State-space Filters Research Articles

Floating point arithmetic and digital filters

The roundoff noise properties of floating point digital filters are examined. To make the analysis tractable, a high level model to deal with the errors in the inner product operation is developed. This model establishes a broad connection between coefficient sensitivity and roundoff noise. Along with the model, an efficient procedure to keep track of the addition scheme used in the inner product, and to compute the statistics of the errors is introduced. A systematic procedure based on the model is then developed to derive general expressions for the roundoff noise of FIR, direct form IIR, and state-space filters. The expressions in the context of state space filters are explored in some detail. Optimality issues are considered, and it is shown that when double precision accumulation is used, the optimal filters are similar in nature to those derived in the context of fixed point arithmetic with the essential difference that they also do depend on the spectrum of the input signal. Optimality with respect to addition schemes, and second-order filters are also examined in some detail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Synthesis of 2-D state-space filter structures with low frequency-weighted sensitivity

Based on the Roesser (1975) local state-space (LSS) model, new measures are introduced to describe over a particular frequency region the sensitivities of a two-dimensional (2-D) transfer function with respect to state-space parameters. The overall sensitivity derived from these measures is called the frequency weighted sensitivity and is evaluated using 2-D generalized controllability and observability Gramians that are newly defined for 2-D state-space digital filters. Then, 2-D filter structures that minimize the frequency weighted sensitivity are synthesized for two cases of no constraint and scaling constraints on the state variables. These 2-D filter structures can also be optimized with respect to the scaling parameters in the latter case. An example is given to illustrate the utility of the proposed technique.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

16-bit microprocessor in second-order state-space digital filter design and implementation

An efficient hardware and software designs for the implementation of an infinite impulse response (IIR) digital filter using the most advanced semiconductor technology and a single card microprocessor are introduced. Noise introduced to the digital filter due to quantization processes shall be considered. Also, a new technique to replace ordenary digital multiplication will be introduced. The new technique is based on the properties of the logarithm function. The required values for this technique are evaluated off-line and stored in a look-up-table. The filter implemented in this paper is a second-order state-space filter derived from the Butterworth filter using the optimization technique of Jackson et al. The experimental results obtained from the optimal state-space filter with the use of the logarithm technique have proved to be very successful and promising.

High speed architectures for two-dimensional state-space recursive filtering

A description is given of 2-D state-space filter algorithms, and two VLSI architectures for high-speed applications are proposed. The advanced state update architecture trades increased throughput rate and a simple input-output scheme for increased hardware. The technique is based on block processing. The state update matrix is changed by recursive multiplications, and the advanced state vector can be calculated. This architecture is a two-dimensional systolic array. It can process one pixel during each multiplication/addition clock period with a regular line-by-line scanning method. To achieve very high speed, the global speedup architecture is utilized by exploiting the inherent spatial concurrency. This architecture consists of a number of column array processors. Based on the computational independence between the samples along the diagonal lines in a first-quadrant 2-D system, it works on multiple columns of images concurrently. The architecture is very modular and flexible. The throughput rate can be very high and is adjustable by changing the number of column processors. Different high-speed architectures for image processing are compared. >

Adaptive recursive state-space filters using a gradient-based algorithm

To aid in the search for better adaptive filter structures, a method is presented to obtain the gradients required to adapt general state-space filters. Unfortunately, the number of computations for this general case is quite high. To reduce the number of computations, two new state-space adaptive filters are introduced. One application where these new structures are shown to be useful is in oversampled filtering where an estimate of the final pole locations is known and the adaptive filter is required only to fine-tune the transfer function. It is shown that for this type of application, the new adaptive structures can have much improved adaptation rates and roundoff noise performance as compared to the corresponding direct-form realizations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Filtered and Predicted States for Discrete-Time Adaptive Control

The paper considers the adaptive control of discrete-time, noise corrupted, systems whose characteristics can be represented by means of an ARMA model, this being posed within a canonical state-space framework. The effects of using different state vector forms, on self-tuning controller performance, are investigated with regard to filtered and predicted state representations when applied to various system types. It is shown how only the optimal observer for the state-space filter model is truly optimal in the sense of being a constituent part of an optimal controller.The important aspect of these results is that the state-space prediction model, also known as the innovations model, is that which is widely used in practice, despite the fact that overall optimality conditions cannot be obtained with a state-feedback controller based on this state-space model. The suggestion is made that the state space filter model, also known as the noise-free measurement model, is a much more viable alternative.

Realization of 2-D state-space filters with fewer multipliers

The authors illustrate that, under certain controllability and observability conditions on the one-dimensional block diagonal subsystems, they can considerably reduce the number of multipliers required for hardware implementation of two-dimensional filters. Although these conditions require some additional properties from the filter parameters, the saving in cost of hardware realization is judged to be well worth the extra effort to verify initially whether these conditions are met. A systematic procedure for obtaining the coefficients of a minimal number of multipliers, when the filter satisfies the proposed conditions, is presented. A numerical example is considered to illustrate the accuracy of the proposed technique. >

Design of orthogonal biquad digital filters

Complex digital filter techniques are briefly discussed. Three designs of a special class of complex filters called orthogonal filters are presented: the orthogonal direct form, minimum norm, and optimal state-space filters. The direct-form structure has previously been used in the design of transmultiplexers, whereas the minimum-norm and optimal state-space structures are new. The noise and stability performance of each structure are discussed and compared.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A stability analysis of two-dimensional nonlinear digital state-space filters

Two different methods for a stability analysis of 2-D nonlinear discrete state-space systems under zero input conditions are provided. The first method reduces the task of testing a 2-D discrete nonlinear or shift-varying system to a single 2-D linear stability test of a system matrix with nonnegative system matrix entries. The second method is based on a number of norm tests for products of extreme matrices, and can be considered the 2-D counterpart of the method for 1-D systems described by K.T. Erickson and A.N. Michel (1985). Both of the introduced methods are based on the sector description of the nonlinearity and can be used to analyze digital filter stability under finite-word-length effects. >

Pipeline interleaving and parallelism in recursive digital filters. II. Pipelined incremental block filtering

For pt.I see ibid., vol.37, no.7, p.1099 (1989). Block implementation and fine-grain pipelined block implementation of recursive digital filters are discussed. A new technique of incremental output computation is introduced which requires a linear complexity in block size. Based on the clustered look-ahead and incremental output computation approaches, incremental block-state structure is derived for block implementation of state-space filters of multiplication complexity linear in block size. The incremental block-state structure is also extended for the multirate recursive filtering case. The techniques of scattered look-ahead, clustered look-ahead, decomposition, and incremental output computation are combined to introduce several pipeline stages inside the recursive loop of the block filter. Deeply pipelined block filter structures are derived for implementation of direct-form and state-space-form recursive digital filters. The multiplication complexity of these pipelined block filters is linear with respect to the block size and logarithmic with respect to the number of loop pipeline stages, and the complexities due to pipelining and block processing are additive.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

State-space approximation of multi-input multi-output systems with stochastic exogenous inputs

Instead of the usual AR(MA)X- or VAR (vector autoregressive) modelling, procedures will be described to obtain approximate balanced state-space models and steady-state Kalman filters with prewhitened inputs. These state-space models and Kalman filters can be used for prediction and also for control whenever the output and input variables are target and control variables respectively.

Relationship between input-scaling and stability of the forced response of recursive digital filters

The stability of forced response of a general recursive digital filter operating in the inherent presence of overflow nonlinearities is examined. A theorem is derived that relates forced-response stability to the degree of input-scaling. The technique is applied to the investigation of the stability properties of state-space digital filters, resulting in a criterion for the forced-input stability of Nth-order state-space filters. Subsequently, it is applied to the second-order case. The resultant stability criterion is applied to the investigation of optimal state-space and normal filters. Finally, the stability properties of wave digital filters are discussed. >

More economical state-space digital-filter structures which are free of constant-input limit cycles

A theorem that establishes conditions for the elimination of constant-input limit cycles is used as the basis for a procedure for the synthesis of state-space structures. Three different second-order structures are obtained, and in each, granularity and overflow limit cycles can be eliminated under zero- and constant-input conditions. Detailed procedures are then presented for the application of one of the proposed structures to the design of parallel and cascade filters. The paper concludes with extensive comparisons which show that more economical low-noise and low-sensitivity state-space filter implementations are possible through the use of one of the proposed structures.

Strays-insensitive state-space switched-capacitor filters

Strays-insensitive realizations for switched-capacitor (SC) filters are presented. Here, the discrete-time transfer function is realized in the biquadratic state-space configuration implementing strays-insensitive SC biquads. Design equations that directly relate the feedback and feedforward coefficients to that of the transfer function are de'ived. Sensitivity analysis of the proposed structure is carried out and it is shown that the transfer function sensitivities with respect to the capacitance ratios are minimized if the capacitance ratios assume their lowest possible values. A design example is presented and its sensitivity performance is carried out by Monte Carlo simulations.

Quantization noise analysis of a general digital filter

A new, precise modeling technique is presented for the exact representation of digital filters, discrete systems, or computational algorithms. It leads to a method for the determination of the noise variance of the filter output due to quantization errors occurring at multipliers, summers, and the A/D converter. The noise formulae are in either infinite series or closed form and through their use a precise evaluation of an arbitrary filter structure may be obtained. The formulas can be easily computer implemented and require no filter analysis by the designer. The special case of state-space filters is examined in detail. Appropriate numerical examples are included.

Low-noise state-space realisation of bandpass active filter

An active 2nd-order bandpass state-space filter realisation known as the symmetric-form filter has been studied theoretically and practically and has been found to have desirable qualities, particularly those of low noise and good linearity. The noise and linearity performance of this realisation has been compared with that of the companion-form filter (feedforward filter).5.7 The dynamic range of the symmetric form, expressed as a power ratio, is shown to be 1000 times greater than that of the companion form having the same transfer function and using the same type of operational amplifier.

Generation of low-sensitivity state-space active filters

This paper investigates the use of the similarity tranformation in the generation of high-order state-space active filters having minimized sensitivities. A sensitivity performance measure similar to the Rosenblum-Ghausi multiparameter sensitivity figure is defined, and a frequency independent relationship between the sensitivity measures of equivalent systems is derived. The results of optimizing an all-pole bandpass filter and a low-pass elliptic filter are presented, along with a detailed analysis of some of the attributes of the optimized realizations. The paper then examines the possibility of simplifying these optimized systems while maintaining good sensitivity behavior, by slightly altering the systems about their optimized values.