MatLab Project for Signal Course
ahmedbataweel
project_rectangular_wave_function.pdf
1
EE 329: Signals and Systems II Spring 2020 Schmid
Project
Distributed: Wednesday, February 12, 2020 Due: Wednesday, March 4, 2020
In this project you will work with two Fourier series and will analyze the complex Fourier series coefficients. At the end, you will evaluate the period of a strong pulsar by applying Fourier Transform to noisy radio astronomy data. Let )(tx be a periodic function with period 0T . The function is shown below:
We represent )(tx on its period as a complex exponential Fourier series:
𝑥(𝑡) = ∑ 𝑎 exp(𝑗2𝜋𝑓 𝑘𝑡), (1) where
00
0
)( 1
0 0
Tt
t
dttx T
a and
00
0
0 0
2exp)( 1
Tt
t
k dtktfjtxT a . (2)
Alternatively, we represent )(tx as a trigonometric Fourier series:
1
000 )}2sin()2cos({)( k
kk ktfcktfbbtx , (3)
where
00
0
)( 1
0 0
Tt
t
dttx T
b ,
00
0
)2cos()( 2
0 0
Tt
t
k dtktftxT b (4)
and
00
0
)2sin()( 2
0 0
Tt
t
k dtktftxT c . (5)
Note that 0t can be selected such that the evaluation of the integrals is simplified.
ca b
dd
……
t e
x(t)
2
1. Find the Fourier series coefficients ka for the function shown above.
2. Determine and plot the magnitude and phase spectrum of the Fourier series coefficients ,ka that is, plot || ka as a function of discrete frequencies kf0 and ka as a function
of discrete frequencies kf0 .
3. Find kb and kc for the function shown above in terms of ka . Note that kb and kc are
real valued coefficients. 4. Plot an approximation )(~ txN obtained by truncating the series (3) to N terms, that is,
N
k kkN ktfcktfbbtx
1 000 )}2sin()2cos({)(
~ .
Set 𝑁 = 1, 3, 5, 15 and 100.
For each case of 𝑁 find the value of the average residual power.
5. Write a short summary covering items 1 through 4. 6. Find the period 0T of a pulsar by analyzing data in Fourier Transform domain. Use Matlab
to read the data, then apply fft function. Analyze the magnitude spectrum. Data files intensity.mat and time.mat will be sent in a separate e-mail.
