Examples of Discrete Transforms In this chapter we discuss the most important fixed discrete transforms. We start with the z-transform, whichis a fundamental tool for describing the input/output relationships in linear time-invariant (LTI) systems. Then we discuss several variants of Fourier series expansions, namely the discrete-time Fourier transform, thediscrete Fourier transform (DFT), and the Fourier fast transform (FFT). Theremainder of this chapter is dedicated to other discrete transforms that are of importance in digital signal processing, such as the discrete cosine transform, the discrete sine transform, the discrete Hartley transform, and the discrete Hadamard and Walsh-Hadamard transform. . | Signal Analysis Wavelets Filter Banks Time-Frequency Transforms and Applications. Alfred Mertins Copyright 1999 John Wiley Sons Ltd Print ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4 Chapter 4 Examples of Discrete Transforms In this chapter we discuss the most important fixed discrete transforms. We start with the z-transform which is a fundamental tool for describing the input output relationships in linear time-invariant LTI systems. Then we discuss several variants of Fourier series expansions namely the discrete-time Fourier transform the discrete Fourier transform DFT and the fast Fourier transform FFT . The remainder of this chapter is dedicated to other discrete transforms that are of importance in digital signal processing such as the discrete cosine transform the discrete sine transform the discrete Hartley transform and the discrete Hadamard and Walsh-Hadamard transform. The z-Transform The z-transform of a discrete-time signal rc n is defined as oo x z 52 x n z n- n oo 4-1 Note that the time index n is discrete whereas z is a continuous parameter. Moreover z is complex even if x n is real. Further note that for z the z-transform is equal to the discrete-time Fourier transform. 75 76 Chapter 4. Examples of Discrete Transforms In general convergence of the sum in depends on the sequence x ri and the value z. For most sequences we only have convergence in a certain region of the z-plane called the region of convergence ROC . The ROC can be determined by finding the values r for which oo .r n r oo. n oo Proof. With z r we have mi oo n z n n oo yy x n r n n oo 4-3 oo n r . n oo Thus X z is finite if x n r n is absolutely summable. The inverse -transform is given by x n _L Xtz z dz. J27T Jc The integration has to be carried out counter-clockwise on a closed contour C in the complex plane which encloses the origin and lies in the region of convergence of X z . Proof of 4-4 . We multiply both sides of with zk 1 and integrate over a .
