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1.4.1 Sampling of Analog Signals T h ere are m any ways to sam ple an analog signal.W e lim it our discussion to periodic o r uniform sampling, which is the type of sam pling used m ost often in practice.

1.4.1 Sampling of Analog Signals
T h ere are m any ways to sam ple an analog signal. W e lim it our discussion to
periodic o r uniform sampling, which is the type of sam pling used m ost often in
practice. T his is described by the relation
w here x(n) is the discrete-tim e signal obtain ed by “taking sam p les” of the analog
signal x aU) every T seconds. This p ro ced u re is illustrated in Fig. 1.16. T he tim e
interval T betw een successive sam ples is called the sampling period or sample
interval and its reciprocal 1 /7 — Fs is called the sampling rate (sam ples p e r second)
o r the sampling frequency (hertz).
P eriodic sam pling establishes a relatio n sh ip betw een th e tim e variables t and
n o f co n tinuous-tim e and discrete-tim e signals, respectively. Indeed, these variables are linearly re la te d through the sam pling p erio d T or, equivalently, through
the sam pling rate Fs — l / 7 \ as
A s a con seq u en ce o f (1.4.2), th ere exists a relatio n sh ip betw een the frequency
variable F (or Q) for analog signals and the frequency variable / (or co) for
d iscrete-tim e signals. T o establish this relationship, consider an analog sinusoidal
signal of the form
x(n) = xa(nT). — o c < n < cc (1.4.1)
(1.4.2)
x B{t) = A cos(2tt F t + 8) (1.4.3)
24 Introduction Chap. 1
Analog (n) = xa(nT) Discrete-time
signal Fs = 1 IT signal
Sampler
Figure 1.16 Periodic sampling of an analog signal.
which, w hen sam pled periodically at a ra te Fs — 1 /7 sam ples p er second, yields
xa(nT) == x{n) = A cos(27t F n T -f 9)
/2 n n F \ (1-4.4)
= A cos ( — + 9

If we com pare (1.4.4) with (1.3.9). we note th at the freq u en cy variables F
and / are linearly related as
(1.4.5)
Is
or, equivalently, as
co = Q T (1.4.6)
T he relatio n in (1.4.5) justifies th e nam e relative o r n orm alized fr e q u e n c y , w hich is
som etim es used to describe th e frequency variable / . A s (1.4.5) im plies, we can use
/ to d eterm in e the frequency F in h ertz only if th e sam pling freq u en cy Fs is know n.
W e recall from Section 1.3.1 th at th e range of the frequency variable F or £2
for continuous-tim e sinusoids are
—oc < F < oo
(I.4 .7)
—oc < £2 < 00
H ow ever, the situation is differen t for discrete-tim e sinusoids. F ro m Section 1.3.2
we recall th a t
c / <

2 2 (1.4.8)
—n < co < n
By substituting from (1.4.5) an d (1.4.6) in to (1.4.8), w e find th a t the frequency
of the continuous-tim e sinusoid w hen sam pled at a ra te Fs = 1 / 7 m ust fall in
Sec. 1.4 Analog-to-Digitai and Digital-to-Analog Conversion
the ran ae
or, equivalently.
----------- T . - = — n F, < Q < Ti F, _ = —T
25
(1.4.9)
(1.4.10)
T h ese relations are su m m arized in T able 1.1.
TABLE 1.1 RELATIONS AMONG FREQUENCY VARIABLES
Continuous-time sienals Discrete-time sienals
Q = F co = 2,t f
radians u .. radians cycles
sec

sample sample
u> = nT,f=F/F, \ 1
1A
5
IA
/ ' ‘ - /
/ q = to/T.F~f- Fs
\ ..................- .......

-o c < fi < oc — tt/T < Q < ,T /r
— oc < f-' < oc -f-2,/2 5 F < /-;«/-
From these relations we observe that the fu n d am en tal difference betw een
contin u o u s-tim e and discrete-tim e signals is in th eir range of values of the frequency variables F and / , or Q and w. P eriodic sam pling of a continuous-tim e
signal im plies a m apping of th e infinite frequency range for the variable F (or £2)
into a finite frequency range for the variable / (o r a>). Since the highest frequency
in a discrete-tim e signal is co — tt or / = •, it follow s th at, w ith a sam pling rate
Fs, th e co rresp o n d in g highest values o f F and £2 are
H _ J_ Y ~ 2 T (1.4.11)
^max — ft Fs —
T h erefo re, sam pling intro d u ces an am biguity, since the highest frequency in a
contin u o u s-tim e signal th at can be uniquely distinguished w hen such a signal is
sam pled at a rate Fs = l / T is Fm&li — F J 2 , or Qmax = nF s. T o see w hat happens
to frequ en cies ab ove F J 2, let us consider the follow ing exam ple.