Understanding Convolution

I am trying to understand the process of constructing a time-area hydrograph i.e. a Runoff Computation by Convolution (linear superposition and addition)

Qn = k=1,n An-k+1 ik (ik = rainfall excess)

Q1 = A i1

Q2 = A2 i1 + A1 i2

Q3 = A3 i1 + A2 i2 + A1 i3

Q4 = A4 i1 + A3 i2 + A2 i3 + A1 i4 . . . etc.

I don't want to use the build-in function viz. convol(x,y) which is part of the Signal Processing Pack but rather develop my own algorithm or function which will help me understand the convolution process that I further explain in the attached worksheet. The best and simplest procedure that I have come across is to develop a function using the summation procedure viz Sum(xk,yn-k) from which I have constructed a function called CONVOLUTE. However this procedure when compared to the inbuilt function convol(x,y) only produces half the necessary ordinates (to see I show the output of each function and have graphed both solutions for comparison). Does anyone know a more general and/or more adroit way to emulate the output of the "convol" built-in function?

Your help will be much appreciated.

Regards, Mark Buckton