Chapter 6. Sampling and Pulse ModulationHusheng LiThe University of Tennessee

Chopper Sampling

We introduce a switching function such that x_s(t)=x(t)s(t), where

Nyquist CriterionThe 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 ReconstructionWe consider the ideal sampling function:

The spectrum of the sampled signal is given by

Reconstruction of SignalWhen the Nyquist criterion is satisfied, the

signal can be reconstructed by using interpolation filter:

Homework Deadline: Nov. 11, 2013

Sampling in PracticeWe 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 filteredusing a RC LPF antialiasingfilter with bandwidthB>>W.

The shaded area representsthe aliased componentsthat have spilled over the filter’spassband.

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 PAMMore popular than the chopper circuit, we can

use a sample-and-hold technique for PAM.

Spectrum of Flat-top SamplingThe spectrum of the flat-top sampling is the

convolution of the ideal sampling spectrum and the spectrum of pulse:

Aperture EffectThe 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 PAMWe 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 ModulationThe 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/PPMPDM 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 PPMUsing the distribution theory of impulses, we

have

PPM with nonuniform sampling is a combination of linear and exponential carrier modulation.