CHAP. 11 SIGNALS AND SYSTEMS 2 1 (a) The sequence x(n), illustrated in Fig. 1-8(a), is a linearly decreasing sequence that begins at index n = 0 and ends at index n = 5. The first sequence that is to be sketched, yl(n) = x(4 - n), is found by shifting x(n) by four and time-reversing. Observe that at index n = 4, yl(n) is equal to x(0). Therefore, yl(n) has a value of 6 at n = 4 and decreases linearly to the left (decreasing values of n) until n = - 1, beyond which y (n) = 0. The sequence y (n) is shown in Fig. 1-8(b). Fig. 1-8. Performing signal manipulations. (b) The second sequence, y2(n) = x(2n - 3), is formed through the combination of time-shifting and down- sampling. Therefore, y&~) may be plotted by first shifting x(n) to the right by three (delay) as shown in SIGNALS AND SYSTEMS [CHAP. 1 Fig. 1-8(c). The sequence y2(n) is then formed by down-sampling by a factor of 2 (i.e., keeping only the even index terms as indicated by the solid circles in Fig. 1-8(c)). A sketch of yn(n) is shown in Fig. I-8(d). (c) The third sequence, y3(n) = x(8 - 3n), is formed through a combination of tirne-shifting, down-sampling, and time-reversal. To sketch y3(n) we begin by plotting x(8 - n), which is formed by shifting x(n) to the left by eight (advance) and reversing in time as shown in Fig. 1 -8(e). Then, y3(n) is found by extracting every third sample of x(8 - n), as indicated by the solid circles, which is plotted in Fig. 1-8( f). (4 Finally, y4(n) = x(n2 - 2n + 1) is formed by a nonlinear transformation of the time variable n. This sequence may be easily sketched by listing how the index n is mapped. First, note that if n 2 4 or n 5 -2, then n2 - 2n + 1 2 9 and, therefore, y4(n) = 0. For -I 5 n 5 3 we have The sequence y4(n) is sketched in Fig. 1-8(g). 1.8 The notation ~((n))~ is used to define the sequence that is formed as follows: ~((n))~ = x(n modulo N) where (n modulo N) is the positive integer in the range [0, N - 11 that remains after dividing n by N. For example, ((3))g = 3, ((12))g = 4, and ((-6))d = 2. If x(n) = (i)%in(nn/2)u(n), make a sketch of (a) x((n))3 and (b) x((n - 2))3. (a) We begin by noting that ((n))3, for any value of n, is always an integer in the range [O, 21. In fact, because ((n))3 = ((n + 3k)h for any k, Therefore, x((n))3 is periodic with a period N = 3. It thus follows that~((n))~ is formed by periodically repeating the first three values of x(n) as illustrated in the figure below: (b) The sequence x((n - 2))3 is also periodic with a period N = 3, except that the signal is shifted to the right by no = 2 compared to the periodic sequence in part (a). This sequence is shown in the figure below: 1.9 The power in a real-valued signal x(n) is defined as the sum of the squares of the sequence values: Suppose that a sequence x(n) has an even part x,(n) equal to CHAP. I] SIGNALS AND SYSTEMS 23 If the power in x(n) is P = 5, find the power in the odd part, x,(n), of x(n). This problem requires finding the relationship between the power in x(n) and the power in the even and odd parts. By definition, x(n) = x,(n) + x,(n). Therefore, Note that x,(n)x,(n) is the product of an even sequence and an odd sequence and, therefore, the product is odd. Because the sum for all n of an odd sequence is equal to zero, Thus, the power in x(n) is m m which says that the power in x(n) is equal to the sum of the powers in its even and odd parts. Evaluating the power in the even part of x(n), we find m m 2n PC = ): (ynl = -I + 2 ): (f) = f n=-m n =O Therefore, with P = 5 we have 10 P, = 5 - P, = T 1.10 Consider the sequence Find the numerical value of Compute the power in x(n), W If x(n) is input to a time-varying system defined by y(n) = nx(n), find the power in the output signal (i.e., evaluate the sum) This is a direct application of the geometric series With the substitution of -n for n we have Therefore, it follows from the geometric series that SIGNALS AND SYSTEMS [CHAP. 1 (h) To find the power in x(n) we must evaluate the sum Replacing n by -n and using the geometric series, this sum becomes (c) Finally, to find the power in y(n) = nx(n) we must evaluate the sum In Table 1- I there is a closed-form expression for the sum but not for C:,n2an. However, we may derive a closed-form expression for this sum as follow^.^ Differenti- ating both sides of Eq. (1.19) with respect to a, we have Therefore. we have the sum Using this expression to evaluate Eq. (1.18). we find 1.11 Express the sequence I 1 n=O 2 n=l .r(n) = 3 n=2 0 else as a sum of scaled and shifted unit steps. In this problem, we would like to perform a signal decomposition, expressing x(n) as a sum of scaled and shifted unit steps. There are several ways to derive this decomposition. One way is to express x(n) as a sum of weighted and shifted unit samples, x(n) = S(n) + 2S(n - I) + 3S(n - 2) and use the fact that a unit sample may be written as the difference of two steps as follows: Therefore, x(n) = u(n) - u(n - I) + 2[u(n - I) - u(n - 2)] + 3[u(n - 2) - u(n - 3)] which gives the desired decomposition: "his method is very useful and should be remembered CHAP. I] SIGNALS AND SYSTEMS 25 Another way to derive this decomposition more directly is as follows. First, we note that the decomposition should begin with a unit step, which generates a value of I at index n = 0. Because x(n) increases to a value of 2 at n = 1, we must add a delayed unit step u(n - 1). At n = 2, x(n) again increases in amplitude by 1, so we add the delayed unit step u(n - 2). At this point, we have Thus, all that remains is to bring the sequence back to zero for n > 3. This may be done by subtracting the delayed unit step 3u(n - 3), which produces the same decomposition as before. Discrete-Time Systems 1.12 For each of the systems below, x(n) is the input and y(n) is the output. Determine which systems are homogeneous, which systems are additive, and which are linear. (a) If the system is homogeneous, y(n) = T[cx(n)] = cT[x(n)] for any input x(n) and for all complex constants c. The system y(n) = log(x(n)) is not homogeneous because the response of the system to xl(n) = cx(n) is which is not equal to c log(x(n)). For the system to be additive, if yl(n) and y2(n) are the responses to the inputs and xz(n), respectively, the response to x(n) = xl(n) + x2(n) must be y(n) = yl(n) + y2(n). For this system we have T[xl(n) +xhN = log[x~(n) + x2(n)l # log[x~(n)l+ log[x2(n)l Therefore, the system is not additive. Finally, because the system is neither additive nor homogeneous, the system is nonlinear. (b) Note that if y(n) is the response to x(n). the response to xl(n) = cx(n) is which is not the same as y1 (n). Therefore, this system is not homogeneous. Similarly, note that the response to x(n) = x,(n) + x2(n) is which is not equal to yl(n) + y2(n). Therefore, this system is not additive and, as a result, is nonlinear. SIGNALS AND SYSTEMS [CHAP. 1 This system is homogeneous, because the response of the system to xl(n) = cx(n) is The system is clearly, however, not additive and therefore is nonlinear. Let y,(n) and yz(n) be the responses of the system to the inputs x,(n) and x2(n), respectively. The response to the input x(n) = axl(n) + bxz(n) y(n) = x(n) sin ( y) = [axl (n) + bx2(n)] sin - r; 1 Thus, it follows that this system is linear and, therefore, additive and homogeneous. Because the real part of the sum of two numbers is the sum of the real parts, if y,(n) is the response of the system toxl(n), and yz(n) is the response to x2(n), the response to y(n) = yl(n) + yz(n) is Therefore the system is additive. It is not homogeneous, however, because unless c is real. Thus, this system is nonlinear. For an input x(n), this system produces an output that is the conjugate symmetric part of x(n). If c is a complex constant, and if the input to the system is xl(n) = cx(n), the output is Therefore, this system is not homogeneous. This system is, however, additive because 1.13 A linear system is one that is both homogeneous and additive. (a) Give an example of a system that is homogeneous but not additive. (b) Give an example of a system that is additive but not homogeneous. There are many different systems that are either homogeneous or additive but not both. One example of a system that is homogeneous but not additive is the following: x(n - I)x(n) ~(n) = x(n + I) Specifically, note that if x(n) is multiplied by a complex constant c, the output will be cx(n-l)cx(n) x(n-I)x(n) ~(n) = = c cx(n + I) x(n + 1) which is c times the response to x(n). Therefore, the system is homogeneous. On the other hand, it should be clear that the system is not additive because, in general, {xl(n - 1) + XZ(~ - l)J(x~(n) + xz(n)I x~(n - l)x~(n) xdn - l)xz(n) x~(n + 1) +xAn + I) + x,(n + 1) + xz(n + 1) CHAP. I] SIGNALS AND SYSTEMS An example of a system that is additive but not homogeneous is Additivity follows from the fact that the imaginary part of a sum of complex numbers is equal to the sum of imaginary parts. This system is not homogeneous, however, because 1.14 Determine whether or not each of the following systems is shift-invariant: (a) Let y(n) be the response of the system to an arbitrary input x(n). To test for shift-invariance we want to compare the shifted response y(n - no) with the response of the system to the shifted input .r(n - nu). With we have. for the shifted response. Now, the response of the system to xl(n) = x(n - no) is Because yl(n) = y(n - no), the system is shifl-invariant. (6) This system is a special case of a more general system that has an input-output description given by where f (n) is a shift-varying gain. Systems of this form are always shift-varying provided f (n) is not a constant. To show this, assume that f (n) is not constant and let nI and nz be two indices for which f (n,) # f (nz). With an input .rl(n) = S(n - nl), note that the output yl(n) is If, on the other hand, the input is x2(n) = 6(n - n2), the response is Although .t.,(n) and xZ(n) differ only by a shift, the responses yl(n) and y2(n) differ by a shift and a change in amplitude. 'Therefore, the systcm is shift-varying. (c) Let be the response of the system to an arbitrary inpul .r(n). The response of the system to the shifted input .rl(n) = x(n - no) is Because this is equal to v(n - no), the system is shift-invariant. SIGNALS AND SYSTEMS [CHAP. 1 (d) This system is shift-varying, which may be shown with a simple counterexample. Note that if x(n) = S(n), the response will be y(n) = 6(n). However,ifxl(n) = 6(n-2). the response will be yl(n) = xl(n2) = 6(n2-2) = 0, which is not equal to y(n - 2). Therefore, the system is shift-varying. (e) With y(n) the response to x(n), note that for the input xl(n) = x(n - N), the output is which is the same as the response tox(n). Because yl (n) # y(n- N), ingeneral, this system isnot shift-invariant. (f) This system may easily be shown to be shift-varying with a counterexample. However, suppose we use the direct approach and let x(n) be an input and y(n) = x(-n) be the response. If we consider the shifted input, xl (n) = x(n - no), we find that the response is However, note that if we shift y(n) by no, which is not equal to yl (n). Therefore, the system is shift-varying. 1.15 A linear discrete-time system is characterized by its response hk(n) to a delayed unit sample S(n - k). For each linear system defined below, determine whether or not the system is shift-invariant. (a) hk(n) = (n - k)u(n - k) (6) hk(n) = S(2n - k) S(n - k - 1) k even 5u(n - k) k odd (a) Note that hk(n) is a function of n - k. This suggests that the system is shift-invariant. To verify this, let y(n) be the response of the system to x(n): The response to a shifted input, x(n - no), is With the substitution 1 = k - no this becomes From the expression for y(n) given in Eq. (1.24, we see that which is the same as yl(n). Therefore, this system is shift-invariant. CHAP. I] SIGNALS AND SYSTEMS 29 (h) For the second system, hI(n) is nor a function of n - k. Therefore, we should expect this system to be shift- varying. Let us see if we can tind an example that demonstrates that it is a shift-varying system. For the input ~(11) = 6(11), the response is Because gl(n) # y(n - I ), the system is shift-varying. (c) Finally, for the last system, we see that although hk(n) is a function of n - k fork even and a function of (n - k) fork odd, 11k(n) # hk-~(n - 1) In other words, the response of the system to 6(n - k - 1) is not equal to the response of the system to 6(n - k) delayed by 1. Therefore. this system is shift-varying. 1.16 Let Tr.1 be a linear system, not necessarily shift-invariant, that has a response hk(n) to the input 6(n - k). Derive a test in terms of kk(n) that allows one to determine whether or not the system is stable and whether or not the system is causal. (a) The response of a linear system to an input ~(n) is Therefore. the output may be hounded as follows: If x(n) is bounded, Ix(n)l 5 A < W, lywi i A 2 IM~)I As a result. if the output will be bounded, and the system is stable. Equation (1.23) is a necessary condition for stability. To establish the sufficiency of this condition, we will show that if this summation is not finite, we can find a bounded input that will produce an unbounded output. Let us assume that hk(n) is bounded for all k and n [otherwiue the system will be unstable. because the response to the bounded input S(n - k) will be unbounded]. With hi(tl) bounded for all k and n, suppose that the sum in Eq. (1.23) is unbounded for some n, say n = no. Let x(n) = sgn(h,(n~)l that is, For this Input, the response at time n = no is which, by assumption, is unbounded. Therefore, the system is unstable and we have established the sufficiency of the condition given in Eq. (1.23). SIGNALS AND SYSTEMS [CHAP. 1 (b) Let us now consider causality. For an input x(n), the response is as given in Eq. (1.22). In order for a system to be causal, the output y(n) at time no cannot depend on the input x(n) for any n > no. Therefore, Eq. (1.22) must be of the form ,I y(n) = x hk(n)x(k) k=-m This, however, will be true for any x(n) if and only if which is the desired test for causality. Determine whether or not the systems defined in Prob. 1 .I5 are (a) stable and (b) causal. (a) For the first system, hk(n) = (n - k)u(n - k), note that hk(n) grows linearly with n. Therefore, this system cannot be stable. For example, note that if x(n) = S(n), the output will be which is unbounded. Alternatively, we may use the test derived in Prob. 1 .I6 to check for stability. Because this system is unstable. On the other hand, because h,(n) = 0 for n < k, this system is causal. (b) For the second system, hk(n) = S(2n - k), note that hl(n) has, at most, one nonzero value, and this nonzero value is equal to I. Therefore, for all n, and the system is stable. However, the system is not causal. To show this, note that if x(n) = &(n - 2), the response is y(n) = h2(n) = 6(2n - 2) = &(n - I) Because the system produces a response before the input occurs, it is noncausal. (c) For the last system, note that cm n = x Su(n - k) = 1 5 A= A= u A add h odd which is unbounded. Therefore, this system is unstable. Finally, because hk(n) = 0 for n < k, the system is causal. Consider a linear system that has a response to a delayed unit step given by That is, sk(n) is the response of the linear system to the input x(n) = u(n - k). Find the response of this system to the input x(n) = 6(n - k), where k is an arbitrary integer, and determine whether or not this system is shift-invariant, stable, or causal. Because this system is linear, we may find the response, hk(n), to the input &(n - k) as follows. With &(n - k) = u(n - k) - u(n - k - I), using linearity it follows that which is shown below: [...]... depends on the value of r(2), which occurs after time n = - I , this system is noncausal CHAP 11 (e) SIGNALS AND SYSTEMS The output of this system at time n is the product of the values of the input x ( n ) at times n - 1, , n Therefore, because the output depends only on past values of the input signal, the system is causal 33 -N ( f ) This system is not causal, which may be seen easily if we rewrite... which is u(n) = y ( n ) - y ( n - I) To show that this is the inverse system, note that n-l (e) Invertibility must hold for complex as well as real-valued signals Therefore, this system is noninvertible because it discards the imaginary part real-valued signals 1.23 0 ' x(n) 1 One could state, however, that this system is invertible over the set of Consider the cascade of two systems S I and S2 (a) If...CHAP 11 SIGNALS AND SYSTEMS From this plot, we see that the system is not shift-invariant, because the response of the system to a unit sample changes in amplitude as the unit sample is advanced or delayed However,... that the response of the system to a unit sample x ( n ) = S(n) is infinite for all values of n except n = 1 (c) Because Icos(x)l 5 1 for all x, this system is stable (d) This system corresponds to a digital integrator and is unstable Consider, for example, the step response of the system With x ( n ) = u ( n ) we have, for n 2 0 , Although the input is bounded, (x(n)l 5 1, the response of the system... system at time n = 0 is which is unbounded Alternatively, because the input-output relation is one of convolution, this is a linear shift-invariant system with a unit sample response h ( n ) = cos (7) SIGNALS AND SYSTEMS [CHAP 1 Because a linear shift-invariant system will be stable only if we see that this system is not stable 1.22 Determine which of the following systems are invertible: To test for... clearly unbounded 1.19 Consider a system whose output y ( n ) is related to the input x ( n ) by Determine whether o r not the system is (a) linear, (b) shift-invariant, ( c ) stable, (d) causal [CHAP I SIGNALS AND SYSTEMS (a) The first thing that we should observe about y(n) is that it is formed by summing products of r(n) with shifted versions of itself For example, Xi y(O) = r2(k) I=-w We expect, therefore,... 2 ( n )due to the linearity of S, With this as the input to S2, the response will be, again by linearity, a y , ( n ) hy7(n) Therefore, if both S I and S2 are linear, the cascade will be linear + + + SIGNALS AND SYSTEMS CHAP 11 35 Similarly, for shift-invariance, if x ( n - no) is input to S , , the response will be w ( n - no) In addition, because S2 is shift-invariant, the response to w ( n - n o... is 1.25 The convolution of two finite-length sequences will be finite in length 1s it true that the convolution of a finite-length sequence with an infinite-length sequence will be infinite in length? SIGNALS AND SYSTEMS [CHAP I It is not necessarily true that the convolution of an infinite-length sequence with a finite-length sequence will be infinite in length It may be either Clearly, if x ( n )... - k ) to the right, we obtain the values of y ( n ) for n > 0 , which are Another way to perform the convolution is to use the fact that * x ( n ) S(n - no) = x ( n - n o ) Writing h ( n ) as CHAP I ] SIGNALS AND SYSTEMS we may evaluate y ( n ) as follows y(n) = 2x(n + 3 ) - 2 x ( n + 1 ) + 2 x ( n - I) - 2x(n - 3 ) Making a table of these shifted sequences and adding down the columns, we obtain the... evaluate the sum, we have 1.28 A linear shift-invariant system has a unit sample response Find the output if the input is x ( n ) = -n3"u(-n) Shown below are the sequences x ( n ) and h ( n ) 38 [CHAP I SIGNALS AND SYSTEMS Because x ( n ) is zero for n > -1, and h ( n ) is equal to zero for n > - 1 , the convolution will be equal to zero for n z -2 Evaluating the convolution sum directly, we have Because . real-valued signal x(n) is defined as the sum of the squares of the sequence values: Suppose that a sequence x(n) has an even part x,(n) equal to CHAP. I] SIGNALS. 0. The sequence y (n) is shown in Fig. 1-8(b). Fig. 1-8. Performing signal manipulations. (b) The second sequence, y2(n) = x(2n - 3), is formed