#### Signals And Systems

1) For signal x(t) below, generate the corresponding graph using the following operation: y(t) = x(2t - 4)

1. State whether the given systems are stable, marginally stable or unstable. Also explain why. A) (D²+3D+2)D²y(t)=(D+3)x(t) B) (D+3)y(t)=(D²+D+1)x(t) C) (D²+D+1) (D²+D+1)y(t)=Dx(t) D) (D²+5D+4)y(t)= (D²+D+1)x(t) E) (D²-5D+4)y(t)= (D+5)x(t)

The following simultaneous ordinary differential equations present a mathematical model for a system:

The continuous time signal xc(t)=cos(49.7mt). is sampled with a sample period T. z[n]=cos(in),-oo<n<oo where A=9 Choose the smallest possible value of T in milliseconds/sample consistent with this information. Provide a number as your answer with an accuracy of two decimal digits.

The continuous time signal xc (t)=sin(23t)+cos(k23mt)where k =6.is sampled with a sample period I to obtain the discrete-time signalc [n]=sin()+cos()where A =25Choose the smallest possible value of T in milliseconds/sample consistentwith this information.Provide a number as your answer with an accuracy of two decimal digits.

Exercise 1. FOURIER SERIES REPRESENTATION OF A DISCRETE PERIODIC SIGNAL (i) Determine the Fourier series coefficients a, for the following signal, and write the signal z[n] in terms of its Fourier series coefficients (hint: solve "by inspection"):

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?

Determine the signal r(t) for the given Fourier series coefficients given that the period T = 4. (You can apply Fourier series properties to examples done in the lectures rather than re-computing the entire problem from scratch.)

Problem 2 [15 points] For the given signals 1. r(t) = 4 sin(20πt) 2. r₂(t) 4 sin(80nt) 3. T3(t) = -4 sin(40πt) Answer the following questions a. [6 points] Using the MATLAB command stem to plot the signals formed by sampling the above three functions over the interval 0 < t <1 with sampling rate of 30 Hz. b. [6 points] Determine the Nyquist sampling rate for each signal. c. [3 points] What do you observe?

6. Textbook 1.11