`
`SAMSUNG 1020
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`1
`
`

`

`Library of Congress Cataloging in Publication Data
`OPPENHEIM, ALAN V.
`(date)
`Signals and systems.
`eesignal processing series)
`1. ‘Seaat
`2.
`Signal
`thi
`ana! a i
`eory
`(Telecommunication,me Willsky, Allan S.
`a aoe t,
`Il. Title. WV."Serial.
`Qadoe
`003
`81-22652
`BN 013-809731.3
`AACR2
`
`Editorial production/supervision
`by Gretchen K. Chenenko
`Chapter opening design
`by Dawn Stanley
`Cover executed by Judy Matz
`Manufacturing buyers: Joyce Levatino
`and Anthony Caruso
`
`© 1983 by Alan V. Oppenheim, Alan S. Willsky, and Ian T. Young
`
`All rights reserved. No part of this book
`may be reproduced in any form or
`by any means without permission in writing
`from the publisher.
`
`Printed in the United States of America
`
`10 9
`
`LSBN O-13-8609731-3
`
`Prentice-Hall International, Inc., London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`Prentice-Hall Canada Inc., Toronto
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of Japan, Inc., Tokyo
`Prentice-Hall of Southeast Asia Pte. Ltd., Singapore
`Whitehall Books Limited, Wellington, New Zealand
`
`2
`
`

`

`7
`Modulation
`
`447
`
`7.0
`TA
`Wd
`7.3
`7.4
`A)
`7.6
`dsb
`
`447
`Introduction
`Continuous-Time Sinusoidal Amplitude Modulation
`Some Applications of Sinusoidal Amplitude Modulation
`Single-Sideband Amplitude Modulation
`464
`Pulse Amplitude Modulation and Time-Division Multiplexing
`Discrete-Time. Amplitude Modulation
`473
`Continuous-Time Frequency Modulation
`479
`Summary
`487
`Problems
`487
`
`449
`
`459
`
`469
`
`8
`Sampling
`
`513
`
`8.0
`8.1
`
`8.2
`
`513
`Introduction
`Representation of a Continuous-Time Signal by Its Samples:
`The Sampling Theorem
`514
`Reconstruction of a Signal
`521
`from Its Samples Using Interpolation
`527
`The Effect of Undersampling: Aliasing
`Discrete-Time Processing of Continuous-Time Signals
`Sampling in the Frequency Domain
`540
`Sampling of Discrete-Time Signals
`543
`Discrete-Time Decimation andInterpolation
`Summary
`553
`Problems
`555
`
`548
`
`531
`
`The
`
`Laplace Transform 573
`
`573
`Introduction
`573
`The Laplace Transform
`The Region of Convergence for Laplace Transforms
`The Inverse Laplace Transform
`587
`Geometric Evaluation of the Fourier Transform
`from the Pole-—Zero Plot
`590
`
`379
`
`Content
`
`3
`
`

`

`
`
`4
`
`

`

`a continuous-time signal to a discrete-time signal. After processing the discrete-time
`signal using a discrete-time system, we can then convert back to continuoustime.
`In the following discussion, wefirst introduce and develop the concept of sam-
`pling and the process of reconstructing a continuous-time signal from its samples.
`Wethen explore the processing of continuous-time signals that have been converted
`
`to discrete-time signals through sampling. Next we consider theae conceptto time-
`
`domain sampling, specifically sampling in the frequency domaix. Finally, we develop
`the concept and someapplications of sampling applied to discrete-time signals.
`
`8.1 REPRESENTATION OF A CONTINUOUS-TIME SIGNAL
`BY ITS SAMPLES: THE SAMPLING THEOREM
`
`In general, we could not expect that in the absence of any additional conditions or
`information, a signal could be uniquely specified by a sequence of equally spaced
`samples. For example, in Figure 8.1 we illustrate three different continuous-time
`signals, all of which haveidentical values at integer multiples of T, thatis,
`-x,(KT) = x,(kT) = x3(kT)
`
`X3(t)
`
`x, (t)
`
`—*x,(t)
`
`Figure 8.1 Three continuous-time signals with identical values at integer mul-
`tiples of T.
`
`In general, there are an infinite number of signals that can generate a given set of
`samples. As we will see, however, if a signal is bandlimited and if the samples are
`taken sufficiently close together, in relation to the highest frequency present in the
`signal, then the samples uniquely specify the signal and we can reconstructit perfectly.
`Thebasic result was suggested in Section 7.4 in the context of pulse amplitude modu-
`lation. Specifically, if a bandlimited signal x(t) is amplitude-modulated with a periodic
`pulse train, corresponding to extracting equally spaced time segments, it can be
`recovered exactly by lowpassfiltering if the fundamental frequency of the modulating
`pulse train is greater than twice the highest frequency present in x(t). Furthermore,
`the ability to recover x(t) is independentof the time duration of the individual pulses.
`Thus, as suggested by Figures 8.2 and 8.3 as this duration becomesarbitrarily small,
`pulse amplitude modulationis, in effect, representing x(t) by instantaneous samples
`equally spaced in time. In the pulse-amplitude-modulation system in Figure 8.2, we
`
`514
`
`Sampling
`
`Chap. 8
`
`5
`
`

`

`p(t)
`
`x(t)
`
`-
`
`y(t)
`
`a FOSee
`
`
`t
`
`p(t)
`
`y(t)
`
`ie 7 >|
`
`i
`A
`
`be
`A
`
`0
`
`:
`
`t
`
`t
`
`Figure 8.2 Pulse amplitude modulation. As A — 0,p(t) approachesan impulse
`train.
`:
`
`have scaled the amplitude of the pulse train to be inversely proportional to the pulse
`width A, In anypractical pulse-amplitude-modulation system,it is particularly impor-
`tant as A becomes small to maintain a constant time-average power in the modulated
`signal. Asillustrated in Figure 8.3, as A approaches zero the modulated signal then
`becomesan impulse train for which the individual impulses have values corresponding
`to instantaneous samples of x(t) at time instants spaced T seconds apart.
`
`8.1.1 Impulse-Train Sampling
`
`In a manneridentical to that used to analyze the more general case of pulse ampli-
`tude modulation, let us consider the specific case of impulse-train sampling depicted
`in Figure 8.3. The impulse train p(t) is referred to as the sampling function, the period
`T as the sampling period, and the fundamental frequency of p(t), w, = 22/T, as the
`sampling frequency. In the time domain we have
`X(t) = xp)
`
`(8.1a)
`
`where
`
`(8.1b)
`p(t) = DELC —nT)
`x,(t) is an impulse train with the amplitudes of the impulses equal to the samples
`of x(t)at intervals spaced by 7,thatis,
`
`Sec. 8.1
`
`Representation of a Continuous-Time Signal by Its Samples
`
`515
`
`6
`
`

`

`p(t)
`
`x(t)
`
`x, (t)
`
`
`
`Figure 8.3. Pulse amplitude modulation with an impulsetrain.
`
`x0) = ¥2 x(nT) d(t — nT)
`
`From the modulation property [Sec. 4.8],
`X,(@) = LIX@) * Po)
`
`and from Example 4.15,
`:
`
`so that
`
`2x S&S
`P@) =F Ls O(@ — ka,)
`k=-00
`
`(8.2)
`
`(8.3)
`
`(8.4)
`
`X,(@) = $ S. X(@ — ko,)
`That is, X,(@) is a periodic function of frequency consisting of a sum of shifted
`replicas of X(q@), scaled by 1/7 as illustrated in Figure 8.4.
`In Figure 8.4(c),
`Oy < (@, — My) or equivalently @, > 2m,,, and thus there is no overlap between
`the shifted replicas of X(@), whereas in Figure 8.4(d) with w, < 2@,,, there is
`overlap. For the case illustrated in Figure 8.4(c), X(@) is faithfully reproduced at
`integer multiples of the sampling frequency. Consequently, if @, > 2@,,, x(t) can be
`recovered exactly from x,(t) by means of a lowpassfilter with gain T and a cutoff
`
`(8.5)
`
`516
`
`Sampling
`
`Chap. 8
`
`7
`
`

`

`X(w)
`
`
`
`(d)
`
`(wy, -— wy)
`
`Figure 8.4 Effect in the frequency domain of sampling in the time domain: (a)
`spectrum of original signal; (b) spectrum of sampling function; (c) spectrum of
`sampled signal with a, > 2@.4; (d) spectrum of sampled signal with a; < 2mm.
`
`frequency greater than w,, and less than w, — @,, as indicated in Figure 8.5. This
`basic result, referred to as the sampling theorem, can be stated as follows:t
`
`{This important and elegant theorem was available for many years in a variety of forms in
`the mathematics literature. See, for example, J. M. Whittaker, “Interpolatory Function Theory,”
`Cambridge Tracts in Mathematics and Mathematical Physics, no. 33 (Cambridge, 1935), chap. 4.
`It did not appear explicitly in the literature of communication theory until the publication in 1949 of
`the classic paper by Shannon entitled “Communication in the Presence of Noise” (Proceedings of the
`IRE, January, 1949, pp. 10-21). However, H. Nyquist in 1928 and D. Gaborin 1946 hadpointed out,
`based on the use of Fourier Series, that 2TW numbersare sufficient to represent a function of time
`duration T and highest frequency W. [H. Nyquist, “Certain Topics in Telegraph Transmission
`Theory,” AIEE Transactions, 1946, p. 617; D. Gabor, “Theory of Communication,” Journal of IEE
`93, no. 26 (1946): 429.]
`
`Sec. 8.1
`
`Representation of a Continuous-Time Signal by Its Samples
`.
`
`517
`
`8
`
`

`

`foo
`z 6(t-—nT)
`
`p(t) it
`
`xlt}
`
`(x)
`
`Xp (t)
`
`x,(t)
`
`X(w)
`
`1
`
`—wy
`
`mu
`
`w
`
`X, (w)
`
`1H
`
`w, > 2wy
`
`-w,
`
`—wy
`
`wm
`
`,
`
`w
`
`H(w)
`
`
`
`—wy
`
`Ou
`
`w
`
`Figure 8.5 Exact recovery of a continuous-time signal from its samples using
`an ideal lowpass filter.
`
`518
`
`9
`
`

`

`Sampling Theorem:
`Let x(t) be a bandlimited signal with X(@) = 0 for |@| > @y. Then x(t)
`is uniquely determined by its samples x(mT), n = 0, +1, +2,...if
`@, > 20,4
`
`where
`
`(@, — @,). The resulting output signal will exactly equal x(t).
`
`2n
`QO, =F
`
`Given these samples, we can reconstruct x(t) by generating a periodic
`impulse train in which successive impulses have amplitudes that are successive
`sample values. This impulse train is then processed through an ideal lowpass
`filter with gain T and cutoff frequency greater than m,, and less than
`
`The sampling frequency a,is also referred to as the Nyquist frequency. The frequency
`2@,,, which, under the sampling theorem, must be exceeded by the sampling
`frequency, is commonlyreferred to as the Nyquistrate.
`
`8.7.2 Sampling with a Zero-Order Hold
`
`The sampling theorem establishes the fact that a bandlimited signal is uniquely repre-
`sented by its samples, and is motivated on the basis of impulse-train sampling. In
`practice, narrow large-amplitude pulses, which approximate impulses, are relatively
`difficult to generate and transmit, and it is often more convenient to generate the
`sampled signal in a form referred to as a zero-order hold. Such a system samples x(t)
`at a given sampling instant and holdsthat value until the succeeding samplinginstant,
`as illustrated in Figure 8.6. Reconstruction of x(¢) from the output of a zero-order hold
`
` 4frere ; x a
`
`7
`
`“XN
`
`
`
`
`
`
`
`x(t)
`
`Zero-order
`hold
`
`Xo (t)
`
`Figure 8.6 Sampling utilizing a zero-order hold.
`
`can again be carried out by lowpassfiltering. However, in this case, the required filter
`no longer has constant gain in the passband. To develop the required filter charac-
`teristic, we first note that the output x,(t) of the zero-order hold can in principle be
`generated by impulse-train sampling followed by an LTI system with a rectangular
`impulse responseas depicted in Figure 8.7. To reconstruct x(t) from x,(t), we consider
`processing x,(t) with an LTI system with impulse response ,(t) and frequency
`response H,(q@). The cascade of this system with the system of Figure 8.7 is shown in
`Figure 8.8, where we wish to specify H,(@) so that r(t) = x(t). Comparing the system
`in Figure 8.8 with that in Figure 8.5, we see that r(t)=x(t) if the cascade combination
`of h,(t) and A,(t) is the ideal lowpass filter H(@) used in Figure 8.5. Since, from
`\ det as { Fano\ os » \
`Representation of a Continuous-Time Signal by Its Samples
`
`Sec. 8.1
`
`519
`
`10
`
`10
`
`

`

`Xo (t)
`
`x(t)
`
`
`Xq (t)
`
`Figure 8.7 Zero-order hold as impulse train sampling followed by convolution
`with a rectangular pulse.
`
`r(t)
`
`x(t)
`
`Figure 8.8 Cascade of the representation of a zero-order hold (Figure 8.7) with
`a reconstructionfilter.
`
`520
`
`11
`
`11
`
`

`

`Example 4.10 and the time-shifting property 4.6.3
`
`Hoa) = eter{28OT/?))
`
`(8.6)
`
`This requires that
`__eTH(@)
`
`H(@) = eee
`
`@
`
`(8.7)
`
`For example with the cutoff frequency of H(w) as @,/2, the ideal magnitude and
`phase for the reconstruction filter following a zero-order hold is that shown in
`Figure 8.9,
`
`| H, (co) |
`
`2
`
`Ws
`Qi
`
`construction filter for zero-order hold.
`
`Figure 8.9 Magnitude and phase forre-
`
`In manysituations the zero-order hold is considered to be an adequate approxi-
`mation to the original signal without any additional lowpassfiltering and in essence
`represents a possible, although admittedly very coarse, interpolation between the
`sample values. In the next section we explore in more detail the general concept
`of interpreting the reconstruction of a signal from its samples as a process ofinter-
`polation.
`
`8.2 RECONSTRUCTION OF A SIGNAL FROM ITS SAMPLES
`USING INTERPOLATION
`
`Interpolation is a commonly used procedure for reconstructing a function either
`approximately or exactly from samples. One simple interpolation procedure is the
`zero-order hold discussed in Section 8.1. Another simple and useful form of inter-
`polation is Jinear interpolation, whereby adjacent sample points are connected by
`a straight line as illustrated in Figure 8.10. In more complicated interpolation for-
`
`Sec. 8.2
`
`Reconstruction of a Signal from Its Samples Using Interpolation
`
`521
`
`12
`
`12
`
`

`

`
`
`Figure 8.10 Linear interpolation between sample points. The dashed curve
`represents the original signal and the solid curve the linear interpolation.
`
`mulas, sample points may be connected by higher-order polynomials or other mathe-
`matical functions.
`Aswe haveseen in Section 8.1, for a bandlimited signal, if the sampling instants
`are sufficiently close, then the signal can be reconstructed exactly, i.e., through the
`use of a lowpassfilter exact interpolation can be carried out between the sample
`points. The interpretation of the recon
`tion of x(t) as a process of interpolation
`becomes evident when weconsiderthe effect in the time domain of the lowpassfilter
`in Figure 8.5. In particular, the output x,(¢) is
`x,(t) = x,(0) * ACO
`.
`or with x,(t) given by eq. (8.2),
`x,(t) = 3x(nT)A(t — nT)
`
`(8.8)
`
`Equation (8.8) represents an interpolation formula since it describes how to fit a
`continuous curve between the sample points. For the ideal lowpass filter H(@) in
`Figure 8.5, A(t) is given by
`,
`
`so that
`
`ts eos
`
`h(t) = Te sine (2+)
`
`x)= 3 x(nT)T2 sinc (ey
`
`(8.9)
`
`(8.10)
`
`The reconstruction according to eq. (8.10) with w, = @,/2 is illustrated in Figure
`8.11.
`
`Interpolation using the sinc function as in eq. (8.10) is commonly referred to
`as bandlimited interpolation, since it implements exact reconstruction if x(t) is band-
`limited and the sampling frequencysatisfies the conditions of the sampling theorem.
`Since a very good approximation to an ideal lowpassfilter is relatively difficult to
`implement, in many casesit is preferable to use a less accurate but simpler filter (or
`equivalently interpolating function) A(t). For example, as we previously indicated, the
`zero-order hold can be viewed as a form of interpolation between sample values in
`which the interpolating function A(t) is the impulse response /,(t) depicted in Figure
`8.7. In that sense, with x,(¢) in Figure 8.7 corresponding to the approximationto x(f),
`the system /,(t) represents an approximation to the ideal lowpassfilter required for
`the exact interpolation. Figure 8.12 shows the magnitudeofthe transfer function of
`the zero-order-hold interpolating filter, superimposed on the desired transfer func-
`tion ofthe exact interpolatingfilter. Both from Figure 8.12 and from Figure8.7 wesee
`
`522
`
`Sampling
`
`Chap. 8
`
`13
`
`13
`
`

`

`x(t)
`
`(a)
`
`
`lation using the sinc function.
`
`Figure 8.11
`
`Ideal bandlimited interpo-
`
`(b)
`
`
`
`Ideal interpolating
`filter
`
`Zero-order
`hold
`
`
`
`
`—@;
`
`Ws
`2
`
`Ww,
`2
`
`Figure 8.12 Transfer function for the zero-order hold and for the ideal inter-
`polatingfilter.
`
`that the zero-order hold is a very rough approximation, although in somecasesitis
`sufficient. For example, if, in a given application, there is additional lowpassfiltering
`that is naturally applied, this will tend to improve the overall interpolation. This
`is illustrated in the case of pictures in Figure 8.13. Figure 8.13(a) showsa picture with
`“impulse” sampling(i.e., sampling with spatially narrow pulses). Figure 8.13(b) is the
`result of applying a two-dimensional zero-order hold to Figure 8.13(a) with a resulting
`mosaic effect when viewed at close range. However, the humanvisual system inherently
`
`Sec. 8.2
`
`Reconstruction of a Signal from Its Samples Using Interpolation
`
`523
`
`14
`
`14
`
`

`

`
`
`
`
`(a) 7
`
`(b)
`
`(a) The original pictures of Figs. 2.2 and 4.2 with impulse sam-
`Figure 8.13.
`pling;
`(b) zero-order hold applied to the pictures in (a). The visual system
`naturally introduces lowpassfiltering with a cutoff frequency that increases with
`distance. Thus, when viewed at a distance, the discontinuities in the mosaic in
`Figure 8.13(b) are not resolved; (c) result of applying a zero-order hold after
`impulse sampling with one-half the horizontal and vertical spacing used in (a)
`and (b).
`
`524
`
`15
`
`15
`
`

`

`(c) Figure 8.13 (cont.)
`
`imposes lowpassfiltering, and consequently when viewed at a distance, the discon-
`tinuities in the mosaic are not resolved. In Figure 8.13(c) a zero-order hold is again
`used, but here the sample spacing in each direction is half that in Figure 8.13(a). With
`normal viewing, considerable lowpassfiltering is naturally applied although, partic-
`ularly with a magnifying glass, the mosaic effect is still somewhat evident.
`Another approximate form of interpolation often used is linear interpolation,
`for which the reconstructed signal is continuous, althoughits derivative is not. Linear
`interpolation, sometimes referred to as a first-order hold, wasillustrated in Figure
`8.10 and can also be viewed as an interpolation in the form of Figure 8.5 and eq.
`(8.8) with A(t) triangular,as illustrated in Figure 8.14. The associated transfer function
`H(q@)is also shown in Figure 8.14 and is given by
`_ 1
`fsin (@7T/2)
`
`(8.11)
`
`H(o) = 7|
`
`7?
`
`The transfer function ofthefirst-order hold in Figure 8.14 is shown superimposed on
`the transfer function for the ideal interpolating filter. Figure 8.15 corresponds to the
`same pictures as in Figure 8.13 but withafirst-order hold applied to the sampled
`picture.
`
`Sec. 8.2
`
`Reconstruction of a Signal from Its Samples Using Interpolation
`
`525
`
`16
`
`16
`
`

`

`x_(t)
`
`
`
`h(t)
`
`
`
`
`
`Ideal interpolating
`filter
`
`
`First-order
`
`hold
`
`
`—w
`Ws
`0
`w,
`WO, W
`
`Figure 8.14 Linear interpolation (first-order hold) as impulse-train sampling
`followed by convolution with a triangular impulse response.
`
`526
`
`17
`
`17
`
`

`

`
`
`Figure 8.15 Figure 8.13 with a first-order hold applied to the sampled pictures.
`
`8.3 THE EFFECT OF UNDERSAMPLING: ALIASING
`
`In the discussion in previous sections, it was assumed that the sampling frequency
`was sufficiently high so that the conditions of the sampling frequency were met. As
`wasillustrated in Figure 8.4, with w, > 2@,,, the spectrum of the sampled signal
`consists of exact replications of the spectrum of x(t), and this forms the basis for the
`sampling theorem. When @, < 2@,,, X(@), the spectrum of x(t), is no longer repli-
`cated in X,(@) and thus is no longer recoverable by lowpassfiltering. This effect, in
`which the individual terms in eq. (8.5) overlap, is referred to as aliasing, and in this
`section we explore its effect and consequences,
`Clearly, if the system of Figure 8.5 is applied to a signal with wm, < 2@,,, the
`reconstructed signal x,(t) will no longer be equal to x(t). However, as explored in
`Problem 8.4 the original signal and the signal x,(t) which is reconstructed using
`bandlimited interpolation will always be equal at the sampling instants; that is,
`for any choice of @,,
`
`(8.12)
`n=0, +1, +2,...
`x,(nT) = x(nT),
`Someinsight into the relationship between x(t) and x,(t) when @, < 2@,, is
`provided by considering in more detail the comparatively simple case of a sinusoidal
`signal. Thus, let x(t) be given by
`
`(8.13)
`x(t) = COS Wot
`with Fourier transform X(q@) as indicated in Figure 8.16(a). In this figure, we have
`graphically distinguished the impulse at @, from that at —q@, for convenience as
`the discussion proceeds. Let us consider X,(@), the spectrum of the sampled signal
`and focusin particular on the effect of a change in the frequency w, with the sampling
`frequency @,fixed. In Figure 8.16(b) — (e) weillustrate X,(@) for several values of
`
`Sec. 8.3
`
`The Effect of Undersampling: Aliasing
`
`527
`
`18
`
`18
`
`