I have attached the project’s needed documents. Keep in mind there are some parts ( Q1-Q5, I think)  of the project need calculation, and need to be scanned. The professor wants to see our calculation and our code. I have the parameters need for this project; the parameters are:

a = 0, b = 1, c = 5, d = 1, e = -1 

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:

?(?) = ∑ ? exp(?2?? ??), (1)
where






0

0

0

)(
1

0
0

T

t

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 ? = 1, 3, 5, 15 and 100.

For each case of ? 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.