The discrete-time bounded-real lemma in digital filtering

The Lossless Bounded-Real lemma is developed in the discrete-time domain, based only on energy balance arguments. The results are used to prove a discrete-time version of the general Bounded-Real lemma, based on a matrix spectral-factorization result that permits a transfer matrix embedding process. Some applications of the results in digital filter theory are finally outlined

On equalization of channels with ZP precoders

In communication systems which used filter bank precoders with zero padding (ZP) at the transmitter, the effect of an FIR channel can be equalized without the use of IIR equalizers. In this paper a number of observations are made with regard to the noise gain created by the equalizer at the receiver. If the number of received samples per block actually utilized in equalization is reduced to the number of transmitted samples per block, then the noise gain can be very large for channels with zeros outside the unit circle. As the number of utilized received samples increases the situation improves. Most importantly, it is shown that when all the redundant samples in each block are utilized for estimation of transmitted symbols then the noise gain is not sensitive to whether the channel zeros are inside, on, or outside the unit circle, and depends only on the FIR channel autocorrelatio

On power-complementary FIR filters

Conditions are derived, under which two linear-phase FIR filter transfer functions H(z)and G(z) have the power-complementary property, i.e., |H(e^{j omega})|^{2} + |G(e^{jomega})|^{2} = 1. It is shown that, the constraint of linear phase on the transfer functions strongly restricts the class of frequency responses that can be realized by a power-complementary pair

On error-spectrum shaping in state-space digital filters

A new scheme for shaping the error spectrum in state-space digital filter structures is proposed. The scheme is based on the application of diagonal second-order error feedback, and can be used in any arbitrary state-space structure having arbitrary order. A method to obtain noise-optimal state-space structures for fixed error feedback coefficients, starting from noise optimal structures in absence of error feedback (the Mullis and Roberts Structures), is also outlined. This optimization is based on the theory of continuous equivalence for state-space structures

On maximally-flat linear-phase FIR filters

An implementation for maximally-flat FIR filters is proposed that requires a much smaller number of multiplications than a direct form structure. The values of the multiplier coefficients in the implementation are conveniently small, and do not span a huge dynamic range, unlike in a direct form implementation

Efficient and multiplierless design of FIR filters with very sharp cutoff via maximally flat building blocks

A new design technique for linear-phase FIR filters, based on maximally flat buildiing blocks, is presented. The design technique does not involve iterative approximations and is, therefore, fast. It gives rise to filters that have a monotone stopband response, as required in some applications. The technique is partially based on an interpolative scheme. Implementation of the obtained filter designs requires a much smaller number of multiplications than maximally flat (MAXFLAT) FIR filters designed by the conventional approach. A technique based on FIR spectral transformations to design new multiplierless FIR filter structures is then advanced, and multiplierless implementations for sharp cutoff specifications are included

A new breakthrough in linear-system theory: Kharitonov's result

Given a real coefficient polynomial D(s), there exist several procedures for testing whether it is strictly Hurwitz (i.e., whether it has all its zeros in the open left-half plane). If the coefficients of D(s) are uncertain and belong to a known interval, such testing becomes more complicated because there is an infinitely large family of polynomials to which D(s) now belongs. It was shown by Kharitonov that in this case it is necessary and sufficient to test only four polynomials in order to know whether every polynomial in the family is strictly Hurwitz. An interpretation of this result in terms of reactance functions (i.e., LC impedances) was recently proposed. These results were also extended recently for the testing of positive real property of rational transfer functions with uncertain denominators. In this paper we review these results along with detailed proofs and discuss extensions to the discrete-time case

Perfect reconstruction QMF banks for two-dimensional applications

A theory is outlined whereby it is possible to design a M x N channel two-dimensional quadrature mirror filter bank which has perfect reconstruction property. Such a property ensures freedom from aliasing, amplitude distortion, and phase distortion. The method is based on a simple property of certain transfer matrices, namely the losslessness property

Theory of optimal orthonormal subband coders

The theory of the orthogonal transform coder and methods for its optimal design have been known for a long time. We derive a set of necessary and sufficient conditions for the coding-gain optimality of an orthonormal subband coder for given input statistics. We also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time. Several theoretical properties of optimal compaction filters and optimal subband coders are then derived, especially pertaining to behavior as the number of subbands increases. Significant theoretical differences between optimum subband coders, transform coders, and predictive coders are summarized. Finally, conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain

Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself