Periodicity of Discrete Frequencies

x(t) = cos(2π f t).

Let w : Reals → Reals be such that for all t ∈ Reals,

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

Sampler_T (x) = Sampler_T (w).

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

z(n) = cos(2π( f + f_s) nT) = cos(2π f nT + 2π f_s nT)

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

where y = Sampler_T (x).

Periodicity:

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

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