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A variable delay is required for beam steering in an adaptive antenna array. This chapter presents an efficient design for approximating such a delay for broadband applications by using only one phase shifter and real coefficients. Exact mathematical formulas for determining the required weighting coefficients have also been derived.

A general, nth order, transfer function (TF) is derived, whose time-domain response approximates optimally that of an ideal differentiator, optimally criterion chosen being the maximisation of the first n derivatives of the ramp response at t-0+. It is shown that transformerless, passive, unbalanced realisability is ensured for n 3, the TF is unstable. For n=3, the TF is not realisable, however, near optimum results can be obtained by perturbation of the pole locations.Optimum TFs are also derived for the additional constraint of inductorless realisability. It is shown that TFs for n⩾2 are not realisable. For all n, however, near optimum results can be achieved by small perturbations of the pole locations; this is illustrated in this paper for n=2.Network realisations, for a variety of cases, are also given.

Implementation of discrete sine transform (DST) is proposed, by folding the input data. This results in recursive structure, besides saving in the overall computation cycles. A simplified architecture to derive DST kernel has also been suggested.

In this paper, designs of linear-phase finite impulse response (FIR) filters approximating the ideal frequency responseH (ω,r) Δ|1/ω r r=1, 2, 3,..., by using maximally flat error criteria have been proposed. Exact weights required for the realizations have been derived and are shown to be independent of the filter orderN. The suggested designs are particularly suitable for operation in the midband frequency range of 0.10π–0.90π radians.

An efficient design of linear phase, FIR structure for amplitude response, approximating [1/ω] (an integrating Hilbert transformer), suitable for the midband frequency range has been proposed. Mathematical formulas for computation of weights have been derived. These weights turn out to be invariant to the order of the filter, and are, therefore, universal in nature. This property is expoited to harness the proposed design with facility of variable orders of the sturcture maintaining the optimality criteria for all possible orders. A versatile structure for such a purpose has also been given.

In many signal processing situations, the desired (ideal) magnitude response of the filter is a rational function: $$\tilde H(\omega ) = |1/\omega |$$ (a digital integrator). The requirements of a linear phase response and guaranteed stable performance limit the design to a finite impulse response (FIR) structure. In many applications we require the FIR filter to yield a highly accurate magnitude response for a narrow band of frequencies with maximal flatness at an arbitrary frequency? 0 in the spectrum (0, ?). No techniques for meeting such requirements with respect to approximation of $$\tilde H(\omega )$$ are known in the literature. This paper suggests a design by which the linear phase magnitude response $$|\tilde H(\omega )|$$ can be approximated by an FIR configuration giving a maximally flat (in the Butterworth sense) response at an arbitrary frequency ?0, 00

In this paper a novel analytical technique for designing linear-phase, FIR integrators of first degree has been proposed by taking the optimality criteria of maximal flatness (in the Butterworth sense) of the amplitude response, at midband frequency:ω=π/2. Mathematical formulas to compute exact weights for the design of first-degree integrators (i.e., frequency response 1/(jω)) have been derived. Thereafter, the design has been extended to approximate linear-phase integrators for any generic degreer, r>1. The performance characteristics of the first-, second- and third-degree integrators have been given to highlight the efficacy of the designs. The suggested designs are particularly suitable for the midband frequency range of operations.

In this paper, Bernstein polynomials have been used to derive an explicit formula for the coefficients of linear phase FIR notch filters which are maximally flat at ω=0 and π. The approach is relatively simple and enables us to design the filter for a specific notch frequency and bandwidth. © 1997 by John Wiley & Sons, Ltd.

Interrelationships between seven types of digital FIR filters, viz. the differentiators (DD), Hilbert transformers, half-band lowpass/highpass filters, bandpass/bandstop filters and frequency discriminators, have been established. It has been shown that all the aforementioned filters can be derived from the design of a DD. If we use the optimality criteria of maximal linearity of the frequency response of a differentiator, the proposed relations yield maximally flat/linear frequency responses for the remaining six filters. Similarly, an equiripple relative error design of a DD gives equiripple filters through the use of the suggested interrelations. A versatile, multipurpose, FIR configuration has also been proposed which yeilds optimal frequency response for all the seven types of digital filters. The suggested design is shown to be particularly useful for operation over the frequency ranges 0 ⪕ ω ⪕ 0.50π (0.5π ⪕ ω ⪕ π) for lowpass (highpass) filters and 0.25π ⪕ ω ⪕ 0.75π for the remaining ones. Mathematical formulas for computing the weights of aforementioned filters have also been derived. Possible uses of the suggested design have been given.

We propose in this paper an efficient design of FIR digital differentiators of degree k, k= 1, 2, 3,…, and their implementation in a novel structure which allows the order to be varied without having to alter the weights used in it. More precisely, we show that by using these ‘universal weights’ in a structure of order Nmax, the order can be varied from N= 3 to Nmax while keeping the weights fixed and retaining the optimality of performance. Mathematical formulae have been derived for these ‘universal weights’ for differentiators performing in the low- or midband-frequency ranges.

Digital differentiators of orders greater than unity are often required in processing various types of data in a number of practical applications. In many of these, the vital information in the signal is contained in the low frequency range. We propose, in this paper, a new and efficient design of FIR digital differentiators of second and higher orders for the low frequency range. Mathematical relations have been established between the weighting coefficients of the maximally linear (first order) FIR digital differentiators and those of the proposed (second and higher order) differentiators. Recursive as well as explicit formulas for the weighting coefficients of the second order differentiators have also been derived. Extremely high accuracies are obtainable from the proposed differentiators, for attractively low orders of the filters.

This paper proposes implementation of the Modified Discrete Sine Transform (MDST) and Inverse MDST (IMDST) using recursive structures. The formulae required for recursive structures have been derived. Discrete Hartley Transform (DHT), a real-valued transform, is closely related to Discrete Fourier Transform (DFT) of a real-valued sequence and hence its use as an alternative to the Fourier Transform avoids complex arithmetic. This paper presents Modified Discrete Hartley Transform (MDHT)/Inverse MDHT (IMDHT) using Modified Discrete Cosine Transform (MDCT)/ Inverse MDCT (IMDCT) and MDST/Inverse MDST (IMDST) recursive structures. The proposed structures are used for simultaneous computation of MDCT/MDST/MDHT of length N (divisible by four) and their Inverse (IMDST/IMDHT). The proposed structures are parallel, simple, regular and therefore highly suitable for VLSI implementation.

In this paper, novel very-large-scale integration (VLSI) algorithms for computing the Discrete Sine Transform (DST) and its inverse (IDST) are proposed. By respectively folding the inputs, the DST kernels are derived to realize it through Infinite-impulse response (IIR) structure. Single folding algorithm provides data throughput two times of that achieved by the conventional methods. The double folding recursive algorithms for DST and IDST have been suggested, which reduce the computation cycles to one forth of that in the single folding method. Moreover, the regular and modular properties of the proposed recursive structures are easily amenable to VLSI implementation.

Exact mathematical formulas, recursive as well as explicit, have been derived for computing a sequence x(n) from its moments. A scheme for reducing the number of multiplications has also been suggested.

In the paper, efficient designs for second and higher degree digital differentiators for midband frequency ranges are proposed. Exact mathematical formulas for computation of the required weighting coefficients are also derived. The designs are capable of achieving extremely low relative errors (±1%–±0.001%) in the frequency range 0.25π≤ω≤0.75π with attractively low-order filters.

Interrelationships between the digital differentiator (DD), the digital Hilbert transformer (DHT), and the half-band low-pass (1/2-LPF) are brought out. A number of important properties confirming the close proximity of these filters are highlighted. Theoretical results are substantiated by transforming minimax relative error DDs to equiripple DHTs and equiripple 1/2-LPFs. Relations connecting their impulse responses and their frequency responses are brought out. The precise frequencies of ripple extrema and the magnitudes of their peaks are shown to be simple related to the corresponding values for the minimax relative error differentiators. >

Une nouvelle classe de derivateurs et de transformateurs numeriques de Hilbert sont proposes. Une formule mathematique pour le calcul des coefficients ponderes de ces circuits est derivee

The frequency response of conventional FIR digital differentiators (DD), using only integral delays, is poor towards the high frequency end, and becomes identically zero at w= pi . To overcome this problem, a half-sample delay, z/sup -1/2/, is used in the minimax relative error DDs. However, the order of such DDs grows exponentially when accuracies better than 99.0% are desired. The authors propose maximally linear DDs, using one half-sample delay, which can achieve a frequency response accuracy better than 99.9%, with attractively low orders for 0.5 pi >