1. Frequency domain analysis and Fourier Transform
Compiled from:
S. Narasimhan
Njegos Nincic
Other 1

2. How to Represent Signals?
Option 1: Taylor series represents any function using polynomials.
Polynomials are not the best - unstable and not very physically meaningful.
Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc). 2

3. Jean Baptiste Joseph Fourier (1768-1830)
Had crazy idea (1807):
Any periodic function can be rewritten as a weighted sum of Sines and Cosines of different frequencies.
Don’t believe it?
Neither did Lagrange, Laplace, Poisson and other big wigs
Not translated into English until 1878!
But it’s true!
called Fourier Series
Possibly the greatest tool
used in Engineering 3

4. Time and Frequency
example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) 4

5. Time and Frequency
example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = + 5

6. Frequency Spectra example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t)6

7. Frequency Spectra
Usually, frequency is more interesting than the phase 7

8. Frequency Spectra = + = 8

9. Frequency Spectra = + = 9

10. Frequency Spectra = + = 10

11. Frequency Spectra = + = 11

12. Frequency Spectra = + = 12

13. ringing near discontinuity
Gibbs phenomenon:
ringing near discontinuity
Source:

14. Frequency Spectra = 14

15. DIRICHLET CONDITIONS Suppose that
f(x) is defined and single valued except possibly at finite number of points in (-l,+l)
f(x) is periodic outside (-l,+l) with period 2l
f(x) and f’(x) are piecewise continuous in(- l,+l)

16. Then the Fourier series of f(x) converges to
f(x) if x is a point of continuity
b)[f(x+0)+f(x-0)]/2 if x is a point of discontinuity

17. METHOD OF OBTAINING FOURIER SERIES OF1. 2. 3. 4.

18. (7.3)
Expressing cos nx and sin nx in exponential form, we may rewrite Eq.(7.1) as
(7.4)
in which
(7.5)
and
(7.6)

19. COMPLEX FORM OF FOURIER SERIES
Recall from earlier that we can write a Fourier series in the complex form
in which
and
From the earlier definitions of we can show that

20. Real and Complex Sinusoids
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 20

21. Fourier Transform
We want to understand the frequency of our signal. So, let’s reparametrize the signal by  instead of x:
f(x)
F()
Fourier
Transform
For every  from 0 to inf, F() holds the amplitude A and phase of the corresponding sine
How can F hold both? Complex number trick!
F()
f(x)
Inverse Fourier
Transform 21

22. Transforms Transform: Can be thought of as a substitution
In mathematics, a function that results when a given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits. (Britannica)
Can be thought of as a substitution

23. Transforms Example of a substitution:
Original equation: x + 4x² – 8 = 0
Familiar form: ax² + bx + c = 0
Let: y = x²
Solve for y
x = ±√y 4

24. Transforms
Transforms are used in mathematics to solve differential equations:
Original equation:
Apply Laplace Transform:
Take inverse Transform: y = Lˉ¹(y)

25. Fourier Transform Property of transforms: Fourier Transform:
They convert a function from one domain to another with no loss of information
Fourier Transform:
converts a function from the time (or spatial) domain to the frequency domain

26. Fourier Series to Fourier Transform
For periodic signals, we can represent them as linear combinations of harmonically related complex exponentials
To extend this to non-periodic signals, we need to consider aperiodic signals as periodic signals with infinite period.
As the period becomes infinite, the corresponding frequency components form a continuum and the Fourier series sum becomes an integral (like the derivation of CT convolution)
Instead of looking at the coefficients a harmonically –related Fourier series, we’ll now look at the Fourier transform which is a complex valued function in the frequency domain

27. Fourier Series to Fourier Transform

28. Effect of increasing period T
a/T
a/T
a/T

29. Frequency Spectra 29

30. FT: Just a change of basis
M * f(x) = F() * = . 30

31. IFT: Just a change of basis
M-1 * F() = f(x) = * . 31

32. Fourier Transform – more formally
Represent the signal as an infinite weighted sum
of an infinite number of sinusoids
Note:
Arbitrary function
Single Analytic Expression
Spatial Domain (x)
Frequency Domain (u)
(Frequency Spectrum F(u))
Inverse Fourier Transform (IFT) 32

33. Fourier Transform Also, defined as: Note:
Inverse Fourier Transform (IFT) 33

34. Review of Cauchy Principal Value Integralsx
1/x
Recall for real integrals,
but a finite result is obtained if the integral interpreted as
because the infinite contributions from the two symmetrical shaded
parts shown exactly cancel. Integrals evaluated in this way are said to be
(Cauchy) principal value integrals (or “deleted” integrals) and are often written as 34

35. Delta function
Definition
Area for any  > 0
Sifting property
since

36. Dirac delta Function
This allows an arbitrary sequence x(n) or continuous-time function f(t) to be expressed as:

37. Example 1: Decaying Exponential
Consider the (non-periodic) signal
Then the Fourier transform is:
a = 1

38. Example 2: Single Rectangular Pulse
Consider the non-periodic rectangular pulse at zero
The Fourier transform is:
Note, the values are real
T1 = 1

39. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
The Sinc Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 39

40. Example 3: Impulse Signal
The Fourier transform of the impulse signal can be calculated as follows:
Therefore, the Fourier transform of the impulse function has a constant contribution for all frequencies 
X(j)

41. Example 4: Periodic Signals
A periodic signal violates condition 1 of the Dirichlet conditions for the Fourier transform to exist
However, lets consider a Fourier transform which is a single impulse of area 2 at a particular (harmonic) frequency =0.
The corresponding signal can be obtained by:
which is a (complex) sinusoidal signal of frequency 0. More generally, when
Then the corresponding (periodic) signal is
The Fourier transform of a periodic signal is a train of impulses at the harmonic frequencies with amplitude 2ak

42. Fourier Transform Pairs (I)
Note that these are derived using
angular frequency ( ) 42

43. Fourier Transform Pairs (I)
Note that these are derived using
angular frequency ( ) 43

44. Fourier Transform and Convolution
Let
Then
Convolution in spatial domain
Multiplication in frequency domain 44

45. Fourier Transform and Convolution
Spatial Domain (x)
Frequency Domain (u)
So, we can find g(x) by Fourier transform FT
IFT 45

46. Properties of Fourier Transform
Spatial Domain (x)
Frequency Domain (u)
Linearity
Scaling
Shifting
Symmetry
Conjugation
Convolution
Differentiation
Note that these are derived using
frequency ( ) 46

47. Properties of Fourier Transform47

53. Exercícios

56. Example use: Hearing mechanism
Human ears do not hear wave-like oscilations, but constant tone
Often it is easier to work in the frequency domain

57. Example use: Smoothing/Blurring
We want a smoothed function of f(x)
Let us use a Gaussian kernel
Then
H(u) attenuates high frequencies in F(u) (Low-pass Filter)! 57

58. Image Processing in the Fourier Domain58

59. Low-pass Filtering
Let the low frequencies pass and eliminating the high frequencies.
Generates image with overall
shading, but not much detail 59

60. High-pass Filtering
Lets through the high frequencies (the detail), but eliminates the low frequencies (the overall shape). It acts like an edge enhancer. 60