By Hsu H.P.
ISBN-10: 007163472X
ISBN-13: 9780071634724
ISBN-10: 0071634738
ISBN-13: 9780071634731
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Extra info for Schaum's outline of signals and systems
Example text
A) Since the output value at n depends on only the input value at n, the system is memoryless. (b) Since the output does not depend on the future input values, the system is causal. (c) Letx[n] = a 1x 1[n] + a 2x2[n]. 68) is satisfied and the system is linear. (d) Lety 1[n] be the response tox 1[n] = x[n - n0 ]. Then y 1[n] = T{x[n - n 0 ]} = nx[n - n 0 ] y[n - n0 ] = (n - n0) x[n - n0 ] # y 1[n] But Hence , the system is not time-invariant. (e) Letx[n] = u[n]. Then y[n] = nu[n]. Thus , the bounded unit step sequence produces an output sequence that grows without bound (Fig.
_n 8 +~)=cos(OJof + ~ )--+ Wo = 1 x(t) is periodic with fundamental period T0 = 2n I w 0 = 2n. 2n (b) x(t)=sm-t--+w0 3 2n =- 3 x(t) is periodic with fundamental period T0 = 2n I OJo = 3. (c) x(t) = cos~t + sin~t = x 1 (t) + x 2 (t) 3 4 where x 1(t) = cos(n/3)t = cos w/ is periodic with T1 = 2n/w 1 = 6 and xp) = sin(n/4)t = sin Wl is periodic with T2 = 2n/w2 = 8. Since T1 /T2 = ~ = %is a rational number, x(t) is periodic with fundamental period T0 = 4T1 = 3T2 = 24. CHAPTER 1 Signals and Systems (d) x(t) =cost+ sin [it= x 1(t) + xz(t) where x 1(t) =cost= cos w 1t is periodic with T1 = 2n/w 1 = 2nand x2(t) = sin flt= sin w2t is periodic with T2 = 2n/w2 = f l n.
Then by the result from Prob. _ iTo[x(t)]2 dt = Wo To i2nlwo AZ cos2(aJo! + 8)dt 2:n: o o A 2w 2nlwo 1 A2 = __ o -[l + cos(20Jot + W)]dt = - < oo 2:n: Jo 2 2 r Thus, x(t) is a power signal. Note that periodic signals are, in general, power signals. CHAPTER 1 Signals and Systems E = Jim (c) T-oo . J T/2 - T/2 1 P = hm T- oo T 2 lx(t)I dt = Jim JT/2 - T/2 (T /2) 3 t dt = Jim - - = oo T- oo 3 JT/2 2 0 T-oo 2 . 1 lx(t)I dt = hm T-oo T JT/2 t 2 . 1 (T /2) 3 dt = hm - - - = oo T-oo T 3 0 Thus, x(t) is neither an energy signal nor a power signal.
Schaum's outline of signals and systems by Hsu H.P.
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