1 The Z- transform The Z-transform performs the transformation of signal from the discrete time- domain into complex frequency-domain which is called z-domain. It represents discrete time signals as a superposition of complex geometric (exponential) functions. The unilateral or one-sided Z-transform of a real discrete time casual signal fn which is a sequence of real or complex number is defined by the analysis formula as; 0nnFzfnfnzZ [1.1] Wherenis non-negative integer, Fz denotes Z-transform offn , and the z is complex frequency variable which can be expressed in polar form as jjTzre re [1.2] where r is magnitude of z and Tis the angle or phase (complex argument) of z in radian. The range of values of the complex variable z, for which the sum or power series converges and Fz exists, is called the region of convergence.2 Signal Processing The bilateral or two-sided Z-transform of discrete-time signal or sequence fn is defined by the analysis formula as; nnFzZfn fnz [1.3] where nis an integer, Fz denotes Z-transform offn , the z is complex frequency and can be expressed in polar form as jTzre where r is magnitude of z and Tas the angle or phase of z. The range of values of the complex variable z for which the Z-transform converges is called the region of convergence. The Z-transform is called two-sided because the signalfn exists forn    . The bilateral and unilateral transforms are equivalent when0 for 0fn n. The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges. :nnROC z f n z [1.4] The relation between the signalfn and its Z-transform Fz is denoted as; fnFz [1.5] Fz in terms of r and Tcan be expressed as:  njjT jTnF z F re F re f n re   [1.6] or jnjnFre fnr e [1.7] ThejFre is the discrete-time Fourier transform of the sequence fn multiplied by a real exponentialnr. jnFre f n rF [1.8]The z-transform 3 The exponential weighting nr will decay or grow with increasing n depending on whether r is greater than or less than unity. The Z-transform will be reduced to Fourier transform for1r: jjFzFefnzeF [1.9] The Z-transform for discrete-time signal reduces to discrete-time Fourier transform on the contour in the complex Z-plane corresponding to a circle with a radius of unity. In other words, the Z-transform reduces to the discrete-time Fourier transform when the magnitude of the transform variable z is unity as shown in figure 1-1. Re zIm zjTzreTr Figure 1-1: Z-plane Example 1-1 Find the Z-transform of the causal sequencenfnaun . Solution: 100nnnnnnnnFz aunz az az   [1.10] The infinite geometric progression 21......nsrr r    [1.11]4 Signal Processing converges if 1r  2011 ... for 11nnnsrr r r rr      [1.12] Hence provided that 11az, then 11011nnzFz azzaaz [1.13] For the convergence of Fz 10nnaz [1.14] The ROC is the range of the value for which 11azequivalently za as shown in figure 1.1 Fz has one zero at 0z  (the roots of the numerator polynomial ) and one pole za (the roots of the denominator polynomial. ) xo01aRe zIm za1xo1a Re zIm za1xo10a Re zIm za-1xo1aRe zIm za-1 Figure 1-1: The region of convergence of nfnaunThe z-transform 5 Example 1-2 Find the Z-transform of the anticasual sequence 1nfn au n Solution: 1nnnFzaunz [1.15]   11nnFz u n az [1.16]   111 1101nn nnn nFzaz az az         [1.17] If 11az or otherwise zathe sum in equation 1.17 converges. note: 11011nnazaz [1.18] then 101nnazazaz za  [1.19] The ROC is the range of the value for which 11azor equivalently za as shown in figure 1.2 Fz has one zero at 0z  (the roots of the numerator polynomial ) and one pole za (the roots of the denominator polynomial. )6 Signal Processing xo01aRe zIm za1xo1a Re zIm za1xo10a Re zIm za-1xo1aRe zIm za-1 Figure 1-2: The region of convergence of 1nfn au n The Z-transform of the Singularity functions Example 1-3 Find the Z-transform of the unit impulse sequencefnn . Solution: 10nnnFz Z n nz zn   [1.20] The Z-transform is independent of the complex variable z and the transform converges for all values ofz . The ROC is everywhere including zero and infinite. Example 1-4 Find the Z-transform of the delayed unit impulse sequence1fn n .The z-transform 7 Solution: 1111nnnFzZ n n z z zn   [1.21] The Z-transform is well defined except at its pole0z. The ROC is everywhere on the Z-plane including zexcept at 0z. Example 1-5 Find the Z-transform of unit step sequence fnun Solution: In example 1.2, Z-transform of nfnaun found to be zz a. Let 1a  1zFz unzZ [1.22] The inverse Z-transform The original discrete time signal can be recovered from Fzby the contour integral 1112ncfn Z Fz Fzz dzj [1.23] whereCis a counterclockwise closed path encircling the origin and entirely in the region of convergence. The contour C must encircle all of the poles of Fz . A special case of this contour integral occurs when Cis the unit circle. The synthesis formula involves integration in the complex z domains,