Digital Signal Processing

Discrete Fourier Transform Inverse Discrete Fourier Transform

Properties of DFT DFT has the same number of datapoints as the signal The signal is assumed to be periodic with a period of N X[k] corresponds to the amplitude of the signal at frequency f=k/(NT) The frequency resolution of the DFT is  f=1/(NT), i.e. the # of samples determines the frequency resolution

Steps for Calculating DFT Determine the resolution required for the DFT, establish a lower limit on the # of samples required, N. Determine the sampling frequency to avoid aliasing Accumulate N samples Calculate DFT

Matlab Example of FFT

Digital Filtering a 1 *y(n) = b 1 *x(n) +b 2 *x(n-1) b nb+1 x(n-nb) - a 2 *y(n-1) -... – a na+1 *y(n-na) A=[a 1, a 2,..., a na+1 ] B=[b 1, b 2,..., b nb+1 ] X=[x(n-nb),..., x(n-1), x(n)]: input signal Filter parameters Y=[y(n-na),..., y(n-1), y(n)]: filtered signal

Ideal Filters Low pass filter High pass filter Bandpass filter Bandstop filter

Common Filters Butterworth filter: Chebyshev filter:

Comparison of Common Filters

MATLAB example of Filtering

MATLAB Example of Undersampling