The Discrete-Time Fourier Transform is a variant of the Fourier transform applied to a discrete-time signal.

This transform replaces the integral in the continuous-time Fourier transform with a summation to handle the discrete nature of the signal.

Consider a discrete-time finite-duration sequence. A periodic sequence is formed when N approaches infinity and duplicates this sequence over larger intervals.

The Fourier spectrum of the discrete signal has an interesting property of periodicity. This means it can be expanded using a Fourier series.

The periodic nature also enables the computation of its inverse, called the Inverse Discrete-Time Fourier Transform (IDFT).

The DTFT and the IDTFT form a pair, symbolizing a one-to-one relationship between the discrete signal and its spectrum.

The existence or convergence of X(Ω) depends on whether the discrete signal is summable.

It's important to note that while the discrete signal is quantized, X(Ω) is a function of a continuous variable, indicating a bridge between the discrete and continuous domains.

DTFT is pivotal in designing digital filters in audio/video systems, communication devices, and biomedical applications.