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Coefficient inaccuracy effects in the frequency response of finite-impulse-response (FIR) switched-capacitor filters implemented in direct form are investigated. No assumption is made with respect to the filter coefficients, so that the results of the analysis are valid for both linear and nonlinear phase FIR filters. Assuming that the errors in the capacitor ratios realizing the tap coefficients are uncorrelated and have identical Gaussian distributions, an exact probability distribution function is derived for the error in the frequency response. It is shown that this distribution can be characterized by a Rayleigh distribution, which is then used to derive an upper bound for the expected stopband attenuation. Extensive simulation results are shown to support the analysis. >

A digitally programmable switched-capacitor equalization filter is presented. The proposed structure uses only two nonoverlapping clock phases and is fully programmable through the use of three independent capacitor arrays. Effects of the finite gain of the operational amplifiers are investigated in detail, and simulation results are presented, verifying the models involved in the analysis. Also shown is a comparison of the performance of the proposed equalizer with that of other equalizers reported in the literature. Experimental results obtained from a prototype circuit built with discrete components are also included. >

A new realization of recursive digital filters based on a cascade of complex all-pass structures is proposed. The cascade realization ensures low stopband sensitivity, whereas the all-pass sections, being structurally bounded, guarantee low passband sensitivity. The proposed structure can exactly realize any elliptic transfer function, and it is easy to derive the multiplier coefficients associated with this structure from the original transfer function expression. To reduce the computational complexity, some of the coefficients are rounded to integer values and an optimum approximation to the original transfer function is obtained by using a minimax optimization procedure. Several examples show that the proposed structure requires significantly fewer bits than other known cascade-form realizations. Another advantage of the proposed structure is that it is free of overflow oscillations.

A model for discrete-time dynamical systems is discussed in which the future values of the internal variables depend on the present and K - 1 previous time instants, where K is the order of the model. We outline transition matrix and impulse response calculations, equivalences between model and conventional state variable models, methods of constructing such a model, and means for calculating certain quantities relevant to the design of cost-optimal low-roundoff noise digital filters.

Complex coefficient digital filters, with applications for processing real sequences, are examined. The method is quite general, and allows any real rational transfer function to be expressed in terms of a complex rational transfer function of reduced order. When implemented in complex hardware form, the reduction of filter order can provide an increase in computational efficiency and speed. Conventional filter structures, such as parallel, cascade, lattice, and state-space forms, are extended to the complex domain. Illustrative examples of complex coefficient filter synthesis are included, along with coefficient sensitivity comparisons between the complex coefficient filters and their real coefficient counterparts.

A digital lossless bounded-real (LBR) transfer function, more commonly known as a stable allpass function, maintains the LBR property under multiplier coefficient quantization when implemented by networks that are structurally LBR. A systematic method is outlined for generating all possible such realizations for second-order transfer functions containing a minimum number of multipliers and delays, allowing the designer to take advantage of the flexibility offered by a catalog of such realizations. Illustrative examples are included. >

The well-known Richards' Theorem of the continuous-time filter theory is reformulated in the digital domain in a convenient manner, leading to a simple derivation of cascaded lattice digital filter structures, realizing lossless bounded transfer functions. The theorem is also extended to the matrix case, leading to a derivation of m -input p -output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary p \times m digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable p \times m transfer matrices in the form of such cascaded lattices is also outlined. The derivation also places in evidence a means of testing the stability of an arbitrary p \times m transfer matrix of a discrete-time linear system.

Lattice structures are developed for the realization of m -input p -output discrete-time all-pass transfer matrices H(z) , given in the form of a right matrix-fraction description (MFD): H(z)= N(z)D^{-l}(z) . The procedure is based on the generation of a sequence of all pass matrices of successively decreasing order, by matrix LBR two-pair extraction. Two cases are distinguished: the first case is when none of the intermediate allpass matrices is degenerate. For this case, the resulting structures are in the form of a cascade of matrix two-pairs separated by vector delays, with each two-pair being a multi-input multi-output digital filter structure characterized by an orthogonal transfer matrix of dimension (m + p) \times ( m + p) . The structures are in general either completely controllable or completely observable, depending upon the location of the delay elements. The synthesis technique also leads to a procedure for obtaining the greatest common right divisor between the polynomial matrices involved in the MFD. The results are extended to the cascaded-lattice synthesis of arbitrary stable transfer matrices by an embedding process. The developments of this paper automatically place in evidence a procedure for testing the stability of a transfer matrix. A special case of the resulting structures when p = m =1 gives rise to the well-known Gray-Markel digital lattice structures, whereas another special case with p = 2 and m = 1 leads to certain recently reported orthogonal digital filters. The second case, where some of the intermediate allpass matrices are degenerate, is handled separately, leading to a modified form of cascaded-multivariable lattice structures.

Several new structures for the block implementation of HilR digital filters are proposed. The relation between the pole locations of the block structure to that of the original scalar transfer function is derived. A method to obtain the scalar transfer function from a given block structure is described.

A theory is developed for spectral transformations for two-dimensional digital filters. It is proved that these transformations take the form of stable two-dimensional all-pass functions and that the result of spectral transformation is stable if the original transfer function is stable. Emphasis is placed on formulating the transformations in a way which makes them easy to use for design, and it is shown that twodimensional spectral transformations offer a means of quickly obtaining new designs from previous ones. Included is a discussion of applications such as the design and realization of two-dimensional digital filters with tunable characteristics.

A procedure for checking the stability of 2-D digital filters is proposed. The key part of the test involves construction of related all-pass sections of reduced order, the stability of which is shown to be equivalent to the stability of the original function. This is a direct extension of a similar approach, developed recently (1987) by P.P. Vaidyanathan and S.K. Mitra, in detail for the 1-D case. The results are shown to be equivalent to some known stability testing methods in 2-D, but have the advantage of an additional insight gained from the circuit interpretation. Explicit formulas for checking the stability in terms of the filter coefficients are derived for some special cases, and numerical examples are included to clarify the application of the proposed approach. >

The all-digital N -path filter structure for the processing of discrete-time signals is introduced. The structure is made of N parallel paths having identical time-invariant digital filters. The filters are connected to a common input signal path through input modulators, and a common output signal path through output modulators. The modulators produce identical periodic sequences but are staggered in phase from filter to filter. A detailed time-domain and frequency-domain analysis of the N -path filter is provided. It is shown that for certain bandlimited modulating sequences, or by appropriately bandlimiting the input and output signals, the overall time-varying structure can be made to appear as a timeinvariant network. This approach leads to filters with adjustable characteristics, and it is shown that by time-sharing the time-invariant digital filter path N times, it is possible to design filters with complex characteristics quite economically. A number of useful modulating functions are described, and some possible applications are discussed.

The fundamental requirements for low-sensitivity analog filter structures are shown to be the bounded-real (BR) property of the transfer function and its implementation by a structure that "preserves" the BR property for incremental changes in the parameters of the network. Using these properties, a low-sensitivity realization of any stable BR analog transfer function is developed. The final structure is in the form of a singly constrained analog active RC "two-pair" implemented as a cascade of two-pairs where each two-pair is characterized by a lossless bounded-real (LBR) transfer matrix. The proposed theory is shown to include the class of low sensitivity wave active RC structures related to the doubly terminated lossless two-ports. The new theory has been verified by computer simulation.

Starting with a prescribed configuration realizing the denominator of a given transfer function, this paper presents a systematic procedure for obtaining the noise-optimal realization, where the basic configuration is kept invariant. This procedure, in turn, is based on the identification of the maximal independent node sets of the given configuration. Thus a major objective of this paper is to develop effective algorithms for determining the independent sets of nodes and maximal independent sets of nodes of any arbitrary digital filter configuration. The entire low-noise design method is illustrated by means of an example.

The authors provide an approximate analytical characterization for wideband low-pass filters using the frequency masking approach introduced by Y.C. Lim (ibid., vol.CAS-33, no.4, p.357-64, Apr. 1968). They extend this realization to the case of wideband bandpass filters. Approximate expressions of the optimal value of the interpolating factor L are derived that minimize the total arithmetic hardware used in the overall realization. Expressions for the number of adders and multipliers required are also derived. The proposed two-branch realization is computationally more efficient than that conventional direct-form realization, with a moderate increase in the number of delays. The noise performance of these wideband designs compares very favorably with those of equivalent direct-form realizations. >

The sectioning approach to the implementation of two-dimensional convolution operations is reviewed. An optimal sectioning algorithm is presented and its computational requirements (CPU and I/O) completely characterized. Procedures are then given to enable the optimal choice of sectioning parameters for an arbitrary computer system. This procedure is Illustrated for a large scientific compUter (CDC 7600), and the sectioning implementation compared with the traditional FFT approach. Computer programs are included for both the sectioning algorithm and the determination of optimal sectioning parameters.

It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions G(z) (for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H(z) (for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of G(z) and H(z) have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined.

A computationally efficient realization of a symmetrical bandpass FIR filter is derived. The realization is composed of two branches, each consisting of a cascade of two FIR sections. In each branch, the first FIR section has a sparse impulse response with only every L th sample being nonzero. The second section generates the remaining samples via interpolation. The method is applicable to both linear and nonlinear phase cases. Approximate expressions for the optimal value of L and the corresponding number of multipliers and adders required in the overall realization are derived. In narrow-band implementations, the number of multipliers and adders is approximately proportional to 1/\sqrt{{\Delta}F} , where {\Delta}F is the desired relative transition bandwidth. Typically, the required number of multipliers and adders is 1/2 to 1/4 th of those in the conventional linear or nonlinear phase direct-form implementations. The total number of delays is only slightly larger than that required in the conventional implementations. The structure allows simple tuning of the center frequency of the bandpass filter and has good finite wordlength properties.