Solutions - Midterm Exam

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1 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 PROBLEM (5 PT) Given he following LTI sysem: oluions - Miderm Exam a) kech he impulse response h[n]. b) Given he following inpu signal x[n]: 2 x[n] n - -2 Carefully skech he oupu y[n]. c) Idenify and find he proper Fourier represenaion (DTF, F, FT, or DTFT) of x[n] and h[n]. d) Find he frequency response of he oupu y[n]. a) Impulse response: We apply he dela funcion o he sysem: 2 h[n] n b) x[n] x[k] n = -2: x[k] h[-k+n] => y[-2] = 2 n-3 n-2 n- n n = -: x[k] h[-k+n] => y[-] = 5 n-3 n-2 n- n n = 0: x[k] h[-k+n] => y[0] = -2 n-3 n-2 n- n n = : x[k] h[-k+n] => y[] = -2 n-3 n-2 n- n h[n] h[-k+n] n-3 n-2 n- n n = 2: x[k] h[-k+n] => y[2] = n-3 n-2 n- n n = 3: x[k] => y[3] = -5 h[-k+n] n-3 n-2 n- n n = 4: x[k] => y[4] = -2 h[-k+n] n-3 n-2 n- n n = 5: x[k] => y[5] = -2 h[-k+n] n-3 n-2 n- n

2 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 y[n] n - -2 c) are no periodic DTFT -5 d) PROBLEM 2 (5 PT) a) Obain he FT for he following signal: b) Use he properies of Fourier represenaion (e.g., ime-differeniaion, convoluion, ime-shif, frequency-shif) o find he FT of:, ' ' denoes convoluion. a), converges for a > 0 b) Differeniaion Propery of FT: If Then: Convoluion Propery of FT: Then:, hen:

3 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 Knowledge of a common FT pair: Then, if, hen: Now:, where:, and Time hif Propery of FT: If Then:, hen: Finally: PROBLEM 3 (0 PT) Given he following sysem: a) Deermine wheher he sysem is i) memoryless, ii) causal, iii) linear, and iv) ime-invarian. Jusify your answers. b) If, hen evaluae. c) Le's call your response in (b):. Is? In oher words, can we evaluae he oupu of he sysem in response o any inpu using convoluion:? Yes or no? Why? a) The sysem is memoryless: I does no depend on fuure samples nor pas samples. The sysem is causal: I does no depend on fuure samples. Lineariy: If he inpu o he sysem is ax A [n] + bx B [n], where a,b, are real numbers, hen he oupu should be ay A [n] + by B [n], where y A [n], y B [n] are he responses o x A [n] and x B [n] respecively. If he inpu o he sysem is ax A [n] + bx B [n], hen: y[n] = n (ax A [n] + bx B [n]) = n (ax A[ n]) + n (bx B [n]) Now: ay A [n] + by B [n] = a n x A [n]u[n] + b n x B [n] We see ha y[n] = a(y A [n]) + b(y B [n]). Thus, he sysem is linear. Time invariance: We have a sysem y[n] = H(x[n]). The response of he sysem o a shifed inpu x[n-k] should be he same as if he oupu y[n] has been shifed by k, i.e., y[n-k]: x[n-k] x[n] k H y [n]= n x[n-k] Response of sysem o a shifed inpu x[n-k]: y [n] = n x[n-k] Oupu y[n] shifed by k: y[n-k] = n-k x[n-k] We see ha y[n-k] y [n]. Thus, he sysem is NOT ime invarian. b) If, hen y[n]= x[n] H n x[n] k y[n-k]= n-k x[n-k] c). While his is he impulse response of he sysem, he sysem is NOT linear an ime invarian, herefore we CANNOT apply he convoluion of an inpu wih he impulse response in order o ge he oupu.

4 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 PROBLEM 4 (0 PT) a) The oupu of a discree-ime sysem is relaed o is inpu x[n] as follows: where a, b, c, are real values i. Under wha condiion (if any) of he values a, b, c is he sysem causal? ii. Under wha condiion (if any) of he values a, b, c is he sysem memoryless? b) An LTI sysem is described by he following impulse response: i. Deermine wheher he sysem is i) memoryless, ii) causal, and iii) sable. ii. If, wha is he inpu? Provide he equaion of. a) b) i. For he sysem o be causal, we need b = 0, so ha:. ii. For he sysem o be memoryless, we need: b = 0, c = 0, so ha:. i. The sysem is NOT memoryless:. The sysem is causal: The sysem is sable: ii. PROBLEM 5 (20 PT) The impulse response of an LTI sysem and an inpu signal are depiced below: h() a) Obain he oupu y() (or carefully skech i). b) Idenify and obain he proper Fourier represenaion (DTF, F, FT, or DTFT) of and h(). Do no forge o specify he Fourier represenaion values when he frequency variable is 0. c) Obain he frequency response of he oupu y(). Do no forge o specify he Fourier represenaion value when he frequency variable is 0. a) CT convoluion:,,,, y()

5 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 h() h(+) < < < < = =5 5 < < b) nonperiodic FT =2 =6-4 c)

6 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 PROBLEM 6 (0 PT) The following represenaion of an LTI sysem (called H) is called Direc Form I. We can hink of he sysem as he cascade of wo sysems Ha and Hb, each wih ha[n] and hb[n] as impulse responses. x[n] Ha 3 w[n] Hb y[n] -2 /3 /4 /2 a) Ge y[n] in erms of x[n] and pas samples of y[n]. b) The enire sysem H relaes y[n] o x[n], and is impulse response is h[n]. How would you express h[n] in erms of ha[n] and hb[n]? c) Draw he Direc Form II represenaion of he sysem H. Wrie down he procedure. a) H: w[n] = 3x[n] - 2x[n-] x[n-2] H2: y[n] = (/3)y[n-] + 0.5y[n-2] + w[n] y[n] = (/3)y[n-] + 0.5y[n-2] + 3x[n] - 2x[n-] x[n-2] b) c) H2: f[n] = x[n] + (/3)f[n-] + 0.5y[n-2] H: y[n] = 3f[n] - 2f[n-] f[n-2] x[n] f[n] 3 y[n] /3-2 /2 /4

7 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 PROBLEM 7 (20 PT) a) For he following periodic signal, idenify and ge he proper Fourier represenaion (i should only depend on he frequency variable). Is he Fourier represenaion periodic? If so, wha is he period? e -3 4 b) For he following signal x[n], idenify and ge he proper Fourier represenaion (i should only depend on he frequency variable). Is he Fourier represenaion periodic? If so, wha is he period? c) You are provided wih he DTFT of a signal x[n]. Assume. Find he signal x[n]. Do no forge o specify he value of x[n] when n=0. Wha is he period of? X(e j ) -W W a), is periodic (T=4) F. b). x[n] is periodic DTF We rewrie he signal: If we remember he equaion: We noice ha: The Fourier represenaion is periodic (N=6 samples), and i is given by: Alernaive mehod (applying he DTF formula):

8 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 c) You are provided wih he DTFT of a signal x[n]. Assume. Find he signal x[n]. Do no forge o specify he value of x[n] when n=0. Wha is he period of? The DTFT is periodic wih period.

9 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 BONU PROBLEM (+ 5 PT) The oupu of an LTI sysem in response o an inpu is. Noe ha is an ineger number, and Find he frequency response of he impulse response, as well as he impulse response of his sysem. Time hif Propery of DTFT: Therefore: Alernaive way: Wih he ime shif propery of he DTFT and he knowledge of he DTFT of he impulse funcion, we can demonsrae ha: Then, given ha: We can use convoluion propery o express he impulse response as: By performing his convoluion, we find ha:. This is he same as Verifying ha our resul is correc: Wih he formula:, we hen demonsrae ha:

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