1. CH#3 Fourier Series and Transform 1 st semester 1436-1437 King Saud University College of Applied studies and Community Service 1301CT By: Nour Alhariqi 1nalhareqi_2015

2. Outline  Introduction  Fourier Series  Fourier Series Harmonics  Fourier Series Coefficients  Fourier Series of Even Functions  Fourier Series of Odd Functions  Fourier Series for Some Periodic Signals  Example  Fourier Series- complex form 2nalhareqi_2015

3. 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 3nalhareqi_2015

4. 4 A composite periodic signal

5. nalhareqi_20155 A nonperiodic composite signal Frequency domain

6. Fourier Series  Fourier proved that a composite periodic signal f(t) could be broken down into an infinite series of simple sinusoids which, when added together, would construct the exact form of the original waveform.  A function is periodic, with fundamental period T, if the following is true for all t: f(t+T)=f(t) 6nalhareqi_2015

7. Fourier Series Trigonometric Form  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 7nalhareqi_2015 be a periodic function with period The can be represented by a trigonometric series as: DC Part Even Part Odd Part

8. Fourier Series Trigonometric Form 8nalhareqi_2015

9. Fourier Series Harmonics Fourier Series = a sum of harmonically related sinusoids fundamental frequencythe k th harmonic frequencythe 2 nd harmonic frequency fundamental the k th harmonic term 9nalhareqi_2015

10. Fourier Series Harmonics ωωω ωωω 10nalhareqi_2015

11. 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 11nalhareqi_2015

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14. nalhareqi_201514 Function s(t) (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series.

15. Even functions can solely be represented by cosine waves because, cosine waves are even functions. A sum of even functions is another even function. 100 5 0 5  Fourier Series of Even Functions 15nalhareqi_2015

16. Odd functions can solely be represented by sine waves because, sine waves are odd functions. A sum of odd functions is another odd function. 100 5 0 5  Fourier Series of Odd Functions 16nalhareqi_2015

17. 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. Fourier Series of Even/Odd Functions 17nalhareqi_2015

18. Fourier Series for Some Periodic Signals 18nalhareqi_2014

19. Example The Fourier series representation of the square wave Single term representation of the periodic square wave 19nalhareqi_2014

20. 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 20nalhareqi_2014

21. Example Fourier representation to contain up to the eleventh harmonic 21nalhareqi_2014

22. 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 22nalhareqi_2014

23. 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. 23nalhareqi_2014

24. Fourier Series- complex form  The Fourier series can be expressed using complex exponential function The coefficient c n is given as 24nalhareqi_2014

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26. Fourier Transform nalhareqi_201526

27. Outline nalhareqi_201527  Fourier transform  Inverse Fourier transform  Basic Fourier transform pairs  Properties of the Fourier transform  Fourier transform of periodic signal

28. Fourier Transform nalhareqi_201528  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 decomposes any function into a sum of sinusoidal basis functions. Each of these basis functions is a complex exponential of a different frequency.

29. Fourier Transform nalhareqi_201529  the Fourier Transform of a function x(t) is defined by:  The result is a function of ω ( frequency). gives how much power x(t) contains at the frequency f or ω. the spectrum of x(t)

30. Inverse Fourier Transform nalhareqi_201530  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:

31. Example nalhareqi_201531  Let  The called the unit impulse signal :  The Fourier transform of the impulse signal can be calculated as follows  So,  X()X() t x(t)

32. Basic Fourier Transform pairs nalhareqi_201532  Often you have tables for common Fourier transforms

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34. Example nalhareqi_201534  Consider the non-periodic rectangular pulse at zero with duration τ seconds  Its Fourier Transform is:

35. nalhareqi_201535 ω f

36. Properties of the Fourier Transform nalhareqi_201536  Linearity:  Left or Right Shift in Time:  Time Scaling:

37. properties of the Fourier Transform nalhareqi_201537  Time Reversal:  Multiplication by a Complex Exponential ( Frequency Shifting) :  Multiplication by a Sinusoid (Modulation):

38. Example: Linearity nalhareqi_201538  Let x(t) be  Find the Fourier Transform of x(t)

39. Example: Linearity nalhareqi_201539 The Fourier Transform of x(t) will be : Note that x(t) is :

40. Example: Time Shift nalhareqi_201540 Let x(t) be : Find the Fourier Transform of x(t)

41. Example: Time Shift nalhareqi_201541 The Fourier Transform of x(t) will be :

42. Example: Time Scaling nalhareqi_201542 time compression  frequency expansion time expansion  frequency compression

43. Example: Multiplication by a Sinusoid nalhareqi_201543 Let x(t) be : The Fourier Transform of x(t) will be :

44. Examples nalhareqi_201544

45. Fourier Transform for periodic signal nalhareqi_201545  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 c n is given as

46. Fourier Transform for periodic signal nalhareqi_201546  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 n th harmonic frequency nω 0 is 2π times n th the Fourier series coefficient c n  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

47. Example nalhareqi_201547  Let  The Fourier series representation of  The Fourier series coefficients  The Fourier transform of  So,

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49. Example nalhareqi_201549  Let, find its Fourier transform ?  The Fourier series representation of is  The Fourier series coefficients  The Fourier transform of is

50. Example nalhareqi_201550  Let find its Fourier transform ?  The complex Fourier series representation of is  The Fourier series coefficients  The Fourier transform of is