A general and efficient method for scaling window sidelobe magnitude has been reported by Lim et al. Although the method always converges in practice, a rigorous proof of convergence is unavailable. In this paper, we introduce a new technique with guaranteed convergence for window sidelobe magnitude reduction. Without further modifications, the convergence speed of this new algorithm is quite the same as that of the previous one. Modifications aimed at speeding up convergence while maintaining the guaranteed convergence property are also presented.

A rectangular-window sidelobe magnitude reduction method by widening the main lobe width was proposed by Saramäki. A technique for trading off main lobe width against sidelobe magnitude for any arbitrary window was reported by Lim et al.; a fast convergence algorithm for its implementation was proposed by the same authors in another article, where the derivatives of the window function are expressed in Chebyshev polynomials which have high arithmetic complexity. All the coefficients of a rectangular-window are equal; this special property is exploited, in this paper, for deriving the window function’s derivatives without the use of Chebyshev polynomials resulting in a great reduction in the arithmetic complexity.

A technique for trading off the main lobe width against sidelobe magnitude for any arbitrary window was reported in Lim et al. and subsequently, a fast convergence method for its implementation was proposed by the same authors. These methods require the computation of derivatives involving the evaluation of trigonometric and hyperbolic functions. In this paper, we show that the derivatives can be computed without evaluating trigonometric and hyperbolic functions if the window function is a sum of exponentials such as a Fourier series.

Many types of windows have been designed in the past decades for various applications. Each window type has its own specific characteristics. In this letter, we present a general technique for trading off main lobe width against sidelobe magnitude for any arbitrary window while keeping the number of sidelobe peaks and their relative magnitudes unchanged although their exact locations and magnitudes are changed.

Windows such as Dolph-Chebyshev window and Kaiser window are adjustable, whereas windows such as Hamming window and Blackman window are traditionally not adjustable. In [1], a technique for trading off main lobe width against sidelobe magnitude for any arbitrary window, including the traditionally non-adjustable windows, was presented. However, the method in [1] requires a large number of iterations if the specification is very tight. Developed based on a new perspective on the window adjustment principle, a new method to achieve the same adjustment capability as in [1], but at a very much fewer iterations is presented in this letter. Our new method is particular useful if the specification is very tight.

The design of perfect-reconstruction cosine-modulated filter banks is revisited by applying new linearization techniques for the time-domain quadratic perfect reconstruction constraints. The new techniques are analyzed to explain why they can provide improved approximation accuracy hence improved performance with insignificant increase in complexity. A design example is presented for performance demonstration and comparison.

A very efficient technique for significantly reducing the number of multipliers and adders in implementing narrow transition band linear-phase finite-impulse response (FIR) digital filters is to use the frequency-response masking (FRM) approach. This paper studies the construction of the masking filter pair in the FRM technique using a common part for further reducing the arithmetic complexity. Extensive simulations are included showing that the use of the common part always considerably decreases the arithmetic complexity at the expense of a slight increase in the overall filter order.

This paper proposes an improved version of sub-band adaptive notch (SAN) filters for detecting and eliminating multiple unknown sinusoids embedded in the broadband (in fact, white) signals. The proposed SAN filters enhance both the convergence speed and estimation accuracy especially when sinusoids have close angular frequencies, under which circumstances the original SAN filters suffer from convergence and accuracy problems. Several simulations are included illustrating the superiority of the proposed SAN filters over their original counterparts, in terms of considerably better estimation accuracy and faster convergence speed.

A highly optimized translation of the core discrete Remez part of the Parks-McClellan (PM) algorithm from its original FORTRAN code to its MATLAB counterpart has recently been proposed by the authors. The optimization was achieved by first figuring out that the search for the "real" extremal points of the weighted error function formed based on the "trial" extremal points can be compressed into two compact search techniques and, second, by using the MATLAB strength of vectors and matrices calculations whenever possible. Most importantly, this achievement revealed that the search technique in the original PM algorithms does not follow the fundamental principle of the Remez multiple exchange (RME) algorithm. That is, if there are more candidate "real" extremal points than required, then the desired points should be selected to retain as many largest absolute values of the weighted error function as possible subject the condition that the sign of this function alternates at the consecutive points. This paper modifies the earlier MATLAB implementation of the core discrete Remez part of PM algorithm to exactly follow the above-mentioned search principle. This modification results in a highly optimized MATLAB code which outperforms the very original MATLAB code in, terms of the code compactness, the required number of iterations and CPU execution time, as is illustrated by means of several examples.

This article presents a highly optimized translation of the core discrete Remez multiple exchange (RME) part of the Parks-McClellan (PM) algorithm from its original FORTRAN code to its MATLAB counterpart. The optimization reduces the CPU execution time and code complexity. For achieving these goals, first, according to a thorough study of the existing FORTRAN code of the PM algorithm, the search in the core part for the ldquorealrdquo extremal points of the weighted error function, which is generated based on the ldquotrialrdquo extremal points, is compressed into only two compact basic search techniques. Secondly, vectors and matrices are used whenever possible due to many fast built-in operations in the MATLAB. Several examples are included to illustrate the superiority of the proposed MATLAB version of the PM algorithm over the existing function firpm, which is mostly based on a direct translation of the original FORTRAN code.

Hilbert transformers are one of the very important special classes of finite impulse response (FIR) filters used in signal processing applications. A method is presented to synthesize Hilbert transformers by using wideband linear-phase FIR filters with a piecewise-polynomial-sinusoidal impulse response. The proposed method is based on merging the earlier synthesis scheme proposed by the authors with the method proposed by Chu and Burrus and by modifying it by using an arbitrary number of separately generated center coefficients instead of none used in the method by Chu-Burrus. The desired impulse response is created by using a parallel connection of several filter branches and by adding an arbitrary number of center coefficients to form it. The arithmetic complexity of these filters is proportional to the number of branches, the common polynomial order for each branch and the number of separate center coefficients. The filter coefficients are optimized by using linear programming. An example shows the benefits of these filters compared to the frequency-response masking approach (FRM) with regard to reducing the number of coefficients as well as arithmetic complexity.

This paper proposes a fast design scheme for optimizing a novel family of multi-carrier transmission (MCT) systems that are built up based on critically-sampled cosine-modulated filter banks (CMFBs). This approach extends an earlier-proposed windowing-method (WM)-based technique for optimizing prototype filters for CMFBs such that, in addition to the cut-off frequency of the ideal filter, three terms in a four-term window function are used as unknowns, thereby leading to an optimization problem with only four adjustable parameters. Such an optimization problem is very efficiently solvable also when long prototype filters and many subchannels are required. What makes the resulting MCT systems novel is that the optimization concentrates on minimizing directly a weighted sum of the inter- symbol and inter-channel interferences in these MCT systems, without directly considering the prototype filter. If the weight values in this sum are fixed, then the values of the unknowns can be optimized for the given overlapping factor. After tabulating these values, a closely optimum solution for any number of channels is obtained by simply using the WM together with these tabulated values.

A method is presented to synthesize wideband linear-phase FIR filters with a piecewise-polynomial-sinusoidal impulse response. The proposed method is based on merging the earlier synthesis scheme proposed by the authors with the method proposed by Chu and Burrus and by modifying it by using an arbitrary number of separately generated center coefficients instead of only one used in the method by Chu-Burrus. The desired impulse response is created by using a parallel connection of several filter branches and by adding an arbitrary number of center coefficients to form it. The arithmetic complexity of these filters is proportional to the number of branches, the common polynomial order for each branch, the number of separate center coefficients, and the number of complex multipliers. The filter coefficients are optimized by using linear programming. An example shows the benefits of these filters with regard to reducing the number of coefficients as well as the arithmetic complexity.

A method for optimizing FRM-based FIR filters with optimum coefficient sensitivity is presented. This technique can be used in conjunction with nonlinear optimization techniques to design very sharp filters that do not only have very sparse coefficient values but also very low coefficient sensitivity.

This paper introduces an efficient filter structure for implementing finite-impulse response (FIR) filters with an adjustable fractional delay. In this structure the first two subfilters are the same as in the modified Farrow structure, whereas the remaining ones are generated by properly combining these two subfilters with some additional very short filters, pure delay terms, adders, and multipliers. For significantly reducing the number of multipliers, the three-step synthesis scheme proposed by Yli-Kaakinen and Saramaki in the case of the modified Farrow structure is followed. First, the number of subfilters and their orders are determined such that the given criteria are sufficiently exceeded. Second, an initial filter is determined using a simple design scheme. This filter serves as a start-up solution for further optimization being performed using a constrained nonlinear optimization algorithm. Third, those coefficient values of the subfilters having a negligible effect on the overall system performance are fixed to be zero-valued. Both the performance and complexity of the proposed adjustable digital filters are compared with those of some existing adjustable FIR filters proposed in the literature. This comparison shows that, in the case of stringent amplitude and phase delay specifications, the number of multipliers for the proposed filters is less than 80 percent when compared with the corresponding optimized modified Farrow structure.

This paper considers how to efficiently implement linear-phase FIR filters for providing a sampling rate conversion by an arbitrary rational factor of M/L, where L(M) is the up-sampling (down-sampling) factor to be implemented before (after) the actual filter. In the proposed implementation, the coefficient symmetry of the linear-phase FIR filter is exploited as well as possible when taking into account the following facts. When increasing (decreasing) the sampling rate by the factor of L(M), only every Lth input sample has a nonzero value (only every Mth output sample has to be evaluated). The proposed implementation is, first, presented by two illustrative examples and, then, guidelines are given on how to efficiently implement a sampling rate converter having an arbitrary rational sampling rate factor M/L

Two computationally efficient structures to design and implement linear-phase narrowband FIR filters with a symmetrical piecewise-polynomial impulse response have been proposed by Saramaki and Mitra. The efficiency of these structures is based on implementing the overall transfer function as a parallel connection of a few branches of the form Gl(zL)Fl(z), where each Fl(z) requires no real multipliers. These structures have been generated in an ad-hoc manner. This paper introduces straightforward approaches to generate such Fl(z)s for both structures

This paper considers designing in the minimax sense complementary low-pass/high-pass approximately linear-phase recursive filters with variable magnitude characteristics. A filter structure based on the parallel connection of a delay and a variable fractional delay all-pass filter is proposed for implementing these filters. The filter optimization is performed in two basic steps. First, an initial filter is generated using a simple design scheme. Second, this filter is used as a start-up solution for further optimization being carried out by an efficient constrained nonlinear optimization algorithm. Examples are included for illustrating the efficiency of the proposed design scheme. In addition, the performance and the complexity of the proposed variable recursive digital filters are compared with those of the other variable recursive digital filters proposed in the literature. This comparison shows that the number of multipliers for the proposed filters is less than 30 percent compared with the other existing structures

In our earlier investigations on developing multiplierless structures for digital recursive filters, we have shown that utilizing low-sensitivity structures and appropriate transformations it is possible to generate multiplierless implementations of bandpass and bandstop filters. The schemes are quite attractive when we allow marginal deviations in the specifications, or start with a design of marginally stricter specifications than the desired specification without any increase in the filter order leading to quite low requirements of nonzero bits. We present results for the structure that employs the low-sensitivity characteristics of the all-pole type of structure for developing multiplierless implementations of bandpass and bandstop digital recursive filters transformed from all-pole lowpass filters.

We propose an efficient algorithm for designing the prototype filters of oversampled, near perfect reconstruction (NPR), GDFT modulated, biorthogonal filterbanks with arbitrary delay. Given the analysis prototype, we show that the minimization of the stopband energy of the synthesis prototype, subject to NPR constraints on the frequency response of the distortion transfer function, can be expressed as a convex optimization problem. Our algorithm consists of initialization with the prototype of an orthogonal filterbank and then successive optimization of the synthesis and analysis prototypes. We compare our algorithm with previous methods and give several design examples.

The paper explores alternatives for implementing a multiplierless implementation of linear-phase finite-impulse response (FIR) digital filters by converting coefficient values to minimum signed powers-of-two (MNSPT) or canonic signed digit (CSD) forms. Our observation is that if one is willing to accept some deviations in the given specifications, the required number of nonzero bits becomes quite low, making multiplierless implementation feasible. Alternatively, one may start with a filter that exceeds the given criteria, at the expense of a slightly increased filter order, and then quantize the coefficient values into the desired representation forms such that the given overall criteria are still met. In many cases, this results in an overall implementation where the total number of nonzero bits is significantly less than that obtained by using the initial design. A fairly exhaustive investigation suggests that less than three nonzero bits per multiplier are quite sufficient along with a reduction in number of arithmetic operations and an attendant increase in the rate of the data throughput.

Under certain conditions, an odd-order lowpass or highpass recursive digital filter can be decomposed into the sum of two allpass filters with real coefficients. This decomposition has the attractive property that, for its implementation, there exist structures where both the number of delays and the number of multipliers are equal to the filter order, thereby making the overall implementation very efficient. The paper develops some allpass filter structures that combine this advantage with those of some low-sensitivity substitution and transformation blocks for replacing unit delay elements. These combinations enable one to generate multiplierless implementations for odd-order recursive digital filters and even-order bandpass and bandstop filters. Utilizing these structures, along with allowing some marginally insignificant deviations in the specifications, such as in the passband and stopband tolerances, the total number of nonzero bits for multiplier coefficients, i.e., those of shifts and adds and/or subtracts, becomes quite small, making this approach very attractive. Alternatively, the overall filter can be designed with marginally stricter tolerances than the desired specifications in such a manner that it meets the criteria after quantizing the filter coefficients.

This paper introduces an image decimation technique based on the use of a near least-squares criterion that makes a proper compromise between the L/sub 2/ and l/sub 2/ norm minimization cases. The theory of orthogonal projections is related to the derivation of a computationally efficient decimation structure possessing good antialiasing properties. It is shown how this structure can be realized by the transposed Farrow structure when using piece-wise polynomial basis functions. It is shown, by means of examples, that with a considerably lower computational complexity the proposed structure provides practically the same quality for the restored images as the best existing structures.

Some low-sensitivity transformations are investigated, wherein a prototype lowpass filter is transformed into a bandpass (bandstop) filter with low-sensitivity using such transformation. The modified coefficients being quite small, they require few shifts and adds and/or subtracts for implementation when these are expressed in minimum signed powers of two (MNSPT) forms or canonic signed digit (CSD) forms, and we obtain a multiplierless realization. Allowing some marginally insignificant deviation in the specification including the tolerances and the bandedges, the number of shifts and adds and/or subtracts per multiplier becomes quite small to make this approach quite attractive. Alternatively, we can design the prototype lowpass filter with marginally stricter tolerances than the desired specifications. Our analysis confirms this approach to be a viable one for multiplierless realization of bandpass and bandstop filters.

An edge-preserving method for image resizing (decimation and interpolation) is proposed. The decimation is considered as an orthogonal projection with respect to the chosen interpolation basis. The latter one is formed in a spline-like manner as a linear combination of B-splines of different degrees. This combination is optimized in such a way that the small image details are preserved. Considering the strongest edges as step edges, a segmentation procedure preceding the decimation is proposed. It leads to resized images with clearly outlined borders.

A theory for jointly optimizing nonuniform analysis and synthesis FIR filter banks with arbitrary filter lengths and an arbitrary delay through the filter bank is developed. The FIR subband coder is optimized with respect to the minimum mean square error between the output and the input signals under a bit constraint. The subband quantizers are modeled as additive noise sources. Theoretical comparisons are made against a well-known 5-3 wavelets used in a tree-structure. The proposed filter banks, which are both rate- and source-dependent, have a better distortion rate performance. Equations for finding jointly optimized analysis and synthesis filter banks under a power constraint are also presented.

Efficient two-step algorithms are described for optimizing the stopband response of the prototype filter for cosine-modulated and modified DFT filter banks either in the minimax or in the least-mean-square sense subject to the maximum allowable aliasing and amplitude errors. The first step involves finding a good start-up solution using a simple technique. This solution is improved in the second step by using nonlinear optimization. Several examples are included illustrating the flexibility of the proposed approach for making compromises between the required filter lengths and the aliasing and amplitude errors. These examples show that by allowing very small amplitude and aliasing errors, the stopband performance of the resulting filter bank is significantly improved compared to the corresponding perfect-reconstruction filter bank. Alternatively, the filter orders and, consequently, the overall delay can be significantly reduced to achieve practically the same performance.

An efficient iterative method is described for designing orthogonal two-channel perfect-reconstruction FIR filter banks in such a way that the low-pass analysis filter has the given number of fixed zeros at z=-1 and its energy in the given stopband region is minimized. When using the resulting two-channel filter bank for generating discrete-time wavelet banks, the number of vanishing moments is equal to the number of zeros being located at z=-1. The proposed design scheme is fast and the convergence to the optimum solution is independent of the starting-point filter bank. Compared to the two-channel filter bank equivalents designed in the minimax sense as proposed by Rioul and Duhamel (1994), the regularities of the resulting wavelets are increased and the stopband energies of the subfilters are decreased. If there are no constraints on the number of zeros at z=-1, then the resulting banks are useful building blocks in generating frequency-selective multi-channel filter banks and octave filter banks.

An efficient two-step approach is presented for designing two-channel perfect-reconstruction linear-phase FIR filter banks. The first step involves finding a good solution by using an iterative procedure. This iterative procedure is generated by properly modifying the Lagrange-Newton method proposed by Horng and Willson (1992). In the second step, the resulting solution is then used as a good initial solution for further optimization that is carried out by the second algorithm of Dutta and Vidyasagar (1977). Several examples are included illustrating the efficiency of the proposed approach and the resulting two-channel filter banks.

Step-like weighting windows are introduced for various digital signal processing and antenna array applications. The usefulness of these windows lies in the fact that the weight values of the in-normalized windows are piecewise constants and take on only integer values, thereby making the data weighting very effective. In the case of antenna arrays, the weight values take on very few distinct values. The optimization and properties of these windows are considered and they are compared with their Dolph-Chebyshev and Kaiser-Bessel counterparts.

Design and properties of polynomial-based FIR filters with adjustable fractional delay are studied. Given the passband region, the filter parameters are optimized to minimize in the passband the worst-case phase delay deviation from the desired value (the maximum deviation for fractional delays between zero and unity) subject to a given worst-case amplitude deviation from unity in the passband. It is shown that the filter with fractional delay equal to one-half determines the lower limit for the achievable amplitude distortion. Because the filters under consideration are polynomial-based, they can be efficiently implemented using the modified Farrow structure introduced by the authors. The main benefit of the proposed synthesis scheme lies in the fact that it results in the structure where the linear-phase branch filters of the modified Farrow structure are fixed and only one parameter controls the fractional delay.

This paper describes an efficient algorithm for the design of multiplierless approximately linear-phase lattice wave digital (LWD) filters (parallel connection of two all-pass filters). The coefficient optimization is performed in two basic steps. First, a nonlinear optimization algorithm is used for determining a parameter space of the finite-precision coefficients including the feasible space where the filter meets the given amplitude and phase specifications. The second step involves finding the filter parameters in this space such that the resulting filter meets the given criteria with the simplest coefficient representation forms. The proposed algorithm guarantees that the optimum multiplierless finite-wordlength solution can be found. This is illustrated by means of an example.

Interpolation techniques are widely used in many applications of signal processing. This paper is devoted to synthesizing and efficiently implementing a class of generalized spline-interpolators. A new parametric class of generalized B-splines is introduced. It contains as special cases B-splines and alternative B-splines. In the most general case, the proposed spline functions can be represented as a linear combination of the weighted and shifted classic B-splines of different orders. The applicability of the resulting splines to discrete-time interpolation is investigated and compared to classical splines. Furthermore, the implementation of the discrete-time interpolation with the aid of efficient digital filter structures is considered in detail.

A class of approximately linear phase recursive digital filters composed of two allpass sections is introduced. The passband response for these filters is equiripple with the maximum number of alternations as for elliptic filters. By slightly widening the passband region and transferring some zeros close to the poles, the poles are forced to move to locations generating an approximately linear phase in the specified passband. For this class of filters, there exists an analytic formula relating the squared-magnitude response to the passband ripple and the zero locations, making the filter optimization very fast. Several examples illustrate that, especially in narrowband applications, these filters are superior to linear-phase non-recursive filters and phase equalized elliptic filters.>

An approach for the design of limit-cycle-free digital fixed-point filters is proposed. By constraining the transfer function already in the design phase, it is guaranteed that the filter will be free from limit cycles and overflow oscillations in the direct form implementation. The design strategy also guarantees low roundoff noise. The proposed approach is illustrated with design examples. The examples show that typically a 50-100% increase in the filter order (as compared to standard elliptic design) is needed for the limit-cycle-free implementation of a lowpass filter with the direct form (DF) structure. Since the DF structure is very efficient to implement in signal processors, the proposed filters are very attractive for such environment, whereas dedicated filter structures (wave digital, state-space, etc.) easily result in a 3 to 4-fold implementation complexity.>

The class of classical recursive digital filters is generalized such that the resulting filters are implementable using low-sensitivity and low-noise structures originally developed for classical filters. These filters can be synthesized to have several passband and stopband regions and arbitrary weightings can be used in these regions. In addition, some transmission and attenuation zeros can be fixed. An efficient algorithm is constructed for optimizing the proposed filters based on iteratively adjusting the passband and stopband responses of the filter with the aid of a Remez-type algorithm. Several examples are included illustrating the usefulness of the proposed generalized filters as well as the efficiency and flexibility of the algorithm.>

Based on the multistage approach, a design procedure is presented for finding a spectral factor of an mth-band filter and for designing multistage decimation filters. The proposed design method finds spectral factors of mth-band FIR (finite-impulse response) filters without direct computation, and yields filters with much higher attenuation than would be possible by conventional methods. Such mth-band filters are used in filter-bank designs, including perfect-reconstruction systems.>

A property of FIR (finite-impulse response) lossless systems is introduced, leading to substantial improvement in the sign procedure for perfect-reconstruction QMF (quadrature mirror filter) banks. The property enables the designer to initialize the coefficients of a lattice structure (which characterizes the analysis bank), in such a way as to speed up to the convergence. A design example is provided. Compared to other methods, the proposed method is shown to converge faster, and always leads to much improved attenuation characteristics for a given filter length.>

This paper extends the recently introduced class of multistage FIR Nyquist filters to cover also the multi-stage IIR and hybrid IIR/FIR Nyquist filters. The filter stages, as well as the overall system, have zero intersymbol interference. The filter stages are simultaneuosly designed to provide a Chebyshev stopband behavior for the overall filter, thereby band-limiting the pulses optimally. Computationally efficient IIR filter stages are obtained by using the polyphase structure with allpass subfilters. Selecting one of the subfilters to be a pure delay results in zero intersymbol interference. Efficient procedures are presented for simultaneously optimizing the filter stages for both IIR and hybrid IIR/FIR cases. Examples demonstrate how the proposed multistage IIR filters provide significant advantages over other existing IIR designs and a slight reduction in the multiplication rate compared to multistage FIR designs.

A new approach is presented for the design of linear-phase anti-aliasing filters. These filters consist of an approximately linear phase analog filter cascaded with a linear phase nonrecursive digital decimator. The use of the decimation part enables us to use a higher sampling rate after the analog part. This increases the required transition bandwidth of the analog filter and, consequently, decreases the required filter order. The passband variation of the group delay response of the analog part is minimized subject to the given amplitude requirements. The optimization is done by an efficient iterative algorithm constructed using recent advances in constrained minimax optimization. As a decimator, we use a filter consisting of two parts, one working at the higher sampling rate and the other one at the final sampling rate. This filter requires significantly fewer multiplications per output sample than a conventional FIR decimator. Examples demonstrate that by oversampling by a factor of 3, a third order analog filter is sufficient to construct an extremely linear phase highly selective anti-aliasing filter meeting the same amplitude specifications as a seventh order purely analog elliptic filter.

This paper presents a new approach for synthesizing and implementing narrowband linear phase nonrecursive digital filters requiring a small number of arithmetic operations. The approach is based on transformations implemented by replacing subnetworks with transfer function (1+z-2)/2 in a prototype network by a subnetwork whose transfer function is a polynomial of (1+z-2)/2 but can be expressed in a form containing no multipliers. By appropriately designing the prototype network and selecting the subnetwork, the resulting filter implementations require significantly less multiplications and additions per sample than conventional nonrecursive designs, at the expense of an increased filter order. Examples show that even a reduction from 1017 to 17 multipliers is possible. The new filters present also considerable advantages over nonrecursive filters composed of cascaded decimators and interpolators.

In this paper we discuss a new class of hybrid IIR/ FIR digital filters. These filters present a maximally flat group delay in the given passband and an equiripple nature in the magnitude response both in the passband and in the stopband(s). The all-pole IIR component is synthesized so that its group delay exhibits a maximally flat behaviour in the pass-band whereas the linear phase FIR component is used to obtain the desired equiripple nature in the amplitude response both in the passband and stopbands. The filter types handled are lowpass, band-pass and highpass. By suitably optimizing the maximum value in the group delay and the order of the IIR component we obtain filters requiring significantly fewer multipliers than equivalent FIR and group delay equalized elliptic designs when the passband is narrow (of the order of .1π or smaller) and a rather sharp cutoff is required. The performance of these filters is, however, slightly worse that obtainable with corresponding filters having equiripple group delay [1]. Several comparisons between different filter types are presented.

In this paper we present an efficient algorithm for designing recursive digital filters with optimum magnitudes in the Chebychev sense, all zeros on the unit circle, and different order numerators and denominators. This algorithm takes advantage of the well-known relations between the poles and zeros of analog filters having an equiripple amplitude response either in the passband or stopband. The algorithm requires thus only one approximation interval. This makes it more efficient than the algorithm of Martinez and Parks [1],which works separately with the numerator and denominator. The number of multiplications in the resulting filters is discussed and the optimal orders for numerator and denominator polynomials are considered. A simple explanation for the effect of an extra ripple [1] and for the minimum attainable passband ripple is given.

In this paper we discuss a new class of hybrid IIR/ FIR digital filters. These filters present equiripple behaviour both in the magnitude and group delay responses. The all-pole IIR component is synthesized so that its group delay exhibits an equiripple variation in the passband whereas the linear phase FIR component is used to obtain the desired equiripple nature in the amplitude response both in the passband and stopbands. The filter types handled are lowpass, bandpass and highpass. By suitably optimizing the ripple in the group delay and the order of the IIR component we obtain filters requiring significantly fewer multipliers than equivalent FIR and group delay equalized elliptic designs when the pass-band is narrow (of order .1π or smaller) and a rather sharp cutoff is required. Examples illustrating this feature are presented in the paper.

In this paper we discuss the use of linear transformations on z+1/z in designing magnitude squared functionsH(z)H(\frac{1}{z}). It is shown how the design of lowpass, highpass and nonsymmetric bandpass recursive digital filters with given zeroes is transformed to the problem of designing a magnitude squared function having its passband stretched onto the whole unit circle. This magnitude squared function for equal ripple and maximally flat responses is then obtained analytically using the theory of rational approximations.

We discuss the design of digital filters with maximally flat or equiripple passband behavior and transfer functions of the formH(z)= K \frac{(z+1)^{q}\Pi\min{i=1}\max{r}(z-e^{j\omega_{i}})(z-e^{-j\omega_{i}})}{D_{n}(z)}, where q+2r, the number of finite-plane zeros, is allowed to vary from 0 to n, the filter order. Analytic expressions are given for the magnitude squared function. Identification of the stopband edge frequency is treated in detail. The hard-ware requirements of these filters are compared with those of Chebychev and elliptic designs.