Discrete Fourier Transform Continuous Discrete Bandwidth Limited Transforms

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 W 1 W so that all the bandwidth for nonzero F( ) is used (otherwise lose information)

Sampling Frequency • • Sampling frequency: Since W 1 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

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

DFT - Summary Inverse Discrete Fourier Transform

Example 1 : Find DFT of fn=n, with N=3

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