# Sampling Theorem

Sampling Theorem

The top figure shows a sine wave of frequency and its samples. The bottom figure shows the Fourier spectrum of the sampled signal. It consists of two spectral lines at , repeated periodically at integer multiples of the sampling rate .

f

S(f)

±f

f

s

According to the sampling theorem, for , the samples uniquely represent the sine wave of frequency . For , aliasing occurs, because the replicated spectra begin to overlap. In the range , a spectral line appears at the frequency . In the upper figure the sine wave with the corresponding frequency and color appears.

f≤f/2

s

f

f>f/2

s

0≤f≤f/2

s

|f-f|

s

Note that for and , additional lines at and appear in the spectrum.

f=8kHz

s

f>6kHz

-f+2f

s

f-2f

s