Chapter 6. Sampling
and Pulse Modulation
Husheng Li
The University of Tennessee
Chopper Sampling
 We introduce a switching function such that
x_s(t)=x(t)s(t), where
Nyquist Criterion
 The sampling rate
should be at least twice
the bandwidth of the
signal, in order to fully
reconstruct the signal.
 Otherwise, there will be
aliasing effect.
Ideal Sampling and
Reconstruction
 We consider the ideal sampling function:
 The spectrum of the sampled signal is given by
Reconstruction of Signal
 When the Nyquist criterion is satisfied, the signal
can be reconstructed by using interpolation filter:
Homework
 Deadline: Nov. 11, 2013
Sampling in Practice
 We need to consider three factors of sampling in
practice:
 The sampled wave consists of pulses having finite amplitude
and duration, rather than impulses.
 Practical reconstruction filters are not ideal filters.
 The message to be sampled are timelimited signals whose
spectra are not and cannot be strictly bandlimited.
Reconstruction Methods
Aliasing
The signal is filtered
using a RC LPF antialiasing
filter with bandwidth
B>>W.
The shaded area represents
the aliased components
that have spilled over the filter’s
passband.
Pulse-Amplitude Modulation
 If a message waveform is adequately described
by periodic sample values, it can be transmitted
using analog pulse modulation wherein the
sample values modulate the amplitude of a
pulse train. The process is called pulse-amplitude
modulation (PAM).
Flat-top Sampling and PAM
 More popular than the chopper circuit, we can
use a sample-and-hold technique for PAM.
Spectrum of Flat-top
Sampling
 The spectrum of the flat-top sampling is the
convolution of the ideal sampling spectrum and
the spectrum of pulse:
Aperture Effect
 The loss of high-frequency content is called
aperture effect. The larger the pulse duration is,
the larger the effect is.
 The aperture effect can be corrected in the
signal reconstruction:
Unipolar Flat-Top PAM
 We define the unipolar flat-top PAM as
 The resulting constant pulse rate f_s is particularly
important for synchronization in time-division
multiplexing.
 The spectrum of the unipolar PAM is obtained by
replace X(f) with
Pulse-Time Modulation
 The time parameters of a pulse train can also be
modulated:
 PDM: pulse-duration modulation
 PPM: pulse-position modulation
Generation of PDM or PPM
Signal Reconstruction
Properties of PDM/PPM
 PDM and PPM need very sharp rising time t_r.
Then, the required bandwidth satisfies
which could be much larger than PAM.
 The PDM and PPM have the potential for
wideband noise reduction, since the information
resides in the time location of the pulse edges,
not the pulses themselves.
Spectrum of PPM
 Using the distribution theory of impulses, we have
 PPM with nonuniform sampling is a combination
of linear and exponential carrier modulation.