Uniform Sampling 423

7.2 Uniform Sampling 423

Equation (7.7) provides the relation between the continuous frequency  (rad/sec) of x(t) and the discrete

frequency ω (rad) of the discrete-time signal x(nT s ) or x[n] 1 :

ω = T s

[rad/sec] × [sec] = [rad]

Sampling a continuous-time signal x(t) at uniform times {nT s } gives a sampled signal

x X s ( t) = x(nT s )δ( t − nT s )

or a sequence of samples {x(nT s ) }. Sampling is equivalent to modulating the sampling signal

periodic of period T s (the sampling period) with x(t). If X() is the Fourier transform of x(t), the Fourier transform of the sampled signal x s ( t) is given by the equivalent expressions

X s () =

X( s T ) − k

x(nT s ) e −jT s n

X 2π

Depending on the maximum frequency present in the spectrum of x(t) and on the chosen sampling frequency  s (or the sampling period T s ) it is possible to have overlaps when the spectrum of x(t) is shifted and added to obtain the spectrum of the sampled signal. We have three possible situations:

If the signal has a low-pass spectrum of finite support—that is, X() = 0 for || >  max (see Figure 7.2(a)) where  max is the maximum frequency present in the signal—such a signal is called band limited. As shown in Figure 7.2(b), for band-limited signals it is possible to choose  s so that the spectrum of the sampled signal consists of shifted nonoverlapping versions of (1/Ts)X(). Graphically (see Figure 7.2(b)), this can be accomplished by letting  s − max ≥ max , or

 s ≥ 2 max

which is called the Nyquist sampling rate condition. As we will see later, in this case we are able to recover X(), or x(t), from X s () or from the sampled signal x s ( t). Thus, the information in x(t) is preserved in the sampled signal x s ( t).

On the other hand, if the signal x(t) is band limited but we let  s < 2 max , then when creating X s () the shifted spectra of x(t) overlap (see Figure 7.2(c)). In this case, due to the overlap it will not be

1 To help the reader visualize the difference between a continuous-time signal, which depends on a continuous variable t, or a real number, and a discrete-time signal, which depends on the integer variable n, we will use square brackets for these. Thus, η(t) is a

continuous-time signal, while ρ[n] is a discrete-time signal.

C H A P T E R 7: Sampling Theory

No aliasing

FIGURE 7.2 (a) Spectrum of band-limited signal, (b) spectrum of sampled signal when satisfying the Nyquist sampling rate condition, and (c) spectrum of sampled signal with aliasing (superposition of spectra, shown in dashed lines, gives a constant shown by continuous line).

possible to recover the original continuous-time signal from the sampled signal, and thus the sampled signal does not share the same information with the original continuous-time signal. This phenomenon is called frequency aliasing since due to the overlapping of the spectra some frequency components of the original continuous-time signal acquire a different frequency value or an “alias.”

When the spectrum of x(t) does not have a finite support (i.e., the signal is not band limited) sampling using any sampling period T s generates a spectrum of the sampled signal consisting of overlapped shifted spectra of x(t). Thus, when sampling non-band-limited signals frequency aliasing is always present. The only way to sample a non-band-limited signal x(t) without aliasing—at the cost of losing information

provided by the high-frequency components of x(t) — is by obtaining an approximate signal x a ( t) that lacks the high-frequency components of x(t), thus permitting us to determine a maximum frequency for it. This is accomplished by antialiasing filtering commonly used in samplers.

A band-limited signal x(t)—that is, its low-pass spectrum X() is such that

|X()| = 0 for || >  max