‘Public law has grown to meet the needs of justice on the one hand and of good government on the other by holding public administration to as much of its….

## Determine the impulse response for the system: (??2 + 5?? + 6)??(??) = ????(??).

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

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

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

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

- (??

- For the following signal find its Fourier series:

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

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

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

- For the following signal find its Fourier series:

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

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

??_{0}(0) = 2 and ???_{0}(0) = -1.

- For the following signal find its Fourier series:

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

- 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 ??(??) = ??^{(-??)}??(??).

- Noisy Sinewave

- Generate a vector signal with 4 cycles of 1kHz sinewave at a sampling frequency of 44.1kHz and an amplitude of 1V.
- Plot the signal on the screen and label the X and Y axes with the correct labels.
- Convert your Matlab code into a function in an M-file.
- Use ‘help’ to lookup the description of the built-in function
*randn()*. - 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.
- Add the noise to your original signal and plot it.
- Plot all three signals as a combined

- 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)

- 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)

- 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

- repeat all parts of question 16 for the following example: