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

2. Fourier Transform Motivation For Fourier Transform
Energy Signal Definition
Fourier Transform Representation
Example: FT Calculation
Example: Pulse
Inverse Fourier Transform
Fourier Transform Properties
Example: Convolution
Parseval’s Theorem
Relation between X(s) and X(j)
Example: Ramp Function

3. Motivation for Fourier Transform
We need a method of representing aperiodic signals in the frequency domain.
The Fourier Series representation is only valid for periodic signals.
The Fourier Transform will accomplish this task for us. However, it is important to note that the Fourier Transform is only valid for Energy Signals.

4. What is an Energy Signal ?
A signal g(t) is called an Energy Signal if and only if it satisfies the following condition.

5. Fourier Transform Representation
The Fourier Transform of an Energy Signal x(t) is found by using the following formula.
There is a one to one correspondence between a signal x(t) and its Fourier Transform. For this reason, we can denote the following relationship.

6. Example: FT Calculation
x(t) = e-atu(t) 1
a > 0
Note: If a<0, then x(t) does not have a Fourier transform because:

7. Fourier Transform
complex function of w

8. Magnitude Response
Let us now find the Magnitude Response. The expression for the magnitude response of a fraction is calculated as follows.

9. Magnitude Response
Now calculate the Magnitude Response of X(j)

10. Magnitude Response Even function of w
We can now plot the Magnitude Response.
w (rad/sec)
Even function of w

11. Phase Response
The expression for the phase response of a fraction is calculated as follows.

12. Phase Response

13. Phase Response Odd function of w We can now plot the Phase Response.
w(rad/sec)
Odd function of w

14. Fourier Transform Tables
We could go ahead and find the Fourier Transform for any Energy Signal using the previous formula. However, Signals & Systems textbooks usually provide a table in which these have already been computed. Some are listed here. FT

15. Example: Pulse
Find the Fourier Transform of:
x(t) t
-T1 T1 1

16. Example: Pulse

17. Magnitude Response
Note: X(jw) = 0, when
So:

18. Magnitude Response
We can now plot the Magnitude Response.

19. Phase Response
We can now plot the Phase Response.

20. Inverse Fourier Transform
Recall that there is a one to one correspondence between a signal x(t) and its Fourier Transform X(j).
If we have the Fourier Transform X(j) of a signal x(t), we would also like be able to find the original signal x(t).

21. Inverse Fourier Transform
Let X(jw) = FT{x(t)} =
x(t) = FT-1{X(jw)} =
FT-1 is the inverse Fourier Transform of X(jw)

22. Fourier Transform Properties
There are several useful properties associated with the Fourier Transform:
Time Domain Differentiation Property
Linearity Property
Time Scaling Property
Time Domain Integration Property
Duality Property
Time Shifting Property
Symmetry Property
Frequency Shifting Property
Convolution Property
Multiplication by a Complex Exponential

23. Linearity Property
Let:
Then:

24. Time Scaling Property
Let:
Then:
where a is a real constant

25. Duality Property
Let:
Then:

26. Time Shifting Property
Let:
Then:
Note: “a” can be positive or negative

27. Frequency Shifting Property
Let:
Then:

28. Time Domain Differentiation Property
Let:
Then:

29. Time Domain Integration Property
where:

30. Symmetry Property
If x(t) is a real-valued time function then conjugate symmetry exists:
Example:

31. Convolution Property
Let:
Then:
Convolution

32. Example: Convolution Filter x(t) through the filter h(t)
y(t)
LTI
System
where h(t) is the impulse response
Convolution

33. 4/10/2017
Example: Convolution
Knowing
We can write
Note:

34. Multiplication by a Complex Exponential
Let:
Then:

35. Sinusoid Examples
(i)
Amplitude Modulation

36. Sinusoid Examples
(ii)
Amplitude Modulation

37. Amplitude Modulation FT FT-1 y(t)=x(t)cos(wot) x(t) x(t) y(t) t t
Y(jw)
-wo wo w w
X(jw) 1

38. Parseval’s Theorem
Let x(t) be an energy signal which has a Fourier transform X(jw).
The energy of this signal can be calculated in either the time or frequency domain:
Time Domain
Frequency Domain

39. Relation between X(s) and X(jw)
If X(jw) exists for x(t):
assuming x(t) = 0 for all t < 0
Example:
Frequency Response
H(s) is known as the transfer function

40. Example: Ramp Function
The Fourier Transform exists only if the region of convergence includes the j axis.
To prove this point, let us look at the Unit Ramp Function. The Ramp Function has a Laplace Transform, but not a Fourier Transform. t

41. Example: Ramp Function
If we define the Step Function as:
The Unit Ramp Function can now be rewritten as t*u(t)
The Laplace Transform is and the corresponding
Region of Convergence (ROC) is Re(s) > 0
Since the ROC does not include the j axis, this means that the Ramp Function does not have a Fourier Transform