1. Chapter 5
The Fourier Transform

2. Basic Idea
We covered the Fourier Transform which to represent periodic signals
We assumed periodic continuous signals
We used Fourier Series to represent periodic continuous time signals in terms of their harmonic frequency components (Ck).
We want to extend this discussion to find the frequency spectra of a given signal

3. Basic Idea
The Fourier Transform is a method for representing signals and systems in the frequency domain
We start by assuming the period of the signal is T= INF
All physically realizable signals have Fourier Transform
For aperiodic signals Fourier Transform pairs is described as
Fourier Transforms of f(t)
Inverse Fourier Transforms of F(w)
Remember:
notes

4. Example – Rectangular Signal
Compute the Fourier Transform of an aperiodic rectangular pulse of T seconds evenly distributed about t=0.
Remember this the same rectangular signal as we worked before but with T0 infinity! V
-T/2
T/2
notes
All physically realizable signals have Fourier Transforms

5. Fourier Transform of Unit Impulse Function
Example:
Plot magnitude and phase of f(t)

6. Fourier Series Properties
Make sure how to use these properties!

7. Fourier Series Properties - Linearity
Find F(w)

8. Fourier Series Properties - Linearity
Due to linearity

9. Fourier Series Properties - Time Scaling
rect(t/T)
rect(t/(T/2))
Due to Time Scaling Property
Remember:
sinc(0)=1;
sinc(2pi)=0=sinc(pi)

10. Fourier Series Properties - Duality or Symmetry
Example:
Find the time-domain waveform for
Arect(w/2B)
Refer to FTP
Table
Remember we had:
FTP: Fourier Transfer Pair

11. Fourier Series Properties - Duality or Symmetry
Example: find the frequency response
Of y(t)

12. Fourier Series Properties - Duality or Symmetry
Example: find the frequency response
Of y(t)
We know
Using Fourier Transform Pairs
Using duality

13. Fourier Series Properties - Convolution
Proof
Proof

14. Fourier Series Properties - Convolution
Example:
Find the Fourier Transform of x(t)=sinc2(t)
In this case we have B=1, A=1 w w
X1(w)
X2(w)
Refer to Schaum’s
Prob. 2.6

15. Fourier Series Properties - Convolution
Example:
Find the Fourier Transform of x(t)= sinc2(t) sinc(t)
We need to find the convolution of a rect and a triangle function: w
Refer to Schaum’s
Prob. 2.6

16. Fourier Series Properties - Frequency Shifting
Example:
Find the Fourier Transform of g3(t) if g1(t)=2cos(200pt), g2(t)=2cos(1000pt);
g3(t)=g1(t).g2(t) ; that is [G3(w)]
Remember: cosa . cosb=1/2[cos(a+b)+cos(a-b)]

17. Fourier Series Properties - Time Differentiation
Example:

18. More… Read your notes for applications of Fourier Transform.
Read about Power Spectral Density
Read about Bode Plots

19. Schaums’ Outlines Problems
Do problems
Do problems 5.4, 5.5, , 5.8, 5.9, 5.10, 5.14
Do problems in the text