1. Autumn 2005-2006 Analog and Digital Communications Autumn 2005-2006
Sep 13, 2005
Fourier Transforms
Analog and Digital Communications
Autumn
Sep 13, 2005
CS477: Analog and Digital Communications
CS477: Analog and Digital Communications

2. Fourier Series: Example
Autumn
Sep 13, 2005
Fourier Series: Example
Find Fourier Series of ?
Fourier series expansion is:
Amplitude normalized by A/4
is sum of rotating phasors
Sep 13, 2005
CS477: Analog and Digital Communications
CS477: Analog and Digital Communications

3. Sinusoidal Fourier Series
For real signals
and can be expressed in terms of
Sep 13, 2005
CS477: Analog and Digital Communications

4. CS477: Analog and Digital Communications
Parseval’s theorem
Average power of a periodic signal can be obtained from its Fourier coefficients
To prove, write the Fourier series expansion
Sep 13, 2005
CS477: Analog and Digital Communications

5. CS477: Analog and Digital Communications
Fourier Transform
Amplitude normalized by A/4
What if the x-axis in the Fourier
series represents frequency?
The spectral lines appear at integer
multiples of fundamental frequency!
Separation between two consecutive
spectral lines is equal to
How does increasing fundamental period affect the
separation between two consecutive spectral lines?  Fourier transform
Sep 13, 2005
CS477: Analog and Digital Communications

6. CS477: Analog and Digital Communications
Fourier Transform
Fourier series in the limit
Represents spectral components of a signal
Signal uniquely represented in time or frequency domain
Sep 13, 2005
CS477: Analog and Digital Communications

7. Shifting and Scaling Review
Sep 13, 2005
CS477: Analog and Digital Communications

8. Fourier Transform: Example
Sep 13, 2005
CS477: Analog and Digital Communications

9. CS477: Analog and Digital Communications
Properties of FT
Linearity
Scaling
Duality
Time shifting
Frequency shifting (modulation)
Sep 13, 2005
CS477: Analog and Digital Communications

10. CS477: Analog and Digital Communications
Properties of FT
Modulation
Product and convolution
More on this under LTI systems
Sep 13, 2005
CS477: Analog and Digital Communications

11. CS477: Analog and Digital Communications
Properties of FT
Differentiation
Integration
Sep 13, 2005
CS477: Analog and Digital Communications

12. The Dirac Delta Function
A generalized function
Sep 13, 2005
CS477: Analog and Digital Communications

13. CS477: Analog and Digital Communications
FT of 1 and exponentials
Fourier transform of RF Pulse:
Sep 13, 2005
CS477: Analog and Digital Communications

14. CS477: Analog and Digital Communications
Other Functions
Signum Function:
Unit step Function:
Sep 13, 2005
CS477: Analog and Digital Communications

15. CS477: Analog and Digital Communications
FT of Periodic Signals
Sep 13, 2005
CS477: Analog and Digital Communications

16. CS477: Analog and Digital Communications
FT of Periodic Signals
Sep 13, 2005
CS477: Analog and Digital Communications