Discrete Fourier Transform Continuous: Discrete:

Bandwidth Limited Transforms 1 Inverse Let For

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

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)

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’

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

Aliassing Example • • • • • • • • • • • • • • • Example

Inverse Discrete FT Numerical rule: Inverse Discrete FT

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

DFT - Summary Inverse Discrete Fourier Transform

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

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