Periodicity of Discrete Frequencies

Let x : Reals ® Reals be such that for all t Î Reals,

x(t) = cos(2p f t).

Let w : Reals ® Reals be such that for all t Î Reals,

w(t) = cos(2p( f + fs) t)

where fs is the sampling frequency, equal to 1/T. Notice then that

SamplerT (x) = SamplerT (w).

The two sine waves are indistinguishable. To check this, let z = SamplerT (w). Then

z(n) = cos(2p( f + fs) nT) = cos(2p f nT + 2p fs nT)

= cos(2p f nT + 2p n) = cos(2p f nT ) = y(n)

where y = SamplerT (x).

Periodicity:

A sinsoid or complex exponential at frequency f that is sampled at frequency fs is indistinguishable from one with frequency f + N fs for any integer N.

Recall that the DTFT is periodic with period 2p, in radians/sample. This means it is periodic with period 2p fs in radians/second, or with period fs in Hertz. Thus, the fact that f is indistinguishable from f + N fs is not surprising.