Novel Selective Transforms for Non-Stationary Signal Processing

GAČR Project 102/11/1795: Novel Selective Transforms For Non-Stationary Signal Processing

Welcome on the site of our research group, which presents novel DSP algorithm: ADZT (Aproximate Discrete Zolotarev Transform).

An innovative time-frequency transform has been developed in the course of fundamental research in recent decades in the field of function approximations and higher transcendental functions. We have discovered the algebraic form of Zolotarev polynomials, refraining from parametric representation, and have developed an extremely efficient algorithm for evaluating them: M. Vlcek and R. Unbehauen, "Zolotarev Polynomials and Optimal FIR Filters", IEEE Transactions on Signal Processing, vol. 47, no. 3, pp. 717-730, Mar.1999. This method allows computation of expansion coefficients for Zolotarev polynomials of the first kind in terms of power series expansion and expansion into Chebyshev polynomials. In contrast to power series representation, the Chebyshev polynomial approach leads to coefficients valued with a small range. The algorithm is of linear complexity with respect to the polynomial order, and is robust enough to easily generate tens of thousands of degree polynomials. Since 1999 we have used Zolotarev polynomials for notch FIR filter design. In a recent PhD dissertation: R. Spetik, “The Discrete Zolotarev Transform”, PhD. thesis, Czech Technical University in Prague, Faculty of Electrical Engineering 2009, we have developed the same algorithm for a modified Zolotarev polynomial, which completes the set of functions generalizing a complex selective exponential. This marks the birth of a fundamentally new multispectral transform for non-stationary signals.

We have developed the Approximate Discrete Zolotarev Transform (ADZT) – see the links on the publications page. It is based on symmetrical Zolotarev polynomials, which create the basis function of the transform. ADZT is naturally reversible, possessing high, simultaneous frequency and time resolution, and can be treated as a signal adaptive transform. It provides an outstanding representation of the input signal with respect to its time-frequency resolution. ADZT approaches DFT for strictly stationary signals, which are orthogonal to the DFT base vectors. In the case of signals that are non-orthogonal to DFT base vectors, ADZT suppresses the spectral leakage that is usually present in a DFT spectrogram. ADZT requires no specific windowing. Our experiments revealed an interesting property of ADZT, that longer signal segments surprisingly yield better multispectral results.