LABORATORY EXPERIMENT 1
Signals and Systems (TCEE3692)
Fourier Series Expansion of a Pulse Train
Instructions:
a) All work must be typed and submitted as a single pdf document.
b) Each student must do his/her work independently.
c) Orderly presentation and clarity of your work is of high importance.
1.
Introduction
Fourier series theory state that any periodic time domain signal can be represented or reconstructed
by a summation of harmonically related sinusoidal waves. Mathematically this can be written in
exponential form as:

xt    c n exp  j 2nf 0 t  ; where
n 0
T
cn   xt  exp  j 2nf 0 t dt gives the Fourier coefficients.
0
Alternatively the sinusoidal Fourier form is given by:

xt   a 0   a n cos2nf 0 t   bn sin 2nf 0 t  ; where the Fourier coefficients are given by:
n 1
a0 
 xt dt
allT
an 
bn 
2
T
2
T
 xt cos2nf t dt
0
allT
 xt sin 2nf t dt
0
allT
f 0 is the signal’s fundamental frequency
In this laboratory work, you will verify the Fourier series theory using a pulse train signal.
2.
Objectives
2.1 To construct a pulse train from frequency harmonic components based on Fourier series.
2.2 To display the constructed pulse train in time and frequency domain.
2.3 To investigate the effect of number of harmonics on the digital signal bandwidth.
2.4 To investigate the effect of pulse width (and therefore duty cycle) on signal bandwidth.
PART A
3.
Pre-Lab
3.1 Draw a periodic pulse train having a time period of 2 seconds with a duty cycle of 50% and a
peak-to-peak voltage of 5V (0V to 5V). The drawing should span over two time periods.
Clearly indicate the time period, and the amplitude of the signal in the drawing.
3.2 Determine the Fourier series expansion of the pulse train above.
3.3 Draw the amplitude spectrum of the signal in 3.1 for the first 5 frequency harmonics and the
DC component clearly indicating the frequency and amplitude of each component.
3.4 Repeat 3.1 to 3.3 for a pulse trains with the same amplitude and frequency but with a duty
cycle of 25% and 75%.
PART B
4.
Laboratory work 1a: Effect of number of harmonic components
5. Develop an m-file for Matlab to: Laboratory work 1a: Effect of number of harmonic
components
5.1
Develop an m-file for Matlab to:
5.1.1
define and plot the mathematical expression for Fourier series expansion of the
1.
pulse train defined in 3.1 of the pre-lab for N frequency harmonic, where N
is equal to 2000 for two time periods.
5.1.2
Plot the time representation of the signal
5.1.3
Define the frequency spectrum of the pulse train.
5.1.4
Determine the bandwidth of the signal.
5.2
Repeat 4.1.1 to 4.1.4 for N=1600, 800 and 400.
5.3
Discuss the effect of the number of frequency components in the Fourier series expansion
on bandwidth and the shape of the resulting time wave form.
5.4
5.5
requency components in the Fourier series expansion on bandwidth and the shape of the
resulting time wave form.
PART C
6. Laboratory work 1b: Effect of Pulse Train Duty Cycle
6.1
Repeat 4.1.1 to 4.1.4 for the signals defined in 3.4
6.2
Discuss the effect of duty cycle on signal bandwidth.