Southern Methodist University

School of Engineering

Electrical Engineering Department

EE 2170 Design and Analysis of Signals and Systems

Laboratory Notes

Yasser Ghanbari, Carlos Davila, Scott C. Douglas, and Panos E.

Papamichalis,

2011, SMU. All Rights Reserved

Spring 2011

Contents

4. Integral in MATLAB

a. Example: integration in MATLAB

b. Example: convolution of two signals using trapz

c. Example: convolution of two signals using conv

d. Assignment 4: Convolution

5. Exponential Fourier Series

a. Exponential Fourier series coefficients

b. Signal reconstruction using the exponential Fourier series coefficients

c. Side note: summation of vectors

d. Assignment 5: exponential Fourier series in MATLAB

6. Introduction to Simulink

a. Example: Fourier series coefficient in Simulink

b. Example: square wave signal reconstruction in Simulink

c. Assignment 6: exponential Fourier series coefficients in Simulink

7. Fourier Transform

a. Example: Fourier transform of a rectangular function

b. Assignment 7: Fourier Transform

8. First Order RC Lowpass Filters

a. Example: frequency response of the first order LPF

b. Assignment 8: RC Lowpass Filters

9. Filters in LTI Systems

a. Example: lowpass filtering of a noisy EKG signal

b. Assignment 9: Lowpass, Highpass, and Bandstop Filters

10. Audio Signals

a. Example: audio signal analysis in MATLAB

b. Assignment 10: Notch Filter for Interference Removal

11. Laplace Transform

a. Example: Laplace transform visualization in MATLAB

b. Assignment 11: Visualizing the Laplace Transform

12. Image Signals

a. Example: image signal analysis in MATLAB

b. Assignment 12: Image Signal Enhancement and Resizing

7. Fourier Transform

The Fourier transform is defined as the following integral

dtetfjF tj )()( (7-1)

The integral can be implemented using trapz in MATLAB. The only issue that may be raised is

the integration time interval that cannot be defined in the interval of )( t . Therefore,

one should select the time interval wide enough to cover all the nonzero parts of the signal. Since

the term inside the integral is complex, the Fourier transform of a function f(t) would result in a

complex function F(j). Therefore for the implementation purposes one should be careful about

the result and return what is desired (e.g. the Fourier transform magnitude, or phase, or both) for

any specific application. The following example explains it.

Example: Fourier transform of a rectangular function

Suppose that the Fourier transform magnitude, |F(j)|, of a rectangular function is to be

calculated and plotted with respect to the frequency. The following program shows how to define

a rectangular function f(t) in the time interval of ]55[ t (sec), to calculate the

corresponding Fourier transform F(j), and to plot its magnitude |F(j)| versus frequency in the

frequency interval of sec)/(]5050[ rad . The resulting plots are displayed in Figure 7-2.

Figure 7-1. Implementation of a rectangular function and the corresponding Fourier transform.

Figure 7-2. Rectangular signal and its Fourier transform magnitude.

clc; clear all; close all; %-----------Rectangular function----------- t = -5 : 0.001 : 5; f = zeros(size(t)); for i = 1 : length(t) if abs(t(i))

Assignment 7: Fourier Transform

In this assignment, we want to implement the Fourier transform in MATLAB. The goal is to

write a function which calculates the Fourier transform coefficients of a given function, and to

apply the function to a synthetic and a real signal.

a. Write a MATLAB function, [F]= FT(f,t,omega), which gets the values of the input function f(t) in a vector f, the vector of time positions t (t in sec), and the vector of omega

(specifying the frequency resolution for the Fourier transform) as the input arguments so

as to calculate and return the Fourier transform (in a vector F with the same size as

omega). The Fourier transform is defined as below:

dtetfjF tj )()(

For integration purposes, use the trapz command available in MATLAB. The integral is

applied to the expression tjetf )( within the time interval determined by the vector t.

b. Write a MATLAB program (m-file) in which the following periodic functions is defined in the time interval of [0 5] sec with the step size of 0.001.

(i) )50cos()( ttf

(ii) )3

100sin()(

ttg

(iii) )3

100sin()50cos()(

ttth

Then define a vector, omega, in the interval [-200 200] rad/sec with the step-size of 2

and call your function from part (a), FT, to obtain the Fourier transform in the vectors F,

G, and H. Plot |F(j)|, |G(j)|, and |H(j)| with respect to in 3 subplots. From your

plots, find and report all the frequency components (in Hertz) of each signal.

c. Write a MATLAB program (m-file) in which the EKG signal provided in the lab is loaded into the variables x and t. First, subtract the mean of signal from itself. Then

define omega vector in the interval [0 200] with the step-size of 2/100 and call your function from part (a), FT, to obtain the Fourier transform (X) of the EKG signal.

(i) Plot |X(j)| with respect to .

(ii) Find the fundamental frequency of the quasi-periodic EKG signal on the plot and calculate the average heartbeat per minute. Explain your observations on the plot.

(iii) A noticeable peak is seen within the frequency range of [300 400] rad/sec. Calculate and report this peaks corresponding frequency in Hertz. Explain what the source of this component is, and why this frequency peak is happening.