7.2 Uniform Sampling 433
=
∫∞
−∞
[
∑
n
x(nTs)δ(τ−nTs)
]
hlp(t−τ)dτ
=
∑
n
x(nTs)
sin(π(t−nTs)/Ts)
π(t−nTs)/Ts
(7.16)
after replacingxs(τ)and applying the sifting property of the delta function. The recovered signal is
thus an interpolation in terms of time-shifted sinc signals with amplitudes the samples{x(nTs)}. In
fact, if we lett=kTs, we can see that
xr(kTs)=
∑
n
x(nTs)
sin(π(k−n))
π(k−n)
=x(kTs)
since
sin(π(k−n))
π(k−n)
=
{
1 k−n=0 ork=n
0 k6=n
This is because the above sinc function by L’Hopital’s rule is shown to be unity whenˆ k=n, and it is
0 whenk6=nsince the sine is zero at multiples ofπ. Thus, the values att=kTsare recovered exactly,
and the rest are interpolated by a sum of sinc signals.
7.2.5 Sampling Simulation with MATLAB
The simulation of sampling with MATLAB is complicated by the representation of analog signals
and the numerical computation of the analog Fourier transform. Two sampling rates are needed:
one being the sampling rate under study,fs, and the other being the one used to simulate the analog
signal,fsim>>fs. The computation of the analog Fourier transform ofx(t)can be done approximately
using the fast Fourier transform (FFT) multiplied by the sampling period. For now, think of the FFT
as an algorithm to compute the Fourier transform of a discretized signal.
To illustrate the sampling procedure consider sampling a sinusoidx(t)=cos( 2 πf 0 t)wheref 0 = 1
KHz. To simulate this as an analog signal we choose a sampling periodTsim=0.5× 10 −^4 sec/sample
or a sampling frequencyfsim=20, 000 samples/sec.
No aliasing sampling—If we samplex(t)with a sampling frequencyfs= 6000 > 2 f 0 =2000 Hz, the
sampled signaly(t)will not display aliasing in its frequency representation, as we are satisfying the
Nyquist sampling rate condition. Figure 7.7(a) displays the signalx(t)and its sampled versiony(t),
as well as their approximate Fourier transforms. The magnitude spectrum|X()|corresponds to the
sinusoidx(t), while|Y()|is the first period of the spectrum of the sampled signal (recall the spec-
trum of the sampled signal is periodic of periods= 2 πfs). In this case, when no aliasing occurs, the
first period of the spectrum ofy(t)coincides with the spectrum ofx(t)(notice that as a sinusoid, the
magnitude spectrum|X()|is zero except at the frequency of the sinusoid or±1 KHz; likewise|Y()|
is zero except at±1 KHz and the range of frequencies is [−fs/2,fs/2]=[−3, 3] KHz). In Figure 7.7(b)
we show the sinc interpolation of three samples ofy(t); the solid line is the interpolated values or the
sum of sincs centered at the three samples. At the bottom of that figure we show the sinc interpola-
tion, for all the samples, obtained using our functionsincinterp. The sampling is implemented using
our functionsampling.
