An LTIC system is specified by the equation (??² + 4?? + 4) ??(??) = ??x(??)

1. Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of the system.
2. Find ??₀(??), the zero-input component of the response ??(??) for t = 0, if the initial conditions are ??₀(0) = 3 and ???₀(0) = -4

1. An LTIC system is specified by the equation (??² + 5?? + 6) ??(??) = (?? + 1)x(??)
   1. Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of the system.
   2. Find ??₀(??), the zero-input component of the response ??(??) for t = 0, if the initial conditions are ??₀(0) = 2 and ???₀(0) = -1

1. Explain with reasons whether the LTIC systems are asymptotically stable, marginally stable, or unstable?
   1. (??² + 8?? + 12) ??(??) = (??-1)x(??)
   2. D(??² + 3?? + 2) ??(??) = (?? + 5)x(??)

1. For the following signal find its Fourier series:

1. For the following signal find its amplitude, frequency, period, angular frequency, and phase.

??(??) = 45 ??????(2??880?? + ??/5)

1. Figure below shows signal x(t). Sketch and describe mathematically this signal timecompressed by factor 3 and delayed by 5.

1. For the following signal find its Fourier series:

1. For a LTI system with the unit impulse response h(??) = ??^(-10??)??(??), determine the response y(t) for the input ??(??) = ??^(-??)??(??) + ??^(-3??)??(??).
2. Determine the impulse response for the system: (??² + 5?? + 6)??(??) = ????(??).

1. Find y(t) for the system mentioned in question 10 for input ??(??) = 5??^(-2??)??(??) when

??₀(0) = 2 and ???₀(0) = -1.

1. For the following signal find its Fourier series:

1. A continuous-time signal ??(??) is shown. Sketch the signals 3??(0.25?? + 4).

1. The unit impulse response of an LTI system is h(??) = [2??^(-3??)– ??^(-2??)]??(??) . Find the system’s zero-state response y(t) if the input ??(??) = ??^(-??)??(??).

1. Noisy Sinewave

1. Generate a vector signal with 4 cycles of 1kHz sinewave at a sampling frequency of 44.1kHz and an amplitude of 1V.
2. Plot the signal on the screen and label the X and Y axes with the correct labels.
3. Convert your Matlab code into a function in an M-file.
4. Use ‘help’ to lookup the description of the built-in function randn().
5. Generate a normally distributed random noise signal, also at 44.1KHz with the same number of samples as your sine wave. The rms value of the noise should be 0.1V.
6. Add the noise to your original signal and plot it.
7. Plot all three signals as a combined

1. Find Fourier Series of the following signal (this part needs to be done by hand)

Plot the signal in Matlab
T=pi/2;%Time period w=2*pi/T; %angular frequency
t=0:0.01:10 %time we wan to plot the square wave over
F=(2*square(w*t)); % square wave should be multiplied by 2 because our square was has amplitude of 2 plot(t,F)

1. Using Matlab, find the Fourier Series Coefficients (a0, an, bn). syms t

T=pi/2; % Time period of the periodic function
n=1:5; % Number of terms
% int(Your function, Your variable, Lower bound of integration, Upper bound of integration)
a0=(1/T)*int(2,t,0,pi/4)-(1/T)*int(2,t,pi/4,pi/2)
an=(2/T)*int(2*cos(n*t*2*pi/T),t,0,pi/4)-(2/T)*int(2*cos(n*t*2*pi/T),t,pi/4,pi/2) bn=(2/T)*int(2*sin(n*2*pi/T*t),t,0,pi/4)-(2/T)*int(2*sin(n*2*pi/T*t),t,pi/4,pi/2)

1. Write a code in Matlab that reconstructs the initial signal using a0, an, and bn coefficients.

A=a0; for i=1:length(an);
A=A+an(1,i)*cos(i*t)+bn(1,i)*sin(i*t); end

1. repeat all parts of question 16 for the following example: