Relationship to the z-transform 

The z-transform X(z) and Fourier transform X(ω) can be given by

Comparing the 2 we infer the relationship as

The z-transform evaluation on unit circle provides the Fourier transform of the sequence x(n). The z-transform of x(n) can be seen as the Fourier transform of the sequence {x(n) r-n}, that is, x(n) multiplied by the exponential sequence r-n. This can be seen by setting z = r ejω in defining the equation of X(z):

Z-transform of the periodic sequence Consider a sequence x(n) which is periodic with period N such that x(n) = x(n+kN) for any integer value of k. The sequence cannot be represented by the z- transform of it, since there is no value of z for which z-transform will converge.

Example For exponential sequence x(n) = aⁿ u(n),

|a| < 1, the DTFT is

We shall put this in the form X(ω)= Magnitude   

Plots of |X| and ∠X are shown. Note that X(ω) is periodic and the magnitude is the even function of ω and the phase is an odd function. (See below on the notation |X| and ∠X  ).

The value of X (e ^jw ) at ω = 0 is

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