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TITLE : Fourier Series and Frequency Transfer

Function

DATE OF
: 24 th Oct, 2022
EXPERIMENT

Lab Number : 3

NAME : ___Asad Iqbal___

ROLL NO : ___200401033____

Instructor : _Ma’am Saniya Zafar_

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 Objective:
In this lab, we will learn to behavior simulation in Simulink/MATLAB after obtaining:
• Fourier series of periodic signals.
• Frequency transfer function of a linear system/device.
 Introduction:
 Periodic Signals and Fourier Series:
A signal is said to be periodic signal if it has a definite pattern and repeats itself at a regular
interval of time. We represent the definition of a periodic function mathematically as:
f ( t )=f ( t+T 0 ) ,
where T 0 >0 represents the period. Any function that satisfies the above equation is a
periodic signal.
Now, any periodic function f ( t ) with an arbitrary time period T 0 can be represented by a
linear combination of harmonic sinusoids, as follows:
a0 ∞
f ( t )= + ∑ ( a cos ( n ω 0 t ) +b n sin ( n ω 0 t ) ) ,
2 n=1 n
2π
where ω 0= is the fundamental frequency in rad/seconds. The a n and bn are the Fourier
T0
coefficients specify that “how much” of the sinusoids cos ( n ω0 t ) and sin ( n ω 0 t ) are present
in f ( t ). These three coefficients are computed as:
1
a 0= ∫ f ( t ) dt ,
T0 T 0

2
a n= ∫ f ( t ) cos ( n ω0 t ) dt ,
T0 T 0

2
b n= ∫ f ( t ) sin ( n ω 0 t ) dt .
T0 T 0

 Frequency Transfer Function of Linear Systems

If an eigenfunction such as complex sinusoid of frequency ω 0 rad/s—that is, exp( j ω0 t)is
applied at the input of a linear system, the steady-state output is also a complex sinusoid of
the same frequency ω 0 rad/s. The amplitude and phase of output complex sinusoid are
determined by the system’s frequency response. In other words, the output y ( t ) can be
expressed as
y ( t ) =H ( ω0 ) exp( j ω0 t) ,

where H ( ω 0 )is a complex-valued function of the input frequency ω 0. Rewrite y ( t ) as:

y ( t ) =|H ( ω0 )|exp ( j ∠ H ( ω0 ) ) exp ( j ω0 t),

¿|H ( ω 0 )|exp ( j ω 0 t + j ∠ H ( ω0 ) ) .

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Clearly, the magnitude and the phase of the output complex sinusoid y ( t ) are influenced
by |H ( ω 0 )|and ∠ H ( ω0 ) , respectively. The general function H ( ω ) , which is the Fourier
transform of the system impulse response, is called the frequency transfer function.

 Experiments:
i. Consider the Simulink design model developed in Tutorial #2. Keep the same
values for resistor and capacitor.
a. For each of the frequencies ω given in the first column of Table 3.1, calculate
¿ H (ω)∨¿ by using equation and fill in the second column. You may need to change
the simulation run time for every frequency to have a proper display on Scope.
b. Measure the output signal amplitude from the model of the RC low-pass filter in
Simulink and complete the third column of the Table. Is the measured value different
than the calculated value? Why?
Resistance (Ohm) = 500, Capacitance (F) = 1033e-6

 Circuit Diagram:

Observed ¿ H (ω)∨¿ from

Input
Scope
frequency Calculated ¿ H (ω)∨¿
V
ω [rad/s] |H ( ω )|= out
V¿
0 1 1
5 0.36 0.36
10 0.19 0.2
20 0.096 0.2
40 0.048 0.04
80 0.024 0.02
120 0.016 0.01

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 There is a small difference the measured values and the calculated values
because of the inaccuracy of software and technical errors.

 ω =0 rad/s

 ω =5 rad/s

 ω =10 rad/s

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 ω =20 rad/s

 ω =40 rad/s

 ω =80 rad/s

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 ω =120 rad/s

ii. To the simulink model of experiment (i), give the periodic input signal generated in
tutorial #1 (partial-sum Fourier Series) of this lab. Is the output signal peridoic? Why?

 Circuit Diagram:

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 Result:

 The output signal is Periodic as it repeats itself after certain interval of time
as the input is also periodic. Since it blocks the higher frequencies making the
signal smoother, acting as a low pass filter.
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 Conclusions:
From this lab report we conclude that with the help of SIMULINK software we can
compute Fourier series of a periodic signal which can be visualize on the scope. Also,
we saw low pass and high pass filter circuit RC & RL using Simulink.

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