The Z-Transform & Discrete Time LTI Systems

The z transform for a discrete time invariant system with impulse response h[n] , the response y[n] of the system to a complex exponential input of the form zn is

$y[n] = H(n) z^n$

Where , $H(z) = \Sigma_{n=-\infty}^{\infty} h[n] z^{-n}$

For z = ejw with w real , the summation in equation corresponds to the discrete time fourier Transform of h[n] . More generally , where [z] is not restricted in unity , the summation is referred to as the z-transform of h[n]

The Z transform of a general discrete time signal x[n] is defined as

$X(z) = \Sigma_{n=-\infty}^{\infty} x[n] z^{-n}$ ; where z ia complex variable .

For convenience the z transform of x[n] will sometimes be denoted as Z{x[n]} and the relationship between x[n] and its Z-transform indicated as

$x[n] \longleftrightarrow X(z)$

In general , the z-transform of a sequence has associated with it a range of values of z for which X(z) converges , and this range of values is referred to as the region of convergence (ROC)