Fourier analysis of discrete-time signals and systems

For discrete-time Fourier transform some authors use the symbol X(e^jω) while others use the symbol X(ω). The symbol ω is used for the digital frequency (radians per sample or just radians) and the symbol Ω for analog frequency (radians/sec). Some authors, use just the opposite of our convention, i.e., ω for the analog frequency (radians/sec) and Ω for digital frequency (radians).

Discrete-time Fourier transform (DTFT) For continuous-time signal x(t), the Fourier transform is given as follows

The impulse-train sampled version, x_s(t), can be given by

So Fourier transform of x_s(t) can be given by

here the last step follows from sifting property of the δ function.  Replace ΩT by ω the discrete-time frequency variable, i.e., the digital frequency. The Unit of Ω is radians/second, and ω has units of radians (/sample). The change of notation gives discrete- time Fourier transform, X(ω), of the discrete-time signal x(n), can be obtained by sampling x(t), as

This defines the discrete-time Fourier transforms of the discrete-time signal x(n). The transform is there if x(n) satisfies a relation of the type

These conditions are enough to guarantee that the sequence has a discrete-time Fourier transform. As in the case of continuous-time signals there are signals which neither are absolutely summable nor have finite energy, but have a discrete-time Fourier transform still.

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