Signal Processing of Random Physiological Signals - Charles S. Lessard

CONVOLUTION 101

TABLE 10.1: Convolution by Multiplication Method

calculations. The question arises as to where to take the values from the time varying function within a fixed interval. Specifically, the question may be rephrased as “Where should the interval be located?” For example, a function, f (t) = −1λ + 8, can be sampled with some interval of time, T, in the following manner: Let T = 2 and the value be selected at midinterval, as in Fig. 10.26. The general procedure is to take the average value of the function f (t) in the interval, kT, where k is the sample number.

FIGURE 10.26: Graph A of the function, f (t) = −1λ + 8, with the interval T = 2. Note that

the values of the sampled function are 7, 5, 3, and 1 at times t = 1, 3, 5 and 7, respectively

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FIGURE 10.27: Graph B of the function, f (t) = −1λ + 8, with the interval T = 2. Note that

the values of the sampled function are 4, 6, 4, 2, and 0 at times t = 0, 2, 4, 6, and 8, respectively

However, one may elect to use the interval between −T/2 < 0 < +T/2, which then yields slightly different answers as shown by Fig. 10.27. Note that in Fig. 10.27, the midpoints for the interval T = 2 are at t = 1, 3, 5 . . . etc.

Let us convolve the f1 function with the values of the function f (t) = −1λ + 8, as shown in Table 10.1 and Fig. (10.26). The calculations and results of the convolution are given in Table 10.2 and are shown in Fig. (10.28). The results are labeled as A, since the second function is composed of the values from Fig. 10.26.

TABLE 10.2: Results A of Convolution with Fig. 10.26

CONVOLUTION 103

Convolution Results A

FIGURE 10.28: Graph of Results A

Repeating the convolution of the f1 function with the values of the function, f (t) = −1λ + 8, as shown in Table 10.1 and Fig. 10.27. Note the values used in the calculations, which are given in Table 10.3. The results of the convolution are shown in Fig. 10.29. The results are labeled as B, since the second function is composed of the

values from Fig. 10.27.

TABLE 10.3: Results B of Convolution with Fig. 10.27

104 SIGNAL PROCESSING OF RANDOM PHYSIOLOGICAL SIGNALS

Convolution Results B

FIGURE 10.29: Graph of Results B

It should be noted that the procedures and results shown in Tables 10.2 and 10.3 and Figs. 10.28 and 10.29 are comparable, but the procedure that locates discontinuities along one of the steps in the approximation gives better results. In the example, the results labeled A (Table 10.2 and Fig. 10.28) are better for the discontinuity at t = 0.

10.3CONVOLUTION ALGEBRA

Convolution is referred to as a “Linear Process” in the arithmetic sense; therefore, let us examine those mathematical properties that make convolution a linear process. Recall the mathematical properties of the “Real Numbering System” whose properties are

1.associative,

2.commutative, and

3.distributive.

Consider the convolution of two functions, one of the functions being the result of a previous convolution process. First, “Convolution is Associative,” which means that

106 SIGNAL PROCESSING OF RANDOM PHYSIOLOGICAL SIGNALS

If a function is convolved with the derivative of a delta function, the result is the derivative of the original function as shown by (10.28).

It should be noted that the delta function is zero at all other values of time except at the instant of time, t, where the delta function exists. This relation is expressed mathematically

validity of a particular computation. For example, in the convolution of two functions, as in (10.31), the area of a convolution product is equal to the product of the areas of the functions. This means that the area under f3 (t) is equal to the area under the function,

Keep in mind that the area is computed by integrating over the interval, −∞ < t < ∞. Proof of this property follows in (10.32) through (10.36).

∞ ∞ ∞

CONVOLUTION 107

Rewriting (10.33), as moment generating functions, is shown in (10.34) and (10.35). Equation 10.36 shows that the moment about zero of the convolution function f3 is equal to the product of the moments about zero of the two functions f1 and f2.

∞

Another useful relationship deals with the center of gravity (centroid) of the convolution in terms of the center of gravity of the factors (10.37). If you recall, the center of gravity of a waveform was defined in terms of the energy of the signal as t0.

∞

The nth moment of a function or waveform is defined by the general (10.38), and

The first moment of the convolution with infinite limits is given by (10.40).

The net result in moments is given by expression (10.41):

CONVOLUTION 109

is known and the second must be recovered. Even though significant progress has been made in numerical methods, the process of deconvolution is very sensitive to errors and is not used in solution of most problems. There is a tendency for the residual (remainder) to become unstable and diverge in magnitude; hence, caution should be taken in using “Time-domain” deconvolution methods.