S. Samadi

Results on maximally flat fractional-delay systems

The two classes of maximally flat finite-impulse response (FIR) and all-pass infinite-impulse response (IIR) fractional-sample delay systems are thoroughly studied. New expressions for the transfer functions are derived and mathematical properties revealed. Our contributions to the FIR case include a closed-form formula for the Farrow structure, a three-term recurrence relation based on the interpolation algorithm of Neville, a concise operator-based formula using the forward shift operator, and a continued fraction representation. Three types of structures are developed based on these formulas. Our formula for the Farrow structure enhances the existing contributions by Valimaki, and by Vesma and Sarama/spl uml/ki on the subsystems of the structure. For the IIR case, it is rigorously proved, using the theory of Pade approximants, that the continued fraction formulation of Tassart and Depalle yields all-pass fractional delay systems. It is also proved that the maximally flat all-pass fractional-delay systems are closely related to the Lagrange interpolation. It is shown that these IIR systems can be characterized using Thieles rational interpolation algorithm. A new formula for the transfer function is derived based on the Thiele continued fractions. Finally, a new class of maximally flat FIR fractional-sample delay systems that exhibit an almost all-pass magnitude response is proposed. The systems possess a maximally flat group-delay response at the end frequencies 0 and /spl pi/, and are characterized by a closed-form formula. Their main advantage over the classical FIR Lagrange interpolators is the improved magnitude response characteristics.

Exact fractional-order differentiators for polynomial signals

A discrete-time fractional-order differentiator is modeled as a finite-impulse response (FIR) system. The system yields fractional-order derivatives of Riemann-Liouville type for a uniformly sampled polynomial signal. The computation of the output signal is based on the additive combination of the weighted outputs of N cascaded first-order digital differentiators. For differentiators of fractional order with a terminal value equal to zero, the weights are time-varying. The weights are obtained in a closed form involving the Stirling numbers of the first kind. The system tends to a time-invariant integer-order differentiator when the order of the derivative tends to an integer value. It yields exact fractional- or integer-order derivatives of a sampled polynomial signal of a certain order.

Ramanujan sums and discrete Fourier transforms

A special class of even-symmetric periodic signals is introduced. The most distinctive feature of these signals is that their real-valued Fourier coefficients can be calculated by forming a weighted average of the signal values using integer-valued coefficients. The signals arise from number-theoretic concepts concerning a class of functions called even arithmetical functions. The integer-valued weighting coefficients, being sums of complex roots of unity, are the Ramanujan sums and may be computed recursively or through closed-form arithmetical relations. The recursive method of computation is based on the cyclotomic polynomials and is described in detail. If the signal values are integers, the computation of the discrete Fourier transform (DFT) coefficients of this class of signals can be performed in an exact quantization-error-free manner by performing arithmetical operations on integers. The theoretical development is supplemented by concrete examples.

Explicit Formula for Predictive FIR Filters and Differentiators Using Hahn Orthogonal Polynomials

An explicit expression for the impulse response coefficients of the predictive FIR digital filters is derived. The formula specifies a four-parameter family of smoothing FIR digital filters containing the Savitsky-Goaly filters, the Heinonen-Neuvo polynomial predictors, and the smoothing differentiators of arbitrary integer orders. The Hahn polynomials, which are orthogonal with respect to a discrete variable, are the main tool employed in the derivation of the formula. A recursive formula for the computation of the transfer function of the filters, which is the z-transform of a terminated sequence of polynomial ordinates, is also introduced. The formula can be used to design structures with low computational complexity for filters of any order.

Characterization of B-spline digital filters

Digital filters arising in the B-spline signal processing are characterized in a unified manner in this paper. The transfer functions of these filters are the z-transforms of the uniformly sampled central B-splines shifted by an arbitrary value. The transfer functions are cascades of an FIR kernel filter and a simple moving average FIR filter. For certain values of the shift parameter, the filters are identical to those referred to as the B-spline digital filters in the literature. The filters thus form a general family of B-spline digital filters. The kernel part of the B-spline filters may be used for transforming a discrete-time signal to a representation based on the B-spline coefficients. The B-spline filters may also be used to convert a sequence of B-spline coefficients to a discrete-time spline signal. The contributions of the paper are as follows. A unifying recurrence relation enabling the computation of the impulse response coefficients of the B-spline kernel filters is derived. An accompanying recurrence relation is also obtained for the entire transfer function of the kernel filters. The recurrences are valid for arbitrary values of the shift parameter. It is proved that the roots of the transfer functions of the kernel filters are distinct, negative and real. We also prove that the roots of the kernel filters of successive orders interlace. The results regarding the location of the zeros are also valid for arbitrary values of the shift parameter. The relation of the kernel filters to the Eulerian polynomials is discussed. It is shown that for certain choices of the parameters the kernel filters are equivalent to the classical Eulerian polynomials that frequently arise in combinatorics. An alternative closed-form expression for the kernel filters in the Bernstein form is also derived. Besides their importance in unifying the existing results on B-spline filters, the generalized family of B-spline filters studied in this paper find applications in fractional delay of B-spline signals.

Complete characterization of systems for simultaneous Lagrangian upsampling and fractional-sample delaying

We present a complete formulation and an exact solution to the problem of designing systems for simultaneous sampling rate increase and fractional-sample delay in the Lagrangian sense. The problem may be regarded as that of a linear transformation, i.e., scaling, and/or shifting, of the uniform sampling grid of a discrete-time signal having a Newton series representation. It is proved that the solution forms a three-parameter family of maximally flat finite impulse response digital filters with a variable group-delay at the zero frequency. Various properties of the solution, including Nyquist properties and conditions for a linear phase response are analyzed. The solution, obtained in the closed form, is exact for polynomial inputs. We show that it is also suited for processing discrete-time versions of certain continuous-time bandlimited signals and signals having a rational Laplace transform. We then derive a generalization of the solution by augmenting the family with a fourth parameter that controls the number of multiple zeros at the roots of unity. This four-parameter family contains various types of maximally flat filters including those due to Herrmann and Baher. We list specific conditions on the four parameters to obtain many of the maximally flat filters reported in the literature. A significant part of the family of systems characterized by the solutions has been hitherto unknown. Examples are provided to elucidate this part as well.

The world of flatness

In the published literature on circuit theory and signal processing, the maximally flat FIR filters are commonly attributed to O. Herrmann. It is the purpose of this paper to elucidate the status of these filters before and after Herrmanns paper. Our survey shows that both the formula and the shape of the magnitude response of these filters had been known to actuaries and mathematicians before the publication of Herrmanns paper. We provide a broad outline of the contributions made by actuaries and mathematicians on this class and other related FIR filters. Some recent developments and extensions are also reviewed.

Multiplier-free structures for exact generation of natural powers of integers

It is shown that the sequence of an arbitrary natural power of consecutive integers can be generated without any multiplications as the impulse response of a multirate recursive structure. This multistage structure has a regular configuration consisting of multirate building blocks and accumulators. A nonrecursive structure is also developed for the generation of the power sequence for a prescribed finite length. This feedforward structure consists of layers of summation nodes interconnected in a regular manner. The two types of multiplier-free discrete-time structures generate the exact values of natural k/sup th/ powers of consecutive integers in a simple, regular and scalable manner.

Characterization of nonuniform perfect-reconstruction filterbanks using unit-step signal

An algebraic characterization of nonuniform perfect reconstruction (PR) filterbanks with integer decimation factors is presented. The PR property is formulated in the z domain based on the response of the linear multirate systems to the delayed unit-step signals. This leads to a unique class of characterizing formulas that are necessary and sufficient conditions for the PR property and free from the complex roots of unity. Two related characterizations of nonuniform PR systems, in the form of necessary conditions, are also developed based on these formulas. As a concrete example, the results are then used to derive necessary and sufficient conditions for PR nonuniform delay chain filterbanks. The conditions show that nonuniform delay chain filterbanks are the signal processing realizations of the mathematical notion of the exact covering systems of congruence relations. Important results from the mathematics literature on the exact covering systems are introduced. The results elucidate the admissible factors of decimation for the nonuniform PR delay chain systems in settings with maximally distinct decimation factors. A simple test of the PR property for delay chain systems is also presented. The test is based on the divisibility of certain polynomials by the cyclotomic polynomials. Finally, multirate systems based on the Beatty sequences, which are the irrational generalization of the exact covering systems, are briefly discussed.

Z-transform of quantized ramp signal

The z-transform of a ramp signal, with a rational slope, which is quantized by rounding down using the floor function, or rounding up using the ceiling function, is computed explicitly. The principles of rational sampling rate conversion are employed in modeling the signal. A multirate sample rate conversion system and a single-rate infinite impulse response (IIR) digital filter are also proposed for the division-free generation of the signal. The systems involve integer coefficients and operate using integer arithmetic. To produce a quantized ramp signal with a zero-valued intercept, the multirate system requires no form of multiplication.