$\mathcal Z$-Transformation in Discrete Time [closed]

1. When z-transform of a discrete time signal?
2. What will be the ROC of z-transform of the discrete time sequence?
3. Why do we take z-transform in place of discrete time Fourier transform?
4. What is the formula for z-transform?

When z-transform of a discrete time signal?

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-time equivalent of the Laplace transform (s-domain).

What will be the ROC of z-transform of the discrete time sequence?

Properties of ROC of Z-Transforms

ROC does not contain any poles. If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane except at z = 0. If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-plane except at z = ∞.

Why do we take z-transform in place of discrete time Fourier transform?

The Z transform is a generalization of the Discrete-Time Fourier Transform (Section 9.2). It is used because the DTFT does not converge/exist for many important signals, and yet does for the z-transform. It is also used because it is notationally cleaner than the DTFT.

What is the formula for z-transform?

Concept of Z-Transform and Inverse Z-Transform

X(Z)|z=ejω=F. T[x(n)].