Transcript Chapter6
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.