ECE 301: Signals and Systems Homework Assignment #4

Transcription

1 ECE 301: Sigals ad Systems Homework Assigmet #4 Due o October 28, 2015 Professor: Aly El Gamal TA: Xiaglu Mao 1

2 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 1 Problem 1 Let x[] be a real periodic sigal with period N ad Fourier coefficiets a k. (a) Show that if N is eve, at least two of the Fourier coefficiets withi oe period of a k are real. (b) Show that if N is odd, at least oe of the Fourier coefficiets withi oe period of a k is real. We have Noe that which is real if x[] is real. a k = 1 N a 0 = 1 N x[]e j(2π/n)k x[] (a) If N is eve, the a N/2 = 1 N Clearly, a N/2 is also if x[] is real. x[]e jπ = 1 N (b) If N is odd, oly a 0 is guarateed to be real. x[]( 1) 2

3 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 2 Problem 2 Cosider the fuctio a[k] = (a) Show that a[k] = N for k = 0, ±N, ±2N, ±3N,... e j(2π/n)k (b) Show that a[k] = 0 wheever k is ot a iteger multiple of N. (Hit: Use the fiite sum formula.) (c) Repeat parts (a) ad (b) if a[k] = e j(2π/n)k where =< N > meas ay cosecutive N iteger umbers. (a) Let k = pn, p Z. The, a[pn] = (b) Usig the fiite sum formula, we have a[k] = e j(2π/n)pn = e j2πp = 1 ej2πk = 0, if k pn, p Z. 1 ej(2π/n)k 1 = N. (c) a[k] = q+ =q e j(2π/n)k where q is some arbitrary iteger. By puttig k = pn, we may agai easily show that Now, a[pn] = q+ =q Usig part (b), we may argue that e j(2π/n)pn = q+ =q a[k] = e j(2π/n)kq e j2πp = e j(2π/n)k. q+ =q 1 = N. a[k] = e j(2π/n)kq 1 e j2πk = 0, if k pn, p Z. 1 ej(2π/n)k 3

4 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 3 Problem 3 Let x[] be a periodic sigal with fudametal period N ad Fourier series coefficiets a k. I this problem, we derive the time-scalig property { x[ x (m) [] = m ], = 0, ±m, ±2m,... i the textbook. (a) Show that x (m) [] has period of mn. (b) Show that if the x[] = v[] + w[] x (m) [] = v (m) [] + w (m) [] (c) Assumig that x[] = e j2πk0/n for some iteger k 0, verify that x (m) [] = 1 m m 1 Noe that here you may use the results from the Problem 2. l=0 e j2π(k0+ln)/mn (d) Usig the results of parts (a), (b), (c), show that if x[] has the Fourier coefficiets a k, the x (m) [] must have the Fourier coefficiets 1 m a k. (a) Note that x (m) [ + mn] = { x[ m + N], = 0, ±m, ±2m,... Therefore, x (m) [] is periodic with period mn. = { x[ m ], = 0, ±m, ±2m,... = x (m) [] (b) The time-scalig operatio discussed i this problem is a liear operatio. Therefore, if the x[] = v[] + w[], x (m) [] = (c) Let us cosider { x[ m ], = 0, ±m, ±2m,... = { v[ m ] + w[ m ], = 0, ±m, ±2m,... = v (m) []+w (m) []. y[] = 1 m This may be writte as m 1 l=0 e j(2π/mn)(k0+ln) = 1 m 1 m ej(2π/mn)k0 l=0 { e j(2π/mn)k 0, = 0, ±N, ±2N,... y[] = e j(2π/m)l. Problem 3 cotiued o ext page... 4

5 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 3 (cotiued) Now, also ote that by applyig time-scalig o x[], we obtai Therefore, we obtai that y[] = x (m) []. (d) We have { e j(2π/mn)k 0, = 0, ±N, ±2N,... x (m) [] = b k = 1 m x (m) []e j(2π/mn)k. mn We kow that oly every mth value i the above summatio is ozero. Therefore, Note that x (m) [m] = x[]. Therefore, b k = 1 x (m) [m]e j(2π/mn)km mn = 1 x (m) [m]e j(2π/n)k mn b k = 1 x (m) [m]e j(2π/n)k = a k mn m. So that we have show that if x[] has the Fourier coefficiets a k, the x (m) [] must have the Fourier coefficiets 1 m a k. 5

6 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 4 Problem 4 Compute the Fourier trasform of each of the followig sigals (a) [e αt cos(w 0 t)]u(t), a > 0 (b) e 3 t si(2t) (c) x(t) as show i Figure 1. Figure 1: The graph of sigal x(t) i (c). (a) The give sigal is Therefore, (b) We kow that Therefore, we obtai (c) e at cos(w 0 t)u(t) = 1 2 e at e jw0t u(t) e at e jw0t u(t). X(jw) = e 3 t 1 2(α jw 0 + jw) + 1 2(α + jw 0 + jw) CSF T w 2 si(2t) CSF T π j (δ(w w 0) δ(w + w 0 )) X(jw) = 1 2π ( w 2 ) π j (δ(w w 0) δ(w + w 0 )) j3 = 9 + (w + 2) 2 j3 9 + (w 2) 2 X(jw) = 1 x(t)e jwt dt 1 2 = e jwt dt + te jwt dt = 1 jw e2jw 1 jw e 2jw + 1 w 2 e jw 1 w 2 ejw = 2 si(w) (cos(2w) jw w ) e jwt dt 6

7 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 5 Problem 5 Cosider the sigal x(t) i Figure 2. Figure 2: The graph of sigal x(t). (a) Fid the Fourier trasform X(jw) of x(t). (b) Sketch the sigal x(t) = x(t) δ(t 4k). k= (c) Fid aother sigal g(t) such that g(t) is ot the same as x(t) ad x(t) = g(t) k= δ(t 4k). (d) Argue that, although G(jw) is differet from X(jw), G(j πk πk 2 ) = X(j 2 ) for all itegers k. You should ot explicitly evaluate G(jw) to aswer the questio. (a) Note that where x(t) = x 1 (t) x 2 (t) x 1 (t) = Also, the Fourier trasform X 1 (jw) of x 1 (t) is Usig the covolutio property we have { 1, w < 1 2 0, otherwise X 1 (jw) = 2 si(w/2). w X(jw) = X 1 (jw)x 1 (jw) = [2 si(w/2) ] 2 w Problem 5 cotiued o ext page... 7

8 Aly El Gamal ECE 301: Sigals ad Systems Homework Assigmet #4 Problem 5 (cotiued) (b) The sigal of x(t) is as ashow i Figure 3. Figure 3: The graph of sigal x(t). (c) Oe possible choice of g(t) is as ashow i Figure 4. Figure 4: The graph of sigal g(t). (d) Note that X(jw) = X(jw) π 2 This may also be writte as δ(j(w k π 2 )) = G(jw)π 2 X(jw) = π X(jπk/2)δ(j(w k π 2 2 )) = π 2 δ(j(w k π 2 )) G(jπk/2)δ(j(w k π 2 )) Clearly, this is possible oly if G(jπk/2) = X(jπk/2). 8