14.2 MODULATION, SAMPLING, AND MULTIPLEXING 6450 W fs − W fs fs + W 2 fs
(b)(a)f0 W fAmplitudeAmplitudeAma 0 Am a 1 Am/ 2
a 2 Am/ 2Figure 14.2.6(a)Spectrum of low-pass signal.(b)Spectrum of sampled signal.
fs≥ 2 W (14.2.8)none of the translated components falls into the signal range of 0≤f≤W, as seen from Figure
14.2.6(b). Therefore, if the sampled signalxs(t) is passed through a low-pass filter, all components
atf≥fs−Wwill be removed so that the resulting output signal is of the same shape asa 0 x(t),
wherea 0 is given byD/Ts. These observations are summarized in the followinguniform sampling
theorem:
A signal that has no frequency components atf ≥Wis completely described by uni-
formly spaced sample values taken at the ratefs ≥ 2 W. The entire signal waveform
can be reconstructed from the sampled signal put through a low-pass filter that rejects
f≥fs−W.The importance of the sampling theorem lies in the fact that it provides a method of recon-
struction of the original signal from the sampled values and also gives a precise upper bound
on the sampling interval (or equivalently, a lower bound on the sampling frequency) needed
for distortionless reconstruction. The minimum sampling frequencyfs = 2 Wis known as
Nyquist rate.
When Equation (14.2.8) is not satisfied, spectral overlap occurs, thereby causing unwanted
spurious components in the filtered output. In particular, if any component ofx(t) originally at
f′>fs/2 appears in the output at the lower frequency
∣
∣fs−f′
∣
∣<W, it is known asaliasing.Inorder to prevent aliasing, one can process the signalx(t) through a low-pass filter with bandwidth
Bp≤fs/ 2 prior tosampling.
The elements of a typicalpulse modulation systemare shown in Figure 14.2.7(a). The
pulse generator produces a pulse train with the sampled values carried by the pulse ampli-
tude, duration, or relative position, as illustrated in Figure 14.2.7(b). These are then known