Digital Analysis of Geophysical Signals and Waves

4.2 Continuous Convolution

The convolution, y(t) of two signals, s(t) and h(t) is expressed by the convolution integral

Here, the shorthand notation is included that replaces the explicit integration operation by an asterisk between the convolved functions. 

Dissecting the convolution integral, we clearly see that there is some kind of a multiplication by s(t) and h(t) in the integrand and there's an integration ("summation") over the variable t.  But what does t-t mean?

First off, s(t) and h(t) are the same as s(t) and h(t); the variable symbol makes no difference.  h(-t) is h(t) that is flipped or reversed in time and h(t-t) is the function h(-t) time shifted along the t axis by an amount t (Figure 4.2). In Figure 4.2. the shift is to the right, i.e., t>0; if t<0 the shift would be to the left. The time shift, t is the time at which the output, y is determined.  To illustrate these concepts in Figure 4.2, the signals s(t) and h(t) are taken to be rectangle functions of unit length with heights of 1 and 0.5, respectively.

The convolution integral is continuously evaluated at each time shift t by multiplication and integration of s(t) times h(t-t) for all values of t possibly running from -infinity to + infinity.  This process is graphically illustrated in Figure 4.3 by the classical example of the convolution of two rectangle functions (or "boxcar functions") to give a triangle function. For example, the convolution at a single instant t=t₁ (the integration of s(t)h(t₁-t)) is simply the shaded area under the curve since the rectangle function, s(t) has a constant value of 1.  The complete convolution represents the changing area under the product s(t)h(t-t) as t (the time shift) varies.  If the functions to be convolved are of finite lengths, t needs to vary only over the interval of overlap of s(t)h(t-t). Figure 4.3 presents convolution results for -t₁ < t < 5t₁.

Using the view of convolution as a filtering operation, we can say that an input rectangle function has been filtered to yield an output triangle function.  The engineer's view (one that we strongly subscribe to) is that the signal s(t) is the input into a "black box" filter which performs a convolution with h(t) yielding an output y(t). 

It is very important to realize that the black box is completely characterized by h(t) which, for very good reasons, is called the impulse response function of the filter.  The impulse response function, h(t) is literally the response of the black box to an impulse function or Dirac delta function as shown in Figure 4.4. If you know the impulse response function, then you know everything about the "black box" filter. In some geophysical applications such as in seismics and electromagnetics, the Earth is the "black box" and what we ultimately want to know is its impulse response, so its a big deal.

We will repeat this concept below when we look at the convolution of several important functions because this is a major idea.  For now let's review the steps in convolution and consider a few of its properties.

The convolution procedure described above and illustrated in Figures 4.2 and 4.3 is summarized as:

1. Flip (reverse) the function h(t) in time yielding h(-t).
   
   
2. Shift h(-t) by an amount t=t₁ giving h(t₁-t).
   
   
3. Multiply the shifted h(t₁-t) by s(t) obtaining s(t)h(t₁-t).
   
   
4. Integrate s(t)h(t₁-t) to find the area under the product to obtain the value of the convolution at the single time t₁.
   
   
5. Complete steps 1- 4 for all minus to plus values of t where overlap of s(t)h(t-t) occurs, -infinity to +infinity if necessary.

Note that steps 1 and 2 can be interchanged, i.e., you can shift before you flip.  Another interchange that can be made, which is not so obvious, is that either h(t)  or s(t) can be flipped and shifted (or shifted and flipped).  That is, the convolution operation is commutative, i.e.,

Let's look at the convolution between two very important functions that we developed earlier in our web-site.

Convolution with a Dirac delta function

The sifting property (equation 2.3.4) enables us to understand convolution by a delta function.  First, let's remember that a δ-function is nonzero only where its argument is equal to zero so δ(t-t) "exists" only where t=t.  Therefore, the convolution of a signal, s(t) with δ(t)  is:

Thus, convolution with a delta function initially centered at the origin identically returns back the original function, s(t). This is easily visualized as a result of a unit-amplitude spike sliding past a signal so the multiplication (and integration) of the two functions simply leaves the original signal unchanged. If the delta function was not initially at the origin, e.g., δ(t-t₀), a delta function at t=t₀, the convolution integral gives

Here, we see that convolution by a delta function with arbitrary argument ends up placing its argument in the original argument of the function s(t).  This is a shifting operation.

Convolution with a Dirac comb

Convolution of a signal, s(t) with a Dirac comb (equation 2.3.5) is

The result follows the previous result above yielding

So, convolution with a Dirac comb yields an infinite series of replicas of the original function with period Δt, the spacing of the teeth of the comb (Figure 4.5). Would you believe that this result (convolution with a Dirac comb) is the basis for really understanding aliasing? Check out Appendix A to be convinced. Basically, aliasing occurs when the teeth of the Dirac comb are closer together than the width of s(t) and the replicas overlap.