Signals and Systems - Electrical Engineering

7.3 The Nyquist-Shannon Sampling Theorem 437

xr = xr+x(k)∗sinc(t−(k−1));

end

xx(1:100:N∗100) = x(1:N);

xx = [xx zeros(1,99)];

NN = length(xx)

t = 0:NN−1;t = t∗Ts/100;

7.3 The Nyquist-Shannon Sampling Theorem

If a low-pass continuous-time signalx(t)is band limited (i.e., it has a spectrumX()such thatX()= 0 for

||> max, wheremaxis the maximum frequency inx(t)), we then have:

n x(t)is uniquely determined by its samplesx(nTs)=x(t)|t=nTs,n=0,±1,±2,···, provided that the

sampling frequencys(rad/sec) is such that

s≥ 2 max Nyquist sampling rate condition (7.17)

or equivalently if the sampling ratefs(samples/sec) or the sampling periodTs(sec/sample) are given by

fs=

1

Ts

≥

max

π

(7.18)

n When the Nyquist sampling rate condition is satisfied, the original signalx(t)can be reconstructed by

passing the sampled signalxs(t)through an ideal low-pass filter with the following frequency response:

H()=

{

Ts − 2 s<  < 2 s

0 elsewhere

The reconstructed signal is given by the following sinc interpolation from the samples:

xr(t)=

∑

n

x(nTs)

sin(π(t−nTs)/Ts)

π(t−nTs)/Ts

(7.19)

Remarks

n The value 2 maxis called theNyquist sampling rate. The values/ 2 is called thefolding rate.
n The units of the sampling frequency fsare samples/sec and as such the units of Tsare sec/sample.
Considering the number of samples available, every second or the time at which each sample is available
we can get a better understanding of the data storage requirements, the speed limitations imposed by
real-time processing, and the need for data compression algorithms. For instance, music being sampled at
44, 000samples/sec, with each sample represented by 8 bits/sample, for every second of music we would
need to store 44 × 8 = 352 Kbits/sec, and in an hour of sampling we would have 3600 × 44 × 8 Kbits.
If you want better quality, let’s say 16 bits/sample, then double that quantity, and if you want more fidelity
increase the sampling rate but be ready to provide more storage or to come up with some data compression
algorithm. Likewise, if you were to process the signal you would have a new sample every Ts=0.0227
msec, so that any real-time processing would have to be done very fast.