Representations of Linear Dual-Rate System Via Single SISO LTI Filter, Conventional Sampler and Block Sampler

In this brief, it is proved that a linear dual-rate system can be represented via a series cascade of: 1) a conventional expander, a single-input single-output (SISO) linear time-invariant (LTI) filter and a block decimator, or 2) a block expander, an SISO LTI filter and a conventional decimator. Hence, incompatible nonuniform filter banks could achieve perfect reconstruction via LTI filters, conventional samplers and block samplers without expanding the input-output dimension of a subsystem of linear dual-rate systems or converting the nonuniform filter banks to uniform filter banks. The main advantage of the proposed representations is to avoid complicated design of the circuit layout caused by connecting subsystems with large input-output dimension or a lot of subsystems together.     

Design of multichannel nonuniform transmultiplexers using generalbuilding blocks

The paper considers design of multichannel, nonuniform-band transmultiplexers. It is well known that using traditional building blocks-up and downsamplers and linear time-invariant (LTI), causal filters-nonuniform transmultiplexers typically do not achieve perfect reconstruction. To alleviate this, we propose to build nonuniform transmultiplexers using general dual-rate structures that provide more design freedom, and hence, perfect reconstruction can be achieved. Such general transmultiplexers have a new source of error called aliasing distortion in addition to the traditional cross-talk, magnitude, and phase distortions. We propose a composite error criterion that captures all four distortions in one. Using this error criterion as reconstruction performance measure. We develop an optimal design procedure and apply it to a three-channel nonuniform example, yielding an FIR transmultiplexer that has good frequency-limiting properties in the synthesis end and is very close to perfect reconstruction     

2-Norm Based Recursive Design of Transmultiplexers with Designable Filter Length

In this paper, motivated by the facts that shorter length filters offer considerable improvements in computation and hardware implementation and that designable filter length brings more design flexibility, we propose a novel transmultiplexer design method that can not only achieve close-to-perfect reconstruction but also obtain designable filter length. The proposed method is based on a composite distortion measure: the 2-norm of the error transfer matrix of the transmultiplexer. Central to the development is providing an efficient means to recursively design the filters in the transmultiplexer and evaluating the composite distortion measure, which greatly improves the computation efficiency. The simulation results show the effectiveness of the proposed algorithms.     

Filter design for MIMO sampling and reconstruction

We address the problem of finite impulse response (FIR) filter design for uniform multiple-input multiple-output (MIMO) sampling. This scheme encompasses Papoulis' generalized sampling and several nonuniform sampling schemes as special cases. The input signals are modeled as either continuous-time or discrete-time multiband input signals, with different band structures. We present conditions on the channel and the sampling rate that allow perfect inversion of the channel. Additionally, we provide a stronger set of conditions under which the reconstruction filters can be chosen to have frequency responses that are continuous. We also provide conditions for the existence of FIR perfect reconstruction filters, and when such do not exist, we address the optimal approximation of the ideal filters using FIR filters and a minmax l/sub 2/ end-to-end distortion criterion. The design problem is then reduced to a standard semi-infinite linear program. An example design of FIR reconstruction filters is given.     

Minimax deconvolution design of multirate systems with channelnoises: a unified approach

The conventional signal reconstruction problem of multirate systems with channel noises can be cast as a robust multirate deconvolution design problem. We investigate a unified minimax approach for the robust deconvolution design of multirate systems. We discuss two typical multirate systems: the multirate filter bank system and the transmultiplexer system. We consider transmission noises resulting from quantization coding errors or external noises. The deconvolution filters for these systems that we derive are all IIR filters. The keypoint is converting the original robust deconvolution design problem to an equivalent minimax matching problem via polyphase decomposition and noble identities. Then, in spite of the presence of input signals and channel noises, we can solve this minimax matching problem by an optimization technique. The proposed method can be interpreted as designing an optimal multirate deconvolution filter such that the worst-case multirate system reconstruction error is minimized over all possible inputs and noises from the energy perspective. Therefore, our proposed design method is more robust than the conventional design method for multirate systems in the presence of uncertain input signals and channel noises. We present several numerical examples that show the good performance of our design method     

Optimal design of nonuniform multirate filter banks

The design of general nonuniform filter banks is studied. Contrary to uniform filter banks, in nonuniform filter banks, it may not be possible to achieve perfect reconstruction, but in some cases by using optimization techniques, we can design acceptable filter banks. Here, the initial finite impulse response (FIR) analysis filters are designed according to the characteristics of the input. By the design procedure, the FIR synthesis filters are found so that theH-norm of an error system is minimized over all synthesis filters that have a prespecified order. Then, the synthesis filters obtained in the previous step are fixed, and the analysis filters are found similarly. By iteration, theH-norm of the error system decreases until it converges to its final value. At each iteration, the coefficients of the analysis or synthesis filters are obtained by finding the least squares solution of a system of linear equations. If necessary, the frequency characteristics of the filters can be altered by adding penalty terms to the objective function.This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Mixed H2/H∞ filtering design inmultirate transmultiplexer systems: LMI approach

A mixed H₂/H_∞ filter design is proposed for multirate transmultiplexer systems with dispersive channel and additive noise. First, a multirate state-space representation is introduced for the transmultiplexer with the consideration of channel dispersion. Then, the problem of signal reconstruction can be regarded as a state estimation problem. In order to design an efficient separating filterbank for a transmultiplexer system with uncertain input signal and additive noise, the H_∞ filter is employed for robust signal reconstruction. The H₂ filter design is considered to be a suboptimal approach to achieve the optimal signal reconstruction in transmultiplexer system under unitary noise power. Finally, a mixed H₂/H_∞ filter is proposed to achieve a better signal reconstruction performance in transmultiplexer systems. These design problems can be transformed to solving the eigenvalue problems (EVP) under some linear matrix inequality (LMI) constraint. The LMI Matlab toolbox can be applied to efficiently solve the EVP by convex optimization technique     

Some results in the theory of crosstalk-free transmultiplexers

The crosstalk-free transmultiplexer (CF-TMUX) focuses on crosstalk cancellation (CC) rather than on suppressing it. The authors present an analysis of the CF-TMUX based on the polyphase component matrices of the filter banks used in TDM→FDM and FDM→TDM conversions, respectively. Thus a necessary and sufficient condition for complete CC is obtained. It is shown that the filters for a CF-TMUX are the same as the filters for a 1-skewed alias free QMF bank. In addition, if the QMF bank satisfies the perfect reconstruction (PR) property, then the TMUX also satisfies PR. The relation between CF-TMUX filters and alias-free QMF banks is used to obtain a direct design procedure for CF-TMUX filters (both FIR and IIR). It is also shown that approximately crosstalk-free TMUX filters can be obtained from any approximately alias-free QMC bank. Design examples and comparison tables are included     

Two-channel perfect reconstruction (PR) quadrature mirror filter (QMF) bank design using logarithmic window function and spline function

In this work, two-channel perfect reconstruction quadrature mirror filter (QMF) bank has been proposed based on the prototype filter using windowing method. A novel window function based on logarithmic function along with the spline function is utilized for the design of prototype filter. The proposed window has a variable parameter ‘\(\alpha \)’, which varies the peak side lobe level and rate of fall-off side lobe level which in turn affects the peak reconstruction error (PRE) and amplitude distortion (\(e_{am}\)) of the QMF bank . The transition width of the prototype is controlled by the spline function using the parameter ‘\(\mu \)’. The perfect reconstruction condition is satisfied by setting the cutoff frequency (\(\omega _{c}\)) of the prototype low-pass filter at ‘\(\pi /2\)’. The performance of the proposed design method has been evaluated in terms of mean square error in the pass band, mean square error in the stop band, first side lobe attenuation (\(A_{1}\)), peak reconstruction error (PRE) and amplitude error (\(e_{am}\)) for different values of ‘\(\alpha \)’ and ‘\(\mu \)’. The results are provided and compared with the existing methods.     

Robust filtering for discrete-time systems with saturation and its application to transmultiplexers

This paper considers the problem of robust filtering for discrete-time linear systems subject to saturation. A generalized dynamic filter architecture is proposed, and a filter design method is developed. Our approach incorporates the conventional linear H/sub 2/ and H/sub /spl infin// filtering as well as a regional l/sub 2/ gain filtering feature developed specially for the saturation nonlinearity and is applicable to the digital transmultiplexer systems for the purpose of separating filterbank design. It turns out that our filter design can be carried out by solving a constrained optimization problem with linear matrix inequality (LMI) constraints. Simulations show that the resultant separating filters possess satisfactory reconstruction performance while working in the linear range and less degraded reconstruction performance in the presence of saturation.     

Optimal sub-Nyquist nonuniform sampling and reconstruction formultiband signals

We study the problem of optimal sub-Nyquist sampling for perfect reconstruction of multiband signals. The signals are assumed to have a known spectral support ℱ that does not tile under translation. Such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching Landau's (1967) lower bound equal to the measure of ℱ. For signals with sparse ℱ, this rate can be much smaller than the Nyquist rate. Unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In a previous study, we derived bounds on the error due to mismodeling and sample additive noise. Adopting these bounds as performance measures, we consider the problems of optimizing the reconstruction sections of the system, choosing the optimal base sampling rate, and designing the nonuniform sampling pattern. We find that optimizing these parameters can improve system performance significantly. Furthermore, uniform sampling is optimal for signals with ℱ that tiles under translation. For signals with nontiling ℱ, which are not amenable to efficient uniform sampling, the results reveal increased error sensitivities with sub-Nyquist sampling. However, these can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate     

近似完全重构IIR余弦调制滤波器组的优化设计

提出一种新的近似完全重构因果稳定的IIR余弦调制滤波器组的设计方法。基于预先给定的极点值,IIR原型滤波器的设计问题可以简化成一个凸极大值极小化的优化问题,从而采用二阶锥规划法求解。所得余弦调制滤波器组具有良好的频率特性和合理的完全重构误差。所设计的原型滤波器是因果稳定的,并且其多相因子分母相同,简化了完全重构条件,可以用来进一步优化得到的完全重构系统。     

A new approach to the design of biorthogonal triplet half-band filter banks using generalized half-band polynomials

This paper presents a novel approach to design a class of biorthogonal triplet half-band filter banks based on the generalized half-band polynomials. The filter banks are designed with the help of three-step lifting scheme (using three kernels). The generalized half-band polynomial is used to construct these three kernels by imposing the number of zeros at \(z=-1\) . The maximum number of zeros imposed for the three kernels is half of the order of half-band polynomial ( \(K/2\) for \(K\) order polynomial). The three kernels give a set of constraints on the coefficients of half-band polynomial by imposing the zeros. In addition to structural perfect reconstruction and linear phase, the proposed filter banks provide better frequency selectivity, more similarity between analysis and synthesis filters (measure of near-orthogonality), and good time–frequency localization. The proposed technique offers more flexibility in the design of filters using two degrees of freedom. Some examples have been presented to illustrate the method.     

一种基于遗传算法的混合滤波器组设计方法

针对可满足近似完全重构的双通道混合滤波器组,其中高阶数的模拟滤波器一般不容易设计优化。采用遗传算法设计5阶模拟分解滤波器,并基于逆快速傅里叶变换实现数字综合滤波器的设计优化以滤除掉镜像频谱,保证近似完全重构。文中设计了由5阶模拟分解滤波器和32阶数字综合滤波器组成的混合滤波器组,仿真结果表明:可以实现的最大失真误差为4.761 8×10-11dB,平均失真误差为-9.2×10-14dB,最大混叠误差为-154 dB,平均混叠误差为-200 dB,可满足24 bits的模数转换器系统的要求。     

filtering for discrete-time linear systems with bounded time-varying parameters

In this paper, the problem of linear parameter varying (LPV) filter design for time-varying discrete-time polytopic systems with bounded rates of variation is investigated. The design conditions are obtained by means of a parameter-dependent Lyapunov function and extra variables for the filter design, expressed as bilinear matrix inequalities. An LPV filter, which minimizes an upper bound to the performance of the estimation error, is obtained as the solution of an optimization problem. A convex model to represent the parameters and their variations as a polytope is proposed in order to provide less conservative design conditions. Robust filters for time-varying polytopic systems can be obtained as a particular case of the proposed method. Numerical examples illustrate the results.     

Time-domain design of FIR multirate filter banks

A new time-domain methodology for designing FIR multirate filter banks is proposed. The conditions for perfect reconstruction systems can only be met by a limited number of systems, and consequently one of the major problems is to design analysis and synthesis filters which reduce the reconstruction error to a minimum. A recursive technique is proposed which uses the synthesis filters from one iteration to update the analysis filters for the next. The Letter shows that this is computationally simpler than previously proposed time-domain methods and produces filter banks in which the reconstruction error is reduced to practically acceptable levels.<>     

Biorthogonal Recombination Nonuniform Cosine-Modulated Filter Banks and their Multiplier-Less Realizations

This paper studies the theory, design and multiplier-less (ML) realization of a class of perfect reconstruction (PR) low-delay biorthogonal nonuniform cosine-modulated filter banks (CMFBs). It is based on a recombination (or merging) structure previously proposed by the authors. By relaxing the original CMFB and the recombination transmultiplexer (TMUX) in the recombination structure to be biorthogonal, nonuniform CMFBs with lower system delay can be obtained. This also increases the possible choices of the prototype filters to meet different design objectives. A matching condition is introduced to suppress the spurious response resulting from the mismatch in the transition bands of the two biorthogonal CMFBs. A complete factorization of biorthgonal CMFB using the lifting scheme is employed to obtain structurally PR biorthogonal nonuniform filter banks (FBs), which are robust to coefficient quantization. In addition, by approximating the lifting coefficients and the modulation matrices by the sum of powers-of-two (SOPOT) coefficients, ML realization with very low implementation complexity is obtained. Design examples and comparison are given to illustrate the effectiveness of the proposed method. S. C. Chan received his B.Sc. (Eng) and Ph.D. degrees in electrical engineering from the University of Hong Kong, Hong Kong, in 1986 and 1992, respectively. He joined City Polytechnic of Hong Kong in 1990 as an assistant Lecturer and later as a University Lecturer. Since 1994, he has been with the department of electrical and electronic engineering, the University of Hong Kong, Hong Kong, and is now an associate Professor. He was a visiting researcher in Microsoft Corporation, Redmond, USA and Microsoft China at 1998 and 1999, respectively. Dr. Chan is currently a member of the Digital Signal Processing Technical Committee of the IEEE Circuits and Systems Society. He was Chairman of the IEEE Hong Kong Chapter of Signal Processing from 2000 to 2002. His research interests include fast transform algorithms, filter design and realization, multirate signal processing, communications signal processing, and image-based rendering. X. M. Xie received the M.S. degree in electronic engineering from Xidian University in 1996, and the Ph.D degree in electrical & electronic engineering from the University of Hong Kong in 2004. She is now with the school of electronic engineering, Xidian University. Her research interests are in digital signal processing, multirate filter bank and wavelet transform.     

Reconstruction of Nonuniformly Sampled Bandlimited Signals Using a Differentiator–Multiplier Cascade

This paper considers the problem of reconstructing a bandlimited signal from its nonuniform samples. Based on a discrete-time equivalent model for nonuniform sampling, we propose the differentiator–multiplier cascade, a multistage reconstruction system that recovers the uniform samples from the nonuniform samples. Rather than using optimally designed reconstruction filters, the system improves the reconstruction performance by cascading stages of linear-phase finite impulse response (FIR) filters and time-varying multipliers. Because the FIR filters are designed as differentiators, the system works for the general nonuniform sampling case and is not limited to periodic nonuniform sampling. To evaluate the reconstruction performance for a sinusoidal input signal, we derive the signal-to-noise-ratio at the output of each stage for the two-periodic and the general nonuniform sampling case. The main advantage of the system is that once the differentiators have been designed, they are implemented with fixed multipliers, and only some general multipliers have to be adapted when the sampling pattern changes; this reduces implementation costs substantially, especially in an application like time-interleaved analog-to-digital converters (TI-ADCs) where the timing mismatches among the ADCs may change during operation.      

Optimal filter banks for signal reconstruction from noisy subbandcomponents

Conventional design techniques for analysis and synthesis filters in subband processing applications guarantee perfect reconstruction of the original signal from its subband components. The resulting filters, however, lose their optimality when additive noise due, for example, to signal quantization, disturbs the subband sequences. We propose filter design techniques that minimize the reconstruction mean squared error (MSE) taking into account the second order statistics of signals and noise in the case of either stochastic or deterministic signals. A novel recursive, pseudo-adaptive algorithm is proposed for efficient design of these filters. Analysis and derivations are extended to 2-D signals and filters using powerful Kronecker product notation. A prototype application of the proposed ideas in subband coding is presented. Simulations illustrate the superior performance of the proposed filter banks versus conventional perfect reconstruction filters in the presence of additive subband noise     

Sampling theorems for uniform and periodic nonuniform MIMO sampling of multiband signals

We examine a multiple-input multiple-output (MIMO) sampling scheme for a linear time-invariant continuous-time MIMO channel. The input signals are modeled as multiband signals with different spectral supports, and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets, with possibly different but commensurate intervals on the different outputs. This scheme encompasses Papoulis' generalized sampling and several nonuniform sampling schemes as special cases. We derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or, equivalently, perfect inversion of the channel. From an implementation viewpoint, we note that it is desirable that the reconstruction filters have continuous frequency responses. We derive necessary and sufficient conditions that guarantee this continuity property. The frequency responses of the reconstruction filters are specified as solutions to a system of linear equations. Finally, we demonstrate that perfect reconstruction may be possible, even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone. In certain instances, this average rate can achieve the recently presented fundamental bounds on MIMO sampling density.