4.3 Discrete Convolution

A to D conversion produces digital signals sampled at a particular sampling interval, Δt (or Δx).  Assuming both s(t) and h(t) are digital functions with a sampling interval of unity, the convolution operation is defined as

(4.3.1)

 

for values of j = ... -3, -2, -1, 0, 1, 2, 3, ...

The summation holds for all products for which the product of sk and h j-k is nonzero. If:

(4.3.2a)

 

and

(4.3.2b)

 

then, the convolution yj has

(4.3.2c)

 

The digital convolution with sample interval Δt = 1 is summarized as:

  1. Flip (reverse) one of the digital functions.

  2. Shift it along the time axis by one sample, j.

  3. Multiply the corresponding values of the two digital functions.

  4. Summate the products from step 3 to get one point of the digital convolution at j.

  5. Repeat steps 1-4 to obtain the digital convolution at all times, j where the digital functions overlap.

For example, let sk = 2, -2, 1 and hk = 1, 3, 1/2, -1. Convolution of these two discrete signals equals 2, 4, -4, 0, 2 1/2, -1, i.e.

(4.3.3)

The convolution animation in Figure 4.6 details the step by step process that produces this result. Step through the animation to convince yourself how descrete convolution works.

Figure 4.6. Action convolution of discrete functions (2, -2, 1) and (1, 3, 1/2, -1) yields (2, 4, -4, 0, 2 1/2, -1).