5) Find the system response to the input x(t), given the impulse response given by h(t), and as described in the figures below (hint: Graphical Convolution), where: x(t) = t & h(t) = t

A signal was acquired at 200 Hz and the spectrum showed a peak at 70 Hz. No anti- aliasing filter was used. Find the 5 lowest possible frequencies that the original signal may have. (Hint: Create and use the folding diagram) Provide the frequency values as integers.

Problem 1 [15 points, 5 points each] Find the inverse Laplace transform of the following functions by using Partial Fraction Expansion. Verify your results using MATLAB's ilaplace function.

Problem 3 [28 points, 4 points each] Consider the continuous-time linear time-invariant system with transfer-function: Answer the following questions: a. What is the differential equation associated with the above transfer-function? b. Calculate the poles and zeros of G(s). c. Is G(s) asymptotically stable? d. Use the bilinear transformation (aka Tustin transformation) S = 2 z 1 Tsz + 1 to calculate the corresponding discretized transfer-function Ga(z). e. Calculate the poles and zeros of Ga(z). f. Is Ga(z) asymptotically stable? (A discrete-time system is asympotically stable if the poles satisfy |pi|< 1.) g. Use a computer program or calculator to sketch the magnitude and the phase of the frequency response G(jw). Now sketch the magnitude and phase of Gd(es) as a function of w when Ts = {0.01, 0.1, 1}s. Compare all the obtained responses. What is the role of Ts?

Topics: Complex Exponentials and LTI Systems Exercise 1. CONTINUOUS-TIME SYSTEM RESPONSE GIVEN COMPLEX EXPONENTIAL INPUT (i) Consider an LTI system with impulse response h(t)=e5u(t). What is the output y(t) when the input is r(t) = ³³ (ii) For the same system, find the output y(t) if the input is more generally r(t) = e. Note that this is a function of w. Also, find the magnitude ly(t)] and sketch as a function of w. (iii) For the same system, suppose now the input is r(t) = cos(2t) sin(t). What is the output y(t)? (Hint: use the linearity of the system.)

Topics: Signal recovery from Fourier coefficients Exercise 2. CONTINUOUS-TIME SIGNAL RECOVERED FROM FOURIER COEFFICIENTS (i) The Fourier series coefficients of a continuous-time signal that is periodic with period 4 is: (ii) A continuous-time periodic signal z(t) is real-valued and has a fundamental period T Fourier series coefficients for this signal are specified as: = 8. The nonzero

Exercise -5: Design a 2nd order bandpass digital filter with center frequency at f=1.6kHz and a 3-dB bandwidth of 400Hz.

Exercise 4. FOURIER SERIES REPRESENTATION OF LTI SYSTEMS Consider a continuous-time LTI system with impulse response

Problem 1 [15 points] Determine the Fourier Transform of the following waveform for A = 12 and T = 3 s.

14.2 A sinusoid with 1-V peak amplitude is applied at the input of a filter having the transfer function Find the peak amplitude and the phase (relative to that of the input sinusoid) of the output sinusoid if the frequency of the input sinusoid is (a) 1 kHz, (b) 10 kHz, (c) 100 kHz, and (d) 1 MHz.