lead to DFT-modulated filter banks with smaller reconstruction error owing to that the two distortions are jointly optimized. Nevertheless, the algorithms in [2, 9, 13] are all based on SDP, which leads to a heavy computational complexity. In this paper, we consider design of oversampled DFT-modulated filter banks by an iterative updating algorithm. First, the frequency-domain PR condition of DFT modulated filter banks is transferred to its time-domain form, a set of quadratic equations with respect to the PF. Then, the design problem is formulated as an unconstrained optimization problem that involves PR condition and stopband energy. The gradient vector of the objective function is derived. With the gradient vector, an iterative algorithm is proposed to solve the optimization problem, in which the PF is updated with linear matrix equations. A filter bank with small reconstruction error and high stopband attenuation can be fast obtained by the proposed algorithm. Also, it is derived that the proposed algorithm is a modified Newton s method, and the 2013 SEMAR GROUPS TECHNICAL SOCIETY. All rights reserved.

2 M.RAJESH, M.SREELATHA convergence of the algorithm is proved. Numerical examples and comparison with other methods are included to show the effectiveness of the algorithm. Section III, the MDFT filter bank is applied with almost PR utilizing the pseudo-quadrature mirror filter (QMF) principle. In Section IV, it is shown that the MDFT filter bank can even be designed to provide PR. Although the MDFT filter bank is particularly suitable for processing complex-valued signals, there also exist mappings for real input signals into complex subband signals which are described in Section V. Finally, in Section VI, the MDFT filter bank is compared to the cosine-modulated filter bank. Fig. 1 shows an M-channel DFT filter bank. H k (z) and F k (z) are the analysis and synthesis filters, respectively. Describes a decimation of the sampling rate by the factor M and the corresponding expander. Fig.1. Structure of an M-channel DFT-modulated filter bank with decimation factor K. II. MDFT FILTER BANKS MDFT filter banks are modified complex modulated, critically sub sampled filter banks based on DFT filter banks. They combine key features of DFT filter banks as linear phase analysis and synthesis filters and an efficient realization with almost PR or PR. In this section we will first recall the characteristics of DFT filter banks and then show how MDFT filter banks can be derived by introducing of some modifications. Due to these modifications a structure inherent alias cancellation is obtained yielding almost PR or PR. i) Modulated Linear-Phase Analysis and Synthesis Filters One of the key features of DFT filter banks is the ability to realize linear-phase analysis and synthesis filters by means of an appropriate complex modulation of a low-pass prototype filter. To achieve this, we derive the modulated filters h k (n) and f k (n) in two steps. In a first step, from a real-valued zero-phase low-pass prototype filter p (n) with a transition band from to we derive complex modulated zero-phase filters (1) A. DFT Filter Banks Digital filter banks have found wide applications and are particularly useful in subband coding and multiple carrier data transmission. In both applications, critically decimated M-channel analysis and synthesis filter banks are used. Among a great number of different filter-bank approaches, cosine modulated filter banks have become very popular and have been developed to provide perfect reconstruction (PR), e.g., [1] [5]. Another popular filter bank, namely the discrete Fourier transform (DFT) polyphase filter bank [6], providing highest computational efficiency, suffers from the fact that it is not able to cancel alias components caused by subsampling the subband signals. This disadvantage has been overcome by introducing a certain modification which leads to the modified DFT (MDFT) filter bank [7] [9]. In this paper, essential features of the MDFT filter bank are presented and compared to the cosine-modulated filter bank. In Section II, it is shown that the MDFT filter bank is able to provide linear phase in both, analysis and synthesis part, independently. Furthermore, it is shown that the MDFT filter bank has a structure inherent alias cancellation: all odd alias spectra are automatically compensated. In Fig.2. Frequency response of the filters. With corresponds to Substituting z by of which, in the z domain, (2) we obtain the frequency representation From (3), it is obvious that the filters are frequency shifted versions of the prototype filter (seefig. 2). In order to obtain causal analysis and synthesis filters, the impulse responses are delayed by samples. Hence, the time-domain representation of the analysis filters is (3)

3 Design of NPR DFT-Modulated Filter Banks Using Modified Newton s Method (4) The synthesis filters to the analysis filters are supposed to be identical Mapping (4) and (5) into the z domain and into the frequency domain yields (5) (6) Fig.4. Modified DFT filter bank. From (7), we see directly that all analysis and synthesis filters are linear phase if the low-pass prototype satisfies the zero-phase property. ii) Realization as a DFT Polyphase Filter Bank: Bellanger et al. have shown in [6] that complex modulated filter banks can be realized efficiently by DFT polyphase filter banks. Fig. 3 shows the polyphase realization of the analysis filter bank. Denotes the the type-1 polyphase filter of the low-pass filter H 0 (z) and reads in the time domain The factor in the kth band of the filter bank in Fig. 3 is due to the phase of the modulation caused by the (7) (8) Time delay of the prototype filter by samples [see (4)]. The inverse DFT (IDFT) only realizes the modulation by. B. Structure-Inherent Alias Cancellation of MDFT Filter Banks Although DFT polyphase filter banks as shown in Fig. 3 are of highest computational efficiency and, therefore, particularly useful for practical implementations, this kind of filter bank is not suitable to subband coding applications since no aliasing is cancelled within the DFT filter bank. This disadvantage will be overcome by introducing the following modifications for the MDFT filter bank. The MDFT filter bank is a complex modulated M-channel filter bank, with a two-step decimation of the subband signals. Upon decimating the sampling rate by the factor M/2 (M must be even), another decimation by two is realized with and without a delay of one sampling period. Fig.3. DFT polyphase realization of a complex modulated analysis filter bank. In the subbands, either the real or the imaginary part is used, respectively. Fig. 4 shows a detail of this filter bank. The impulse responses of the linear-phase analysis and synthesis filters and are given in (4) and (5), respectively. The MDFT filter bank is critically sampled. In case of complex valued input data a sequence of M input samples is mapped into M real and M imaginary subband signals at a sampling rate reduced by M. For real input signals, half of the subband signals are either complex conjugates of the others or zero as will be shown in Section V, and therefore remain unused. In this section, we demonstrate that not only adjacent alias spectra but all

4 M.RAJESH, M.SREELATHA odd alias spectra are compensated within the MDFT filter bank. The alias cancellation is independent of the prototype filter design and only depends on the structure of the MDFT filter bank. This property will be used later on to derive pseudo-qmf and PR conditions on MDFT filter banks. In order to describe aliasing terms and distortions produced by the MDFT filter bank we have to express the reconstructed signal in terms of the input signal. First, the input signal is divided into subband signals and. Within the analysis filter bank, we especially look at the effects of the modifications introduced by the MDFT filter bank: III. ITERATIVE DESIGN OF GDFT FILTERBANKS Subband processing has attracted considerable attention in the past decade, and several studies [4, 5, 9, 8, 1] have shown that good performances can be achieved with nearly perfect reconstruction (NPR) oversampled filter banks. To reduce the implementation cost, modulated uniform filter banks are commonly employed. Complex modulation is especially interesting, since it allows aliasing elimination for any (sufficiently high) down-sampling factor. In the M- channel filter bank (FB) from Fig. 1, the analysis filters Hk(z) and synthesis filters Fk(z) are obtained via generalized DFT (GDFT) modulation from two prototype filters, H(z) and F(z), respectively. The impulse responses of the filters are (9) Where h[n], f[n] is the impulse responses of the prototypes and D is the overall delay of the filterbank. The frequency responses of the filters are like. If the input signal is real, then only M=2 filters are necessary in each bank. We assume that the prototype filters are FIR, of order Nh and Nf. In an NPR FB, in the absence of subband processing, the output y[n] is approximately equal to the delayed copy x[n D] of the input. Real-time processing imposes bounds on the delay of the filterbank. As orthogonal1 GDFT FBs have a delay equal to prototypes order, we are interested in biorthogonal FBs, for which the delay is arbitrary. Certainly, the case of interest is when D Nh;Nf. Design algorithms for biorthogonal FBs are proposed in [4, 3, 1, 2]. The algorithm we have proposed in [2] differs from the oth- ers by offering almost complete control on the PR error. Although it has an iterative nature, as in [1], our algorithm uses a completely different initialization (and also the optimization criterion is different). In this paper we aim to refine the work from [2]. As initially proposed, our algorithm is relatively fast for orders below 100, but becomes unacceptably slow for orders greater than 200. The main burden is the initialization step, where an orthogonal FB is designed with a technique similar to that from [10]. So, the main subject of this paper is to study fast initialization techniques. We review the basic algorithm from [2]. It can present two alternative initialization methods, based on interpolation of lower order prototypes and on the window method [6], respectively. Section 4 is dedicated to experiments, from which we derive the best approach to the design of high order prototypes. The optimization problems are convex: semidefinite programming (SDP) in second-order cone programming (SOCP). The complexity of the SDP problem grows faster with the order and dominates the computation when the order is in the hundreds. We present the average times necessary for initialization (step 1) and one iteration of the algorithm, for orders Nh = Nf used in the examples from the experimental section. All times are measured on a PC with Pentium IV at 1GHz. It is clear that the initialization becomes too costly for orders situated between 100 and 200 and even prohibitive (or impossible) for larger orders. This is the main reason for searching fast initialization techniques. A. Initialization Techniques The algorithm presented in the previous section is convergent, but it does not solve a convex optimization problem, although each step is based on convex optimization. So, the solution may depend strongly on the initialization, moreover if only one or few iterations are performed. Our experience from [2] shows that the initialization with an orthogonal FB is good and robust, in the sense that similar values of the average stopband energy are obtained for a relatively large range of initial orders N0. However, as shown in this initialization may be too complex. Our aim is to explore the possibilities of producing initializations of similar quality with significantly less computational effort. We propose two distinct initialization techniques: The window method [6] is used for designing approximately orthogonal GDFT FBs. The method is fast, as only one parameter is optimized, namely the cuto frequency of a Kaiser window. However, the PR error is not directly controlled and the stopband edge is not guaranteed; moreover, there is no optimization of the stopband energy. The initial analysis prototype is obtained by interpolation from the prototype of a shorter length NPR GDFT filterbank. As very good quality small order FBs can be designed easily with the algorithm from [2], this is a very appealing method. We mention that interpolation was proposed in [7] for designing orthogonal M-channels FBs starting from two-channels FBs. However, the PR error of the resulting FBs is out of control, as well as the stopband

5 Design of NPR DFT-Modulated Filter Banks Using Modified Newton s Method properties of the filters. So, we suggest its use mainly for initialization. IV. NUMERICAL EXAMPLES Numerical examples are given to demonstrate the effectiveness of the proposed algorithm. As we know, when the synthesis PF is time-inversed version of the analysis PF, Additionally, the parameter α serves to obtain trade-off between reconstruction error and stopband attenuation. In order to demonstrate this, we also design another filter bank whose specifications are the same as the previous two filter banks except that the weight α is set to be 0.1 by the proposed algorithm. For comparison, its performance measures are also listed in Table 1. It can be seen that, when the weight α is increased, the reconstruction property of the filter bank is obviously degraded while the stopband level is improved. Therefore, it can be concluded that the great improvement of the reconstruction error is achieved at the expense of relatively slight degradation of stopband attenuation and in band aliasing, similar to that of the BISOCP algorithm [9]. Table1: Comparison of the performance measures of the filter banks designed by the SDP algorithm[13] and the proposed method, where the bolded fonts denote the best one in each column Fig.5. Design example 1: (a) normalized magnitude response of the PF designed by the SDP method [13] and (b) normalized magnitude response of the PF by the proposed algorithm with α = the system delay of the DFT-modulated filter bank is equivalent to the order of the analysis PF [3]. Therefore, the maximum transfer function distortion can be measured by Et = maxω{ T0(ω) e j (N 1)ω }. Other performance measures of the filter banks includes the maximum aliasing distortion Ea = maxω{ Tl(ω), l = 1, 2,..., K 1}, the reconstruction error Er, the stopband level SL, and the inband aliasing IA. Numerical example 1 Consider two oversampled DFTmodulated filter banks with M = 8,K = 4,N = 33. The first filter bank is designed by the SDP method with εr = Then, starting from the PF of the first filter bank, the iterative updating algorithm is used to design the second filter bank, in which the weight is taken as α = in (22) and the threshold is set to ε = It only took four iterations for the algorithm to terminate. The magnitude responses of the PF designed by the SDP method and that by the proposed algorithm are shown in Fig. 5. The performance of the two methods is contrasted in Table 1. It is observed that compared with the SDP method, the proposed algorithm can lead to a filter bank with much less reconstruction error at the expense of slight degradation in inband aliasing. In addition, the methods are performed at Matlab platform equipped on a 3.0-GHz PC with 2.5-GB memory. The total CPU time of the algorithms are calculated, which is listed in Table 1. It is shown that the two algorithms have similar computational cost. Fig.6. Design example2: (a) normalized magnitude response of the PF designed by the SDP method [13] and (b) normalized magnitude response of the PF by the iterative updating algorithm. In the first example, the channel number M is a multiple of the decimation factor K, which results in a simple timedomain PR condition. In fact, when M and K is relatively prime, the time-domain PR condition becomes complicated. In this case, the iterative updating algorithm can also work well, as shown in the next example. Numerical example 2 Consider two DFT-modulated filter banks with M = 7, K = 3,N = 22. The first filter bank is designed by the SDP method with εr = The second filter bank is designed by the proposed method with α = 0.04, ε = , where the initial filter is the PF of

6 M.RAJESH, M.SREELATHA the first filter bank. It only took two iterations for the proposed algorithm to terminate. The normalized magnitude responses of the two PFs are shown in Fig. 6. The performance of the two filter banks are shown in Table 2. Note that the proposed algorithm provides about 11 db reconstruction error improvements at the expense of about 1 db loss at stopband level over the SDP method. In the following example, we will contrast the performance of the proposed algorithm, the BI-QP algorithm, and the BI-SOCP algorithm. Numerical example 3 In the Example 1 of [9], a 16- channel DFT-modulated filter bank is designed with the BI-QP and BI-SOCP methods, respectively. Here, we apply the proposed method to design this filter bank. First, the initial PF is designed by the SDP method with εr = , Table2: Comparison of the performance measures of the filter banks designed by the SDP method [13] and the proposed method, where the bolded fonts denote the best one in each column iterative methods. Next, starting from the initial PF, the iterative updating algorithm with α = and ε = is used to solve the PF. It only took three iterations for the algorithm to terminate. The normalized magnitude responses of the PF and the analysis filters designed by the proposed method are depicted in Fig. 7. The performance measures of the filter banks designed by the three methods are compared in Table 3, where Sa and Sb denote the stopband attenuation of the analysis and synthesis PFs, respectively. It is observed that compared with the BI-QP method, the BI-SOCP and the proposed algorithms can lead to a filter bank with much less reconstruction error at a slightly degradation in stopband attenuation. Note that the filter bank designed by the proposed algorithm has poorer reconstruction property than that by the BI-SOCP algorithm, which is partially due to that the BI-SOCP algorithm has double degrees of design freedom. Additionally, it is worth noting that the proposed method has much lower computational cost than other two methods due to that it optimizes the PF by solving linear matrix equations while other two bi-iterative methods generate PFs by iteratively solving convex optimizations. V. CONCLUSIONS In this paper, an iterative updating algorithm is proposed to design the NPR DFT modulated filter banks. The design algorithm is based upon the time domain PR condition. Compared with the SDP method, the proposed algorithm can lead to a filter bank with great improvement on reconstruction error at the expense of relatively small degradation on stopband attenuation and inband aliasing. Also, the proposed method is of low computational cost, a merit over several existing bi-iterative methods. The performance of the proposed algorithm is partially dependent on the initial point. The proposed algorithm is verified by a kind of modified Newton s method, and its convergence is proved. VI. REFERENCES Fig.7. Design example 3: magnitude response of (a) the prototype filter, (b) the analysis filters Table3: The performance and CPU time of the BI-QP method [2], the BI-SOCP method [9], and the proposed method, where the bolded fonts denote the best one in each column [1] Junzheng Jiang Shan Ouyang Fang Zhou, Design of NPR DFT-Modulated Filter Banks via Iterative Updating Algorithm, Received: 1 November 2011 / Revised: 19 October [2] H.H. Dam, S. Nordholm, A. Cantoni, J.M. de Haan, Iterative method for the design of DFT filter bank. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 51(11), (2004). [3] B. Dumitrescu, R. Bregovi, T. Saramki, Simplified design of low-delay oversampled NPR GDFT filterbanks. EURASIP J. Appl. Signal Process. 2006, (2006). [4] Y.T. Fong, C.W. Kok, Iterative least squares of DCleakage free paraunitary cosine modulated filter banks.

7 Design of NPR DFT-Modulated Filter Banks Using Modified Newton s Method IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 50(5), (2003). [5]. X.Q. Gao et al., An efficient digital implementation of multicarrier CDMA system based on generalized DFT filter banks. IEEE J. Sel. Areas Commun. 24(6), (2006). [6]. T.Q. Nguyen, Digital filter bank design quadraticconstrained formulation. IEEE Trans. Signal Process. 43(9), (1995). [7]. I.W. Selesnick, R.G. Baraniuk, N.G. Kingsbury, The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22(6), (2005). [8]. P.L. Shui, Image denoising using 2-D oversampled DFT modulated filter banks. IET Image Process. 3(3), (2009). [9]. P.L. Shui, J.Z. Jiang, X.L.Wang, Design of oversampled double-prototype DFTmodulated filter banks via bi-iterative second-order cone program. Signal Process. 90(5), (2010). [10]. W.Y. Sun, Y.X. Yuan, Optimization Theory and Methods:Nonlinear Programming (Springer, Berlin, 2006).