EE120 Midterm 1 - Spring 2004Problem 1 (Short Questions) 20 PointsFor each of the following statements, if you believe it is true, give a justification. If you believe it is false, givea counterexample.(a) A linear causal continuous-time system is always time-invariant.(b) The system with (real-valued) input x(t) and output given byy(t) = (1+(x(t))^2)^(cos(t))is stable.(c) The discrete-time signal x[n] = cos(n) is a periodic signal.(d) For an otherwise completely unknown system, it is known that when the input is given byx(t) = cos(t) + cos(2t),the output isy(t) = .5(1+cos(t)+cos(2t)+cos(3t)).This system cannot be a linear time-invariant (LTI) system.Problem 2 (Convolution) 20 PointsThe continuous-time signals x(t) and y(t) are given in Figure 1. In the figure, draw the signal z(t) given byz(t) = (x * y)(t).Carefully label both axes. |x(t) |y(t) | | 1|_______ 1|_______ | | | | | | | | ________-1______|_______|________ ________________|_______|__________ | | 1 t | 1 t | | | |_______|-1 | | | | |Figure 1: Convolution: z(t) = (x * y)(t)EE120 Midterm 1 - Spring 2004Problem 1 (Short Questions) 20 Points 1Problem 3 (Inverse discrete-time Fourier Transform.) 15 PointsA discrete-time signal h[n] has discrete-time Fourier transformH(e^(jw)) = (1+e^(-jw))/(1-.5e^(-jw)).Find the signal h[n].Problem 4 (A linear time-invariant system.) 30 PointsA linear time-invariant system with input x(t) and output y(t) satisfies(a^2)y(t) + 2a(dy(t)/dt) + d^2(y(t))/dt^2 = x(t).(a) (10 Points) Find the frequency response H(jw) of the considered system.(b) (10 Points) For a=1/2, sketch the magnitude of the frequency response H(jw). Is the system ratherhigh-pass or rather low-pass? Justify your answer.(c) (10 Points) For what values of a is the system stable? Justify your answer. Remark: If you cannot solve themath, don't worry. Just describe clearly and concisely how you would proceed, and you will get partial credit.Problem 5 (Filtering.) 15 pointsThe signal x(t) with spectrum X(jw) as shown in Figure 2 is passed through a linear time-invariant (LTI)system with impulse responseh(t) = 2sinc(2t),where, as defined in class,sinc(t) = sin(pi*t)/(pi*t).Denote the output fo the system by y(t). Calculate the error between x(t) and y(t), given byintegral from -infinity to +infinity of |x(t) - y(t)|^2 dt. |X(jw) ______ |2 ______ | | | | | | | | | | | |_____ |1 _____| | | | /\ /|\ /\ | |EE120 Midterm 1 - Spring 2004Problem 3 (Inverse discrete-time Fourier Transform.) 15 Points 2| | / \ / | \ / \ | | _______|___________|/____\/__|__\/____\|___________|________ -3pi -9pi/4 -pi pi 3pi/2 9pi/4 3pi w -3pi/2Figure 2: The spectrum of the signal x(t).EE120 Midterm 1 - Spring 2004Problem 5 (Filtering.) 15 points