Discrete-time Fourier transform of (non-periodic) sequences  

The Fourier transform of a discrete-time sequence tells us that the frequency content of the signal.

Definition The Fourier transform X(ejω) of the sequence x(n) can be given by

The inverse Fourier transform can be given by

The equations (A) and (B) are known as Fourier transform pair for a sequence x(n) with X(ω) thought of as the frequency content of sequence x(n). Equation (A) is analysis of the equation and the equation (B) is synthesis equation. As X(ω) is a periodic function of ω, we can think of x(n) as Fourier coefficients in Fourier series representation of X(ω). i.e., equation (A), actually, expresses X(ω) in the form of a Fourier series.

The sketch below sums up the relationship in between the time and frequency domains. The periodicity of it is 2π. From relation ω = ΩT we can infer that at the point ω = 2π on the horizontal axis Ω = ω/T = 2π/T = 2πFs = Ωs. Or we can say that, in terms of the analog frequency variable the point ω = 2π corresponds to Ω = Ω_s or F = F_s.

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