The discrete-time Fourier transform (DTFT)—not to be confused with the discrete Fourier transform (DFT)—is a special case of such a Z-transform obtained by restricting z to lie on the unit circle.The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges.It can be considered as a discrete-time equivalent of the Laplace transform.
In the case where the ROC is causal (see Example 2), this means the path C must encircle all of the poles of is stable, that is, when all the poles are inside the unit circle.
With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle: The Z-transform with a finite range of n and a finite number of uniformly spaced z values can be computed efficiently via Bluestein's FFT algorithm.
The following substitution is used: from the Z-domain to the Laplace domain.
Through the bilinear transformation, the complex s-plane (of the Laplace transform) is mapped to the complex z-plane (of the z-transform).
Solution − Taking Z-transform on both the sides of the above equation, we get $$S(z)Z^2-3S(z)Z^1 2S(z) = 1$$ $\Rightarrow S(z)\lbrace Z^2-3Z 2\rbrace = 1$ $\Rightarrow S(z) = \frac=\frac = \frac \frac$ $\Rightarrow S(z) = \frac-\frac$ Taking the inverse Z-transform of the above equation, we get $S(n) = Z^[\frac]-Z^[\frac]$ $= 2^-1^ = -1 2^$ Find the system function H(z) and unit sample response h(n) of the system whose difference equation is described as under $y(n) = \fracy(n-1) 2x(n)$ where, y(n) and x(n) are the output and input of the system, respectively.