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.