1. Digital Signal Processing

2. Discrete Fourier Transform Inverse Discrete Fourier Transform

3. 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

4. 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

5. Matlab Example of FFT

6. 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

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

8. Common Filters Butterworth filter: Chebyshev filter:

9. Comparison of Common Filters

10. MATLAB example of Filtering

11. MATLAB Example of Undersampling