.-H
`
`.
`
`'
`
`.
`
`.
`
`_
`
`.
`
`_
`
`.
`
`I’__, m_,,...'—u-qu- .,_____,,
`In I, ‘. I
`_ mu-.—.,..-‘,.n........--d3nu.‘nn:unn-nun:
`fi.._,;,,-_.-...__..,..-.. Zk u¢!
`. _=g;-;gw-—- r
`..
`rcDV"""'
`‘A.-wu-t
`..
`.
`.
`‘T
`
`Page 1 of 19
`
`ZTE EXHIBIT 1019
`
`ZTE EXHIBIT 1019
`
`Page 1 of 19
`
`

`
`Library of Congress Cataloging-in-Publication Data
`
`Oppenheim, Alan V.
`Discrete-time signal processing f Alan V. Oppenhe1m, Ronald W.
`Schafer.
`p.
`
`cm.- (Prentice Hall signal processing series)
`
`Bibliography: p.
`Includes index.
`ISBN 0-13-216292-X
`1. Signal processing- Mathematics. 2. Discrete-time systems.
`I. Schafer, Ronald W.
`II. Title.
`III. Series.
`TK5102.5.02452 1989
`621.38'043- dc 19
`
`88-25562
`CIP
`
`Editorial/production supervision: Barbara G. Flanagan
`Interior design: Roger Brower
`Cover design: Vivian Berman
`Manufacturing buyer: Mary Noonan
`
`© 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 0-13-2162~2-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
`
`Page 2 of 19
`
`

`
`iystems
`
`Chap. 2
`
`Sec. 2.2
`
`Discrete-Time Systems
`
`17
`
`2.2 DISCRETE-TIME SYSTEMS
`
`A discrete-time system is defined mathematically as a transformation or operator that
`maps an input sequence with values x[n] into an output sequence with values
`y[n]. This can be denoted as
`
`y[n] = T{x[n]}
`
`(2.2D
`
`and is indicated pictorially in Fig. 2.6. Equation (2.21) represents a rule or formula for
`computing the output sequence values from the input sequ,ence valu~s. It .should be
`emphasized that the value of the output sequence at each value of the index n may be a
`function ofx[n] at all vall!e~ of n. The following examples illustrate some simple ah9
`useful ~ystems.
`·
`
`Example 2.1 The Ideal Pelay Sys!em
`The ideal de!ay sy~tem is defined by the equation
`
`- oo < n < oo,
`
`(2.22)
`
`where n4 is a fixed positive integer called the delay of the system. In words, the ideal delay
`system simply shifts the input sequence to the right by 114 samples to form the output. If
`in Eq: (2.22) n4 is ·a. fixed negative integer, then the system would shift the input to the left
`by n4 samples ~rresponding to a time adv.ance.
`·
`
`In Example 2.1 only one sample of the input sequence is involved in determining
`a certain· output sample. In the following example, this is not the case.
`
`Examvle 2.2 Moving Average
`The general m.oving average system is defined by the equation
`
`1
`1 L x[n - k]
`y[n] = M M
`I + 2 + A• - Mt
`
`M,
`
`=
`
`.
`1
`{x[n + M tJ + x[n + M 1 - 1] + · · · + x[n]
`Ml +M2 +
`1
`+ x[n- 1] + ·· · + x(n - M 2]}.
`
`(2.23)
`
`This system computes the nth sample of the output sequence as the average of
`(M 1 + M 2 + 1) samples of the input sequence around the nth sample. Figure 2.7 shows
`an input sequence plotted as a function of a dummy index k and the samples involved
`in the computation of the output sample y[n] for 11 = 7, M 1 = 0, anci M 2 = 5. The
`?Utput sample y[7] is equal to ! times the sum of all the ~amples between the
`vertical dotted lines. To compute y[8], both dotted lines would move one sample to
`the right.
`.
`.
`·
`.
`
`Figure 2.6 Representation of a discrete(cid:173)
`time system, i.e., a transformation that
`~
`maps an input sequenCQ x[n] into a
`~
`unique output sequence y[n).
`
`10 11 for several different
`ro0 increases from zero
`.-d), the sequenCe:
`tpidly. As ro0 increases
`·ts d- a), the oscillations
`
`Page 3 of 19
`
`

`
`18
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`x(kl
`
`k
`
`0
`
`Figure 2.7 Sequence values involved in computing a causal moving average.
`
`Classes of systems are defined by placing constraints on the properties of the
`transformation T{·}. Doing so often leads to very general mathematical representa(cid:173)
`tions, as we will see. Of particular importance to consider are the system constraints
`and properties, discussed in Sections 2.2.1-2.2.5.
`
`2.2. 1 Memoryfess Systems
`
`A system is referred to as memoryless if the output y[n] at eyery value of n depends
`only on the input x[n] at the same value of n.
`
`Example 2.3
`An example of a memoryless system is one for which x[n) and y(nJ are related by
`
`y[n) = (x(nJ)2
`
`for each value of n.
`
`(2.24)
`
`The system in Example 2.1 is not memoryless unless n4 = 0. In particular it is referred
`to as having "memory" whether n4 is positive (time delay) or negative (time advance).
`The system in Example 2.2 is not memorylcss unless M1 = M 2 = 0.
`
`2.2.2 Linear Systems
`
`The class of linear systems is defined by the principle of superposition. If y 1[n] and
`y2[n] are the responses of a system when x 1[n] and x 2[n] are the respective inputs,
`then the system is linear if and only if
`T{x 1[n] + x2[n]} = T{x 1[n]} + T{x 2[n]} = y 1[n] + y 2[n]
`T{ax[n]} = aT{x[n]} = ay[n].
`
`and
`
`(2.25a)
`
`(2.25b)
`
`where a is an arbitrary constant. The first property is called the additivity property and the
`second is called the homogeneity or scaling property. These two properties can be combined
`into the principle of superposition, stated as
`
`(2.26)
`
`Page 4 of 19
`
`

`
`;terns
`
`Chap. 2
`
`Sec. 2.2
`
`Discrete-Time Systems
`
`19
`
`x(k)
`
`for arbitrary constants a and b. This can be generalized to the superposition of many
`inputs. Specifically, if
`
`I
`I
`I
`I
`I
`
`I
`I
`I
`
`I rr
`
`then
`
`x[n] = L;akxk[n],
`k
`
`y[n] = L;akyk[n],
`
`k
`
`(2.27a)
`
`(2.27b)
`
`where yk[n] is the system response to the input xk[n].
`By using the definition of the principle of superposition, we can easily show that
`the systems of Examples 2.1 and 2.2 .are linear system,s. (See Problem 2.2.) An
`example of a nonlinear system is the system in Example 2.3.
`
`Example 2.4
`The system referred to as an accumulator and defined by the relation
`"
`l: x[k]
`y[n] =
`A: •- oo
`
`(2.28)
`
`is a linear system. The output at each sample instant n is equal to the accumulated sum of
`the value at nand all previous values of the input. Comparing Eq. (2.28) and Eq. (2.8), we
`note that if the input x[n] is a unit impulse 8[n], then the output y[n] is a unit step u[n].
`
`terage.
`
`properties of the
`ttical representa(cid:173)
`·stem constraints
`
`tlue of n depends ·
`
`2.2.3 Time-Invariant Systems
`
`are related by
`
`(2.24)
`
`:icular it is referred
`ve (time advance).
`0.
`
`ion. If y 1[n] and
`respective inputs,
`
`(2.25b)
`
`y property and the
`:scan be combined
`
`(2.26)
`
`A time-invariant system (equivalently often referred to as a shift-invariant system) is
`one for which a time shift or delay of the input sequepce causes a corresponding shift
`in the output sequence. Specifically, suppose that a system transforms the input
`sequence with values x[n] into the output sequence with values y[n]. The system is
`said to be time-invariant if for all n0 the input sequence with values x 1[n] = x[n - n0]
`produces the output sequence with values y 1[n] = y[n- n0].
`All of the systems in Examples 2.1-2.4 are time-invariant. The following
`example illustrates a system that is not time-invariant.
`
`Example 2.S
`The system defined by the relation
`
`-oo < n < oo,
`(2.29)
`y[n] = x[Mn],
`with M a positive integer, is called a compressor. Specifically, it discards (M - 1)
`samples out of M; i.e., it creates the output sequence by selecting every Mth sample. It
`should be clear simply from the description of its operation that this system is not time(cid:173)
`invariant. We can see this mathematically by considering the response y 1[n] to the input
`x 1[n] defined as x1[n] = x[n - n0 ]. Then
`y1[n] = x 1[Mn] = x[Mn - n0].
`
`(2.30)
`
`But
`
`y[n- n0 ] = x[M(n - n0 )] ~ y 1[n].
`With x 1[11] equal to x[n] shifted by n0 , y 1[n] is not y[n] shift~ by n0 , so the
`downsampler is not time-invariant, unless of course M = 1.
`
`(2.31)
`
`Page 5 of 19
`
`

`
`Page 6 of 19
`
`Page 6 of 19
`
`

`
`/Stems
`
`Chap. 2
`
`Sec. 2.3
`
`Unear Time-Invariant Systems
`
`21
`
`lue at index n =
`; implies that if
`is, the system is
`and is noncausal
`'2 ~ 0; otherwise
`liator of Example
`ICe y[l) = x[M).
`
`(2.32)
`
`Jsal. However, the
`
`(2.33)
`
`:nse if and only if
`:. The input x[n]
`
`(2.34)
`
`ositive finite value
`
`(2.35)
`
`of Example 2.4 is
`put for which the
`ounded. For this
`
`(2.36)
`
`nded, i.e., there is
`n.
`ed in this section
`•e may be able to
`
`find inputs for which the properties hold, but the existence of the property for some
`inputs does not mean that the system has the property. For the system to have the
`property, it must hold' for all inputs. For example, an unstable system may have some
`bounded inputs for which the output is bounded, but for the system to have the
`property of stability, it must be true that for all bounded inputs the output is
`bounded. If, as we were able to do with the accumulator system of Example 2.4, we
`can find just one input for which the system property does not hold, then we have
`shown that the system does not have that property.
`
`'
`
`,; ·~-· :· .. ' ..
`
`,
`
`·.
`2.3 LINEAR TIME-INVARIANT SYSTEMS
`A particularly important class of systems consists of those that are linear and time(cid:173)
`invariant. These two properties in combination lead to especially convenient repre(cid:173)
`sentations for these systems. Most important, this class of systems has significant
`signal processing applications. The class of linear systems is defined by the principle
`of superposition in Eq. (2.26). If the linearity property is combined wit)l the
`representation of a general sequence as a linear combination of delayed impulses as in
`Eq. (2.6), it follows that a linear system can be completely characterized by its impulse
`response. Specifically, let hk[n] be the response of the system to o[n - k], an impulse
`occurring at n = k. Then from Eq. (2.6),
`y[n] = T L-~ oo x[k]o[n - k] }-
`
`(2.37)
`
`(2.38)
`
`From the principle of superposition in Eq. (2.26), we can write
`y[n] = I x[k]T{o[n- k]} =
`I x[k]hk[n].
`
`00
`
`k = - oo
`
`a)
`
`k = -oo
`
`According to Eq. (2.38), the system response to any input can be expressed in terms of
`the response of the system to o[n - k]. If only linearity is imposed, hk[n] will depend
`on both n and k, in which case the computational usefulness of Eq. (2.38) is
`limited. We obtain a more useful result if we impose the additional constraint of time
`in variance.
`The property of time invariance implies that if h[n] is the response to o[n], then
`the response to o[n - k] is h[n - k]. With this additional constraint, Eq. (2.38)
`becomes
`
`a)
`
`y[n] = L x[k]h[n - k].
`
`k • - oo
`
`(2.39)
`
`As a consequence of Eq. (2.39), a linear time-invariant system (which we will
`sometimes abbreviate as LTI) is completely characterized by its impulse response,
`h[n], in the sense that, given h[n], it is possible to use Eq. (2.39) to compute the output
`y[n] due to any input x[n].
`
`Page 7 of 19
`
`

`
`22
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`Equation (2.39) is commonly called the convolution sum. If y[n] is a sequence
`whose values are related to the values of two sequences IJ[n] and x[n] as in Eq. (2.39),
`we say that y[n] is the convolution of x[n] with h[n] and represent this by the
`notation
`
`(2.40)
`
`y[n] = x[n] • h[n].
`The operation of discrete-time convolution takes two sequences x [n] and IJ[n] and
`produces a third sequence y[n]. Equation (2.39) expresses each sample of the output
`sequence in terms all of the samples of the input and tpe impulse response sequences.
`The derivation ofEq. (2.39) suggests the interpretation that the input sample at
`n = k, represented as x[k]<5[n - k], is transformed by the system into an output
`sequence x[k]IJ[n - k], - oo < n < oo and that these sequences for each k are
`superimposed to form the overall output sequence. This interpretation is illustrated
`in Fig. 2.8, which shows an impulse response, a simple input sequence having three
`nonzero samples, the individual outputs due to each sample, and the composite
`output due to all the samples in the input sequences. Specifically, x[n] can be
`decomposed as the sum of the three sequences x[ - 2]<5[n + 2], x[0]<5[n], and
`x[3]o[n - 3] representing the three nonzero values in the sequence x[n]. The
`sequences x[ - 2]h[n + 2], x[O]h[n], and x[3]h[n - 3] are the system responses to
`x[ -2]o[n + 2], x[O]o[n], and x[3]<5[n - 3], respectively. The response to x[n] is
`then the sum of these three individual responses.
`Although the convolution-sum expression is analogous to the convolution
`integral of continuous-time linear system theory, the convolution sum should not be
`thought of as an approximation to the convolution integral. The convolution integral
`plays mainly a theoretical role in continuous-time linear system theory; we will see
`that the convolution sum, in addition to its theoretical importance, often serves as an
`explicit realization of a discrete-time linear system. Thus it is important to gain some
`insight into the properties of the convolution sum in actual calculations.
`The preceding interpretation of Eq. (2.39) emphasizes that the convolution sum
`is a direct result of linearity and time in variance. However, a slightly different way of
`looking at Eq. (2.39) leads to a particularly useful computational interpretation.
`When viewed as a formula for computing a single value of the output sequence, Eq.
`(2.39) dictates that y[n] (i.e., the nth value of the output) is obtained by multiplying
`the input sequence (expressed as a function of k) by the sequence whose values are
`h[n - k], - oo < k < oo, and then, for any fixed value of n, summing all the values of
`the products x[k]h[n - k], with k a counting index in the summation process. The
`operation of convolving two sequences thus involves doing the computation for all
`values of n, thus generating the complete output sequence y[n], - oo < 11 < oo.
`The key to carrying out the computations of Eq. (2.39) to obtain y[n] is under(cid:173)
`standing how to form the sequence h[n - k], - oo < k < oo, for all values of n that are
`of interest. To this end, it is useful to note that
`
`h[11 - k] = h[ - (k- n)].
`
`(2.41)
`
`The interpretation of Eq. (2.41) is best done with an example . .
`
`Page 8 of 19
`
`

`
`stems
`
`Chap. 2
`
`n] is a sequence
`I as in Eq. (2.39),
`sent this by the
`
`(2.40)
`
`n] and h[n] and
`ple of the output
`'onse sequences.
`l input sample at
`into an output
`for each k are
`:ion is illustrated
`nee having three
`d the composite
`lly, x[n] can be
`I, x[O]c5[n], and
`Jence x[n]. The
`tern responses to
`ponse to x[n] is
`
`the convolution
`tm should not be
`tvolution integral
`eory; we will see
`'ften serves as an
`:ant to gain some
`tions.
`convolution sum
`y different way of
`al interpretation.
`'ut sequence, Eq.
`:d by multiplying
`whose values are
`g all the values of
`:ion process. The
`mputation for all
`I, - oo < n < oo.
`n y[n] is under(cid:173)
`alues of n that are
`
`(2.41)
`
`sec. 2.3
`
`Unear Time-Invariant Systems
`
`23
`
`x[n)
`
`-2
`
`0
`
`n
`
`h[n)
`
`... .'II
`
`0
`
`t • • •
`
`n
`
`I x _2 [n) • x[-2) 6(n + 2)
`
`• • • • • • •
`
`0
`
`n
`
`•
`
`• •
`
`- 2
`
`• • •
`
`n
`
`I I
`
`0
`
`y_2 [n) = x[ - 2)h[n + 2)
`t • • • • •
`
`x0<.(nl • x[0)6[n)
`
`0
`
`n
`
`x3 [n) = x(3)6[n - 3)
`
`• • • • • ... I ..
`
`0
`
`n.
`
`Yo[n) = x[O)h[n)
`
`• • • n
`
`y3 (n) • x(3]h[n- 3)
`• • • • • • •
`
`3
`
`0
`
`x [n) • x _2 [n) + x0 (n) + x 3 (n)
`
`y[n) = y _2 (n) +Yo [n) + y 3 (n)
`
`- 2
`
`0
`
`n
`
`3
`
`n
`
`Figure 2.8 Representation of the output of a linear time-invariant system as the
`superposition of responses to individual samples of the input.
`
`Page 9 of 19
`
`

`
`24
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`I I I I
`.. • • • I I
`I I I t
`
`h(k)
`
`• •
`
`k
`
`6
`
`- 3
`
`0
`
`(a)
`
`hl-k!• h(O - k)
`
`I I I I
`• ' t I I
`I t • • • • •
`I I I I
`• • • • • t I I I
`I t •
`
`-6
`
`0
`
`(b)
`
`3
`
`h(n - k!• h(- (k - nil
`
`n - 6
`
`0
`
`n
`
`n+3
`
`k
`
`k
`
`(c)
`
`Figure 2.9 Forming the sequence h[n - k]. (a) The sequence h[k] as a function of
`k. (b) The sequence h( -k] as a function of k. (c) The sequence h( 11 - k] as a funet.ion
`of k for n = 4. ·
`·
`·
`
`Example 2.7
`SupP,Ose h[k] is the s~quence shown in Fig. 2.9(a). First consider h[ - k] plotted against
`k; h[ -k] is simply h[k] reflected or "flipped" about k = 0, as shown in Fig. 2.9(b). Re(cid:173)
`placing k by k - n_, where n is a fixed integer, leads to a shift of the origin of'the sequence
`h[ ....: k] to k = n, as shown in Fig. 2.9(c) for the case n = 4.
`
`Generalizing E~ample 2.7, it should be clear that in general the sequence h[n - k],
`- oo < k ~· oo, is obtained by
`
`1. reflecting h[k] about the origin to obtain h[ - k];
`2. shifting the origin of the reflected sequence to k = n.
`
`To implement discrete-time convolution, the two sequences x[k] and h[n - k] are
`multiplied together for - oo < k < oo and the products are summed to compute the
`output sampl~ y[n]. To obtain another output sample, the origin of ~he sequence
`
`Page 10 of 19
`
`

`
`stems
`
`Chap. 2
`
`sec. 2.3
`
`Linear Time-Invariant sysi:ems
`
`25
`
`• •
`
`k
`
`h[O -k)
`
`t
`n+3 •
`
`k
`
`·unction of
`a function
`
`- k] plotted against
`in Fii. 2.9(b). Re(cid:173)
`gin ofthe ~equence
`
`quence h[n- k],
`
`and h[n - k] are
`d to ~ompute the
`1 of the sequenee.
`
`Ji[ - k] is shifted to the new. sample position and . the process is repeated. This
`computational procedure applies whether the computations are carried out numeri~
`cally on sampled data or analytically with sequences for which the sample values have
`simple formulas. The following example illustrates discrete-time convolution for the
`latter case.
`
`Example 2.8
`Consider a system with impulse response
`
`The input is
`
`lz[n] = u[iz] - u[n - N]
`
`= {1,
`
`0,
`
`0 ~ n ~ N -1,
`otherwise.
`
`x[n] = a"u[11].
`
`To find the output at a p;trticuhir index 11, we must form the sums over all k of the product
`x[k]h[11 - k]. . . In· this case, we can find formulas for y[11] for different. sets of values of
`11. For ex!lmple, Fig. 2.iO(a) shows the sequences x[k] and h[11- k] plot~ed for n.a
`negatiye integer. Clearly, all negative vaiues of 11 give a similar picture; i.e., the nonzero
`portions of the sequences x[k] and h[11 - k] do not overlap, so
`
`II < 0.
`y[11] = 0,
`Figure :i.10(b) illustrates the two sequences when 0 ~ 11 and n - N + 1 ~ 0. T~ese two
`conditions can be combined into the single condition b ~ 11 ~ N - 1. By considering
`Fig. 2.10(b), we see that since
`
`then
`
`x[k]h[n - k] = a1
`
`,
`
`for 0 ::;; n ::;; N - 1.
`
`(2.42)
`
`The limits oti. tlie suin are determined directly from Fig. 2.10(1>). Equation (2.42) shows
`that y[n] is the sum of n + 1 terins of a geometric secles where the ratio of terms is a. This
`sum cim be expressed in closed form using the general formula
`
`N2
`
`ri.Hs _
`L;ak =-- - -
`1 - a
`t • N,
`
`(XN1 + l
`
`(2.43)
`
`Applying trus to Eq. (2.42), we obtain
`
`1 - a•+ 1
`y[n] = 1
`- a
`Finally, Fig. 2.10(c) shows the two sequences when 0 < n - N + 1 or N - 1 < n. As
`before,
`
`(2.44)
`
`,
`
`x[k]h[n - k] = al,
`
`n -N + 1 <k ~ n,
`
`Page 11 of 19
`
`

`
`26
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`X
`
`X
`
`0 h(n - k)
`
`X x(k)
`
`n -
`
`(N - 1)
`
`(a)
`
`n
`
`X
`
`n-(N - 11
`
`n
`
`(b)
`
`(c) n- (N - 1)
`
`ll I ,,,,
`1 I I I_ I I I·r r r 1
`
`• • • • • • • • • • •
`
`0
`
`k
`
`k
`
`k
`
`n
`
`(d)
`
`N - 1
`
`Figure 2.10 Sequences involved in computing a discrete convolution. (a)- ( c) The
`sequences x[k] and lt[n - k] as a function of k for different values or n. (Only nonuro
`samples arc shown.) {d) Corresponding output sequence as a function of n.
`
`but now the lower limit on the sum is n - N + 1, as seen in Fig. 2.10(c). Thus
`"
`y[n] = L
`ak
`lt • • - N+ 1
`
`for N - 1 < 11.
`
`(2.45)
`
`Using Eq. (2.43) we obtain
`
`a"- 11+ 1 -a"+l
`y[n] = -
`--:-1- - (cid:173)
`- a
`
`Page 12 of 19
`
`

`
`'Stems
`
`Chap. 2
`
`---k
`
`1_ .
`k
`
`n
`
`1) -(c) The
`ly nonzero
`II.
`
`)(c). Thus
`
`(2.45)
`
`Properties of Linear Time-Invariant Systems
`
`Sec. 2.4
`
`or
`
`1- aN)
`y[n] = a•- N+t - - .
`(
`1 - a
`
`27
`
`(2.46)
`
`Thus, because of the piecewise-exponential nature of both the input and the unit sample
`response, we have been able to obtain the following closed-form expression for y[n] as a
`function of the index n:
`
`y[n] =
`
`0,
`l - a•+ 1
`'
`1 - a
`n·N+I(l - aN)
`1- a '
`a
`
`II <0,
`
`0 ~II~ N -
`
`I,
`
`N - 1 <n.
`
`(2.47)
`
`This sequence is shown in Fig. 2.10(d).
`
`Example 2.8 illustrates how the convolution sum can be computed analytically
`when the input and the impulse response are given by simple formulas. In such cases,
`the sums may have a compact form that may be derived using the formula for the sum
`of a geometric series or other "closed-form" formulas.t When no simple form is
`available, the convolution sum can still be evaluated numerically using the technique
`illustrated in Example 2.8 whenever the sums are finite, which will be the case if either
`the input sequence or the impulse response is of finite length, i.e., has a finite number
`of nonzero samples. We cannot overemphasize that neatly drawn and labeled
`sketches such as those in Fig. 2.10 are invaluable to ensure that the limits on the sum
`are correctly determined.
`
`2.4 PROPERTIES OF LIN EAR TIME-INVARIANT SYST EMS
`
`Since all linear time-invariant systems are described by the convolution sum of Eq.
`(2.39), the properties of this class of systems are defined by the properties of discrete(cid:173)
`time convolution. Therefore the impulse response is a complete characterization of
`the properties of a specific linear time-invariant system.
`Some general properties of the class of linear time-invariant systems can be
`found by considering properties of the convolution operation. For example, the
`convolution operation is commutative:
`x[n] * h[n] = h[n] * x[n].
`This can be shown by applying a substitution of variables to Eq. (2.39). Specifically,
`with m = n- k,
`
`(2.48)
`
`y[n] = L x[n - m]h[m],
`
`00
`
`m=- oo
`
`(2.49)
`
`t Such results are discussed, for example, in Grossman (1982). A very useful 11ook of sums is Jolley
`(1961).
`
`Page 13 of 19
`
`

`
`28
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`x[n)•l ~W•h2W I y(~)
`
`Figure 2.11 Three linear time-invariant
`systems with identical impulse responses.
`
`so the roles of x[n] and h[rt] in the summation are interchanged. That is, the order of
`the sequences in a convolution is unimportant, and hence the system output is the
`same if the roles of the input and impulse response are reversed. A linear time(cid:173)
`invariant system with input x[n] and impulse response h[n] will have the same output
`as a linear time-invariant system with input lt[n] and impulse response x[n]. The
`convolution operation also distributes over addition, i.e.,
`x[n] • (h 1[n] + h2(n]) = x[n] • h1 [n] + x[n] •1t2[n].
`This follows in a straightforward way from Eq. (2.39).
`In a cascade connection of systems, the output of the first system is the input to
`the second, the output of the second is the input to the third, etc. The output of the
`last system is the overall output. Two linear time-invariant systems in cascade
`correspond to a linear time-invariant system with an impulse response that is the
`convolution of the impulse responses of the two systems. This is illustrated in Fig.
`2.11. In the upper block diagram, the output of the first system will be h1[n] if
`x[n] = b[n]. Thus the output of the second system (and the overall impulse response
`of the system) will be
`
`(2.50)
`
`As a consequence of the commutative property of convolution, it follows that the
`impulse response of a cascade combination of linear time-invariant systems is
`independent of the order in which they are cascaded. This result is summarized in Fig.
`2.11, where the three systems all have the same impulse response.
`In a parallel connection, the systems have the same input and their outputs are
`summed to produce an overall output. It follows from the distributive property of
`convolution that two linear time-invariant systems in parallel are equivalent to a
`single system whose impulse response is the sum of the individual impulse responses,
`i.e.,
`
`This is depicted in Fig. 2.12.
`The constraints of linearity and time in. variance define a class of systems with
`very special properties. Stability and causality represent additional properties, and it
`is often important to know whether a linear time-invariant system is stable and
`whether it is causal. Recall from Section 2.2.5 that a stable system is one for which
`
`(2.51)
`
`Page 14 of 19
`
`

`
`;tems
`
`Chap. 2
`
`Sec. 2.4
`
`Properties of Linear Time-Invariant Systems
`
`29
`
`ear time-invariant
`impulse responses.
`
`tt is, the order of
`m output is the
`A linear time(cid:173)
`the same output
`oonse x[n]. The
`
`n is the input to
`:1e output of the
`ems in cascade
`onse that is the
`ustrated in Fig.
`will be h1 [n] if
`npulse response
`
`(2.50)
`
`'ollows that the
`iant systems is
`omarized in Fig.
`
`heir outputs are
`tive property of
`equivalent to a
`pulse responses,
`
`(2.51)
`
`of systems with
`roperties, and it
`n is stable and
`s one for which
`
`(o)
`
`~-
`(a) Parallel combination of
`Figure 2.12
`X[~)
`linear time-invariant systems. (b) An
`(b)
`equivalent system.
`
`every bounded input produces a bounded ouput. Linear time-invariant systems are
`stable if and only if the impulse response is absolutely summable, i.e., if
`
`s =
`
`00
`
`I lh[kJI < oo.
`
`k • - oo
`
`This can be shown as follows. From Eq. (2.49),
`
`ly[n]l = lk~~}[k]x[n- k] I :s:; k~~«>lh[k]llx[n - k] l.
`
`If x[n] is bounded so that
`
`then
`
`ly[n]l :s:; Bx L lh[k]l.
`
`«>
`
`Jc • - a)
`
`(2.52)
`
`(2:53)
`
`(2.54)
`
`Thus y[n] is bounded if Eq. (2.52) holds; i.e., Eq. (2.52) is a sufficient condition for
`stability. To show that it is also a necessary condition, we must show that if S = oo,
`then a bounded input can be found that will cause an unbounded·output. Such an
`input'is the sequence with values
`
`h•[ - n]
`lh[- n] l '
`
`x[n] =
`
`{
`
`0,
`
`h[n]. :f. 0,
`
`h[n] = 0,
`
`(2.55)
`
`where h• [n] is the complex conjugate of h[n]. The sequence x[n] is clearly bounded
`by unity. However, the value of the output at n = 0 is
`
`Thus if S = oo, it is possible for a bounded input sequence to produc~ an unbounded
`output sequence.
`
`(2.56)
`
`Page 15 of 19
`
`

`
`30
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`The class of causal systems was defined in Section 2.2.4 as those systems for
`which the output y[n0] depends only on the input samples x[n], for n ~ n0 . It follows
`from Eq. (2.39) or Eq. (2.49) that this definition implies the condition
`
`h[n] = 0,
`
`n < 0,
`
`(2.57)
`
`for causality of linear time-invariant systems. (See Problem 2.12.) For this reason it is
`sometimes convenient to refer to a sequence that is zero for n < 0 as a causal sequence,
`meaning that it could be the impulse response of a causal system.
`To illustrate how the properties of linear time-invariant systems are reflected in
`the impulse response, let us reconsider some of tlie systems defined in Examples
`2.1-2.6. First note that only the systems of Examples 2.1, 2.2, 2.4, and 2.6 are linear
`and time-invariant. Although the impulse response of nonlinear or time-varying
`systems can be found, it is generally of limited interest since the convolution-sum
`formula and Eqs. (2.52) and (2.57), expressing stability and causality, do not apply to
`such systems.
`First, let us find the impulse responses of the systems in Examples 2.1, 2.2, 2.4,
`and 2.6. We can do this by simply computing the response of each system to <5[n]
`using the defining relationship for the system. The resulting impulse responses are as
`·
`follows:
`
`Jdeal Delay (Exafl'!ple 2.1)
`
`h[n] = <5[n - n4 ],
`
`n4 a positive fixed integer
`
`(2.58)
`
`I'
`
`Moving Average (Example 2.2)
`
`M2
`
`1
`L: o[n - kJ
`h[nJ =
`Mt + M2 + 1 k = -M,
`= {M1 + ~2 + 1'
`
`0,
`
`otherwise ·
`
`Accumulator (Example 2.4)
`
`"
`h[nJ = L: o[kJ
`k • - co
`= {1,
`0,
`= u[n]
`
`n~O
`n<O
`
`Forward Difference (Example 2.6)
`h[n] = o[n + 1] - <5[n]
`
`(2.59)
`
`(2.60)
`
`(2.61)
`
`Page 16 of 19
`
`

`
`;terns
`
`Chap. 2
`
`tose systems for
`~ n0 . It follows
`
`(2.57)
`
`r this reason it is
`. causal sequence,
`
`ts are reflected in
`ted in Examples
`nd 2.6 are linear
`or time-varying
`~onvolution-sum
`, do not apply to
`
`tples 2.1, 2.2, 2.4,
`h system to <5[n]
`: responses are as
`
`(2.58)
`
`(2.59)
`
`(2.60)
`
`(2.61)
`
`Sec. 2.4
`
`Properties of linear Time-Invariant Systems
`
`Backward Difference (Example 2.6)
`
`la[n] = <5[n] - <5[n - 1]
`
`31
`
`(2.62)
`
`Next let us test the stability of each system by computing the sum
`
`GO
`
`S = L lh[n)l.
`
`,.. - co
`
`For the ideal delay, moving average, forward difference, and backward difference
`examples, it is clear that S < oo since the impulse response has only a finite number of
`nonzero samples. Such systems are called finite-duration impulse response (FIR)
`systems. Clearly FIR systems will always be stable as long as each of the impulse
`response values is finite in magnitude. The accumulator, however, is unstable because
`
`00
`
`S = L u[n] = <X>.
`• • 0
`
`In Section 2.2.5 we also demonstrated the instability of the accumulator by giving an
`example of a bounded input (the unit step) for which the output is unbounded.
`The impulse response of the accumulator is infinite in duration. This is an
`example of the class of systems referred to as infinite-duration impulse response (IIR)
`systems. An example of an IIR system that is stable is one having the impulse
`response h[n] = a"u[n] with lal < 1. In this case
`
`S = L Ia I".
`• • 0
`If Ia I < 1, the formula for the sum of the terms of an infinite geometric series gives
`
`(2.63)
`
`00
`
`1
`S = 1 - lal < oo.
`
`(2.64)
`
`If, on the other hand, Ia I ~ 1, the sum is infinite and the system is unstable.
`To test causality of the linear time-invariant systems in Examples 2.1, 2.2, 2.4,
`and 2.6, we can check to see if h[n] = 0 for n < 0. As discussed in Section 2.2.4, the
`ideal delay (n4 ~ 0 in Eq. 2.22) is causal. If n4 < 0, the system is noncausal. For the
`moving average, causality requires that - M 1 ~ 0 and M 2 ~ 0. The accumulator and
`backward difference systems are causal, and the forward difference system is noncau(cid:173)
`sal.
`
`The property of convolution can be used to describe and analyze interconnec(cid:173)
`tions of linear time-invariant systems. Consider the system of Fig. 2.13(a), which
`consists of a forward difference system cascaded with an ideal delay of one sam(cid:173)
`ple. According to the commutative property of convolution, the order in which
`systems are cascaded does not matter as long as they are linear and time-invariant.
`Therefore we obtain the same result when we compute the forward difference of a
`sequence and delay the result (Fig. 2.13a) as when we delay the sequence first and then
`compute the forward difference (Fig. 2.13b). Also, it follows from Eq. (2.50) that the
`
`Page 17 of 19
`
`

`
`32
`
`Discrete-Time Signals and Systems
`
`Chap. 2
`
`I
`.J
`
`(b)
`
`(c)
`
`Figure 2.13 Equivalent systems round
`by using the commutative property of
`convoluiion.
`
`overall impulse response of each cascade system is the convolution of the individual
`impulse responses. Therefore
`h[n] = (o[n + 1] - o[n]) * o[n - 1]
`. = o[n - 1] * (o[n + 1] .:.:. o[n])
`= o[n] - o[n - 1].
`
`(2.65)
`
`Thus h[n] is identical to the impulse response .of the backward difference system, i.e.,
`the cascaded systems of Figs. 2.13(a) and 2.13(b) can be replaced by a backward
`difference system, as shown in Fig. 2.13(c).
`Note that the noncausal forward difference systems in Figs. 2.13(a) and (b) have
`been converted to causal systems by cascading them ~ith a delay. In general, any
`noncausal FIR system can be made causal by cascading it with a sufficiently long
`delay.
`Another example of cascaded systems introduces the concept of an inverse
`system. Consider the cascade of systems in Fig. 2.14. The impulse response of the
`cascade system is
`
`h[n] = u[n] * (o[n] - o[n- 1J)
`= u[n] - u[n - 1]
`
`= o[n].
`
`(2.66)
`
`That is, the cascade combination of an accumulator followed by a backward
`difference (or vice versa) yields a system whose overall impulse response is the
`impulse. Thus the output of the cascade system will always be equal to the input since
`
`Accumulator
`system
`
`y[n)
`
`x[n)
`
`Backward
`difference
`system
`
`x(n]
`
`Figure 2.14 An accumulator in cascade
`with a backward difference. Since the
`backward difference is the inverse system
`ror the accumulator, the cascade
`combination i