DISCRETE-TIME
`SIGNAL
`PROCESSING
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`SONOS EXHIBIT 1039
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`SONOS EXHIBIT 1039
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`SONOS EXHIBIT 1039
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`SECOND EDITION
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`DISCRETE-TIME
`SIGNAL
`PROCESSING
`
`M ASSACHUSETTS I NSTITUT E OF 'l'ECHNOLOCY
`
`ALAN V. OPPENHEIM
`R ONALD w. S CHAFER
`
`GEORGIA INSTITUT E OF TECHNOLOGY
`
`WITH
`
`JOHN R. B UCK
`
`U NIVERSITY OF MASSACHUSETTS DARTMOUTH
`
`PRENTICE HALL
`UPPE R SADDLE RIVER, N EW J ERSEY 07458
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`SONOS EXHIBIT 1039
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`Oppenheim, Alan V.
`D iscrete-time signal processing / A lan V. O ppe nheim, Ronald W.
`Schafer, with John R. Buck. - 2nd ed.
`p.
`cm.
`Includes bibliographical references and index.
`ISBN 0-13-754920-2
`1. Signal processing- Mathematics. 2. Discrete-time systems.
`I. Schafer. Ronald W.
`II. Buck, John R.
`Ill. Tille.
`TK5102.9.067 1998
`621.382'2-dc21
`
`98-50398
`CIP
`
`Acquisitions editor: Tom Robbins
`Production service: Interactive Composition Corporation
`Editorial/production supervision; Sharyn Vitrano
`Copy editor: Brian Baker
`Cover design: Vivian Berman
`Art director: Amy Rosen
`Managing editor: Eileen Clark
`Editor-in-Chief: Marcia Horton
`Director of production and manufacturing: David W. Riccardi
`Manufacturing buyer: Pat Brown
`Editorial assistant: Dan De Pasquale
`
`© 1999, 1989 Alan V. Oppenheim, Ronald W. Schafer
`Published by Prentice-Hall. Inc.
`Upper Saddle River, New Jersey 07458
`
`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.
`
`The author and publisher of this book have used their best efforts in preparing this book. These efforts include
`the development, research, and testing of the theories and programs to determine their effectiveness. The
`author and publisher make no warranty of any kind, expressed or implied, with regard to these programs
`or the documentation contained in this book. The author and publisher shall not be liable in any event for
`incidentaJ or consequential damages in connection with, or arising out of, the furnishing, performance, or use
`of these programs.
`
`Printed in the United Sta tes of America
`10 9 8 7 6 5 4
`ISBN
`
`0-13-754920-2
`
`Prentice-Hall International (UK) Limited, London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada Inc., Toronto
`Prentice-H all 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-H all do Brasil, Ltda., Rio de Janeiro
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`4
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`S AMPLING OF
`CONTINUOUS-TIME SIGNALS
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`4.0 INTRODUCTION
`
`Discrete-time signals can a rise in many ways, but they most commonly occur as repre(cid:173)
`sentations of sampled continuous-time signals. It is remarkable that unde r reasonable
`constraints, a continuous-time signal can be quite accurately represented by samples
`taken at discrete points in time. In this chapter we discuss the process of periodic sam(cid:173)
`pling in some detail. including the phenomenon of aliasing, which occurs when the
`signal is not bandlimitcd or when the sampling rate is too low. Of particular impor(cid:173)
`tance is the fact that continuous-time signal processing can be implemented through a
`process of sampling, discrete-time processing, and the subsequent reconstruction of a
`continuous-time signal.
`
`4. 1 PERIODIC SAMPLING
`
`Although other possibilities exist (sec Steiglitz. 1965: Oppenheim and Johnson , 1972).
`the typical method of obtaining a discrete-time representation of a conti nuous-time
`signal is through periodic sampling, where in a sequence of samples, x[n] , is obtained
`from a continuous-time signal xc(t) according to the relation
`
`x[n] = Xc(nT).
`In Eq. ( 4.1), T is the sampling period, and its reciprocal. f, - 1/ T. is the sampling
`
`- oo < n < oo.
`
`(4.1)
`
`140
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`SONOS EXHIBIT 1039
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`Sec. 4.1
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`Periodic Sampling
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`141
`
`CID
`' - - - . - - - - - - ' x [ n] = Xe ( Tl T)
`
`T
`
`Figure 4.1 Block diagram
`representation of an ideal
`continuous-to-discrete-time (CID)
`converter.
`
`frequency, in samples per second. We also express the sampling frequency as Q s = 2rr / T
`when we want to use frequencies in radians per second.
`We refer to a system that implements the operation of Eq. ( 4. I) as an ideal
`continuous-to-discrete-time (CI D) converter, and we depict it in block diagram form
`as indicated in Figure 4.1. As an example of the relationship between xc(t) and x [n].
`in Figure 2.2 we illustrated a continuous-time speech waveform and the corresponding
`sequence of samples.
`In a practical setting, the operation of sampling is implemented by an analog-to(cid:173)
`digital (AI D ) converter. Such systems can be viewed as approximations to the ideal
`CI D converter. Important considerations in the implementation or choice of an AID
`converter include quantization of the output samples, linearity of quantization steps,
`the need for sample-and-hold circuits, and limitations on the sampling rate. The effects
`of quantization are discussed in Sections 4.8.2 and 4.8.3. Other practical issues of AID
`conversion are electronic circuit concerns that are outside the scope of this text.
`The sampling operation is generally not invertible; i.e., given the output x [n ],
`it is no t possible in general to reconstruct x,.(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 a fundamental issue in signal processing. Fortunately, it is
`possible to remove the ambiguity by restricting the input signals that go into the sampler.
`It is convenient to represent the sampling process mathematically in the two stages
`depicted in Figure 4.2(a). The stages consist of an impulse train modul ator followed
`by conversion of the impulse train to a sequence. Figure 4.2(b) shows a continuous(cid:173)
`time signal Xc(L ) and the results of impulse train sampling for two different sampling
`rates. Figure 4.2(c) depicts the corresponding output sequences. The essential difference
`between Xs (t) and xlnl is that Xs(l) 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
`normalization; i.e., the sequence of numbers x [n] contains no explicit information about
`the sampling rate. Furthermore, the samples of Xc(t) are represented by finite numbers
`in x [n] rather than as the areas of impulses, as with xs(t).
`It is important to emphasize that Figure 4.2(a) is strictly a mathematical represen(cid:173)
`tation that is convenient for gaining insight into sampling in both the time domain and
`frequency domain. It is not a close 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 Figure 4.2(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 the approach leads to
`a number of important insights that arc difficult to obtain from a more formal d erivatio n
`based on manipulation of Fourier transform formulas.
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`142
`
`Sampling of Continuous-Time Signals
`
`Chap. 4
`
`CID converter
`,------------------~
`I
`I
`I
`s (1)
`I
`I
`. -- -- -~ 1
`:
`Conversion from
`:
`impulse train
`( I
`to discrete-time
`X,. I) (
`sequence
`1
`' - - - - - -~ I
`I
`L-----------------~
`(a)
`
`(
`Xs I)
`
`I -----
`
`xfn] "- x,.(nT)
`
`T = T 1
`
`T = 2T1
`
`/
`
`x, (t)
`\..
`
`x , (1)
`
`,,,
`
`/ - -- ..... .....
`
`- 2T-T O T 2T
`
`/ Tr'tT1
`
`- 2T
`
`- T
`
`()
`
`T
`
`2T
`
`(b)
`
`Figure 4.2 Sampling with a periodic impulse train followed by conversion to
`a discrete-time sequence. (a) Overall system. (b) x5(t) tor two sampling rates.
`(c) The output sequence for the two different sampling rates.
`
`4.2 FREQUENCY-DOMAIN REPRESENTATION OF SAMPLING
`
`To derive the frequency-domain rela tion be tween the input and o utput o f an ideal CID
`converter, le t us first consider the conversion o f xc(r) to xs(t) through modu lation of
`the periodic impulse train
`
`X
`
`s(r) = L 8(t - nT).
`
`II= X
`
`(4.2)
`
`where 8(t) is the unit impulse func tion, or Dirac delta function. We modulate s(t) with
`xc(i), obtaining
`
`Xs(t) = Xc(t)s(t)
`oc
`= Xc(r) L o(t - nT).
`
`11 = - X
`
`(4.3)
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