EEC 3073 Signals & Systems

ASSIGNMENT 4
Part A (Laplace Transforms)
Define Laplace transforms. What is the motivation of using Laplace transforms in engineering?
Find the Laplace Transforms of an exponential function
y(t)=e^(-10t)
Given a differential equationwith initial condition y(0)=1:
2 dy/dt-y=sint
Solve using Laplace transforms.
Find transfer function.

In the following circuit, assuming that the initial capacitor voltage is zero and initial inductor current is zero.

Determine the circuit transfer function H(s) from the input vin(t) to the output vc(t).for the values of the components shown in the figure.
Determine the time response vc(t) of the circuit if  , i.e. 1.2 impulse input.

Determine the Laplace transform of the following signals.

Determine the Laplace transform of  .
(Use  )

Determine the inverse Laplace transform of  . Use tables and the MATLAB symbolic ilaplace() functions as necessary.

Part B (Z-Transform)
What is Z-transform?What is the need to use Z-transform if Laplace transform can be used to deal with boundary value problem quite successfully?
An electrical system has the following transfer function:
G(s)= (1?RC)/(S+1?RC)
If R=20 K-Ohm and C=200 micro-Farad find the Z-transform of the G(s).
Find the output of the above system to an impulse input of 4 Volts assuming a sampling time of 1 second.
Find the output of the above system to a step input of 4 Volts assuming a sampling time of 1 second.

Using the Z-transform, the following system transfer function was determined from the signal flow diagram:

Find the corresponding difference equation in discrete time?

Determine the Z-transform of  .

If  , find   given the following equation.

A linear time-invariant, discrete-time system is described by the input/output difference equation:

Determine the system transfer function.
Use the graph below to plot at least one cycle of the function:

Function maximum value
Function period

Number the x-axis and y-axis appropriately.
The signal, x(t), is sampled by T = 2.5ms. Use ‘*’ to indicate the sampled values of the signal    in the chart above.
Determine the sampled signal by using the fact t = nt
Describe the following signal, x(t), as the sum of two cosine waves.

Draw the spectrum (amplitude only) for the signal, x(t). Label the axes.

Find the step response of the system represented by the following transfer function for zero initial condition.

Part C (Fourier Transforms)
1.    If  and  , use the Fourier Transform properties to determine the following.

(x convolved with y)

2. Use the Fourier Transform tables and properties to determine the following.

Prove that the trigonometric, complex exponential and Cosine with phase form of Fourier series are equivalent and derivable from each other.
Using trigonometric form Find the Fourier series of:
x(t)=sin??(2?ft)? Where f=2
And derived the same results using complex exponential form.
Prove that the integral:
?_0^T?e^(j2?(n-m)t/T)  dt
=0,if n?m

=T,if n=m
Define the Fourier and inverse Fourier Transforms and Prove that the Fourier transforms are a special case of Laplace Transforms.
Find the Fourier Transforms of the following signals:
x(t)=e^(-at) u(t)
x(t)=p_a (t)
=1 if |t|<a
=1 if |t|>a

Part D (Convolution)
Define convolution. Explain steps of convolution with an example.
Compute the output y(t) for a continuous time LTI system whose impulse response h(t) and the input x(t) are given by:
h(t)=e^(-at) u(t)and
x(t)=e^at u(-t)
For a>0.
For a continuous time LTI system whose impulse response h(t) and the input x(t) are given by:
h(t)=e^(-at) u(t)and
x(t)=u(t)
For a>0. Prove. that convolution is commutative