6.003 Homework #10 Solutions

Transcription

1 6.3 Homework # Solutions Problems. DT Fourier Series Determine the Fourier Series coefficients for each of the following DT signals, which are periodic in N = 8. x [n] / n x [n] n x 3 [n] n x 4 [n] / n x [n] = sin n 4 = e j 4 e j 4 j = <N> a k e j N kn j k = a k = a k+8 = j k = k =,, 3, 4 b k = b k+8 = x[n]e j N kn N <N> = ( + e j k 4 + e j k e j 3k 4 e jk 5k j e 4 + 3k e j + ) 7k e j 4 8 c k = c k+8 = N x[n]e j N kn = <N> e j k 4 e j k 4 4 = j sin k 4 j 6 sin k j 3k sin e j k e j k + 3k e j 4 e j 3k 4 Because x 3 [n] is real-valued and an odd function of n, the series is purely imaginary. 8 x 4 [n] = x [n ] d k = d k+8 = e j j e j/4 k = N ak = d k = j e j/4 k = k =,, 3, 4

2 6.3 Homework # Solutions / Fall. Inverse DT Fourier Series Determine the DT signals with the following Fourier series coefficients. Assume that the signals are periodic in N = 8. / a k k b k k a k = cos k 4 = e jk 8 + e jk 8 x [n] = 4δ[n ] + 4δ[n + ] = N x[n]e j N kn <N> for n < 5. Since x [n] is periodic, a more general expression is x [n] = 4δ[n + 8k] + 4δ[n + + 8k] x [n] = k= k=<n> b k e j N kn = e j 8 n + e j 8 n Notice that x [n] has imaginary components, because the Fourier series coefficients are not conjugate symmetric (b k b k ).

3 6.3 Homework # Solutions / Fall 3 3. Impulsive Input Let the following periodic signal x(t) = δ(t 3m) + δ(t 3m) δ(t 3m) m= be the input to an LTI system with system function H(s) = e s/4 e s/4. Let b k represent the Fourier series coefficients of the resulting output signal y(t). Determine b 3. The period of x(t) is T = 3. Therefore the period of y(t) is also T = 3. The fundamental frequency of x(t) (and y(t)) is = T = 3. Let a k represent the Fourier series coefficients for x(t). Then a k = 3 (δ(t) + δ(t ) δ(t )) e j 3 kt dt = ( + e j 3 k e j k) a 3 = 3 The frequency response of the system is given by Therefore H(j) = e j/4 e j/4 = j sin 4 b k = H (j 3 ) ( k a k = j sin k ) a k and b 3 = ( j sin ) 3 = j 3.

4 6.3 Homework # Solutions / Fall 4 4. Fourier transform Part a. Find the Fourier transform of x (t) = e t. X (j) = = = e t e jt dt = e ( j)t dt + j j = + Part b. Find the Fourier transform of x (t) = + t. Hint: Try duality. By duality, if then Therefore e t + + t e X (j) = e e t e jt dt + e ( j)t dt = e( j)t j e t e jt dt + e( j)t j We can check this result by inverse transforming X (j): x (t) = = e e jt d = e (+jt) d + = + jt + jt = + t e e jt d + e ( +jt) d = e (+jt) + jt e e jt d + e ( +jt) + jt

5 6.3 Homework # Solutions / Fall 5 5. Fourier transform Part a. Determine x (t), whose Fourier transform X (j) has the following magnitude and angle. X (j) 9 X (j) / / Express x (t) as a closed-form and sketch this function of time. X (j) = X (j) e j X (j) = { 3j < otherwise Notice that X (j) is 3j times X a (j) defined as { X a (j) = < otherwise Thus x (t) = 3 d dt x a(t) where Thus x a (t) = sin t t. x (t) = 3 (t cos t sin t) t t Part b. Determine x (t), whose Fourier transform X (j) has the following magnitude and angle. X (j) 9 X (j) / /

6 6.3 Homework # Solutions / Fall 6 Express x (t) as a closed-form and sketch this function of time. X (j) can be expressed as a difference: 9 9 Thus sin t x (t) = 9 6 sin t t t t Part c. What are important similarities and differences between x (t) and x (t)? How do those similarities and differences manifest in their Fourier transforms? Both x (t) and x (t) are real functions of time. However, x (t) is an odd function of time and x (t) is an even function of time. Taken together, these features mean that X (j) is an odd function of that is purely imaginary, and X () is an even function of that is purely real. Both X (j) and X (j) are zero for >. infinite extents in time. Therefore, both x ( t) and x (t) have Both X (j) and X (j) are discontinuous functions of. Thus, the magnitudes of x (t) and x (t) both decrease as t for large t.

7 6.3 Homework # Solutions / Fall 7 6. Fourier Transforms The magnitude and angle of the Fourier transform of a signal x(t) are given in the following plots. X(j) X(j) Five signals are derived from x(t) as shown in the left column of the following table. Six magnitude plots (M-M6) and six angle plots (A-A6) are shown on the next page. Determine which of these plots is associated with each of the derived signals and place the appropriate label (e.g., M or A3) in the following table. Note that more than one derived signal could have the same magnitude or angle. signal magnitude angle dx(t) dt (x x)(t) ( x t ) x(t) x (t) M5 M3 M M4 M6 A4 A A A3 A

8 6.3 Homework # Solutions / Fall 8 M A M A M3 A3 M4 A4 M5 A5 4 M6 A6