If the signalx(t)to be sampled is band limited with Fourier transformX()and maximum frequency
max, by choosing the sampling frequency s to satisfy the Nyquist sampling rate condition, or
s>2max, the spectrum of the sampled signal xs(t) displays a superposition of shifted versions of the spectrum ofx(t), multiplied by 1/Ts, but with no overlaps. In such a case, it is possible to recover the original analog signal from the sampled signal by filtering. Indeed, if we consider an ideal low-pass analog filterHlp(j)with magnitudeTsin the pass-band−s/2< < s/2, and zero elsewhere—that is,
Hlp(j)=
(Ts −s/2< < s/2
0 elsewhere (7.14)
the Fourier transform of the output of the filter isXr()=Hlp(j)Xs()or Xr()=
(X() −s/2< < s/2 0 elsewhere
which coincides with the Fourier transform of the original signalx(t). So that when sampling a band- limited signal, using a sampling periodTsthat satisfies the Nyquist sampling rate, the signal can be recovered exactly from the sampled signal by means of an ideal low-pass filter.
7.2 Uniform Sampling 429
Bandlimited or Not?
The following, taken from David Slepian’s paper “On Bandwidth” [66], clearly describes the uncertainty about bandlimited signals:
The Dilemma—Are signals really bandlimited? They seem to be, and yet they seem not to be.
On the one hand, a pair of solid copper wires will not propagate electromagnetic waves at optical frequencies and so the signals I receive over such a pair must be bandlimited. In fact, it makes little physical sense to talk of energy received over wires at frequencies higher than some finite cutoffW, say 1020Hz. It would seem, then, that signals must be bandlimited.
On the other hand, however, signals of limited bandwithWare finite Fourier transforms,
s(t)=
W
Z
−W
e2πiftS(f)df
and irrefutable mathematical arguments show them to be extremely smooth. They possess derivatives of all orders.
Indeed, such integrals are entire functions oft, completely predictable from any little piece, and they cannot vanish on anytinterval unless they vanish everywhere. Such signals cannot start or stop, but must go on forever. Surely real signalsstart and stop, and they cannot be bandlimited!
Thus we have a dilemma: to assume that real signals must go on forever in time (a consequence of bandlimit- edness) seems just as unreasonable as to assume that real signals have energy at arbitrary high frequencies (no bandlimitation). Yet one of these alternatives must hold if we are to avoid mathematical contradiction, for either signals are bandlimited or they are not: there is no other choice. Which do you think they are?
Remarks
n In practice, the exact recovery of the original signal may not be possible for several reasons. One could be that the continuous-time signal is not exactly band limited, so that it is not possible to obtain a maximum frequency causing frequency aliasing in the sampling. Second, the sampling is not done exactly at uniform times—random variation of the sampling times may occur. Third, the filter required for the exact recovery is an ideal low-pass filter, which in practice cannot be realized; only an approximation is possible. Although this indicates the limitations of sampling, in most cases where: (1) the signal is band limited or approx- imately band limited, (2) the Nyquist sampling rate condition is satisfied in the sampling, and (3) the reconstruction filter approximates well the ideal low-pass filter, the recovered signal closely approximates the original signal.
n For signals that do not satisfy the band-limitedness condition, one can obtain an approximate signal that satisfies that condition. This is done by passing the non-band-limited signal through an ideal low-pass filter. The filter output is guaranteed to have as maximum frequency the cut-off frequency of the filter (see Figure 7.4). Because of the low-pass filtering, the filtered signal is a smoothed version of the original signal—high frequencies of the signal have been removed. The low-pass filter is called anantialiasing filter, since it makes the approximate signal band limited, thus avoiding aliasing in the frequency domain.
n In applications, the cut-off frequency of the antialiasing filter is set according to prior knowledge. For instance, when sampling speech, it is known that speech has frequencies ranging from about 100 Hz to
FIGURE 7.4 Anti-aliasing filtering of non-band-limited signal.
−Ωc
−Ωc
Ωc
Ωc
Ω
1
X(Ω) Xa(Ω)
H(s)
H(jΩ)
Ω Ω
about 5 KHz (this range of frequencies provides understandable speech in phone conversations). Thus, when sampling speech an anti-aliasing filter with a cut-off frequency of 5 KHz is chosen and the sampling rate is then set to 10,000 samples/sec. Likewise, it is also known that an acceptable range of frequencies from 0 to 22 KHz provides music with good fidelity, so that when sampling music signals the anti-aliasing filter cut-off frequency is set to 22 KHz and the sampling rate to 44 K samples/sec or higher to provide good-quality music.
Origins of the Sampling Theory—Part 1
The sampling theory has been attributed to many engineers and mathematicians. It seems as if mathematicians and researchers in communications engineering came across these results from different perspectives. In the engineering com- munity, the sampling theory has been attributed traditionally to Harry Nyquist and Claude Shannon, although other famous researchers such as V. A. Kotelnikov, E. T. Whittaker, and D. Gabor came out with similar results. Nyquist’s work did not deal directly with sampling and reconstruction of sampled signals but it contributed to advances by Shannon in those areas.
Harry Nyquist was born in Sweden in 1889 and died in 1976 in the United States. He attended the University of North Dakota at Grand Forks and received his Ph.D. from Yale University in 1917. He worked for the American Telephone and Telegraph (AT&T) Company and the Bell Telephone Laboratories, Inc. He received 138 patents and published 12 technical articles. Nyquist’s contributions range from the fields of thermal noise, stability of feedback amplifiers, telegraphy, and television, to other important communications problems. His theoretical work on determining the bandwidth requirements for transmitting information provided the foundations for Claude Shannon’s work on sampling theory [33].
As Hans D. Luke [44] concludes in his paper “The Origins of the Sampling Theorem,” regarding the attribution of the sampling theorem to many authors:
This history also reveals a process which is often apparent in theoretical problem in technology or physics: first the practicians put forward a rule of thumb, then theoreticians develop the general solution, and finally someone discovers that the mathematicians have long since solved the mathematical problem which it contains, but in
“splendid isolation.”
nExample 7.3
Consider the two sinusoids
x1(t)=cos(0t) − ∞ ≤t≤ ∞ x2(t)=cos((0+s)t) − ∞ ≤t≤ ∞
7.2 Uniform Sampling 431
Show that if we sample these signals usingTs=2π/s, we cannot differentiate the sampled signals (i.e., x1(nTs)=x2(nTs)). Use MATLAB to show the above graphically when0=1 ands=7.
Explain the significance of this.
Solution
Sampling the two signals usingTs=2π/s, we have
x1(nTs)=cos(0nTs) − ∞ ≤n≤ ∞ x2(nTs)=cos((0+s)nTs) − ∞ ≤n≤ ∞
but sincesTs=2π, the sinusoidx2(nTs)can be written as
x2(nTs)=cos((0Ts+2π)n)
=cos(0Tsn)=x1(nTs)
The following script shows the aliasing effect when0=1 ands=7 rad/sec. Notice thatx1(t)is sampled satisfying the Nyquist sampling rate condition (s=7>20=2 rad/sec), whilex2(t)is not (s=7<2(0+s)=16 rad/sec).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Example 7.3 ---Two sinusoids of different frequencies being sampled
% with same sampling period -- aliasing for signal with higher frequency
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all; clf
% sinusoids
omega 0 = 1;omega s = 7;
T = 2∗pi/omega 0; t = 0:0.001:T; % a period of x1 x1 = cos(omega 0∗t); x2 = cos((omega 0 + omega s)∗t);
N = length(t); Ts = 2∗pi/omega s; % sampling period M = fix(Ts/0.001); imp = zeros(1,N);
for k = 1:M:N−1.
imp(k) = 1; % sequence of impulses end
xs = imp.∗x1; % sampled signal plot(t,x1,’b’,t,x2,’k’); hold on
stem(t,imp.∗x1,’r’,’filled’);axis([0 max(t)−1.1 1.1]); xlabel(’t’); grid
Figure 7.5 shows the two sinusoids and the sampled signal that coincides for the two signals. The result in the frequency domain is shown in Figure 7.6: The spectra of the two sinusoids are different
but the spectra of the sampled signals are identical. n
FIGURE 7.5
Sampling of two sinusoids of frequencies0=1and
0+s=8withTs=2π/s. The higher-frequency signal is undersampled, causing aliasing, which makes the two sampled signals coincide.
0 1 2 3 4 5 6
−1
−0.5 0 0.5 1
t x1(t), x2(t), x1(nTs)
x1(t) x2(t) x1(nTs)
FIGURE 7.6
(a) Spectra of sinusoidsx1(t)andx2(t). (b) The spectra of the sampled signalsx1s(t) andx2s(t)look exactly the same due to the
undersampling ofx2(t). (a) (b)
1
8 6
6 1
1
−1
8 8 X1(Ω) X1s(Ω)
X2(Ω) X2s(Ω)
Ω
Ω Ω
−8 Ω −8−6 −1
−8−6
ã ã ã
ã ã ã
ã ã ã
ã ã ã
−1