Synthesis and performance of a new class of implementation forhigh-order recursive digital filters

The authors investigate a new class of approaches to the synthesis of high-order recursive digital filters. These are all characterised by partitioning the overall filter into a set of parallel subfilters, each of which is of moderate order. The subfilters may be any of the classical implementations: direct form, parallel, cascade or lattice. The authors investigate cascade and lattice implementation of the subfilters. The investigation includes selection of the number of subfilters to use, how to derive their coefficients and theoretical noise calculations. This is supported by simulation testing of a variety of real and synthetic filters using maximal length binary sequences (MLS) as input as this provides information on both the linear and nonlinear errors involved. It is found that particularly when filter order is 50 or more, the new structures have reduced quantisation noise and improved tolerance to quantisation than their classical counterparts     

Polyphase FIR Networks Based on Frequency Sampling for Multirate DSP Applications

It is well known that the frequency sampling approach to the design of Finite Impulse Response digital filters allows recursive implementations which are computationally efficient when most of the frequency samples are integers, powers of 2 or null. The design and implementation of decimation (or interpolation) filters using this approach is studied herein. Firstly, a procedure is described which optimizes the tradeoff between the stopband energy and the deviation of the passband from the ideal filter. The search space is limited to a small number of samples (in the transition band), imposing the condition that the resulting filter have a large number of zeros in the stopband. Secondly, three different structures to implement the decimation (or interpolation) filter are proposed. The implementation complexity, i.e., the number of multiplications and additions per input sample, are derived for each structure. The results show that, without taking into account finite word-length effects, the most efficient implementation depends on the filter length to decimation (or interpolation) ratio.     

A new method for efficient convolution in frequency domain bynonuniform partitioning for adaptive filtering

By using block processing, partitioning, and fast Fourier transforms (FFTs), large filters perform efficiently in the frequency domain. For small processing delay the complexity can still be too large for implementation on a digital signal processor (DSP). A solution is to partition the filter into unequal-length subfilters. Application in adaptive filtering yields the nonuniform partitioned block frequency domain adaptive filter (NU-PBFDAF)     

Error-free coefficient realisation of recursive part via input-partitioned filter structure

A new filter structure is developed which allows precise coefficient implementation in the recursive part of the system. This is achieved by partitioning the input sequence into sets of inputs and realising the recursive part of the system into parallel subfilters with precise pole realisation.     

Hardware Architecture of Polyphase Filter Banks Performing Embedded Resampling for SoftwareDefined Radio FrontEnds

In this paper, we describe resource-efficient hardware architectures for software-defined radio (SDR) front-ends. These architectures are made efficient by using a polyphase channelizer that performs arbitrary sample rate changes, frequency selection, and bandwidth control. We discuss area, time, and power optimization for field programmable gate array (FPGA) based architectures in an M -path polyphase filter bank with modified N -path polyphase filter. Such systems allow resampling by arbitrary ratios while simultaneously performing baseband aliasing from center frequencies at Nyquist zones that are not multiples of the output sample rate. A non-maximally decimated polyphase filter bank, where the number of data loads is not equal to the number of M subfilters, processes M subfilters in a time period that is either less than or greater than the M data-load’s time period. We present a load-process architecture (LPA) and a runtime architecture (RA) (based on serial polyphase structure) which have different scheduling. In LPA, N subfilters are loaded, and then M subfilters are processed at a clock rate that is a multiple of the input data rate. This is necessary to meet the output time constraint of the down-sampled data. In RA, M subfilters processes are efficiently scheduled within N data-load time while simultaneously loading N subfilters. This requires reduced clock rates compared with LPA, and potentially less power is consumed. A polyphase filter bank that uses different resampling factors for maximally decimated, under-decimated, over-decimated, and combined up- and down-sampled scenarios is used as a case study, and an analysis of area, time, and power for their FPGA architectures is given. For resource-optimized SDR front-ends, RA is superior for reducing operating clock rates and dynamic power consumption. RA is also superior for reducing area resources, except when indices are pre-stored in LUTs.     

Coefficient-Symmetries for Implementing Arbitrary-Order Lagrange-Type Variable Fractional-Delay Digital Filters

In this paper, we first derive an explicit formula for expressing the coefficients of an arbitrary-order Lagrange-type variable fractional-delay (VFD) digital filter as polynomials of the VFD parameter , and then develop some useful symmetries for both even- and odd-order Lagrange-type VFD filter coefficients. The coefficient-symmetries facilitate the evaluations of VFD filter coefficients as well as variable frequency responses with reduced computational complexity. More importantly, the coefficient-symmetries can be exploited for efficiently implementing both even- and odd-order Lagrange-type VFD filters as Farrow structure and a more efficient one called even-odd structure such that the subfilters have symmetric or anti-symmetric coefficients, which saves the storage for VFD filter coefficients and reduces the number of multiplications required in VFD filtering process by almost 50%. Therefore, exploiting the coefficient-symmetries not only speeds up the VFD filtering, but also reduces implementation cost.     

Design of FIR digital filters using prescribed subfilters

A method for the design of linear-phase digital filters by the tapped cascaded interconnection of identical subfilters is presented. The method is an extension of the method proposed by Saramaki (1987). An example is given to show that the number of distinct multipliers of the filter determined by the proposed method is less than that of filters determined by Saramaki's method (1987). We also consider the case in which the subfilters are determined by multiple use of a single filter. In particular, if we can make the subfilters multiplierless then the number or multiplications per sample required to implement the overall filter is less than that required by the direct-form minimax method. Methods for the design of computationally efficient filters are also developed based on the proposed transformation method. The multiplication rate of the overall filter is the same as that of the prototype filter. It is very low as compared to that designed by the equivalent direct-form minimax method. With the proposed transformation method, methods for the design of a filter having nth-order tangency at both ends (0, π) are also developed. This is an extension of Vaidynathan's method (1985) and the proposed transformation method. The advantages of the method are that the resulting filters have very flat passbands and the stopbands are computationally efficient.     

A Class of Multimode Transmultiplexers Based on the Farrow Structure

This paper introduces multimode transmultiplexers (TMUXs) in which the Farrow structure realizes the polyphase components of general lowpass interpolation/decimation filters. As various lowpass filters are obtained by one set of common Farrow subfilters, only one offline filter design enables us to cover different integer sampling rate conversion (SRC) ratios. A model of general rational SRC is also constructed where the same fixed subfilters perform rational SRC. These two SRC schemes are then used to construct multimode TMUXs. Efficient implementation structures are introduced and different filter design techniques such as minimax and least-squares (LS) are discussed. By means of simulation results, it is shown that the performance of the transmultiplexer (TMUX) depends on the ripples of the filters. With the error vector magnitude (EVM) as the performance metric, the LS method has a superiority over the minimax approach.     

An Efficient Algorithm for the Optimization of FIR Filters Synthesized Using the Multistage Frequency-Response Masking Approach

A very efficient technique to drastically reduce the number of multipliers and adders in narrow transition-band linear-phase finite-impulse response digital filters is to use the one-stage or multistage frequency-response masking (FRM) approach, which has been originally introduced by Lim and further improved by Lim and Lian. In these original synthesis techniques, the subfilters in the overall implementation are separately designed. As shown earlier by the authors of this contribution together with Johansson, the arithmetic complexity in one-stage FRM filter designs can be considerably reduced by using the following two-step technique for simultaneously optimizing all the subfilters. First, a suboptimal solution is found by using a simple design scheme. Second, this solution is used as a start-up solution for further optimization, which is carried out with the aid of an efficient nonlinear optimization algorithm. This paper exploits this approach to synthesizing multistage FRM filters. An example taken from the literature illustrates that both the number of multipliers and the number of adders for the resulting optimized multistage FRM filters are approximately 70 percent compared with those of the filters synthesized using the original multistage FRM filter design schemes. Additional examples are included in order to show the benefits provided by the proposed synthesis scheme over other recently published design techniques, in terms of an improved performance of the resulting solution, a higher accuracy of the solution, and a faster speed required to arrive at the best solution.     

Least-Squares and Minimax Design of Polynomial Impulse Response FIR Filters for Reconstruction of Two-Periodic Nonuniformly Sampled Signals

This paper proposes polynomial impulse response finite-impulse response filters for reconstruction of two-periodic nonuniformly sampled signals. The foremost advantages of using these reconstruction filters are that on-line filter design thereby is avoided and subfilters with fixed dedicated multipliers can be employed in an implementation. The overall implementation cost can in this way be reduced substantially in applications where the sampling pattern changes from time to time. The paper presents two different design techniques that yield optimum filters in the least-squares and minimax senses, respectively. Design examples are included that illustrate the benefits of the proposed filters     

A Multimode Transmultiplexer Structure

This paper introduces a multimode transmultiplexer (TMUX) structure capable of generating a large set of user-bandwidths and center frequencies. The structure utilizes fixed integer sampling rate conversion (SRC) blocks, Farrow-based variable interpolation and decimation structures, and variable frequency shifters. A main advantage of this TMUX is that it needs only one filter design beforehand. Specifically, the filters in the fixed integer SRC blocks as well as the subfilters of the Farrow structure are designed only once. Then, all possible combinations of bandwidths and center frequencies are obtained by properly adjusting the variable delay parameter of the Farrow-based filters and the variable parameters of the frequency shifters. The paper includes examples for demonstration. It also shows that, using the rational SRC equivalent of the Farrow-based filters, the TMUX can be described in terms of conventional multirate building blocks which may be useful in further analysis of the overall system.     

Generalized WLS Method for Designing All-Pass Variable Fractional-Delay Digital Filters

This paper presents an optimal weighted least squares (WLS) method for designing low-complexity all-pass variable fractional-delay (VFD) digital filters. Instead of using a fixed range for the VFD parameter p and same-order constant-coefficient filters (subfilters), both the VFD parameter range p isin [p _Min,p _Max] and subfilter orders are optimized such that a low-complexity all-pass VFD filter can be achieved for the LS design. To suppress the peak errors of variable frequency response, weighting functions are adopted and optimized such that the boundary peak errors can be further reduced but without noticeably increasing the total error energy (integral of squared error) of variable frequency response. After optimizing the variable range of the VFD parameter, weighting functions, and subfilter orders, an all-pass VFD filter can be designed by using a generalized noniterative WLS method, which yields a closed-form solution. Design examples are given to illustrate that utilizing different-order subfilters, along with the optimal range and optimal weighting functions, can yield an all-pass VFD filter with significantly reduced complexity and design errors as compared with existing ones.     

Use Of The Remez Algorithm For Designing FRM Based FIRr Filters

A very efficient technique for drastically reducing the number of multipliers and adders in narrow transition-band linear-phase finite impulse response (FIR) filters is to use the one-stage or multistage frequency-response masking (FRM) approach as originally introduced by Lim. In the original synthesis techniques developed by Lim and Lian, the subfilters in the overall approach were designed using time-consuming linear programming. In order to perform the overall synthesis faster, this paper shows how these subfilters can be designed with the aid of the the Remez multiple exchange algorithm, the most powerful technique for designing arbitrary-magnitude linear-phase FIR filters in the minimax sense. In addition to speeding up the overall procedure, the use of the Remez algorithm enables one to generate a very fast MATLAB program for the overall synthesis so that after being given the filter specifications as well as the number of stages, the program automatically provides the solution with the minimum number of multipliers and adders required in the overall implementation. This is possible because the MATLAB Remez routine is directly available and thus can be used for this purpose after appropriate modifications.     

Design of stable, causal, perfect reconstruction, IIR uniform DFTfilter banks

Design procedures for stable, causal and perfect reconstruction IIR parallel uniform DFT filter banks (DFT FBs) are presented. In particular a family of IIR prototype filters is a good candidate for DFT FB, where a tradeoff between frequency selectivity and numerical properties (as measured by the Weyl-Heisenberg frames theory) could be made. Some realizations exhibiting a simple and a massively parallel and modular processing structure making a VLSI implementation very suitable are shown. In addition, some multipliers in the filters (both the analysis and synthesis) could be made; powers or sum of powers of 2, in particular for feedback loops, resulting in a good sensitivity behavior. For these reasons as well as for the use of low order IIR filters (as compared with conventional FIR filters), the overall digital filter bank structure is efficient for high data rate applications. Some design examples are provided     

Linear-phase FIR interpolation, decimation, and mth-band filters utilizing the farrow structure

This paper introduces novel linear-phase finite-impulse response (FIR) interpolation, decimation, and Mth-band filters utilizing the Farrow structure. In these new overall filters, each polyphase component (except for one term) is realized using the Farrow structure with a distinct fractional delay. The corresponding interpolation/decimation structures can therefore be implemented using only one set of linear-phase FIR subfilters and one set of multipliers that correspond to the distinct fractional delays. The main advantage of the proposed structures is that they are flexible as to the conversion factors, and this also for an arbitrary set of integer factors, including prime numbers. In particular, they can simultaneously implement several converters at a low cost. The proposed filters can be used to generate both general filters and Mth-band filters for interpolation and decimation by the integer factor M. (In this paper, a general filter for interpolation and decimation by M means a filter having a bandwidth of approximately /spl pi//M without the restriction that /spl pi//M be included in the transition band. This is in contrast to an Mth-band filter whose transition band does include /spl pi//M.) In both cases, the overall filter design problem can be posed as a convex problem, the solution of which is globally optimum. Design examples are included in the paper illustrating the properties and potentials of the proposed filters.     

On Low-Delay Frequency-Response Masking FIR Filters

This paper offers two main contributions to the theory of low-delay frequency-response masking (FRM) finite impulse response (FIR) filters. First, a thorough investigation of the low-delay FRM FIR filters and their subfilters or three different structures, referred to as narrow-, wide-, and middle-band filter structures, is given. The investigation includes discussions on delay distribution over the subfilters as well as estimation of the optimal periodicity of the periodic model filter. Second, systematic design procedures are given, with explicit formulas for distribution of the ripples and the delay to the subfilters. For each of the three structures, two design procedures are given that include joint optimization of the subfilters. The first proposal uses partly linear-phase FIR subfilters and partly low-delay FIR subfilters. Thus, it has a lower arithmetic complexity compared to the second proposal, which has exclusively low-delay FIR subfilters. The second proposal is instead more flexible and can handle a broader range of specifications. The design procedures result in low-delay FIR filters with a lower arithmetic complexity compared to previous results, for specifications with low delay and narrow transition band.     

Recursive weighted median filters admitting negative weights andtheir optimization

A recursive weighted median (RWM) filter structure admitting negative weights is introduced. Much like the sample median is analogous to the sample mean, the proposed class of RWM filters is analogous to the class of infinite impulse response (IIR) linear filters. RWM filters provide advantages over linear IIR filters, offering near perfect “stopband” characteristics and robustness against noise. Unlike linear IIR filters, RWM filters are always stable under the bounded-input bounded-output criterion, regardless of the values taken by the feedback filter weights. RWM filters also offer a number of advantages over their nonrecursive counterparts, including a significant reduction in computational complexity, increased robustness to noise, and the ability to model “resonant” or vibratory behavior. A novel “recursive decoupling” adaptive optimization algorithm for the design of this class of recursive WM filters is also introduced. Several properties of RWM filters are presented, and a number of simulations are included to illustrate the advantages of RWM filters over their nonrecursive counterparts and IIR linear filters     

利用直接矩阵求逆和临界采样子带自适应滤波器的快速系统辨识

在许多应用中,子带自适应滤波器结构已经显示了其在计算和性能上的优点。基于最近提出的一个采用临界采样滤波器组的子带自适应结构,该文引入了子带直接矩阵求逆(DMI)算法。在保持了该算法快速收敛优点的同时,利用相关矩阵块三对角的特殊结构,降低了该算法的计算复杂度。理论分析及计算机实验显示,子带直接矩阵求逆算法只需经过较少的更新次数自适应子滤波器自由度的两倍,就能够收敛到高于最小均方误差的3dB附近。     

Suppression of transients in variable recursive digital filterswith a novel and efficient cancellation method

A new method for suppressing transients in recursive infinite impulse response (IIR) digital filters is proposed. The technique is based on modifying the state (delay) variables of the filter when coefficients are changed so that the filter enters a new state smoothly without transient attacks, as originally proposed by Zetterberg and Zhang (1988). In this correspondence, we modify the Zetterberg-Zhang algorithm to render it feasible for efficient implementation. We define a mean square error (MSE) measure for transients and determine the optimal transient suppressor to cancel the transients down to a desired level at the minimum complexity of implementation. The application of the method to all-pole and direct-form II (DF II) IIR filter sections is studied in detail. Time-varying recursive filtering with transient elimination is illustrated for tunable fractional delay filters and variable-bandwidth lowpass filters