# Periodicity of Discrete Frequencies

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

*x*(*t*) = cos(2π *f t*).

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

*w*(

*t*) = cos(2π(

*f + f*)

_{s}*t*)

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*for any integer

_{s}*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*in Hertz. Thus, the fact that

_{s}*f*is indistinguishable from

*f*+

*N*

*f*is not surprising.

_{s}