Saed Samadi

Universal maximally flat lowpass FIR systems

The family of FIR digital filters with maximally flat magnitude and group delay response is considered. The filters were proposed by Baher (1982), who furnished them with an analytic procedure for derivation of their transfer function. The contributions of this paper are the following. A simplified formula is presented for the transfer function of the filters. The equivalence of the novel formula with a formula that is derived from Bahers analytical procedure is proved using a modern method for automatic proof of identities involving binomial coefficients. The universality of Bahers filters is then established by proving that they include linear-phase filters, generalized half-band filters, and fractional delay systems. In this way, several classes of maximally flat filters are unified under a single formula. The generating function of the filters is also derived. This enables us to develop multiplierless cellular array structures for exact realization of a subset of the filters. The subset that enjoys such multiplierless realizations includes linear-phase filters, some nonsymmetric filters, and generalized halfband filters. A procedure for designing the cellular array structures is also presented.

Design and multiplierless realization of maximally flat FIR digital Hilbert transformers

A unified treatment of approximation and realization of type-3 finite impulse response (FIR) linear-phase Hilbert transformers is presented. A simple method based on Bernstein polynomials and half-band filters is proposed to derive the transfer function of the system, and a triangular array realization based on the de Casteljau algorithm is developed from the Bernstein form of the transfer function. It is shown that the array structure, consisting of multiplierless identical modules, can be realized hierarchically using complex and real signal processing techniques.

Explicit formula for improved filter sharpening polynomial

A closed-form formula for the extended amplitude change function in filter sharpening is derived. The derivation is based on the Bernstein form representation of the amplitude change function. The function was originally proposed by Hartnett and Boudreaux-Bartels (see ibid., vol.43, p.2805-10, 1995) to achieve better control on amplitude improvement and is a generalization of the Kaiser-Hamming (1977) filter sharpening function.

Multiplierless and hierarchical structures for maximally flat half-band FIR filters

A simple method to derive a closed-form expression for the transfer function of linear-phase half-band filters with maximally flat amplitude-response characteristics is presented. The method is based on the binomial series. It results in hierarchical and modular structures with low hardware complexity for low-pass and high-pass filters. For a filter of a given order, the structures provide access to ail maximally flat filters of lower orders. Two types of structures are presented. The first enjoys a set of multiplier coefficients with reduced dynamic range, and the second can be realized free of multiplier coefficients in a modular manner. Extension of the order of the filter can be achieved by cascading additional building blocks for the former structure or by adding an extra layer of modules for the latter structure. The proposed structures and formulas are applicable to maximally flat Hilbert transformers as well.

Generalized half-band maximally flat FIR filters

The existence of nonsymmetric generalized half-band lowpass and highpass FIR filters with maximally flat magnitude and group delay characteristics is proved. Like their linear-phase counterparts, approximately half of the impulse response coefficients of these generalized half-band filters are exactly zero. The magnitude response of the filters is not a monotone function in general. However, the filters can be designed to yield improved magnitude response characteristics compared to the linear-phase maximally flat filters. A closed-form formula for the transfer function of the filters is also derived using Bernstein polynomials. The filters may find applications in multirate and wavelet signal processing.

Filter-generating systems

Combining the methods of generating functions and discrete-time linear systems, we can develop multidimensional (M-D) digital systems for systematic generation of entire families of interrelated digital filters. These M-D systems, dubbed filter-generating systems, are mostly of infinite-impulse response (IIR) type and can produce the impulse response sequence of any member of the family. Temporal and spatio-temporal implementation of filter-generating systems are studied. A temporal implementation scheme is devised for causal filter-generating systems in order to implement an arbitrary filter member using (M-D) signal processing techniques. It is shown that this may result in low order recursive filtering structures that can implement arbitrarily high order members of the family. A spatio-temporal implementation scheme is developed for generation of regular signal flow-graphs for the generated family of finite-impulse response (FIR) filters. It is shown that the signal flow-graphs give rise to cellular array structures. Concrete examples are provided for maximally flat FIR filters.

Modular realization of bandstop and bandpass FIR digital filters

The generalized Neville algorithm is used to realize 3-band maximally flat linear phase FIR systems including bandstop and bandpass filters. A structure consisting of three subfilters each of which is a rectangular array of identical modules is derived. Bandstop and bandpass filters can be realized with two subfilters by appropriate ordering of the subfilters.<<ETX>>

Eigensignals of Downsamplers in Time and Transform Domains

As a fundamental building block of multirate systems, the downsampler, also known as the decimator, is a periodically time-varying linear system. An eigensignal of the downsampler is defined to be an input signal which appears at the output unaltered or scaled by a non-zero coefficient. In this paper, the eigensignals are studied and characterized in the time and z domains. The time-domain characterization is carried out using number theoretic principles, while the one-sided z-transform and Lambert-form series are used for the transform-domain characterization. Examples of non-trivial eigensignals are provided. These include the special classes of multiplicative and completely multiplicative eigensignals. Moreover, the locus of poles of eigensignals with rational z transforms are identified.

Design of approximately linear phase lattice digital allpass filters via lattice parameters

The design of lattice allpass subfilters approximating linear phase in passband and stopband is studied. The problem is reduced to a nonlinear optimization inside a hypercube in the lattice parameters space and solved using Fletcher-Powell algorithm. Using the recursive expressions for the phase response of lattice allpass filters, the authors also illustrate a simple method for fast design. This method uses one-variable optimization techniques and enables one to acquire initial values of lattice parameters for the optimization algorithm. It is shown that the method results in filters that are close to the optimum solution, from various design examples.<<ETX>>