Discrete Linear System Summary

1.   Sampling is multiplication by a comb of delta functions:

    

2.   Fourier transform   

    

3.   Alternative representation of FT:

    

         possibility of aliasing.

4.   Finite length sequence  convolution with sinc function.

5.   Windowing   multiply sequence in order to achieve balance between width and sidelobes of convolving function

6.   Leakage  sidelobes of window function may affect appearance of samples in the frequency domain.

7.   Circular convolution  sampling the function in the frequency domain   periodicity in the time domain.

8.   Filtering of sampled data in frequency domain is convolution in the time domain over a periodic function.

9.   Stability determined as for continuous systems: stable if no poles in Right Half-Plane.