Traditional designs for **non**-**uniform** **filter** bank (NUFB) are usually complex; involve complicated nonlinear optimiza- tion with a large number of parameters and lack of linear phase (LP) property. In this paper, we describe a simple de- sign 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 struc- ture 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 de- sign examples are presented to show the superiority of the proposed method over existing one.

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This **Non**-**uniform** **Filter** **Banks** (NUFBs) are used in several applications such as audio and image signal proc- essing, biomedical signal processing, hearing aid instru- ments because of their flexibility in partitioning sub- bands and better performance [1-3]. Based upon the fre- quency spacing between bands, **filter** **banks** are broadly classified into two categories i.e. **uniform** band and **non**-**uniform** band **filter** bank. Most of the surveys focus on **uniform** frequency bandwidth in which incoming sig- nals are split with same sampling rate [4-6]. But in few applications having different incoming data rates and different quality of service **non**-**uniform** **filter** bank is required. For example, to approximate the critical bands sensed by the human ear, **non**-**uniform** designs of **filter** bank using all pass frequency transformation shows bet- ter performance than **uniform** one [7]. Although few methods already have been reported in the literature us- ing constrained or unconstrained optimization techniques, still efficient solutions to the direct design of **non**-uni- form **filter** **banks** have yet to be developed [8]. This nonlinear arrangement of frequency bands is necessary when the signal energy exhibits bandwidth dependent distribution. The basic structure of **uniform** **filter** bank is shown in Figure 1, where the input signal x n is de- composed by M analysis filters H z and then outputs

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The rest of the paper is organized as follows. In Section 3, OFB are reviewed in light of the frame, FB, and channel cod- ing theory. In particular, the results on the equivalence be- tween the projection receiver and the syndrome decoding de- rived for DFT codes in [5] are extended to the case of OFB codes. That is, for paraunitary FBs, it is shown that the sig- nal reconstruction methods derived from the FB theory and from the coding theory are equivalent even in presence of quantization noise. The structures of OFB based on CMFBs that are considered in the sequel are described in Section 4, together with the corresponding frame properties and pack- etization schemes. In Section 5, the PR and the erasure re- covery properties of the two families of codes considered are analyzed. Even though OFBs based on CMFB have simple structures, it is diﬃcult to analytically verify the PR prop- erty for all erasure patterns. For some particular erasure pat- terns, we show that PR depends only on the structure of the code and not on the **filter** coeﬃcients. Section 6 considers the problem of reconstruction in presence of quantization noise. The mean square error (MSE) performance bounds under particular quantization noise distribution assumptions are provided. It is shown that, in presence of quantization noise, the MSE for some erasure patterns does not depend on the **filter** coe ﬃ cients. Section 7 provides performance results in terms of mean-square reconstruction error obtained with the codes studied here in comparison with a DFT code.

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In this work, **Cosine** **Modulated** **filter** bank is used for the design of **filter** bank and then by rounding operation we have tried to reduce the computational complexity for the designed **filter** bank. Among the different classes of multi- channel **filter** **banks**, **cosine**-**modulated** **filter** **banks** are the most frequently used **filter** **banks** due to simpler design, where analysis and synthesis **filter** **banks** are derived by **cosine** modulating a low-pass prototype **filter**. The design of whole **filter** bank thus reduces to that of a single low- pass prototype **filter**. **Cosine** **modulated** **filter** bank (CMFB) finds wide application in different areas of digital signal processing such as equalization of wireless communication channel, sub band coding, spectral analysis, adaptive signal processing, denoising, feature detection and extraction. Several design techniques have been developed for these **filter** **banks** in the past [1], [2], [3]. In CMFB, analysis and synthesis filters are **cosine** **modulated** versions of low pass prototype **filter**. Thus, the design of whole **filter** bank reduces to that of the prototype **filter** and the cost of overall **filter** bank is almost equal to the cost of one **filter** with modulation overheads.

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As an example for discarding channels in the MCAF blocks, assume a scenario with real valued sensor signals and a blocking matrix with columns, whereby the frequency responses of the column vectors are ideal as given in Fig. 9(a). By employing eight-channel oversampled generalized DFT (GDFT) **filter** **banks** [15], [16] with frequency responses shown in Fig. 9(b), each of the blocking matrix outputs , , is decomposed into eight subband signals , . They form, in total, eight subband MCAF blocks, as shown in Fig. 10. Since the input signals are real, we only need to process the first four subband MCAF blocks. From the characteristics of the frequency responses in Fig. 9(a) and (b), it can be seen that some subband channels will have zero power and can be discarded. The pattern for discarding these channels is given in Fig. 10, where the discarded subbands in the last four subband MCAF blocks are also shown for complete- ness. With increasing and , the number of discarded sub- band channels becomes approximately half of the total number of subbands. However, in practice, the frequency responses of the column vectors and the **filter** **banks** are not ideal and finite transition bands are unavoidable, such that the number of dis- carded subbands will be smaller than in the ideal situation.

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The MDFT **filter** bank proposed by Fliege belongs to time-invariant QMF **filter** **banks**. The channel number and subband filters do not change with time. In speech, audio and image processing, the time-varying **filter** bank can be used to adjust the input signal properties in order to get better processing performance [6, 7, 9, 10]. In the past, we have done a systematic research for the general time-varying **filter** bank including time-varying **cosine**-**modulated** **filter** **banks** [13–15]. In this paper, we want to use the established time- varying **filter** bank theory to analyze the time-varying MDFT **filter** bank (TV-MDFT), and investigate the properties of the MDFT **filter** bank in the case that its **filter** coe ﬃ cients change with time. We want to also study the prototype **filter** design for a linear time-varying MDFT **filter** bank. After instruction Section 2 provides the description of the MDFT **filter** bank and its mechanism of aliasing error cancellation in both the time and frequency domain. Section 3 analyzes the time- varying MDFT **filter** bank including its PR condition and the relationship with time-varying **cosine**-**modulated** **filter** bank. In Section 4, the design of the prototype window **filter** for TV-MDFT is discussed. The well-know window switching method used in time-varying **cosine**-**modulated** **filter** bank [13] will be used in the prototype window design for TV- MDFT. Section 5 summarizes the main results of the paper.

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increases linearly. Also the DFT size needs to be increased. Pucker, L. in paper [2] entitled ―Channelization techniques for software defined radio‖ proposed DFT **Filter** **Banks**. DFT **filter** bank is a uniformly **modulated** **filter** bank, which has been developed as an efficient substitute for PC approach when the number of channels need to be extracted is more, and the channels are of **uniform** bandwidth (for example many single standard communication channels need to be extracted). The main advantage of DFT **filter** bank is that, it can efficiently utilize the polyphase decomposition of filters. The limitations of DFTFBs is that the channel filters have fixed equal bandwidths corresponding to the specification of a given standard.R. MAHESH et.al. in paper [3] entitled ―Reconfigurable Low Area Complexity **Filter** Bank Architecture Based on Frequency Response Masking for **Non** **uniform** Channelization in Software Radio Receivers‖ proposed a new reconfigurable FB based on the FRM approach for extracting multiple channels of **non** **uniform** bandwidths. The FRM approach is modified to achieve following advantages: 1) incorporate reconfigurability at the **filter** level and architectural level, 2) improve the speed of filtering operation, and 3) reduce the complexity.

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Abstract: When **filter** **banks** are used in multicarrier modulation the transition bandwidth of the **filter** is usually very narrow. This is because the bandwidth of the filters is determined by the number of subchannels and the greater the number of channels lesser will be the transition bandwidth. The order of the **filter** being inversely proportional to the transition bandwidth is generally very high. Hence computational complexity is also very high. We propose a method that will help reduce this complexity by designing the decimators/interpolators in the subchannels using a multistage approach. We compare the saving obtained in comparison to direct design approaches. We will also compare the performance of a **cosine** **modulated** **filter** bank designed in this way with direct design methods.

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In this paper, the use of **cosine**-**modulated** **filter** **banks** (CMFBs) for multicarrier modulation in the application of very-high-speed digital subscriber lines (VDSLs) is studied. We refer to this modulation technique as **cosine**-**modulated** multitone (CMT). CMT has the same transmitter structure as discrete wavelet multitone (DWMT). However, the receiver structure in CMT is diﬀerent from its DWMT counterpart. DWMT uses linear combiner equalizers, which typically have more than 20 taps per subcarrier. CMT, on the other hand, adopts a receiver structure that uses only two taps per subcarrier for equalization. This paper has the following contributions. (i) A modification that reduces the computational complexity of the receiver structure of CMT is proposed. (ii) Although traditionally CMFBs are designed to satisfy perfect-reconstruction (PR) property, in transmultiplexing applications, the presence of channel destroys the PR property of the **filter** bank, and thus other criteria of **filter** design should be adopted. We propose one such method. (iii) Through extensive computer simulations, we compare CMT with zipper discrete multitone (z-DMT) and filtered multitone (FMT), the two modulation techniques that have been included in the VDSL draft standard. Comparisons are made in terms of computational complexity, transmission latency, achievable bit rate, and resistance to radio ingress noise.

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The conversion between the 40 TVWS channels and the baseband signal required for stage 1 is performed with the help of an oversampled DFT-**modulated** **filter** bank with K 2 (i) channels, operating as an FBMC transmultiplexer. The design is based on a 8 MHz-wide prototype as characterised in Fig. 2, whose transition band depends on the oversampling ratio. In our design, we have decided to sample the TVWS channels at 16 MHz, i.e. they are oversampled a factor of 2. This provides a sufficient transition band, to protect adjacent channels as well as keeping the prototype **filter** as short as possible.

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The low-pass structure **uniform** analysis **filter** bank [6] is shown in Figure 2. The number of the channel is M . The multiple of the decimation is K and M = × K F . The input signal firstly goes through the down-conversion module to become a baseband signal. And then **filter** the baseband signal to get the sub-band signals. At last, we decimate these sub-band signals. When the number of the channel is large, it is very difficult to complete engineering realization. Assume that the or- der of the low-pass **filter** is N , so the total order of this whole structure is M × N . As the number of the channel increases, the total order also increases, so the consumed multiplier and adder resources increase. The complex exponential down-conversion module requires the resource of the multiplier and adder, so when the number of the frequency band increases the required resources will also increase. The decimation module can reduce the sampling rate, but in this struc- ture the decimation module is at the end of the system. So, the down-conversion

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proposed. Conventional FRM ﬁlter is used as the prototype ﬁlter. FRM approach results in a tremendous amount of reduction in the number of multipliers because of the large number of sparse coef- ﬁcients [5]. To further reduce the implementation complexity, the FRM ﬁlter coefﬁcients are represented with minimum number of SPT terms using canonic signed digit (CSD) representation[19]. The design of the FRM ﬁlter in the discrete space can degrade the ﬁlter performance. This calls for the use of **non**-linear optimiza- tion techniques. Meta-heuristic algorithms are used in this paper because classical gradient based techniques cannot be directly ap- plied due to the fact that the search space consists of integers. This results in the proper tuning of the parameters with respect to a spe- ciﬁc design problem, which in turn can lead to global solution [18]. The resulting prototype ﬁlter is multiplierless and hence the design leads to a totally multiplierless **cosine** **modulated** ﬁlter bank. Since multipliers are the most power consuming circuits and occupy large space, the multiplierless realization will reduce complexity, power consumption and silicon area. Such a multiplierless ﬁlter bank is hitherto not reported in the literature.

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else are equal to zero, satisfying the unit energy constraint between [ −1, 1]. NPR condition is set by keeping the distortion to 10 −5 . The maximum generation ( G max ) is set to 2000 generations. The programme was run using MATLAB 7.8 and the obtained coefficients were used for designing NPR **cosine** **modulated** **filter** bank. The amplitude response of the designed **filter** **banks** for 8 channel with 16 and 24 taps, with and without dc leakage condition are plotted in Figure 2 and Figure 3. The result of response indicates the superiority of the longer tap filters as the stopband attenuation increase with the increase in no. of taps. Secondly, incorporating dc leakage deteriorate the response of the **filter** but is good from the image coding point as dc leakage free condition reduce the aliasing errors and all the subband filters except the lowpass one, has at least one zero at z = 1 and thus all other bandpass and highpass filters are dc transmission free. Both the **filter** **banks** are designed giving highest priority to coding gain amongst the multiple objectives. The 8 × 16 CMFB in design Example I, with dc leakage (optimized for pure coding gain); attain 9.3749 dB, which almost equals the coding gain of optimal biorthogonal systems. However, applying dc leakage free constraints on the **filter** coefficients ensure a high level of percep- tual performance in image coding. In design example II, the 8 × 24 case, our CMFB achieves an improvement of 0.001 dB in coding gain. CMFB’s with **filter** length 16 and 24 are shown in Figure 2 and Figure 3. Increasing the length only improves the coding gain marginally, it helps in the case of stopband attenuation (where longer filters are always beneficial), which result in more complexity in design and implementation.

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After a delay chain and downsamplers, the input values com- ing to odd paths are sign-changed. When feeding this mod- ified sequence through the butterflies of the ELT, the input sequence to the transform block is almost correct if com- pared with the sequence obtained from the SMFB structure. The values coming from the even-numbered paths are cor- rect, but the values from the odd-numbered paths have op- posite signs. These opposite signs can be compensated if the DCT-IV matrix is used instead of the DST-IV matrix. After the modulation block all the subband signals are correct, but they are just in the reverse order. Thus, using the above pro- cedure, it is possible to compute **cosine**/sine **modulated** se- quences using only one fast algorithm originally designed for just ELT computing. It should be also pointed out that the sine **modulated** synthesis system is obtained when the above- mentioned steps are done in reverse order.

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A general overview of FBMC system model is shown in Fig. 1. The **filter** **banks** are employed in transmultiplexer configuration. It consists of symbol mapper and an upsampler followed by synthesis filters at the transmitter end. The synthesis **filter** performs the frequency division multiplexing of complex data symbols into parallel subcarriers. The analysis **filter** at the receiver end separates’ the data from each carrier. The synthesis and analysis filters are **cosine** **modulated** versions of low pass prototype **filter**. The prototype **filter** is designed using Kaiser Window with cut off frequency pi/8 and enough stop band attenuation.

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The pioneering work of Chang [7], on the other hand, has received less attention within the signal processing com- munity. Those who have cited [7], have only acknowledged its existence without presenting much details, for example [11, 16, 19, 22]. For instance, Hirosaki who has extensively studied and developed digital structures for implementation of Saltzberg’s method, [11, 16], has made a brief reference to Chang’s method and noted that since it uses VSB modulation and thus its implementation require a Hilbert transforma- tion, it is more complex than that of Saltzberg’s method. He thus proceeds with a detail discussion and development of multirate structures for the Saltzberg’s method only. On the other hand, a vast literature in digital signal processing has studied a class of multicarrier systems that has been referred to as discrete wavelet multitone (DWMT). The initial works on DWMT are [23–25]. In the period of 1995 to 2003 a fair number of contributions from various authors appeared in the literature, for example [26–30]. Reference [30], in particular, did a thorough study of DWMT and noted that this method operates based on **cosine** **modulated** **filter** **banks** which were extensively developed in the 1980’s in the context of compression techniques [31]. The fact that DWMT uses **cosine** **modulated** **filter** **banks** has also been acknowledged by other authors, for example [28]. Reference [30] also greatly simplified the equalizer structure that was originally proposed in [23–25] and widely adopted by others. Moreover, [30] noted that a DWMT signal is synthesized by aggregating a set of VSB **modulated** PAM signal sequences. However, most of the works on DWMT (including [30]) have made no direct reference to Chang’s method. In other words, the Chang’s multicarrier method was re-invented, with a strong multirate signal processing flavor, in the 1990’s. Part of our attempt in this paper is to show this very important relationship between what has been done over 40 years ago, and the independent developments on DWMT/**cosine** **modulated** multicarrier techniques that have been developed in more recent years. We also hope that the tutorial treatment

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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].

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ABSTRACT: The **filter** bank multicarrier (FBMC) transmission system is an enabling technology for the new concepts and, especially, cognitive radio and it results into an enhanced physical layer for conventional networks. In this paper, we present an decision feedback equalizer based on RLS (Recursive Least Square) and LMS (Least Mean Square) algorithms at the sub-channel level for FBMC systems using exponentially **modulated** **filter** **banks**. The input to the FBMC system is offset quadrature amplitude **modulated** (OQAM) input symbols. Simulation results exhibit that, in spite of its increased computational complexity, the FBMC/OQAM transmission technique provides better bit error rate performance.

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For two channel **filter** **banks**, no matter whether they are one-dimensional or two-dimensional, there is not one which can be simultaneously be symmetric (linear phase) [17], compactly supported (FIR) [18] and orthogonal, except the Harr wavelet [19] which is not continuous. However, linear phase, which is good properties for filtering to avoid distortion in signal processing, is often desired. To obtain a FIR **filter** bank, only nearly- orthogonal **filter** bank is possible. In this paper, zeros at aliasing frequency points are directly imposed on the two-dimensional quincunx symmetric **filter**, and optimize it to satisfy the orthogonal condition as closely as possible. Then a nearly-perfect reconstructed (NPR) [20] and nearly-orthogonal (ON) **filter** bank can be obtained, and a corresponding scaling function can be generated. The question is what is the corresponding wavelet and what is the degree of orthogonality of it, since the **filter** bank is not perfect reconstructed. Firstly, we can construct a semi-orthogonal 1D which is perfect reconstructed. We will show that the semi-orthogonal **filter** is very close to the nearly-orthogonal **filter** bank in the sense that the corresponding filters in the two **filter** bank have the exact phase response and almost the same amplitude response. The NPR **filter** bank is regarded as an implementation of the semi-orthogonal **filter** and correlation analysis is carried on the semi-orthogonal **filter** bank. Both theoretical and experimental analysis are given, which show that the semi-orthogonal **filter** bank is nearly-orthogonal. Secondly, a complementary **filter** with the same vanishing moment as the original **filter** is found. The correlation of this perfect reconstructed **filter** bank is also carried out. The analysis show that, in the

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