3) Express the waveform in the following figure in terms of unit step or ramp functions.

Find ao, an's, b's, C's and D's. Also, evaluate an, bn, Cn and D' for n=3 (find a3, b3, C3 and D and _3.) n

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.

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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: 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

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.)

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

Topics: Determining the Fourier Series Coefficients Exercise 3. FOURIER SERIES REPRESENTATION OF PERIODIC SIGNAL (i) Determine the Fourier series coefficients a for the following signal: