19Discrete-TimeSampling

In the previous lectures we discussed sampling of continuous-time signals. Inthis lecture we address the parallel topic of discrete-time sampling, which hasa number of important applications. The basic concept of discrete-time sam-pling is similar to that of continuous-time sampling. Specifically, we multiplya discrete-time sequence by a periodic impulse train, thus retaining every Nthsample and setting the remaining ones to zero (where N denotes the period ofthe sampling impulse train). The consequences in the frequency domain andthe constraints on the bandwidth of the original sequence such that it canbe recovered from its samples parallel those for continuous time. Under theconstraints of the sampling theorem, exact interpolation can again be imple-mented with an ideal lowpass filter.

Closely associated with, but not identical to, the concept of discrete-timesampling is that of decimation or downsampling. After sampling a sequencewith an impulse train, we have obtained a new sequence that is nonzero onlyat multiples of the sampling period N. Consequently, in many practical situa-tions there is no reason to explicitly retain these zero values since they canalways be reinserted. Thus, somewhat distinct from the notion of sampling isthe concept of decimation, whereby a new sequence is generated from theoriginal sequence by selecting every Nth sample. This in effect results in atime compression. Although not typically implemented this way, it can bethought of as a two-step process, the first step consisting of periodic samplingand the second step corresponding to discarding the zero values between thesamples. Decimation is also commonly referred to as downsampling since ifthe original sequence resulted from time sampling a continuous-time signal,the new sequence resulting from decimation would be exactly what wouldhave been obtained had a lower sampling rate been used originally. If, for ex-ample, a continuous-time signal is sampled at or near the Nyquist rate and isthen processed by a discrete-time system that provides some further band-limiting, downsampling or decimation is often used.

The reverse of downsampling is "upsampling," whereby we attempt toreconstruct the original sequence. The process is again best thought of in twostages, the first corresponding to converting the decimated sequence to a

19-1

Signals and Systems19-2

sampled sequence by reinserting the (N- 1) zero values between the samplepoints. The second stage is interpolation with a lowpass filter to construct theoriginal sequence.

The processes of downsampling and upsampling have a number of prac-tical implications. One, as indicated above, is sampling rate conversion afteradditional processing. Another very important one is converting a sequencefrom one sampling rate to another perhaps to generate compatibility betweenotherwise incompatible systems. For example, it is often important to convertbetween different digital audio systems that use different sampling rates.

In this lecture we also briefly discuss the concept of sampling in the fre-quency domain. Frequency-domain sampling typically arises when we wouldlike to measure or explicitly evaluate numerically the Fourier transform. Al-though in general the Fourier transform for both continuous time and discretetime is a function of a continuous-frequency variable, the measurement orcalculation must be made only at a set of sample frequencies. Because of theduality between the time and frequency domains for continuous time, the is-sues, analysis, and concepts related to frequency-domain sampling for con-tinuous-time signals are exactly dual to those of time-domain sampling. Thus,for example, the Fourier transform can exactly be recovered from equallyspaced samples in the frequency domain provided that the time-domain signalis timelimited (the dual of bandlimited). Basically, the same result applies indiscrete time, i.e., the Fourier transform of a timelimited sequence can be ex-actly represented by and recovered from equally spaced samples providedthat the sample spacing in frequency is sufficiently small in relation to thetime duration of the signal in the time domain.

Suggested ReadingSection 8.6, Sampling of Discrete-Time Signals, pages 543-548

Section 8.7, Discrete-Time Decimation and Interpolation, pages 548-553

Section 8.5, Sampling in the Frequency Domain, pages 540-543

Discrete-Time Sampling19-3

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Signals and Systems

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Discrete-Time Sampling19-9

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Signals and Systems19-10

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Discrete-Time Sampling19-11

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TRANSPARENCY19.14Signal recovery afterfrequency-domainsampling.

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Signals and Systems

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