samplingtheorem

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P ROOF OF THE S AMPLING T H E O R E M

William Stallings

Copyright 2003 William Stallings


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A common definition of the sampling theorem is:

Sampling Theorem: If a signalx(t) is sampled at regular intervals of time and ata rate higher than twice the highest signal frequency, then the samples contain allthe information of the original signal. The functionx(t) may be reconstructed

from these samples by the use of a lowpass filter.

The sampling theorem can be restated as follows.

Sampling Theorem: Given:

x(t) is a bandlimited signal with bandwidthfh

.

p(t) is a sampling signal consisting of pulses at intervals Ts = 1/fs, wherefs is the sampling

frequency.xs(t) =x(t)p(t) is the sampled signal.

Thenx(t) can be recovered exactly fromxs(t) if and only iffs 2fh.

Proof

Becausep(t) consists of a uniform series of pulses, it is a periodic signal and can be

represented by a Fourier series:

p t( ) = Pnej2pnf

st

n= -

We have

xs t( ) = x t( )p t( ) = Pnx t( )e

j2pnfst

n=-

Now consider the Fourier transform ofxs(t):

Xs f( ) = xs t( )e-j 2pft

-

dt


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Substituting forxs(t), we have

Xs f( ) = Pnx t( )ej2pnfste-

j 2pft

n=-

-

dt

Rearranging yields

Xs f( ) = Pnn=-

x t( )e-j2p f-nfs( )t

-

dt

From the definition of the Fourier transform, we can write

X f- nfs( ) = x t( )e-j2p f-nfs( )t

-

dt

whereX(f) is the Fourier transform ofx(t). Substituting this into the preceding equation, we have

Xs f( )= PnX f- nfs( )

n= -

This last equation has an interesting interpretation, which is illustrated in Figure 1, where we

assume without loss of generality that the bandwidth ofx(t) is in the range 0 tofh. The spectrum

ofxs(t) is composed of the spectrum ofx(t) plus the spectrum ofx(t) translated to each harmonic

of the sampling frequency. Each of the translated spectra is multiplied by the corresponding

coefficient of the Fourier series ofp(t). Iffs 2fh, these various translations do not overlap, and


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the spectrum ofx(t), multiplied by P0, appears inXs(f). By passingXs(f) through a lowpass filter

withffs, the spectrum ofx(t) is recovered. In equation form,

Xs f( ) = P0X f( )

-fs

2f

fs2

Xs(f)

Figure 1 Demonstration of Sampling Theorem

(b) Spectrum ofxs(t)

fh

fs fh

fh fs

P0

f

P1

2fs

P2

2fs

P2

fs

P1

0

X(f)

(a) Spectrum ofx(t)

fh fh

1

f0

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