practice_problems - EECS 20 Final Exam Practice Problems...

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EECS 20. Final Exam Practice Problems, Spring, 2005. 1. Consider a LTI system with [ A, b, c, d ] representation given by: A = 1 1 0 1 , b = 0 1 , c T = [1 0] , d = 0 . (a) Calculate the zero-input state response when the initial state is s (0) = [ s 1 (0) s 2 (0)] T . (b) Calculate the (zero-state) impulse response, h . (c) Calculate the response y ( n ) , n 0 when the initial state is s (0) = [1 1] T and the input signal is n 0 , x ( n ) = δ ( n - 1) . 2. Consider the difference equation y ( n ) - y ( n - 1) = x ( n ) - 2 x ( n - 1) . (a) Take the state at time n as s ( n ) = [ y ( n - 1) , x ( n - 1)] T and write down the [ A, b, c, d ] representation of the system. Find its zero-state impulse response. (b) Implement the difference equation using two delay elements whose outputs are the two state components. (c) Find another implementation using only one delay element. Write the [ A, b, c, d ] repre- sentation for this implementation. Find its zero-state impulse response. (d) Are the two impulse responses the same? (e) Find the frequency response directly from the difference equation and by taking the DTFT of the impulse response. Are the two frequency responses the same? (f) Sketch the magnitude and phase response. 3. Consider a LTI system with [ A, b, c, d ] representation given by: A = 1 1 1 0 , b = 0 1 , c T = [1 1] , d = 0 . (a) Suppose the initial state is s (0) = [0 0] T . Find an input sequence x (0) , x (1) of length two such that the state at time 2 is s (2) = [1 1] T . (b) Suppose the initial state is s (0) = [ s 1 s 2 ] T . Find an input sequence x (0) , x (1) such that the state at time 2 is s (2) = [0 0] T . (The input sequence will depend on s (0) .) 4. Two SISO systems with representations [ A i , b i , c i , d i ] , i = 1 , 2 are connected in cascade com- position. Find an [ A, b, c, d ] representation for the composiiton. 5. A discrete-time, causal LTI system S produces the output y given by y ( n ) = δ ( n ) + δ ( n - 1) + δ ( n - 2) , 1
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π −π −π/2 π/2 ω X ( ω ) 1 π −π −π/2 π/2 ω Y ( ω ) Figure 1: Signal x for problem 7. in response to the input x given by x ( n ) = δ ( n ) + δ ( n - 2) . Find the impulse response h of S . 6. Evaluate the convolution integral y i = h i * x when x is the continuous-time unit step function , x ( t ) = 0 , t < 0; = 1 , t 0 , and the impulse response h i is as given below, i = 1 , 2 , 3 .
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