Notes on Convolution

 

Discrete convolution assumes two conditions that are difficult to find in the real world:

 

 

To counteract these assumptions:

 

(1) we make the lengths, , of the filter's impulse response and, , of  the input signal the same by padding with zeros the shortest of both signals, and

 

(2) when the signal is not periodic, we consider that the convolution operation will introduce artifacts the in first output entries from the far end of the data and vice versa.  To avoid these, we create a zone of zeroed entries at the end of the input signal. The length of this zone depends on the length of the filter's impulse response, as many as the most positive or negative index for which the filter's impulse response signal is non-zero, whichever is largest. But by doing so a number of entries, totalling the same number of zeroed entries in the filter's impulse response, are corrupted (a) at the beginning and (b) at the end of the resulting signal.  However, the resulting signal is also augmented by this number of entries and these can safely be ignored for any practical purpose.

In this case there will be /2 corrupted entries at the beginning, and /2 or /2+1 corrupted entries at the very end of the resulting signal, depending if the length of the filter is even or odd, respectively.

 

Figure 1.  Suppressed End-Effects Augmented Output