Apple 1025
`
`U.S. Pat. 6,470,399
`
`Apple 1025
`U.S. Pat. 6,470,399
`
`

`
`Disc;i*ete—Timé
`
`Signai Pracessing
`
`

`
`PRENTICE HALL SIGNAL PROCESSING SERIES
`
`Alan V. Oppenheim, Eeiitor
`
`ANDREWS AND HUNT Digital Image Restoration
`BRIG-HAM The Fast Fourier Transform
`BRIGHAM The Fast Fourier Transform and Its Applications
`BURDIC Underwater Acoustic System Analysis
`CASTLEMAN Digital Image Processing
`COWAN AND GRANT Adaptive Filters
`CROCHIERE AND RABINER Multirate Digital Signal Processing
`DUDGEON AND MERSEREAU Multidimensional Digital Signal Processing
`HAMMING Digital Filters, 3/E
`HAYKIN, ED. Array Signal Processing
`JAYANT AND NOLL Digital Coding of Waveforms
`KAY Modern Spectral Estimation
`KINO Acoustic Waves: Devices, Imaging, and Analog Signal Processing
`LEA, ED. Trends in Speech Recognition
`LIM Two-Dimensional Signal and Image Processing
`LIM, ED.
`Speeck Enhancement
`LIM AND OPPENHEIM, EDS. Advanced Topics in Signal Processing
`MARPLE Digital Spectral Analysis with Applications
`MCCLELLAN AND RADER Number Theory in Digital Signal Processing
`MENDEL Lessons in Digital Estimation Theory
`OPPENHEIM, ED. Applications of Digital Signal Processing
`OPPENHEIM, WILLSKY, WITH YOUNG Signals and Systems
`OPPENHEIM AND SCHAFER Digital Signal Processing
`OPPENHEIM AND SCHAFER Discrete-Time Signal Processing
`QUACKENEUSH ET AL. Objective Measures of Speech Quality
`RABINER AND GOLD Theory and Applications of Digital Signal Processing
`RABINER AND SCHAFER Digital Processing of Speech Signals
`ROBINSON AND TREITEL Geophysical Signal Analysis
`STEARNS AND DAVID Signal Processing Algorithms
`TRIBOLET Seismic Applications of Homomorphic Signal Processing
`WIDROW AND STEARNS Adaptive Signal Processing
`
`

`
`Au.-—;—y
`
`
`
`.._.........a-1-—-.-1mw.l>'A»t&9~
`
`Library of Congress Catalogéng-in-Publication Data
`
`Oppenheim, Alan V.
`Discrete-time signal processing / Alan V. Oppenheim, Ronald W.
`Schafer.
`cm.—(Prentice Hall signal processing series)
`p.
`Bibliography:
`p.
`Includes index.
`ISBN 0-13-216292-X
`2. Discrete-time systems.
`1. Signal processing—Mathematics.
`I. Schafer, Ronald W.
`II. Title.
`III. Series.
`TK5102.5.02452
`1989
`621.38’043—dc 19
`
`88-25 562
`CIP
`
`Editorial/production supervision: Barbara G. Flanagan
`Interior design: Roger Brower
`Cover design: Vivian Berman
`Manufacturing buyer: Mary Noonan
`
`nu'|||
`
`© 1989 Alan V. Oppenheim. Ronald W. Schafer
`Published by Prentice-Hall. Inc.
`A Division of Simon & Schuster
`
`Englewood Cliffs, New Jersey 07632
`
`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 8 7 6 5
`
`ISBN D-L3-E]i1:E"lE-X
`
`Prentice-Hall International (UK) Limited, London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada Inc., Toronto
`
`Prentice-Hall Hispanoamericana, S.A., Mexico
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of Japan, Inc., Tokyo
`Simon & Schuster Asia Pte. Ltd., Singapore
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`
`

`
`4
`
`Introduction
`
`Chap. I
`
`response of a certain class of discrete-time filters, the signal value at any time index is a
`linear function of (and thus linearly predictable from) previous values. Consequently,
`efficient signal representations can be obtained by estimating these prediction
`parameters and using them along with the prediction error to represent the signal. The
`signal can then be regenerated when needed from the model parameters. This class of
`signal coding techniques has been particularly effective, in speech coding and is
`described in considerable detail in Jayant and N011 (1984), Markel and Gray (1976),
`and Rabiner and Schafer (1978).
`Another advanced topic of considerable importance is adaptive signal process-
`ing. In this text the emphasis is almost entirely on linear time-invariant systems.
`Adaptive systems represent a particular class of time-varying and, in some sense,
`nonlinear systems with broad application and with established and effective tech-
`niques for their design and analysis. Again, many of these techniques build from the
`fundamentals of discrete-time signal processing covered in this text. Details of
`adaptive signal processing are given by Haykin (1986) and Widrow and Stearns
`(1985).
`These represent only a few of the many advanced topics that extend from the
`topics covered in this text. Others include advanced and specialized filter design
`procedures, a variety of specialized algorithms for evaluation of the Fourier trans-
`form, specialized filter structures, and various advanced multirate signal processing
`techniques. An introduction to many of these advanced topics is contained. in Lim
`and Oppenheim (1988).
`It is often said that the purpose of a fundamental textbook should be to uncover
`rather than cover a subject, and in choosing the topics and depth of coverage in this
`book we have been guided by this philosophy. The preceding brief discussion of
`advanced topics and the Bibliography at the end of the book should be strongly
`suggestive of the rich variety of directions that these fundamentals begin to uncover.
`
`Hlstorlcal Perspective
`
`Discrete~time signal processing has a rich history. It has advanced in uneven steps
`over a long period of time. Since the invention of calculus in the 17th century,
`scientists and engineers have developed models to represent physical phenomena in
`terms of functions of continuous variables and differential equations. Numerical
`techniques have been used to solve these equations when analytical solutions are not
`possible. Indeed, Newton used finite-difference methods that are special cases of some
`of the discrete-time systems that we present in this text. Mathematicians of the 18th
`century, such as Euler, Bernoulli, and Lagrange, developed methods for numerécal
`integration and interpolation of functions of a continuous variable. Interesting
`historical research by Heideman, Johnson, and Burrus (1984) showed that Gauss
`discovered the fundamental principle of the fast Fourier transform (discussed in
`Chapter 9) as early as 1805 — even before the publication of Fourier’s treatise on
`harmonic series representation of functions.
`Until the early 1950s, signal processing as we have defined it was typically done
`with analog systems that were implemented with electronic circuits or even with
`mechanical devices. Even though digital computers were becoming available in
`
`

`
`...-,__..__.,....._-.__._._-‘
`
`
`
`
`w»_ _i\§w--———— ——*-
`
`Introduction
`
`Chap.
`
`1
`
`5
`
`business and scientific laboratories, they were expensive and had relatively limited
`capabilities. About that time, the need for more sophisticated signal processing in
`some application areas created considerable interest in discrete-time signal process-
`ing. One of the first uses of. digital computers in digital signal processing was in oil
`prospecting, where seismic data could be recorded on magnetic tape for later
`processing. This type of signal processing could not generally be done in real time;
`minutes or even hours of computer time were often required to process only seconds
`of data. Even so, the flexibility of the digital computer made this alternative extremely
`inviting.
`Also in the 1950s, the use of digital computers in signal processing arose in a
`different way. Because of the flexibility of digital computers, it was often useful to
`simulate a signal processing system on a digital computer before implementing it in
`analog hardware. In this way, a new signal processing algorithm or system could be
`studied in a flexible experimental environment before committing economic and
`engineering resources to constructing it. Typical examples of such simulations were
`the vocoder simulations carried out at Lincoln Laboratory and at Bell Laboratories.
`In the implementation of an analog channel vocoder, for example, the filter character-
`istics can affect the perceived quality of the coded speeclr: signal in ways that are
`difficult to quantify objectively. Through computer simulations, these filter character-
`istics could be adjusted and the perceived quality of a speech coding system evaluated
`prior to construction of the analog equipment.
`'
`In all of these examples of signal processing using digital computers,
`the
`cornputer offered tremendous advantages in flexibility. However,
`the processing
`could not always be done in real time. Consequently, a prevalent attitude was that the
`digital computer was being used to approximate, or simulate, an analog signal
`processing system. In keeping with that style, early work on digital filteréng was very
`much concerned with ways in which a filter could be programmed on a digital
`computer so that with analog-to-digital conversion of the signal, followed by the
`digital filtering, followed by digital-to-analog conversion, the overall system approxi-
`mated a good analog filter. The notion that digital systems might, in fact, be practical
`for the actual real-time implementation of signal processing in speech communication
`or radar processing or any of a variety of other applications seemed at the most
`optimistic times to be highly speculative. Speed, cost, and size were, of course, three of
`the important factors in favor of the use of analog components.
`As signals were being processed on digital computers, researchers had a natural
`tendency to experiment with increasingly sophisticated signal processing algorithms.
`Some of these algorithms grew out of the flexibility of the digital computer and had no
`apparent practical implementation in analog equipment. Thus, many of these algor-
`ithms were treated as interesting, but somewhat impractical, ideas. An example of a
`class of algorithms of this type was the set of techniques referred to as cepstrum
`analysis and homomorphic filtering. It had been clearly demonstrated on digital
`computers that these techniques could be applied to advantage in speech bandwidth
`compression systems, deconvolution, and echo detection and removal. However,
`implementation of these techniques required the explicit evaluation of the inverse
`Fourier transform of the logarithm of the Fourier transform of the input, and the
`required accuracy and resolution of the Fourier transform were such that analog
`
`

`
`6
`
`lntroduction
`
`Chap.
`
`1
`
`spectrum analyzers were not practical. The development of such signal processing
`algorithms made the notion of all—digital implementation of signal processing systems
`even more tempting. Active work began on the investigation of digital vocoders,
`digital spectrum analyzers, and other all—digital systems, with the hope that eventually
`such systems would become practical.
`The evolution of a new point of view toward discrete-time signal processing was
`further accelerated by the disclosure by Cooley and Tukey (1965) of an efficient
`algoréthm for computation of Fourier transforms. This class of algorithms has come
`to be known as the fast Fourier transform, or FFT. The FFT was significant for
`several reasons. Many signal processing algorithms that had been developed on
`digital computers required processing times several orders of magnitude greater than
`real time. Often this was because spectrum analysis was an important component of
`the signal processing and no efficient means were available for implementing it. The
`fast Fourier transforrn algorithm reduced the computation time of the Fourier
`transform by orders of magnitude, permitting the implementation of increasingly
`sophisticated signal processing algorithms with processing times that allowed interac-
`tive experimentation with the system. Furthermore, with the realization that the fast
`Fourier transform algorithms might, in fact, be implementable in special-purpose
`digital hardware, many signal processing algorithms that previously had appeared to
`be impractical began to appear to have practical implementations.
`Another important implication of the fast Fourier transform algorithm was that
`it was an inherently discrete-time concept. It was directed toward the computation of
`the Fourier transform of a discrete-time signal or sequence and involved a set of
`properties and mathematics that was exact in the discrete-time domain——it was not
`simply an approximation to a continuous-time Fourier transform. This had the effect
`of stimulating a reformulation of many signal processing concepts and algoréthms in
`terms of discrete-time mathematics, and these techniques then formed an exact set of
`relationships in the discrete-time domain. Following this shift away from the notion
`that signal processing on a digital computer was merely an approximation to analog
`signal processing techniques, there emerged a strong interest in discrete-time signal
`processing as an important field of investigation in its own right.
`Another major development in the history of discrete-time signal processing
`occurred in the field of microelectronics. The invention and subsequent proliferation
`of the microprocessor paved the way for low-cost implementations of discrete-time
`signal processing systems. Although the first microprocessors were too slow to
`implement most discrete-time systems in real time, by the mid-1980s integrated circuit
`technology had advanced to a level that permitted the implementation of very fast
`fixed-point and floating-point microcomputers with architectures specially designed
`for implementing discrete-time signal processing algorithms. With this technology
`came the possibility for the first time of widespread application of discrete-time signal
`processing techniques.
`Microelectronics engineers continue to strive for increased circuit densities and
`production yields, and as a result, the complexity and sophistication of microelec-
`tronic systems are continually increasing. As wafer-scale integration techniques
`become highly developed, very complex discrete-time signal processing systems will
`be implemented with low cost, miniature size, and low power consumption. Conse-
`
`

`
`Introduction
`
`Chap.
`
`1
`
`7
`
`the importance of discrete-time signal processing will almost certainly
`quently,
`continue to increase. Indeed, the future development of the field is likely" to be even
`more dramatic than the course of development that we have just described. Discrete-
`time signal processing techniques are already promoting revolutionary advances in
`some fields of application. A notable example is in the area of telecommunications,
`where discrete-time signal processing techniques, microelectronic technology, and
`fiber optic transmission combine to change the nature of communication systems in
`truly revolutionary ways. A similar impact can be expected in many other areas of
`technology.
`While discrete-time signal processing is a dynamic, rapidly growing field, its
`fundamentals are well formulated. Our goal in this book is to provide a coherent
`treatment of the theory of discrete-time linear systems, filtering, sampling, and
`discrete-time Fourier analysis. The topics presented should provide the reader with
`the knowledge necessary for an appreciation of the wide scope of applications for
`discrete-time signal processing and a foundation for contributing to future develop-
`ments in this exciting field of technology.
`
`

`
`\/\/\/\/
`
`Sampling of
`Continuous-Time
`
`
`
`I
`
`-.-'?§,_\._
`
`,
`._
`
`1-
`
`w ~3g_m»;- Q.-tn
`
`-
`
`:-
`
`-;-,:-v_ 59
`
`y-
`
`-
`
`-~ ---~
`
`.
`
`'_-~-
`
`-.
`
`-
`
`-;;z_-_u-u','_-
`
`:2; v‘*s's}-
`
`--
`
`as;
`
`v _.-
`
`--
`
`-
`
`._—.---
`
`3IO
`
`!NTRGDUCTlON
`
`Discrete-time signals can arise in many ways, but they most commonly occur as
`representations of continuous-time signals. This is partly due to the fact
`that
`processing of continuous-time signals is often carried out by discrete—time processing
`of sequences obtained by sampling. It is somewhat remarkable that under reasonable
`constraints, a continuous-time signal can be adequately represented by samples. In
`this chapter we discuss the process of periodic sampling in some detail, including the
`issue of aliasing when the signal is not bandlimited or the sampling rate is not high
`enough. Of particular importance is the fact that continuous-time signal processing
`can be implemented through a process of sampling, discrete—time processing, and
`subsequent reconstruction of a continuous-time signal.
`
`:2:
`-E3
`.-
`.
`-_=»: «~sss;2:.- at“
`-.
`.4: :2; 3
`.—_—..I’ 1i’.$.-...2z mi “*"-at
`F.’-ENRlOB!C SAMPLING
`
`3.1
`
`
`.
`
`Although other possibilities exist (see Steiglitz, 1965; Oppenheim and Johnson, 1972),
`the typical method of obtaining a discrete—time representation of a continuous-time
`signal is through periodic sampling, wherein a sequence of samples x[n] is obtained
`from a continuous-time signal xc(t) according to the relation
`
`x[n] = xC(nT),
`
`—oo < n < 00.
`
`(3.1)
`
`80
`
`

`
`Sec. 3.1
`
`Periodic Sampling
`
`81
`
`
` X["] = x°("T)
`
`Figure 3.1 Block diagram
`representation of an ideal continuous-
`
`T
`
`to-discrete (C/D) converter.
`
`In Eq. (3.1), T is the sampling period, and its reciprocal, fl = 1/T, is the sampling
`frequency, in samples per second.
`We refer to a system that implements the operation of Eq. (3.1) as an ideal
`c0ntinu0us-to-discrete-time (C/D) converter, and we depict it in block diagram form as
`indicated in Fig. 3.1. As an example of tlae relationship between xc(t) and x[n], in Fig.
`2.2 we illustrated a continuous-time speech waveform and the corresponding se-
`quence of samples.
`In a practical setting, the operation of sampling is often implemented by an
`analog-to-digital (A/D) converter. Such systems can be viewed as approximations to
`the ideal C/D converter. Important considerations in the implementation or choice of
`an A/D converter include quantization of the output samples, linearity, the need for
`sample-and-hold circuits, and limitations on the sampling rate. The elfects of
`quantization are briefly discussed in Sections 3.7.2 and 3.7.3. Other practical issues of
`A/D conversion are outside the scope of this text.
`The sampling operation is generally not invertible; i.e., given the output x[n], it
`is not possible in general to reconstruct xc(t), the input to the sampler, since many
`continuous-time signals can produce the same output sequence of samples. The
`inherent ambiguity in sampling is of primary concern in signal processing. Fortu-
`nately it is possible to remove the ambiguity by restricting the class of input signals to
`the sampler.
`It is convenient to mathematically represent the sampling process in the two
`stages depicted in Figure 3.2(a). This consists of an impulse train modulator followed
`by conversion of the impulse train to a sequence. Figure 3.2(b) illustrates two
`continuous—time signals and the results of impulse train sampling. Figure 3.2(c)
`depicts the corresponding output sequences. The essential difference between xs(t)
`and x[n] is that xs(t) is, in a sense, a continuous-time signal (specifically an impulse
`train) that is zero except at integer multiples of T. The sequence x[n] on the other
`hand is indexed on the integer variable n, which in effect introduces a time normaliza-
`tion; i.e, x[n] contains no explicit information about the sampling rate. Furthermore,
`the samples of x,,(t) are represented by finite numbers in x[n] rather than as the areas
`of impulses as in xs(t).
`It is important to emphasize that Fig. 3.2(a) is a mathematical representation of
`sampling, not a representation of any physical circuits or systems’ designed to
`implement the sampling operation. Whether a piece of hardware can be construed to
`be an approximation to the block diagram of Fig. 3.2(a) is a secondary issue at this
`point. We have introduced this representation of the sampling operation because it
`leads to a simple derivation of a key result and because this approach leads to a
`number of important
`insights that are diflicult
`to obtain from a more formal
`derivation based on manipulation of Fourier transform formulas.
`
`

`
`82
`
`Sampling of Continuous-Time Signals
`
`Chap. 3
`
`Cl D converter
`
`
`
`Conversion from
`
`impulse train
`
`to discrete«time
`x[n] = xc(nT)
`sequence
`
`
`
`
`-2T-T0 T2T "
`
`t
`
`-2T
`
`—T
`
`o
`
`T
`
`2T
`
`t
`
`(b)
`
`-4-3-2-101234 n
`
`
`Figure 3.2 Sampling with a periodic impulse train followed by conversion to a
`discrete-time sequence. (a) Overall system. (b) xs(t) for two sampling rates. The
`dashed envelope represents x,(t). (c) The output sequence for the two different
`sampling rates.
`
`g~._:
`3%
`
`__
`
`_x
`
`l_.....%'
`t
`
`»
`
`.
`
`9
`
`..
`
`9'‘
`
`2°:
`
`3.2 FREQUENC7-f3GMA!N REPRESENTATTCN OF SANIFLHNEG
`
`To derive the frequency-domain relation between the input and output of an ideal
`C/D converter, let us first consider the conversion of xc(t) to xs(t) through impulse
`train modulation. The modulating signal s(t) is a periodic impulse train
`
`s(t): i 5(t—nT),
`n=-oo
`
`where 6(t) is the unit impulse function or Dirac delta function. Consequently,
`
`xs(t) = xc(t)S(t)
`
`= xc(t) i an — nT).
`n=-oo
`
`(3.2)
`
`(3.3)
`
`

`
`Sec. 3.2
`
`Frequency-Domain Representation of Sampling
`
`83
`
`Through the “sifting property” of the impulse function, xs(t) can be expressed as
`CD
`
`xs(t) = 2 xc(nT)6(t — nT).
`n=-oo
`
`(3.4)
`
`Let us now consider the Fourier transform of xs(t). Since from Eq. (3.3), x,(t) is
`the product of xc(t) and s(t), the Fourier transform of x,(t) is the convolution of the
`Fourier transforms Xc(jfl) and S(jQ). The Fouréer transform of a periodic impulse
`train is a periodic impulse train (Oppenheim and Willsky, 1983). Specifically, S(jQ) is
`
`309) =
`
`35 i am — kfls),
`T 2. =-no
`
`where Q, = 27r/ T is the sampling frequency in radians/s. Since
`
`.
`1
`.
`.
`XS(.]Q) = E XC(19) * S09),
`
`where as denotes the operation of convolution, it follows that
`CO
`
`me) =
`
`Xnn — kjQs)-
`
`(3.5)
`
`(3.6)
`
`Equation (3.6) provides the relationship between the Fourier transforms of the
`input and the output of the impulse train modulator in Fig. 3.2(a). We see from Eq.
`(3.6) that the Fourier transform of xs(t) consists of periodically repeated copies of the
`Fourier transform of x,,(t). The copies of Xc(jQ) are shifted by integer multiples of the
`sampling frequency and then superimposed to produce the periodic Fourier transform
`of the impulse train of samples. Figure 3.3 depicts the frequency-domain representa-
`tion of impulse train sampling. Figure 3.3(a) represents a bandlimited Fourier
`transform where the highest nonzero frequency component in Xc(jQ) is at QN. Figure
`3.3(b) shows the periodic impulse train S(jQ), and Fig. 3.3(c) shows Xs(jfl), the result
`of convolving Xc(jfl) wéth S(jQ). From Fig. 3.3(c) it is evident that when
`
`(2, — QN > (IN, or
`
`(25 > ZQN,
`
`(3.7)
`
`the replicas of Xc(jfl) do not overlap and therefore, when they are added together in
`Eq. (3.6), there remains (to within a scale factor of 1/ T) a replica of Xc(jfl) at each
`integer multiple of Q3. Consequently xc(t) can be recovered from xs(t) with an ideal
`lowpass filter. This is depicted in Fig. 3.4(a), which shows the impulse train
`modulator followed by a linear time-invariant system with frequency response
`H,(j§2). For Xc(jfl) as in Fig. 3.4(b), Xs(jQ) would be as shown in Fig. 3.4(c), where it
`is assumed that Q, > ZQN. Since
`
`X.09) = Hr(jQ)Xs(jQ):
`
`(3-8)
`
`it follows that if H,(jQ) is an ideal lowpass filter with gain T and cutoff frequency S],
`such that
`
`QN < Q: < (gs — QN):
`
`

`
`84
`
`Sampling of Continuous-Time Signals
`
`Chap. 3
`
`Xc (i9)
`
`(Q: - ON)
`
`Figure 3.3 Effect in the frequency domain of sampling in the time domain. (a)
`Spectrum of the original signal. (b) Spectrum of sampling function. (c) Spectrum of
`sampled signal with 95 > ZQN.
`((1) Spectrum of sampled signal with (13 < ZQN.
`
`then
`
`X'09) = XA19),
`
`(3-10)
`
`as depicted in Fig. 3.4(e).
`If the inequality (3.7) does not hold, i.e., if Q, _<_ ZQN, the copies of Xc(jQ)
`overlap so that when they are added together Xct:jfl) is rao longer recoverable by
`lowpass filtering. This is illustrated in Fig. 3.3(d). In this case, the reconstructed
`output x,(t) in Fig. 3.4(a) is related to the original continuous-time input through a
`distortion referred to as aliasing. Figure 3.5 illustrates aliasing in the frequency
`domain for the simple case of a cosine .-signal. Figure 3.5(a) shows the Fourier
`transform of the signal
`
`xC(t) = cos 90 t.
`
`(3.11)
`
`

`
`Sec. 3.2
`
`Frequency-Domain Representation of Sampling
`
`85
`
`s(t)= i an -nT)
`
`
`
`(bl
`
`
`
`(c)
`
`Hrlifll
`
`52,. <9: < (:2, — SIN)
`
`-96
`
`size
`
`S2
`
`(dl
`
`X,“-Q)
`
`1
`
`-9“
`
`QN
`
`5]
`
`(e)
`
`Figure 3.4 Exact recovery of a
`continuous-time signal from its samples
`using an ideal lowpass filter.
`
`Part (b) shows the Fourier transform of xs(t) with 90 < 95/2, and part (c) shows the
`Fourier transform of xs(t) with (20 > Q5/2. Parts ((1) and (e) correspond to the Fourier
`transform of the lowpass filter output for £20 < Q3/2 = 11/T and (20 > 7r/ T, respective-
`ly, with Q = Q5/2. Figures 3.5(c) and (e) correspond to the case of aliasing. With no
`aliasing (b and d), the reconstructed output x,(t) is
`
`With aliasing, the reconstructed output is
`
`x,(t) = cos 90 t.
`
`x,(t) = cos(§2s — §20)t,
`
`(3.12)
`
`(3.13)
`
`

`
`86
`
`Sampling of Continuous-Time Signals
`
`Chap. 3
`
`-(Q5 — Q0)
`
`(Q3 - (10)
`
`SI
`
`(el
`
`Figure 3.5 The effect of aliasing in the
`sampling of a cosine signal.
`
`the higher-frequency signal cos Qot has taken on the identity (alias) of the
`i.e.,
`lower-frequency signal cos(Qs —— Q0)t as a consequence of the sampling and recon-
`struction. This discussion is the basis for the Nyquist sampling theorem (Nyquist,
`1928; Shannon, 1949), stated as follows.
`
`Nyquist Sampling Theorem.
`
`Let xc(t) be a bandlimited signal with
`
`Xc(jQ) = 0
`
`for my > QN.
`
`(3.14a)
`
`Then xc(t) is uniquely determined by its samples x[n] = xc(nT), n = 0, 1-1,
`i2, . .
`. , if
`
`2
`
`:25 = ?” > 2nN.
`
`(3.14b)
`
`

`
`Sec. 3.3
`
`Reconstruction of a Bandlimited Signal from Its Samples
`
`87
`
`The frequency QN is commonly referred as the Nyquist frequency, and the frequency
`ZQN that must be exceeded by the sampling frequency is called the Nyquist rate.
`Thus far we have considered only the impulse train modulator in Fig.
`3.2(a). Our eventual objective is to express X(e’‘‘’), the discrete-time Fourier trans-
`form of the sequence x[n], in terms of Xs(jfl) and XC(jfl). To this end, let us consider
`an alternative expression for Xs(jQ). Applying the Fourier transform to Eq. (3.4), we
`obtain
`
`CD
`
`Xs(jQ) = Z xC(nT)e‘j“”.
`
`x[n] = xc(nT)
`
`X(ej“’) = Z x[n]e‘j“”‘,
`
`Since
`
`and
`
`it follows that
`
`X509) = X(e"")lw=m = X(e’‘")-
`Consequently from Eqs. (3.6) and (3.18),
`
`.
`1
`°°
`X(Wm) = 7. X X.09 _jkQs)s
`k = - 00
`
`or, equivalently,
`
`(3.15)
`
`(3.16)
`
`(3.17)
`
`(3-13)
`
`(119)
`
`(0
`
`1
`
`°‘’
`
`_ C0
`
`27rk
`
`(3.20)
`X<eJ > — — k=X_wX.(; ; -1 7»)
`From Eqs. (3.18)-(3.20) we see that X(ei“’) is simply a frequency-scaled version of
`Xs(jfl) with the frequency scaling specified by to = QT. This scaling can alternatively
`be thought of as a normalization of the frequency axis so that the frequency (2 = Q, in
`Xs(jQ) is normalized to co = 27: for X(e"“’). The fact that there is a frequency scaling
`or normalization in the transformation from Xs(jQ) to X(ej“’) is directly associated
`with the fact that there is a time normalization in the transformation from xs(t) to
`x[n]. Specifically, as we see in Fig. 3.2, xs(t) retains a spacing between samples equal
`to the sampling period T. In contrast, the “spacing” of sequence values x[n] is always
`unity,
`i.e., the time axis is normalized by a factor of T. Correspondingly in the
`frequency domain, the frequency axis is normalized by a factor offi = 1/T.
`
`asccrsisrnucrsoiu or A nAi\n3L|MrrEn Siam:
`FROM !TS SAMPLES
`
`oo
`
`:3.
`
`-a»
`
`According to the sampling theorem, samples of a continuous—time bandlimited signal
`taken frequently enough are sufficient to represent the signal exactly in the sense that
`the signal can be recovered from the samples and from knowledge of the sampling
`
`1-«'35-__.___—~-:—u_'§un$_-,..s.:..
`
`
`
`

`
`88
`
`Sampling of Continuous-Time Signals
`
`Chap. 3
`
`period. Impulse train modulation provides a convenient means for understanding the
`process of reconstructing the continuous-time bandlimited signal from its samples.
`In Section 3.2 we saw that if the conditions of the sampling theorem are met and
`if the modulated impulse train is filtered by an appropriate lowpass filter, then the
`Fourier transform of the filter output will be identical to the Fourier transform of the
`original continuous-time signal xc(t), and thus the output of the filter will be xc(t). If
`we are given a sequence of samples x[n], we can form an impulse train xs(t) in which ,
`successive impulses are assigned an area equal to successive sequence values, i.e.,
`02)
`
`xs(t) = Z x[n]6(t — nT).
`
`(3.21)
`
`The nth sample is associated with the impulse at t = nT, where T is the sampling
`period associated with the sequence x[n]. If this impulse train is the input to an ideal
`lowpass continuous-time filter with frequency response H,(j§E) and impulse response
`h,(t), then the output of the filter will be
`Q3
`
`x,(t) = Z x[n]h,(t — nT).
`n=—oo
`
`(3.22)
`
`A block diagram representation of this signal reconstruction process is shown in Fig.
`3.6(a). Recall that the ideal reconstruction filter has a gain of T (to compensate for the
`factor of 1/ T in Eq. 3.19 or 3.20) and a cutoff frequency QC between QN and
`Q5 — QN. A convenient and commonly used choice of the cutoff frequency is
`Q, = 95/2 = 7r/ T. This choice is appropriate for any relationship between Q3 and QN
`that avoids aliasing (i.e., so long as (25 > 2§2N). Figure 3.6(b) shows the frequency
`response of the ideal reconstruction filter. The corresponding impulse response, h,(t),
`is the inverse Fourier transform of H,(jQ), and for cutoff frequency 71/ T is given by
`
`h,(t) =
`
`sin 1rt/ T
`
`7rt/ T -
`
`(3.23)
`
`This impulse response is shown in Figure 3.6(c). From substituting Eq. (3.23) into Eq.
`(3.22) it follows that
`
`°°
`xr(t) =. Z x[n]
`= _
`
`sin[7I(t — nT)/T]
`7t(t — nT)/T
`
`(324)
`
`From the frequency-domain argument of Section 3.2 we saw that if x[n] =
`xc(nT), where Xc(j£2) =0 for
`|Q| 2 7:/T,
`then x,(t) is equal
`to xc(t). It
`is not
`immediately obvious that this is true by considering Eq. (3.24) alone. However, useful
`insight is gained by looking at Eq. (3.24) more closely. First let us consider the
`function h,(t) given by Eq. (3.23). We note that
`
`h,(0) = 1.
`
`(3.25a)
`
`This follows from l’H6pital’s rule. In addition,
`
`h,(nT) = 0
`
`for n = i 1, i2, . ...
`
`(3.25b)
`
`

`
`Sec. 3.3
`
`Reconstruction of a Bandiimited Signal from its Samples
`
`89
`
`ideal reconstruction system
`
`
`Ideal
`Convert from
`
`reconstruction
`sequence to
`filter
`
`impulse train
`H.012)
`
`3
`I
`I
`|
`Sampling
`:
`period T
`.
`l_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ J
`
`
`
`I :
`
`(al
`
`Hr (in)
`
`T
`
`__7L
`T
`
`1
`T
`
`(b)
`
`Q
`
`
`
`(c)
`
`(a) Block diagram of an ideal
`Figure 3.6
`bandlimited signal reconstruction system.
`(b) Frequency response of an ideal
`reconstruction filter. (c) Impulse response
`of an ideal reconstruction filter.
`
`It follows from Eqs. (3.25) and Eq. (3.22) that if x[n] = x,(n T), then
`
`x,(mT) = xc(mT)
`
`(3.26)
`
`for all integer values of m. That is, the signal that is reconstructed by Eq. (3.24) has the
`same values at the sampling times as the original continuous-time signal, independent
`of the sampling period T.
`In Fig. 3.7 we show a continuous-time signal xc(t) and the _corresponding
`modulated impulse train. Figure 3.7(c) shows several of the terms
`
`x[n]
`
`sin[1r(t — nT)/T]
`
`7t(t — nT)/ T
`
`and the resulting reconstructed signal x,(t). As suggested by this figure, the ideal
`lowpass filter interpolates between the impulses of xs(t) to construct a continuous-time
`signal x,(t). From Eq. (3.26), the resulting signal is an exact reconstruction of xc(t) at
`
`
`
`F...‘..‘_..____._...__:-—-—-——~
`
`

`
`90
`
`SamplingofContinuous-TimeSignals
`
`Chap.3
`
`I
`
`xc(t)
`
`la)
`
`
`
`(c)
`
`Ideal bandlimited
`Figure 3.7
`interpolation.
`
`the sampling times. The fact that the lowpass filter interpolates the correct recon-
`struction between the samples, if there is no aliasing, follows from our frequency-
`domain analysis of the sampling and reconstruction process.
`It is useful to formalize the preceding discussion by defining an ideal system for
`reconstructing a bandlimited signai from a sequence of samples. We will call this
`system the ideal discrete-to-continuous-time (D/C) converter. The desired system is
`depicted in Fig. 3.8. As we have seen,
`the ideal reconstruction process can be
`represented as the conversion of the sequence to an impulse train as in Eq. (3.21)
`followed by filtering with an ideal lowpass filter, resulting in the output given by Eq.
`(3.24). The intermediate step of conversion to an impulse train is a mathematical
`convenience in deriving Eq. (3.24) and in understanding the signal reconstruction
`process. However, once we are familiar with this process, it is useful to define a more
`compact representation, as depicted in Fig. 3.8(b), where the input is the sequence
`x[n] and the output is the continuous-time signal x,(t) given by Eq. (3.24).
`The properties of the ideal D/C converter are most easily seen in the frequency
`domain. To derive