The Nyquist–Shannon Sampling Theorem

Impulse Train

Impulse Train (also known as "Dirac comb") is an infinite series of delta functions with a period of T. Mathematical description of an impulse train is:

( )n

t nT

t

T 2T 3TT2T3T

......

Sampling a Signal

Sampling a signal with a sampling rate of T (which means taking a sample every T seconds) is basically multiplying with an impulse train with a period of T. Mathematically, the sampled signal holds the following:

Now, we want to explore how the signal looks like in the frequency domain. That is, calculate the Fourier transform of

( )t( )t

( )s t

( ) ( ) ( )sn

t t t nT

( )s t

( )s t

Fourier Transform of

Lemma:

The transform:

21( )

ni t

T

n n

t nT eT

2

2 2

(1) (2)

2

(3) (4)

1( ) ( ) ( ) ( ) ( )

1 1ˆ( )

ni t

i f t i f tTs

n n

ni f t

T

n n

F t f t t nT e dt t e e dtT

nt e dt f

T T T

( )s t

The Original and Sampled Signals in The Frequency Domain

By filtering frequencies higher than and lower than - and then multiplying the remaining signal by T we will get the original signal's Fourier transform. From the original signal's Fourier transform we can restore the original signal itself easily, by using the inverse Fourier transform.

This restoration was possible because the original signal was band limited and was big enough so that the different copies didn't overlap.

fsf

0c

f

Fourier transform of the original signal Fourier transform of the sampled signal

0f0f 0f0fsf

......

Fourier transform of the original signal

Fourier transform of the sampled signal

0f 0f

0f

The Nyquist–Shannon Sampling Theorem:

If a signal is bandlimited to B (all frequencies are between –B and B), then in order to have a perfect reconstruction of the original signal, the sampling frequency should be at least 2B.

If a signal is bandlimited to B, but the duplications may overlap each other, and if we'll apply the same filter used to restore the original signal we will get the original signal with distortions (due to the overlap). This phenomenon is called "aliasing".

2sf Bsf