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
1st semester
nalhareqi_2013

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
nalhareqi_2013

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
nalhareqi_2013

4. Fourier Series
Fourier proved that a composite periodic signal with period T (frequency f ) can be decomposed into the sum of sinusoidal functions.
A function is periodic, with fundamental period T, if the following is true for all t: f(t+T)=f(t)
nalhareqi_2013

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 
nalhareqi_2013

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
nalhareqi_2013

7. Fourier Series Harmonicsω ω ω ω ω ω
nalhareqi_2013

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
nalhareqi_2013

9. Fourier Series for Some Periodic Signals
nalhareqi_2013

10. Example The Fourier series representation of the square wave
Single term representation of the periodic square wave
nalhareqi_2013

11. 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
nalhareqi_2013

12. Example Fourier representation to contain up to the eleventh harmonic
nalhareqi_2013

13. 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
nalhareqi_2013

14. 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.
nalhareqi_2013

15. Even Functions
The value of the function would be the same when we walk equal distances along the X-axis in opposite directions. t
Mathematically speaking -
nalhareqi_2013

16. Odd Functions
The value of the function would change its sign but with the same magnitude when we walk equal distances along the X-axis in opposite directions. t
Mathematically speaking -
nalhareqi_2013

17. 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
nalhareqi_2013

18. 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
nalhareqi_2013

19. 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.
nalhareqi_2013

20. Fourier Series- complex form
The Fourier series can be expressed using complex exponential function
The coefficient cn is given as
nalhareqi_2013

21. Fourier Transform
nalhareqi_2013

22. Outline Fourier transform Inverse Fourier transform
Basic Fourier transform pairs
Properties of the Fourier transform
Fourier transform of periodic signal
nalhareqi_2013

23. 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). 
nalhareqi_2013

24. 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:
nalhareqi_2013

25. 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)
nalhareqi_2013

26. Basic Fourier Transform pairs
Often you have tables for common Fourier transforms
nalhareqi_2013

27. Example
Consider the non-periodic rectangular pulse at zero with duration τ seconds
Its Fourier Transform is:
nalhareqi_2013

28. properties of the Fourier Transform
Linearity:
Left or Right Shift in Time:
Time Scaling:
nalhareqi_2013

29. properties of the Fourier Transform
Time Reversal:
Multiplication by a Complex Exponential ( Frequency Shifting) :
Multiplication by a Sinusoid (Modulation):
nalhareqi_2013

30. Example: Linearity Let x(t) be :
The Fourier Transform of x(t) will be :
nalhareqi_2013

31. Example: Time Shift Let x(t) be :
The Fourier Transform of x(t) will be :
nalhareqi_2013

32. Example: Time Scaling time compression  frequency expansion
time expansion  frequency compression
nalhareqi_2013

33. Example: Multiplication by a Sinusoid
Let x(t) be :
The Fourier Transform of x(t) will be :
nalhareqi_2013

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
nalhareqi_2013

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
nalhareqi_2013

36. Example Let The Fourier series representation of
The Fourier series coefficients
The Fourier transform of
So,
nalhareqi_2013

37. nalhareqi_2013

38. Examples
nalhareqi_2013

39. Example 1 Let which has the fundamental frequency
Rewrite x(t) as a complex form and find the Fourier series coefficients ?
Its known from Euler’s relation that:
nalhareqi_2013

40. Solution Rewriting x(t) as a complex form, x(t) will be :
nalhareqi_2013

41. Solution
Thus, the Fourier series coefficients are:
nalhareqi_2013

42. Example 2
Consider a periodic signal x(t) with fundamental frequency π, that has the following Fourier series coefficients
Rewrite x(t) as a trigonometric form?
From the given coefficients, the x(t) in complex form
nalhareqi_2013

43. Solution
rewriting x(t) and collecting each of the harmonic components which have the same fundamental frequency, we obtain
Using Euler’s relation, x(t) can be written as:
nalhareqi_2013

44. Example 3
A periodic signal x(t) with a fundamental period T = 8 has the following nonzero Fourier series coefficients
Express x(t) in the trigonometric form?
The fundamental frequency is
nalhareqi_2013

45. Example 3 Let , find its Fourier transform ?
The Fourier series representation of is
The Fourier series coefficients
The Fourier transform of is
nalhareqi_2013