1. Discrete Fourier Transform
Continuous:
Discrete:

2. Bandwidth Limited Transforms 1
Inverse
Let
For

3. Bandwidth Limited Transforms 2
Discrete t:
Coefficients cn of the exponential form of the FS for F(2x) for

4. Bandwidth Limited Transforms 3
This set of points in the time domain determines F() completely
F() determines f(t) for all time
Must have W1W so that all the bandwidth for nonzero F() is used (otherwise lose information)

5. Sampling Frequency Sampling frequency: Since W1W (Nyquist frequency)
Can sample at a higher frequency but inefficient
If we sample at less than Nyquist rate?
f(t) not completely determined
Can get ‘aliassing’

6. Aliassing Sampling at Nyquist frequency gives correct signal•
Less than Nyquist rate eg
- Gives incorrect wavelength (aliassing)

7. Aliassing
Example • • • • • • • • • • • • • • •
Example

8. Inverse Discrete FT
Numerical rule:
Inverse Discrete FT

9. Forward DFT
Consider
(for some integer m)
But what is m?
DFT

10. DFT - Summary Inverse Discrete Fourier Transform

11. Example1 : Find DFT of fn=n, with N=3

12. Example 2: F is the 12-point DFT of a real signal f of length 12.