1. CH#3 Fourier Series and Transform
King Saud University
College of Applied studies and Community Service
1301CT
By: Nour Alhariqi
CH# Fourier Series and Transform
2nd semester
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2. Outline Introduction Fourier Series Fourier Series Harmonics
Fourier Series Coefficients
Fourier Series for Some Periodic Signals
Example
Fourier Series of Even Functions
Fourier Series of Odd Functions
Fourier Series- complex form
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3. Fourier Series Fourier Transform Introduction
The Fourier analysis is the mathematical tool that shows us how to deconstruct the waveform into its sinusoidal components.
This tool help us to changes a time-domain signal to a frequency-domain signal and vice versa.
Time domain: periodic signal
Frequency domain: discrete
Fourier Series
Time domain: nonperiodic signal
Frequency domain: continuous
Fourier Transform
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4. Fourier Series
Fourier proved that a composite periodic signal with period T (frequency f ) can be decomposed into the sum of sinusoidal functions ( or complex exponentials) .
A function is periodic, with fundamental period T, if the following is true for all t: f(t+T)=f(t)
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5. Fourier Series
A periodic signal can be represented by a Fourier series which is an infinite sum of sinusoidal functions (cosine and sine), each with a frequency that is an integer multiple of f = 1/T 
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6. Fourier Series Harmonics
Fourier Series = a sum of harmonically related sinusoids
fundamental frequency
the 2nd harmonic frequency
the kth harmonic frequency
the kth harmonic
fundamental
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7. Fourier Series Harmonicsω ω ω ω ω ω
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8. Fourier Series Coefficients
Are called the Fourier series coefficients, it determine the relative weights for each of the sinusoids and they can be obtained from
DC component or average value
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9. Function s(x) (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. 
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10. Fourier Series for Some Periodic Signals
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11. Example The Fourier series representation of the square wave
Single term representation of the periodic square wave
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12. Example
The two term representation of the Fourier series of the periodic square wave
The three term representation of the Fourier series of the periodic square wave
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13. Example Fourier representation to contain up to the eleventh harmonic
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14. Example
Since the continuous time periodic signal is the weighted sum of sinusoidal signals, we can obtain the frequency spectrum of the periodic square-wave as shown below
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15. Example
From the above figure we see the effect of compression in time domain, results in expansion in frequency domain.
The converse is true, i.e., expansion in time domain results in compression in frequency domain.
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16. Fourier Series of Even Functions
Even functions can solely be represented by cosine waves because, cosine waves are even functions. A sum of even functions is another even function. 10 5 q
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17. Fourier Series of Odd Functions
Odd functions can solely be represented by sine waves because, sine waves are odd functions. A sum of odd functions is another odd function. 10 5 q
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18. Fourier Series of Even/Odd Functions
The Fourier series of an even function
is expressed in terms of a cosine series.
The Fourier series of an odd function
is expressed in terms of a sine series.
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19. Fourier Series- complex form
The Fourier series can be expressed using complex exponential function
The coefficient cn is given as
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20. Fourier Transform
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21. Outline Fourier transform Inverse Fourier transform
Basic Fourier transform pairs
Properties of the Fourier transform
Fourier transform of periodic signal
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22. Fourier Transform
Fourier Series showed us how to rewrite any periodic function into a sum of sinusoids. The Fourier Transform is the extension of this idea for non-periodic functions.
the Fourier Transform of a function x(t) is defined by:
The result is a function of ω (frequency). 
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23. Inverse Fourier Transform
We can obtain the original function x(t) from the function X(ω ) via the inverse Fourier transform.
As a result, x(t) and X(ω ) form a Fourier Pair:
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24. Example Let The called the unit impulse signal :
The Fourier transform of the impulse signal can be calculated as follows
So ,
x(t) t w
X(w)
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25. Basic Fourier Transform pairs
Often you have tables for common Fourier transforms
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26. Example
Consider the non-periodic rectangular pulse at zero with duration τ seconds
Its Fourier Transform is:
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27. Properties of the Fourier Transform
Linearity:
Left or Right Shift in Time:
Time Scaling:
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28. properties of the Fourier Transform
Time Reversal:
Multiplication by a Complex Exponential ( Frequency Shifting) :
Multiplication by a Sinusoid (Modulation):
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29. Example: Linearity Let x(t) be :
The Fourier Transform of x(t) will be :
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30. Example: Time Shift Let x(t) be :
The Fourier Transform of x(t) will be :
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31. Example: Time Scaling time compression  frequency expansion
time expansion  frequency compression
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32. Example: Multiplication by a Sinusoid
Let x(t) be :
The Fourier Transform of x(t) will be :
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33. Examples
determine the Fourier transform of the following periodic signal
5 𝛿 𝑡+2 +𝛿(𝑡−1)
Given that
Given that x(t) has the Fourier transform X(ω), Express the Fourier transform of the following signal in terms of X(ω)
𝑓(𝑡)=𝑥 1−2𝑡 +𝑥(−3−𝑡)
𝛿(𝑡)↔1
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34. Fourier Transform for periodic signal
We learned that the periodic signal can be represented by the Fourier series as:
We can obtain a Fourier transform of a periodic signal directly from its Fourier series
the coefficient cn is given as
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35. Fourier Transform for periodic signal
The resulting transform consists of a train of impulses in the frequency domain occurring at the harmonically related frequencies, which the area of the impulse at the nth harmonic frequency nω0 is 2π times nth the Fourier series coefficient cn
So, the Fourier Transform is the general transform, it can handle periodic and non-periodic signals. For a periodic signal it can be thought of as a transformation of the Fourier Series
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36. Example Let The Fourier series representation of
The Fourier series coefficients
The Fourier transform of
So,
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37. nalhareqi_2014

38. Example Let , find its Fourier transform ?
The Fourier series representation of is
The Fourier series coefficients
The Fourier transform of is
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39. Example Let find its Fourier transform ?
The complex Fourier series representation of is
The Fourier series coefficients
The Fourier transform of is
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