1. Math Review with Matlab:
Fourier Analysis
Fourier Series
S. Awad, Ph.D.
M. Corless, M.S.E.E.
D. Cinpinski
E.C.E. Department
University of Michigan-Dearborn

2. Fourier Series Periodic Signal Definition
Fourier Series Representation of Periodic Signals
Fourier Series Coefficients
Orthogonal Signals
Example: Orthogonal Signals
Example: Full Wave Rectifier
Complex Exponential Representation
Example: Finding Complex Coefficients
Magnitude and Phase Spectra of Fourier Series
Parseval’s Theorem

3. What is a Periodic Signal ?
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What is a Periodic Signal ?
A Periodic Signal is a signal that repeats itself every period
The Period of a signal is the amount of time it takes for a given signal to complete one cycle.
For example, the normal U.S. AC from wall outlet has a sine wave with a peak voltage of 170 V (110 Vrms)

4. General Sinusoid A general cosine wave, v(t), has the form:
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General Sinusoid
A general cosine wave, v(t), has the form:
M = Magnitude, amplitude, maximum value
w = Angular Frequency in radians/sec (w=2pF)
F = Frequency in Hz
T = Period in seconds (T=1/F)
t = Time in seconds
q = Phase Shift, angular offset in radians

5. General Sinusoid Amplitude = 5 Plot in Blue: /2 Phase Shift
1 Period = 1/60 sec.
= ms.
Plot in Red:

6. AC Wall Voltage Sine Wave
1 Period
1 Period

7. Represent Periodic Signals
For a general periodic signal x(t) shown to the right: T
x(t)
-T/2
T/2 t
...
x(t+nT) = x(t) for all
where n is any integer, i.e. n = 0, ± 1, ± 2,…

8. Frequency of Periodic Signals
The frequency of a signal is defined as the inverse of the period and has the unit “number of cycles/sec.”
T is the period and
is the fundamental frequency.
The frequency of a US standard outlet is 1/T = 60 Hz

9. What is Fourier Series ?
Fourier Series is a technique developed by J. Fourier.
This technique (studied by Fourier) allows us to represent periodic signals as a summation of sine functions of different frequency, amplitude, and phase shift.

10. Represent a Square Wave
Represent the Square Wave at the right using Fourier Series
Notice that as more and more terms are summed, the approximation becomes better

11. Fourier Series Representation of Periodic Signals
Any periodic function can be represented in terms of sine and cosine functions:
This can also be written as:

12. Fourier Series Coefficients
The above
a0, an, and bn are known as the Fourier Series Coefficients.
These coefficients are calculated as follows.

13. Calculating the a0 Coefficient
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Calculating the a0 Coefficient
ao, the coefficient outside the summation, is known as the average value or the dc component
ao is calculated as follows:

14. Calculating the an and bn Coefficients
The an and bn coefficients are calculated as follows:
n = 1, 2,…
n = 1, 2,…

15. Orthogonal Signals
Two periodic signals g1(t) and g2(t) are said to be “Orthogonal” if the the integral of their product over one period is equal to zero.

16. Example: Orthogonal Signals
Show that the following signals are orthogonal:

17. Orthogonal Signals

18. Example: Full Wave Rectifier
Consider the output of a full-wave rectifier:
y(t)=|sin(wot)| t
y=|x| x y
x(t)
T/2
one period T
Note that the rectified wave has a period equal to one-half of the source wave period.

19. Function Characteristics
The period of y(t) = T/2 and the fundamental frequency of y(t) is 2wo (rad/sec).
Now bn=0 since y(t) is an even function.
Thus,

20. Finding ao

21. Finding ao
* Use o = 2pi/T

22. Finding ao

23. Finding an, n = 1, 2, ….

24. Solution for an, n = 1, 2, ….

25. Solution for an, n = 1, 2, …. So: Thus:
Note: We can only obtain an output signal with a nonzero average value by using a nonlinear system with our zero average value input signal

26. Euler’s Identity
We could also say:
and ...

27. Representing Sin  and Cos  with Complex Exponentials
Subtract the equations:
Add the equations:

28. Complex Exponential Representation
The Sine and Cosine functions can be written in terms of complex exponentials.

29. Complex Exponential Fourier Series
From previous slides…
Using the Complex Exponential representation of Sine and Cosine, the Fourier series can be written as:

30. Fourier Series with Complex Exponentials
Noting that 1/j = -j, we can write:

31. Fourier Series with Complex Exponentials
Make the following substitutions:

32. Fourier Series with Complex Exponentials
The Complex Fourier series can be written as:
where:
Complex cn
*Complex conjugate
Note: if x(t) is real, c-n = cn*

33. Line Spectra
Line Spectra refers the plotting of discrete coefficients corresponding to their frequencies
For a periodic signal x(t), cn, n = 0, ±1, ± 2,… are uniquely determined from x(t).
The set cn uniquely determines x(t)
Because cn appears only at discrete frequencies, n(wo), n = 0, ± 1, ± 2,… the set cn is called the discrete frequency spectrum or line spectrum of x(t).

34. Line Spectra The Cn coefficients are in general complex.
The standard practice is to make 2 2D plots.
Plot 1: Magnitude of Coefficient vs. frequency
Plot 2: Phase of Coefficient vs. frequency
The standard practice is to make 2 2D plots.
Plot 1: Magnitude of Coefficient vs. frequency

35. Magnitude of Cn
Recall that the magnitude for a complex number a+jb is calculated as follows:

36. Phase of Cn
Recall that the phase for a complex number a+jb depends on the quadrant that the angle lies in.
Angle(a+jb) =
Quadrant 1:
Quadrant 2:
Quadrant 3:
Quadrant 4:

37. Amplitude Spectrum of Cn
Note: If x(t) is real then |Cn| is of even symmetry.

38. Phase Spectrum of Cn
Note: If x(t) is real then the Phase of Cn is odd

39. Example: Finding Complex Coefficients
x(t) t
-2.5 -2
-1.5 -1
-0.5
0.5 1
1.5 2
2.5
Consider the periodic signal x(t) with period T = 2 sec. Thus:

40. The area under x(t) from -1 to -.5 and from .5 to 1 is zero.
Finding Co(avg)
The area under x(t) from -1 to -.5 and from .5 to 1 is zero.
Co(avg) = 0.5

41. Calculating Cn

42. Factor Evaluation Now it can be shown that:
sin(np/2) = 0 for n = ±2, ±4, …  Cn = 0
sin(2p/2) = sin(p) = 0
sin(-4p/2) = sin(-2p) = 0
etc .
It can be also be shown that:
sin(np/2) = -1 for n = 3, 7, 11,…
sin(np/2) = 1 for n = 1, 5, 9,…
sin(3p/2) = -1
sin(-7p/2) = 1
etc .

43. Factor Evaluation
Recall:
Co(avg) = 0.5

44. Summary of Results Note: Cn=p if Cn is negative Therefore: and
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Summary of Results
Note: Cn=p if Cn is negative
Therefore:
and

45. Plot the Magnitude Response
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Plot the Magnitude Response

46. Plot the Phase Response
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Plot the Phase Response

47. What is Parseval’s Theorem ?
Parseval’s Theorem states that the average power of a periodic signal x(t) is equal to the sum of the squared amplitudes of all the harmonic components of the signal x(t).
This theorem is excellent for determining the power contribution of each harmonic in terms of its coefficients

48. Parseval’s Theorem
Average power of x(t) is calculated from the time or frequency domain by:
Time Domain:
Frequency Domain: