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Idiot's Guide to Discrete-Time Convolution Convolution is complicated and often messy. You need to be very careful to make it come out correctly. When you are learning the technique, it pays to draw accurate diagrams and follow a step-by-step procedure to eliminate (or at least reduce) errors. If you follow the steps below, it should help considerably. Consider the following problem of convolving x{17] with A[n] : str] = xen - ky kee 1. (Optional) Check to see whether x{7] or Alm] is simpler. Exchange the order of x[n] and An] so that A[n] is the simpler function. (Recall this is valid since convolution is commutative). 2. Write expressions for x[k] and h[n —k] 3. Sketch the function x{k]. Make sure to label get confused between mand & avis as a function of &. Itis easy to 4. Sketch the function A{k]. Nc instead of 1 hat sthe same as A{n]except that you are using k 5. Sketch the function A{-k]}. This: ul function A{] reflected about the y- axis. Also note that this 1s th ca n=O. Finally, note that, since the function is plotted with respect to A, 9 n will simply shift the f Changing the value of hat constants shift functions 6. Now sketch the function {n= k] Note that this will look just like Af A] except that the end points will have a + mappended to them. This makes sense since, when n= 0) the praph is exactly as you drew it in step $ make the function s! It is also true that positive values of mt will ift to the right This 1s because this plot has been rotated about the y-anis. In any case, yust remember that increasing nm shifts the function fo the right You can now draw the general graph of Al ~k] by using the graph of A{~k] and adding n to all the values along the 4-axis. I sometimes include a note to myself indicating with an arrow that increasing n shifts the plot to the right You now need to think about multiplying the function you drew in step 3 with the function you drew in step 6 Of course, when you multiply two functions and one of them 1s zero over a particular region. the result will be zero over this same region Hence, the function that goes to zero first will be the controlling function that defines the edze of the product 
