Relation between CTFT and DTFT

The difference is pretty quickly explained: the CTFT is for continuous-time signals, i.e., for functions x(t) with a continuous variable t∈R, whereas the DTFT is for discrete-time signals, i.e., for sequences x[n] with n∈Z.

1. What is the relation between DTFT and DFT?
2. What is the relationship between DTFT and is a transform?
3. How is the DTFT related to the continuous Fourier transform?
4. What is the relationship between continuous-time signals and discrete-time signals?

What is the relation between DTFT and DFT?

DFT (Discrete Fourier Transform) is a practical version of the DTFT, that is computed for a finite-length discrete signal. The DFT becomes equal to the DTFT as the length of the sample becomes infinite and the DTFT converges to the continuous Fourier transform in the limit of the sampling frequency going to infinity.

What is the relationship between DTFT and is a transform?

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.

How is the DTFT related to the continuous Fourier transform?

This means that the sampling frequency in the continuous-time Fourier transform, , becomes the frequency in the discrete-time Fourier transform. The discrete-time frequency corresponds to half the sampling frequency, or . The second key piece of the equation is that there are an infinite number of copies of spaced by .

What is the relationship between continuous-time signals and discrete-time signals?

A continuous-time signal has values for all points in time in some (possibly infinite) interval. A discrete time signal has values for only discrete points in time. Signals can also be a function of space (images) or of space and time (video), and may be continuous or discrete in each dimension.