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Chapter 6: Convolution

The Sum of Weighted Inputs

The characteristics of a linear system are completely described by its impulse response. This is the basis of the input side algorithm: each point in the input signal contributes a scaled and shifted version of the impulse response to the output signal. The mathematical consequences of this lead to the output side algorithm: each point in the output signal receives a contribution from many points in the input signal, multiplied by a flipped impulse response. While this is all true, it doesn't provide the full story on why convolution is important in signal processing.

Look back at the convolution machine in Fig. 6-8, and ignore that the signal inside the dotted box is an impulse response. Think of it as a set of weighing coefficients that happen to be embedded in the flow diagram. In this view, each sample in the output signal is equal to a sum of weighted inputs. Each sample in the output is influenced by a region of samples in the input signal, as determined by what the weighing coefficients are chosen to be. For example, imagine there are ten weighing coefficients, each with a value of one-tenth. This makes each sample in the output signal the average of ten samples from the input.

Taking this further, the weighing coefficients do not need to be restricted to the left side of the output sample being calculated. For instance, Fig. 6-8 shows y[6] being calculated from: x[3], x[4], x[5] and x[6]. Viewing the convolution machine as a sum of weighted inputs, the weighing coefficients could be chosen symmetrically around the output sample. For example, y[6] might receive contributions from: x[4], x[5], x[6], x[7] and x[8]. Using the same indexing notation as in Fig. 6-8, the weighing coefficients for these five inputs would be held in: h[2], h[1], h[0], h[-1] and h[-2]. In other words, the impulse response that corresponds to our selection of symmetrical weighing coefficients requires the use of negative indexes. We will return to this in the next chapter.

Mathematically, there is only one concept here: convolution as defined by Eq. 6-1. However, science and engineering problems approach this single concept from two distinct directions. Sometimes you will want to think of a system in terms of what its impulse response looks like. Other times you will understand the system as a set of weighing coefficients. You need to become familiar with both views, and how to toggle between them.