Fourier Transforms

1.    If  and  , use the Fourier Transform properties to determine the following.

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

3.    Prove that the trigonometric, complex exponential and Cosine with phase form of Fourier series are equivalent and derivable from each other.



4.    Using trigonometric form Find the Fourier series of:



 And derived the same results using complex exponential form.



5.    Prove that the integral:

6.    Define the Fourier and inverse Fourier Transforms and Prove that the Fourier transforms are a special case of Laplace Transforms.




7.    Find the Fourier Transforms of the following signals:







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Laplace Transforms

All these exercises are solved step-by-step. Typed on MS word document.



1.    Define Laplace transforms. What is the motivation of using Laplace transforms in engineering?


2.    Find the Laplace Transforms of an exponential function

3.    Given a differential equation with initial condition y(0)=1:

a.    Solve using Laplace transforms.
b.    Find transfer function.

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

a.    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.

b.    Determine the time response vc(t) of the circuit if  , i.e. 1.2 impulse input.



5.    Determine the Laplace transform of the following signals.



6.    Determine the Laplace transform of.



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



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