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CSUN ECE 351 - Homework #1

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ECE 351 - Linear Systems II Homework #1 1. Sketch the following sequences: a. fk = (1/2)k uk b. fk = (-1/2)k uk+2 c. fk = uk+1 - uk-3 d. fk = δk+2 + δk + δk-2 e. fk = 2 k [ uk+2 – uk-2 ] f. fk = (1/3)ku3-k Answers a. b. c. d. e. f. 2. Express the sequences shown below in terms of the unit step sequence. a. [Answer: fk = uk+3 - uk-4 ] b. Answer: fk= k uk− uk − 4[ ]+ 4uk − 4= kuk− k − 4( )uk − 4 12341/21/41/81/161fkk. . .12-1-21-1/21/44fkk. . .-212-1-2fkk112-1-2fkk112-1-2-4fkk-212-1fkk1/2731/91/313. . .12-1-2fkk1-4-33412-1-2fkk134562344. . .4c. Answer: fk= 3 − k( )uk− uk − 4[ ]+ 3u−1− k 3. Find whether each of the sequences below is periodic or aperiodic. If it is periodic, find its period, N. a. xk = exp{j7πk/9} [Answer: periodic with N = 18 ] b. xk = exp{j7k/9} [Answer: aperiodic] c. xk = exp{j6πk/9} [Answer: periodic with N = 3 ] 4. For the signal f(t) with Fourier Transform F(jω) shown below, what is the maximum interval T at which it can be sampled for perfect reconstruction ? [ Answer: T < 0.25 mS ] 5. For the signal: f(t) = sin 2000πt + cos 6000πt + 3 cos 10000πt what is the maximum sampling interval, T, that will allow for perfect reconstruction of f(t)? [ Answer: T < 0.1 mS ] 6. Find the linear difference equation for the discrete time system shown below. [ Answer: yk + 2= xk+ 2yk + 1− 4 yk⇒ yk + 2− 2yk + 1+ 4 yk= xk 12-1-2fkk1323. . .33f(t)F(j)-400040001unitdelayunitdelay24xkyk+-7. Find the linear difference equation for the discrete time system shown below. [Answer: yk= yk1− yk2⇒ yk+ 2yk − 2= 4 xk− 4 xk − 2 ] 8. Find the linear difference equation for the discrete time system shown below. [Answer: yk= −yk1+ yk2⇒ yk− 2yk −1= xk −1− 2xk] Before completing problems 9 - 13, review the MATLAB Tutorial #1. For each MATLAB problem, turn in your MATLAB code and the graphs. 9. Use MATLAB to graph the sequence in problem 1b for -4 ≤ k ≤ 4. 10. Use MATLAB to graph the sequence in problem 1e for -4 ≤ k ≤ 4. 11. Use MATLAB to graph the sequence in problem 1d for -4 ≤ k ≤ 4. 12. Use MATLAB to graph the sequence in problem 1f for -2 ≤ k ≤ 5. 13. For the three complex exponential functions given in problem 3 for 0 ≤ k ≤ 50: a. Plot the real part of x versus the imaginary part of x. b. Plot the magnitude of x versus k c. Plot the phase of x versus k (in radians)


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