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.
- Slides: 12