[pmath] 1+x+x^2 [/pmath] [pmath size=16] * [/pmath] [pmath] 1+x+x^2 = [/pmath] [pmath]1+2x+3x^2+2x^3+x^4[/pmath]
-
The filter is the sum of the last 3 signal samples.
-
The signal is a pulse set of three ones. The signal arrives 1 sample at a time.
Notice the coefficients of the multiplied polynomials are equal to the signal output values at times 0 through 4.
If instead of using X as our variable we could use Z and we would see that all this is the Z transform. The following principle is true for the above signal and filter. It is true in general.
Signal convolved with Filter [pmath] doubleleftright [/pmath] Transform of signal [pmath size=16] * [/pmath] Transform of filter
Notice that the powers of X perform the function of place holding for location in time. They keep track of what we can tally and these tallies correspond in power of X to time. Time=4 corresponds to powers of 4 of X.