Convolution of time signals using polynomials-The Super Easy Z transform

• 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    Transform of signal   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.