Signals and Systems
Sharif University of Technology
Dr. Hamid Reza Rabiee
November 6, 2010
CE 40-242
Date Due: Aban 29, 1389
Homework 5 (Chapter 5)
Problems
1. Computing the Fourier transform.
5-21. a
5-21. b
5-21. d
5-21. f
5-21. h
5-21. j
5-21. k
2. Determining corresponding signals of the transforms.
5-22. a
5-22. b
5-22. d
5-22. e
5-22. f
5-22. h
3. Problem 5-10
4. Problem 5-31
5. Problem 5-34
6. Suppose the input to system L is:
x(t) =
1
1+t2
and system L has an impulse response whose Fourier transform is
H(ω) =
1, -1 ≤ ω < 1
0,
else
If the output of the system is y(t), find the energy of y(t),
Ey =
R∞
−∞ y
2 (t)dt
and express Ey as a percentage of the energy input to the filter, Ex ,
1
Ex =
R∞
−∞ x
2 (t)dt
7. A continuous-time signal x(t) has the Fourier transform:
X(jω) =
1
b+jω
where b is a constant. Determine the Fourier transform V (jω)of thef ollowingsignals.
(a). v(t) = x(5t − 4)
(b). v(t) = x(t)ej2t
(d). v(t) = x(t)cos4t
(e). v(t) = x2 (t)
Practical Assignment
(a) Provide a simple m-file in MATLAB to compute the discrete Fourier transform of a given
sequence. The inputs of the function are the input sequence x as a row vector and the length
of the transform N. It checks the length of x to be satisfied with N. Then a transformation
matrix W will be formed and the DFT vector X will be produced by a matrix-vector
multiplication. The magnitude of the DFT should be plotted at the end.
(b) Find the Fourier transform of the signals shown below. Plot the magnitude and phase of
the Fourier transform using your m-file. Compare the output of your function to output of
standard function ”fft”.
2