Separate/joint optimization of error feedback and realization for roundoff noise minimization in a class of 2-D state-space digital filters

Techniques for the separate/joint optimization of error-feedback and realization are developed to minimize the roundoff noise subject to l ₂-norm dynamic-range scaling constraints for a class of 2-D state-space digital filters. In the joint optimization, the problem at hand is converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained problem obtained is then solved by applying an efficient quasi-Newton algorithm. A numerical example is presented to illustrate the utility of the proposed techniques.     

An Efficient and Robust Orthogonal Structure for Linear Discrete-Time Systems

An orthogonal structure is derived for system implementation. The expression of roundoff noise propagation gain for this structure is obtained. The proposed structure is an improved version of that reported in the paper by G. Li, M. Gevers, and Y.X. Sun (Performance analysis of a new structure for digital filter implementation, IEEE Trans. Circuits Syst. I 47:474–482, 2000), but is more efficient and robust against the quantization errors. For comparison, an alternative expression of roundoff noise gain for the normalized lattice structure is derived, based on which it is shown that the roundoff noise gain of an Nth order all-pass system, when implemented using the normalized lattice structure, is 4N and that an Nth order system implemented with the structure proposed in the paper by P.P. Vaidyanathan, S.K. Mitra, and Y. Neuvo (A new approach to the realization of low sensitivity IIR digital filters, IEEE Trans. Acoust. Speech Signal Process. ASSP-34(2):350–361, 1986), yields a roundoff noise gain of N+1, smaller than that of the classical optimal roundoff noise state-space realizations. Design examples are presented to illustrate the behavior of the proposed structure and to compare it with a class of existing orthogonal structures and the classical optimal roundoff noise realizations. It is shown that the proposed structure outperforms the others in terms of minimization of roundoff noise as well as implementation efficiency.     

Separate/joint optimization of error feedback and coordinate transformation for roundoff noise minimization in two-dimensional state-space digital filters

This paper is concerned with the minimization of roundoff noise subject to l/sub 2/-norm dynamic-range scaling constraints in two-dimensional (2-D) state-space digital filters. Two methods are proposed, with the first one using error feedback alone and the second one using joint error feedback and coordinate transformation optimization. In the first method, several techniques for the determination of optimal full-scale, block-diagonal, diagonal, and scalar error-feedback matrices for a given 2-D state-space digital filter are proposed. In the second method, an iterative approach for minimizing the roundoff noise under l/sub 2/-norm dynamic-range scaling constraints is developed by jointly optimizing a scalar error-feedback matrix and a coordinate transformation matrix, which may be regarded as an alternative approach to the conventional method for synthesizing the optimal 2-D filter structure with minimum roundoff noise. An analytical method for the joint optimization of a general error-feedback matrix and a coordinate transformation matrix under the scaling constraints is also proposed. A numerical example is presented to illustrate the utility of the proposed techniques.     

一种基于结构优化的FIR滤波器舍入噪声性能改进方案

针对滤波器组系统硬件实现时原型滤波器的有限字长效应问题,该文研究如何改善FIR原型滤波器由信号量化引起的舍入噪声,即降低舍入噪声增益,提出一种FIR滤波器优化结构。通过分析舍入噪声来源,利用多项式参数化方法对舍入噪声增益表达式进行推导。仿真实例证明,在不同字长约束条件下所提结构滤波器的幅频相频响应与理想状态基本吻合;通过与现有算法对比,所提结构具有较小的舍入噪声增益。

     

l 2–l ∞ Filtering for Multirate Systems Based on Lifted Models

For a multirate sample-data system where the output sampling rate is slower than the input updating rate, we study the l ₂–l _∞ filtering problems for fast state estimation by using the lifted model. The filtering problem is handled in the framework of linear matrix inequalities (LMIs) with a nonconvex constraint, which is numerically solved by the product reduction algorithm. Finally, the effectiveness of the proposed method is illustrated and verified by simulation examples. This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Canada Foundation for Innovation and the National Natural Science Foundation of China (60372105 and 60672118).     

Low roundoff noise augmented realizations of IIR filters

The scaling factor in a direct form II infinite impulse response (IIR) digital filter depends on only the system poles, not its zeros. Adding pole-zero cancellation pairs changes the scaling factor and the associated roundoff noise without affecting the transfer function. It is shown that with l₂ scaling, the upper bound of the noise reduction for direct form II IIR digital filters using K pole-zero cancellation pairs is about 6K dB. The authors consider parallel form 3P to be the basic structure, in which each subfilter is augmented with one or two pole-zero cancellation pairs and realized in direct form II. They seek an iterative optimization process for each subfilter to find the optimal pole-zero cancellation pairs that minimize the roundoff noise. Experimental results indicate the global optimal solution for these augmented subfilters can be obtained by using simple criteria to determine the starting points of the iterative processes. Alternative structures, illustrative examples, and overflow limit cycles are examined     

An Improved Lattice Filter Structure with Minimum Roundoff Noise Gain

In this paper, a new filter structure is derived by combining the lattice structure and error feedback technique, which contains a free parameter used for structure optimization. The roundoff noise of the proposed structure is analyzed, and the corresponding mathematical expression is deduced. The problem of how to optimize the free parameter is investigated in terms of minimizing the roundoff noise gain. When the free parameter is expressed as an integer power-of-two, for an Nth-order digital filter, the proposed structure requires \(5N+1\) multipliers, which yields the same implementation complexity as the normalized lattice structure. A numerical example is given to show the finite wordlength performance of the proposed structure.     

The digital all-pass filter: a versatile signal processing buildingblock

The properties of digital all-pass filters are reviewed and a broad overview of the diversity of applications in digital filtering is provided. Starting with the definition and basic properties of a scalar all-pass function, a variety of structures satisfying the all-pass property are assembled, with emphasis placed on the concept of structural losslessness. Applications are then outlined in notch filtering, complementary filtering and filter banks, multirate filtering, spectrum and group-delay equalization, and Hilbert transformations. In all cases, the structural losslessness property induces very robust performance in the face of multiplier coefficient quantization. Finally, the state-space manifestations of the all-pass property are explored, and it is shown that many all-pass filter structures are devoid of limit cycle behavior and feature very low roundoff noise gain     

采用集成滤波器的低噪声D类放大器

This paper investigates the noise sources in a single-ended class D amplifier (SECDA) and suggests corresponding ways to lower the noise. The total output noise could be expressed as a function of the gain and noises from different sources. According to the function, the bias voltage (V_B) is a primary noise source, especially for a SECDA with a large gain. A low noise SECDA is obtained by integrating a filter into the SECDA to lower the noise of the V_B. The filter utilizes an active resister and an 80 pF capacitance to get a 3 Hz pole. A noise test and fast Fourier transform analysis show that the noise performance of this SECDA is the same as that of a SECDA with an external filter.     

Multirate filter banks with block sampling

Multirate filter banks with block sampling were recently studied by Khansari and Leon-Garcia (1993). In this paper, we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices, respectively. We show that multirate filter banks with block sampling studied by Khansari and Leon-Garcia are special vector filter banks where the transfer matrices are pseudocirculant. We present some fundamental properties for the basic building blocks, such as Noble identities, interchangeability of down/up sampling, polyphase representations of M-channel vector filter banks, and multirate filter banks with block sampling. We then present necessary and sufficient conditions for the alias-free property, finite impulse response (FIR) systems with FIR inverses, paraunitariness, and lattice structures for paraunitary vector filter banks. We also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling. As an application of this theory, we present all possible perfect reconstruction delay chain systems with block sampling. We also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling. In this paper, we also present a connection between vector filter banks and vector transforms studied by Li. Vector filter banks also play important roles in multiwavelet transforms and vector subband coding     

A design algorithm for optimal low-pass nonlinear phase FIR digital filters

The transfer function of the low-pass nonlinear phase finite impulse response (NLPFIR) digital filter is decomposed into a nonlinear phase part and a linear phase part. An algorithm is proposed to iteratively design the magnitude of the linear phase part and the squared magnitude of the nonlinear phase part by directly calling the Remez algorithm of McClellan, et al. [1]. In the design of the nonlinear phase part, we assume that the linearity constraint on the phase is dropped but the phase response is not specified. A scheme is incorporated into our algorithm so that it can design the filter with the desired ripple ratio. This approach also leads to a method for finding the minimum ripple ratio for the given orders of the two parts and band edges of the filters. The filters with ripple ratio larger than this minimum value can be designed by our algorithm and neither passband nor stopband ripples are required to be prescribed. Analysis of roundoff noise reveals that the cascade filter implementation usually needs higher wordlengths than its direct for counterpart for the same roundoff noise performance.     

Optimum FIR and IIR multistage multirate filter design

A theorem is introduced which is useful in deriving equivalent multirate filter structures. Frequency responses of multistage multirate filters are derived and defined by deriving their equivalent one-stage filters. A design principle is proposed to reduce filtering requirements at each stage and move the filter operations to low-sampling-rate stages and thus result in a lower arithmetic rate. Optimum FIR and IIR multistage multirate filter designs are developed based on this principle. The new design has a one-point passband specification for each decimator and/or interpolator stage resulting in a wider transition region and lower filter order. Examples are given to explain the design procedure, and comparisons are made to show the superiority of the new filters.     

Simulation of an on-board hierarchical multistage digital FDM demultiplexer for mobile scpc satellite communications

This paper is focused on the finite-precision time-domain simulation of the hierarchical multistage method detailed in References 1 and 2 for demultiplexing of an FDM signal being composed of L = 32 slot signals. This approach to FDM demultiplexing is based on the processing of complex signals by linear-phase FIR filters, where at any stage of processing the respective signals are always oversampled by a factor of two. The simulation results fully confirm the system performance predicted in Reference 1 by modelling the distortions (spectral foldover and quantization noise) inherent in the system: assuming an ideal FDM signal at the input port of the analogue anti-aliasing bandpass filter in front of the analogue-to-digital converter, a minimum signal-to-distortion ratio (S/D) of 30 dB is achieved at all L output ports of the demultiplexer with {w, w_F, w_i} = {11,12,14}, as anticipated in Reference 1. The signal word length w apply to the A/D conversion, w_F to requantization between the filter stages, and w_i to filter internal signal processing.     

Quantization and roundoff noises in fixed-point FIR digital filters

The errors in a fixed-point finite impulse response (FIR) filter due to quantization (analog-to-digital conversion) and roundoff are considered. Expressions for the exact moments of the filter output noise are derived. It is well known that, when the input signal satisfies certain conditions, the popular additive white noise model can be valid in describing the quantization noise. The characteristics of multiplicative roundoff noises, however, differ from what this model predicts, even under conditions where the roundoff noises are white. Hence the additive white noise model does not provide accurate results on the characteristics of the output error in an FIR filter. Using the exact formulas for the moments, the author computes the exact power spectrum of the filter output error. These results agree well with those obtained from simulation     

分数阶傅里叶域两通道滤波器组

 基于两通道滤波器组构建的子带信号处理方法已在图像、语音信号处理中得到广泛的应用.本文从分数阶傅里叶域多抽样率信号处理基本理论和分数阶卷积定理出发,推导了分数阶傅里叶域两通道滤波器组准确重建的基本条件,并基于传统傅里叶域有限长标准正交镜像滤波器组和共轭正交镜像滤波器组的原型滤波器设计了分数阶傅里叶域标准正交镜像滤波器组和共轭正交镜像滤波器组.本文所提出的结论为分数阶傅里叶域滤波器组理论的建立提供了基本依据,同时也为分数阶傅里叶变换在图像、语音信号处理等工程实践中的应用奠定了理论基础.最后,仿真实验验证了所提分数阶傅里叶域滤波器设计方法的有效性.     

A CMOS 1.0-μm two-dimensional analog multirate system forreal-time image processing

Two-dimensional (2-D) filters for video signal processing typically operate at high uniform sampling rates and require very large delay-line (DL) memory blocks. By employing 2-D multirate signal processing techniques to reduce the sampling rate, not only the DL memory blocks can be downsized to save silicon area, but also the memory access time can be increased to save power as well. This is demonstrated in this paper considering a 2-D switched-capacitor multirate image processor that realizes (2×2)nd-order recursive low-pass and high-pass filtering functions employing half of the storage cells that would be needed in a nonmultirate system. Only one type of operational transconductance amplifier with 1-mS transconductance and 120-MHz unity gain bandwidth is needed for both the vertical filter and associated DL memory blocks and the horizontal decimating filter. Fully differential circuit techniques are employed to increase immunity to charge feedthrough injection in the analog storage cells. The complete system has been implemented in a CMOS 1.0-μm double-poly technology. The core active area is only 2.5×3.0 mm², and at 5-V supply and 18-MHz sampling it dissipates 85 mW     

An improved orthogonal digital filter structure

A novel structure is derived for digital filter implementation. This structure is actually an improved version of an existing one in terms of implementation efficiency and reducing finite word length (FWL) effects. Expression of roundoff noise gain is obtained for the proposed structure. Design examples are given to demonstrate the performance of this structure and to compare it with the existing one and the classical minimum roundoff state-space realizations. Numerical examples show that the proposed structure outperforms the others in terms of minimizing roundoff noise as well as implementation efficiency.     

Reconstruction of band-limited periodic nonuniformly sampled signals through multirate filter banks

A band-limited signal can be recovered from its periodic nonuniformly spaced samples provided the average sampling rate is at least the Nyquist rate. A multirate filter bank structure is used to both model this nonuniform sampling (through the analysis bank) and reconstruct a uniformly sampled sequence (through the synthesis bank). Several techniques for modeling the nonuniform sampling are presented for various cases of sampling. Conditions on the filter bank structure are used to accurately reconstruct uniform samples of the input signal at the Nyquist rate. Several examples and simulation results are presented, with emphasis on forms of nonuniform sampling that may be useful in mixed-signal integrated circuits.     

An improved /spl rho/DFIIt structure for digital filters with minimum roundoff noise

In this brief, a new efficient digital filter structure with minimum roundoff noise is derived. This structure can be block-diagrammed as the recently proposed direct-form II transposed (DFIIt) structure in /spl rho/-operator, denoted as /spl rho/DFIIt, in which the first-order /spl rho/ operators are replaced with a set of second-order operators. The corresponding expression for the roundoff noise gain is obtained. The problem of how to choose these second-order operators is investigated in terms of minimizing the roundoff noise gain. Two design examples are given to illustrate the excellent performance of the proposed structure and to confirm the theoretical analysis.     

Roundoff noise and dynamic range of wave digital filters

Roundoff noise and dynamic range of wave digital filters have not been discussed in detail hitherto. In this paper the calculation of the roundoff noise is performed in a manner which is natural and quite simple for wave digital filters, i.e. the great amount of computation in case of such highly recursive systems can be considerably reduced. Scaling is done in order to improve the dynamic range and to make the comparison with other filter structures possible. The resulting signal-to-noise ratio is determined for several filter examples.