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Chapter 6: Convolution

The Delta Function and Impulse Response

The previous chapter describes how a signal can be decomposed into a group of components called impulses. An impulse is a signal composed of all zeros, except a single nonzero point. In effect, impulse decomposition provides a way to analyze signals one sample at a time. The previous chapter also presented the fundamental concept of DSP: the input signal is decomposed into simple additive components, each of these components is passed through a linear system, and the resulting output components are synthesized (added). The signal resulting from this divide-and-conquer procedure is identical to that obtained by directly passing the original signal through the system. While many different decompositions are possible, two form the backbone of signal processing: impulse decomposition and Fourier decomposition. When impulse decomposition is used, the procedure can be described by a mathematical operation called convolution. In this chapter (and most of the following ones) we will only be dealing with discrete signals. Convolution also applies to continuous signals, but the mathematics is more complicated. We will look at how continious signals are processed in Chapter 13.

Figure 6-1 defines two important terms used in DSP. The first is the delta function, symbolized by the Greek letter delta, δ[n]. The delta function is a normalized impulse, that is, sample number zero has a value of one, while all other samples have a value of zero. For this reason, the delta function is frequently called the unit impulse.

The second term defined in Fig. 6-1 is the impulse response. As the name suggests, the impulse response is the signal that exits a system when a delta function (unit impulse) is the input. If two systems are different in any way, they will have different impulse responses. Just as the input and output signals are often called x[n] and y[n], the impulse response is usually given the symbol, h[n] . Of course, this can be changed if a more descriptive name is available, for instance, f[n] might be used to identify the impulse response of a filter.

Any impulse can be represented as a shifted and scaled delta function. Consider a signal, a[n], composed of all zeros except sample number 8, which has a value of -3. This is the same as a delta function shifted to the right by 8 samples, and multiplied by -3. In equation form: a[n] = -3δ[n-8]. Make sure you understand this notation, it is used in nearly all DSP equations.

If the input to a system is an impulse, such as -3δ[n-8], what is the system's output? This is where the properties of homogeneity and shift invariance are used. Scaling and shifting the input results in an identical scaling and shifting of the output. If δ[n] results in h[n], it follows that -3δ[n-8] results in -3h[n-8]. In words, the output is a version of the impulse response that has been shifted and scaled by the same amount as the delta function on the input. If you know a system's impulse response, you immediately know how it will react to any impulse.

Next Section: Convolution