Z-Transform and DFT

1. What is the relation between z-transform and DFT?
2. What is the difference between DFT and Fourier transform?
3. Why do we use z-transform instead of DTFT?
4. How does z-transform contribute to the analysis of DT systems?

What is the relation between z-transform and DFT?

Also, if r = 1, then the discrete time Fourier transform (DTFT) is same as the Z-transform. In other words, the DTFT is nothing but the Z-transform evaluated along the unit circle centred at the origin of the z-plane.

What is the difference between DFT and Fourier transform?

Discrete Fourier Transform (DFT) is the discrete version of the Fourier Transform (FT) that transforms a signal (or discrete sequence) from the time domain representation to its representation in the frequency domain. Whereas, Fast Fourier Transform (FFT) is any efficient algorithm for calculating the DFT.

Why do we use z-transform instead of DTFT?

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.

How does z-transform contribute to the analysis of DT systems?

In the same way, the z-transforms changes difference equations into algebraic equations, thereby simplifying the analysis of discrete-time systems. The z-transform method of analysis of discrete-time systems parallels the Laplace transform method of analysis of continuous-time systems, with some minor differences.