Communication by Sampling
3.8 A SAMPLING SYSTEM
A complete communication system that is based on sampling must have at least the elements shown in Fig. 3-11 a. The PCM approach requires a few additional elements, which are identified in Fig. 3-11b, and are described in Chaps. 4 and 5.
The input filter ensures that the band-limited condition damanded by the sampling theorem is obeyed. The sampling unit interrogates the message signal at regular intervals. T; using-pulses of width T. The output filter separates the baseband message signal from its harmonics.
Earlier we made a comparison between ideal, and natural sampling. The result was obtained that, as one approaches the ideal case (T .... 0), then the amplitude of the harmonics of the message signal are virtually unattenuated at the high frequencies.
Thus, the signal presented to the output filter will contain all the harmonics of the message signal at an equal amplitude. The purpose of the output filter is to remove these harmonics.
Intuitively, it is also clear that the output filter is required. Different waveforms, although of equal amplitude, can in fact be derived from the same samples, as shown
(a)
(b)
Fig. 3-11 The basic elements within a pulse modulation system:
(a) minimum elements required in a sampling system; (b) minimum elements required in a PCM system.
Communication by Sampling 49
in Fig. 3-12. The output filter has a low-pass characteristic and retrieves the message signal without ambiguity by restricting it to a limited frequency range.
A sampling system based on peM, or time division multiplexed PAM, requires an element known as the sample and hold, or holding, circuit. This circuit holds the amplitude value of each impulse for the duration of the sampling interval T.
The sample and hold circuit is used at the transmitter to give a peM encoder sufficient time to perform a series of operations that result in the coded pulse pattern.
At the receiver, the circuit is used to "stretch" the narrow impulses that are provided by demultiplexing the composite time division multiplexed signal (Fig. 3-13). The stretched signal is a rectangular wave which approximates the message waveshape (Fig. 3-13d).
The implementation of the sample and hold process is based on rapidly charging a capacitor to a potential which represents the sample amplitude. The narrow impulses that are derived from a low impedance source are allowed access to the capacitor by a switch, which is quickly open circuited toward the end of each impulse. Provided that the leakage from the capacitor is small, the sample values will be retained during the complete sampling interval T, whereupon the switch is closed once more, and a new impulse value is recorded.
The transfer function of the sample and hold circuit is that of a low-pass filter.
This may be explained by considering a network with the transform H(jw), as shown in Fig. 3-14. The input to this network is a pulse of width T, and amplitude A.
50 Pulse-Code Modulation
Demultiplex /
r----yIJ;-c--f1 (t)
TDM signal circuit
- - - - , . ; ; . . . . - - - . , ; ; . . ; , . , ; , ; . . . . ; . - - - ' ) cr
I
f2(t}\ ;,;C
~f3(t}
I
CHolding circuits (a)
(b)
I // --
3 3 <,~- I
-(c)
(d)
Fig. 3-13 The effect of the sample and hold circuit: (a) basic TDM switch; (b) composite TDM signal; (c) demultiplexed signal
!set); (d) stretched signal after holding circuit !set).
T
r-,---A
~ L---
o I--T----1Fig. 3-14 The transfer function, associated with the sample and hold operation.
This of course represents the transfer characteristics of a perfect sample and hold circuit. The filtration is provided by the component in parentheses, which is the sine function identified earlier in this chapter. Thus, at high frequencies because sine (w T/21T) tends toward zero, there is attenuation. The filter characteristic is identified in Fig. 3-15, where it is compared to the spectrum of the sampled signal.
The two curves are similar in shape since they both contain sine terms in their defining equations; however, the null points are different.
Communication by Sampling 51
Distortion due to sample-hold sin (wT/2)
(wT/2)
O~~~~ _ _ ~L-_ _ _ _ _ f max I
: Frequency ISampling frequency
Fig. 3-15 The filtration characteristic of the sample, and hold function.
Careful analysis of Fig. 3-15 will reveal that there is some distortion in the range
o
to !max, due to the unequal transmission of the spectral components in this range.Usually this distortion is small, and acceptable; however, if required, an equalizer that has the response lIsinc (wTI2r) may be added after the sample and hold circuit.
The equalizer is here a passive circuit, which corrects for the distortion and yields an overall transfer characteristic that is flat.
It is also clear from Fig. 3-15 that when the sample and hold circuit is used at the receiver, the filtration is inadequate for retrieving the message signal information and suppressing the spectral harmonics. Some improvement in the filtration is ob-tained by increasing the sample rate, as shown in the diagram. However, an additional output filter is still required, and this ideally should have a tailored response such that the overall transfer characteristic is optimized. The desired characteristic is rectangular, as shown in Fig. 3-8a; however, this can only be approximated and in practice something less than ideal is acceptable.
We have assumed in the treatment of the sample and hold process that the sampling (switching) operation takes an infinitesimal time. Also, that the charging and dis-charging times of the capacitor are equally rapid. This is obviously not so. In fact, the optimization of a sample and hold circuits is rather complex, and is beyond the scope of this book.
The implementation of the sampling operation has so far been considered as mechan-ical, while in practice electronic devices are used. A transmission circuit is required that has at its output during the sampling interval an exact reproduction of the input wave, and is otherwise zero. Such circuits are the subject of extensive literature, 9 and consequently, will not be duplicated here.
In this chapter we have identified several types of pulse modulation systems, and have shown PCM to be the most useful from the transmission point of view. In subsequent chapters the methods of implementing, and the theoretical limitations imposed by PCM transmission will be described.
8 J. R. Gray and S. C. Kitsopoulous, "A Precision Sample and Hold Circuit with Subnanosecond Switch-ing," IEEE Transactions on Cable Television. vol. CTII. 1964, p. 389.
9 J. Millman and H. Taub. Pulse. Digital and Switching Waveforms. McGraw-Hill, New York. 1965.