1. Fourier Analysis ENGI 4559 Signal Processing for Software Engineers
Dr. Richard Khoury
Fall 2009

2. Fourier Analysis: Overview
ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Analysis: Overview
Continuous Fourier Transform
Discrete-Time Fourier Transform
Sampling Theory
Decimation and Interpolation
Discrete Fourier Transform
Fast Fourier Transform

3. Fourier Jean Baptiste Joseph Fourier (1768-1830)
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Fourier
Jean Baptiste Joseph Fourier ( )
Orphan in pre-revolutionary France
Raised by Benedictine monks
Kept out of scientific corps (reserved for aristocracy)
Schooled in a military academy, where he discovered his talent and love of math
Career as mathematician
1789 (21 years old) he became a math celebrity for his work solving numerical equations
1794: made chair of mathematics at the École Normale Supérieure
1795: made chair of mathematics at the much more prestigious École Polytechnique

4. Fourier Political career “Théorie analytique de la chaleur” (1822)
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Fourier
Political career
Local district organizer in the French Revolution (1789)
Part of Napoleon’s Egypt Expedition (1798)
Made governor of Lower Egypt and secretary of the Institut d’Égypte
Pushed for French-sponsored scientific and mathematical work in Egypt, to limit British cultural influence
1801: French forces surrender to the British in Egypt, Fourier returns to France
“Théorie analytique de la chaleur” (1822)
Showed that Newton’s Law of Cooling could be reduced to a mathematical analysis problem
Required him to prove that any periodic function could be represented as a sum of sinusoids
Led to the postulation of the global greenhouse effect (1824)

5. ENGI 4559 © Dr. Richard Khoury, 2009
Mathematical History
1753: Daniel Bernoulli postulates (but does not prove) that a vibrating string can be represented as a weighted sum of sinusoids
1788: Joseph Lagrange (Fourier’s doctoral advisor) writes “Méchanique Analytique”, a comprehensive book on analytical mechanics
1805: Carl Friedrich Gauss develops a recursive divide-and-conquer mathematical computation algorithm; forgotten
1822: Fourier proves that any periodic function can be represented as a weighted sum of sinusoids (later known as the Fourier Series)
1823: Siméon Poisson (Fourier’s doctoral student) wrote the summation formula linking Fourier Series to the Fourier Transform, generalizing it to non-periodic functions
1829: Johann Dirichlet (Fourier and Poisson’s doctoral student) developed the conditions under which Fourier’s Series doesn’t lose information from the original function
1965: Rediscovery of Fast Fourier Transform computer algorithm

6. Review of Complex Numbers
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Review of Complex Numbers
A complex number
Cartesian form:
Polar form:
Exponential form:
Conversion formulae:
Other operations:
Conjugate:
Complex norm:

7. Fourier Series Any signal can be represented as a sum of sinusoids
ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Series
Any signal can be represented as a sum of sinusoids

8. ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Series
Any signal can be transformed losslessly into a sum of sinusoids
The original signal might be in the time or space domain
The Fourier Series is in the frequency domain
The sinusoids are often easier to model, analyse, and work with than the signal
The transformation can be done very efficiently since the Fast Fourier Transform algorithm
After processing, the resulting series of sinusoids can be losslessly transformed back into a signal

9. Continuous-Time Fourier Series
ENGI 4559 © Dr. Richard Khoury, 2009
Continuous-Time Fourier Series
Fourier series of a continuous periodic signal with period T0
Where:

10. Continuous-Time Fourier Transform
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Continuous-Time Fourier Transform
A continuous periodic signal is a sum of sinusoids that are integer multiples of Ω0
i.e. defined at the harmonics
A continuous non-periodic signal s(t) is also a combination of sinusoids, but defined everywhere, not just at harmonics
So it's an integral

11. Fourier Series Exercise
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Fourier Series Exercise
Compute the Fourier Series of this signal

12. Fourier Series Exercise
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Fourier Series Exercise

13. Fourier Series Exercise
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Fourier Series Exercise

14. Fourier Series Exercise14
ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Series Exercise

15. Fourier Transform Example
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Fourier Transform Example
Compute the Fourier Transform of a rectangular signal

16. Fourier Transform Example
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Fourier Transform Example

17. Discrete-Time Fourier Transform
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Discrete-Time Fourier Transform
Continuous-Time
Discrete-Time

18. Discrete-Time Fourier Transform
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Discrete-Time Fourier Transform
Continuous Fourier Series and Continuous Fourier Transform convert a periodic and non-periodic continuous-time input signal into a continuous frequency function
Discrete-Time Fourier Transform converts a periodic or non-periodic discrete-time input signal into a continuous frequency function

19. Fourier Transform Exercise19
ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Transform Exercise
Compute the CTFT of the signal
And the DTFT of the signal

20. Fourier Transform Exercise
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Fourier Transform Exercise

21. Fourier Transform Exercise
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Fourier Transform Exercise

22. Fourier Transform Exercise
ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Transform Exercise

23. Fourier Transform Exercise
ENGI 4559 © Dr. Richard Khoury, 2009
Fourier Transform Exercise

24. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
Let’s study what happens to the Fourier Transform of a continuous signal when it is sampled
Recall sampling from Lecture 1:
where sc is the original continuous-time signal, ss is the sampled discrete-time signal, Ts is the sampling period and n is an integer

25. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
Starting with the continuous-time signal and its Fourier transform:
We perform the sampling:

26. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
We can break the integral in segments, compute the integral of each segment, and sum them
This is getting messy… let’s offset the frequency and adjust the limits to simplify:

27. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
Notice that the exponent contains , which is always equal to 1!
We know from the last lecture that

28. Sampling Theory Finally, swap the summation and integral
ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
Finally, swap the summation and integral
Compare with the discrete-time Fourier Transform we know:

29. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
Where Ωs is the sampling frequency
The frequency spectrum of the sampled function is the frequency spectrum of the original function, shifted by the sampling frequency, and scaled by the sampling period

30. Sampling Theory Given the frequency spectrum of an original function
ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theory
Given the frequency spectrum of an original function
The frequency spectrum of the sampled function is the same, shifted by the sampling frequency, and scaled by the sampling period

31. Sampling Theorem Recall the sampling theorem:
ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theorem
Recall the sampling theorem:
If the maximum frequency contained in an analog signal is Ωmax = B, then it can be perfectly reconstructed from samples taken at the sampling frequency Ωs = 2B (or more).
Now we see why!

32. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theorem
We can reconstruct the original signal from a sample using a low-pass filter

33. Sampling Theorem What if we set Ωs < 2B ?
ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theorem
What if we set Ωs < 2B ?

34. ENGI 4559 © Dr. Richard Khoury, 2009
Sampling Theorem
We no longer can reconstruct the original signal because of aliasing

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Sampling Rate
Sometimes we need to change the rate at which samples come into the system
Downsampling
Slower sampling rate
Prefiltering + downsampling = decimation
Upsampling
Faster sampling rate
Upsampling + postfiltering = interpolation

36. Downsampling Reduce the sampling of s(n) by a factor of M
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Downsampling
Reduce the sampling of s(n) by a factor of M
Simply keep every Mth sample and discard the others
Example: downsampling by half:

37. Downsampling What is the impact on the frequency spectrum?
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Downsampling
What is the impact on the frequency spectrum?
The frequency spectrum of the downsampled function is the sum of the original scaled by the sampling rate and shifted by a factor of 

38. Downsampling
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39. Downsampling Aliasing can happen in downsampling too!
ENGI 4559 © Dr. Richard Khoury, 2009
Downsampling
Aliasing can happen in downsampling too!
Aliasing in sampling if:
Aliasing in downsampling if
Recall:

40. Downsampling Recall: aliasing occurs when neighbouring signals overlap
ENGI 4559 © Dr. Richard Khoury, 2009
Downsampling
Recall: aliasing occurs when neighbouring signals overlap
Avoid aliasing by preventing overlap
Can be done easily by putting a lowpass filter with width of /M before downsampler
This is decimation
s(n)
f(n)

41. Upsampling Increase the sampling of s(n) by a factor of M
ENGI 4559 © Dr. Richard Khoury, 2009
Upsampling
Increase the sampling of s(n) by a factor of M
Simply add M-1 zeros between each sample
Example: upsampling by three:

42. Upsampling What is the impact on the frequency spectrum?
ENGI 4559 © Dr. Richard Khoury, 2009
Upsampling
What is the impact on the frequency spectrum?
The frequency spectrum of the upsampled function is the original spectrum stretched by the sampling rate

43. ENGI 4559 © Dr. Richard Khoury, 2009
Upsampling

44. Upsampling Aliasing cannot happen with upsampling
ENGI 4559 © Dr. Richard Khoury, 2009
Upsampling
Aliasing cannot happen with upsampling
It compressed the frequency components
However, there are images to eliminate
Notice the interval [-π, π] went from one component to three
Can be done easily by putting a lowpass filter with width of /M after upsampler
This is interpolation
s(n)
f(n)

45. Discrete Fourier Transform
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Discrete Fourier Transform
Continuous-Time
Discrete-Time
Discrete

46. DFT Exercise Compute the Fourier Transform of the discrete signal
ENGI 4559 © Dr. Richard Khoury, 2009
DFT Exercise
Compute the Fourier Transform of the discrete signal

47. DFT Exercise Given the previous values of S2(ω), compute s2 (n)
ENGI 4559 © Dr. Richard Khoury, 2009
DFT Exercise
Given the previous values of S2(ω), compute s2 (n)

48. Fast Fourier Transform
ENGI 4559 © Dr. Richard Khoury, 2009
Fast Fourier Transform
The Fourier Transform is a very useful technique
But too slow to compute: O(N²)
FFT are a family of algorithms that allow for fast computation: O(Nlog2N)
Divide-and-conquer algorithm
Given a signal s1 = s2s3, compute the transform of s2 and s3 separately and sum them
We’ll look at two popular FFT algorithms
Decimation in time FFT
Decimation in frequency FFT

49. Fast Fourier Transform
ENGI 4559 © Dr. Richard Khoury, 2009
Fast Fourier Transform
For simplicity of notation, define the following:
This leads to two equivalences:
The DFT now is:

50. Decimation in Time Assumes that N is a power of two: N = 2a
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time
Assumes that N is a power of two: N = 2a
N-point DFT computed as a set of 2-point DFTs
A 2-point DFT is computed simply as:

51. Decimation in Time Now separate the even and odd DFTs
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time
Now separate the even and odd DFTs
Compute two N/2-point DFTs

52. ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time

53. Decimation in Time Recall the equivalence:
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time
Recall the equivalence:
We only need to compute half the multiplications!

54. ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time

55. ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time
So we divided a DFT over N points into two DFTs over N/2 points
N = 2a
Each one can be recursively divided
1 DFT over N points
2 DFTs over N/2 points computed with P and Q
4 DFTs over N/4 points to compute P and Q …
2a-1 DFTs over 2 points

56. Decimation in Time Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time Example
Example: 8-point DFT

57. Decimation in Time Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time Example
8-point DFT

58. Decimation in Time Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time Example
4-point DFT
4-point DFT

59. Decimation in Time Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time Example
2-point DFT
2-point DFT
2-point DFT
2-point DFT

60. Decimation in Time Example
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Decimation in Time Example

61. Decimation in Time Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Time Example

62. Decimation in Frequency
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Decimation in Frequency
Again, assumes that N is a power of two: N = 2a
Decimation in time divided s(n) into subsets to make the DFT computation easier
Decimation in frequency divides S(k) into subsets instead

63. Decimation in Frequency
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Frequency
Divide the DFT formula into two sets of frequencies

64. Decimation in Frequency
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Decimation in Frequency
Next, separate the odd- and even-numbered frequencies

65. Decimation in Frequency
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Decimation in Frequency
We see a new but familiar graph appear

66. Decimation in Frequency Example
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Decimation in Frequency Example
Example: 8-point DFT

67. Decimation in Frequency Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Frequency Example
8-point DFT

68. Decimation in Frequency Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Frequency Example
4-point DFT
4-point DFT

69. Decimation in Frequency Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Frequency Example
2-point
DFT
2-point
DFT
2-point
DFT
2-point
DFT

70. Decimation in Frequency Example
ENGI 4559 © Dr. Richard Khoury, 2009
Decimation in Frequency Example

71. ENGI 4559 © Dr. Richard Khoury, 2009
Summary
Fourier transform allows us to convert a signal from the time or space domain into a summation of sinusoids in the frequency domain that represent its frequency spectrum
Analysis in the frequency domain gives us a different perspective on the signals, and a deeper understanding of sampling and related phenomena
The Fourier transform are mathematical operations, with interesting properties that you should read about in the textbook

72. Summary Continuous-time Fourier transform
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Summary
Continuous-time Fourier transform
Discrete-time Fourier transform
Discrete Fourier transform

73. ENGI 4559 © Dr. Richard Khoury, 2009
Summary
Two of many possible ways of computing a fast Fourier transform
Decimation in time
Decimation in frequency

74. Textbook Material Readings Recommended problems Chapter 2
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Textbook Material
Readings
Chapter 2
Recommended problems
2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 19, 20, 21, 25, 26, 29, 30, 32, 33, 34, 35, 36, 37, 39, 50, 51, 52, 60