Motivation

Convolution helps to determine the effect a system has on an input signal. It can be shown that a linear, time-invariant system is completely characterized by its impulse response. At first glance, this may appear to be of little use, since impulse functions are not well defined in real applications. However, the sifting property of impulses tells us that a signal can be decomposed into an infinite sum (integral) of scaled and shifted impulses. By knowing how a system affects a single impulse, and by understanding the way a signal is comprised of scaled and summed impulses, it seems reasonable that it should be possible to scale and sum the impulse responses of a system in order to determine what output signal will results from a particular input. This is precisely what convolution does - convolution determines the system's output from knowledge of the input and the system's impulse response.

In the rest of this module, we will examine exactly how convolution is defined from the reasoning above. This will result in the convolution integral (see the next section) and its properties. These concepts are very important in Electrical Engineering and will make any engineer's life a lot easier if the time is spent now to truly understand what is going on.

In order to fully understand convolution, you may find it useful to look at the discrete-time convolution as well. It will also be helpful to experiment with the applets available on the internet. These resources will offer different approaches to this crucial concept.

Convolution Integral

As mentioned above, the convolution integral provides an easy mathematical way to express the output of an LTI system based on an arbitrary signal, xt x t , and the system's impulse response, ht h t . The convolution integral is expressed as

We now present two distinct approaches for deriving the convolution integral. These derivations, along with a basic example, will help to build intuition about convolution.

Derivation I: The Short Approach

The derivation used here closely follows the one discussed in the Motivation section above. To begin this, it is necessary to state the assumptions we will be making. In this instance, the only constraints on our system are that it be linear and time-invariant.

Figure 1: We begin with a system defined by its impulse response, ht h t .

Figure 2: We then consider a shifted version of the input impulse. Due to the time invariance of the system, we obtain a shifted version of the output impulse response.

Figure 3: Now we use the scaling part of linearity by scaling the system by a value, fτ f τ , that is constant with respect to the system variable, tt.

Figure 4: We can now use the additivity aspect of linearity to add an infinite number of these, one for each possible ττ. Since an infinite sum is exactly an integral, we end up with the integration known as the Convolution Integral. Using the sifting property, we recognize the left-hand side simply as the input ft f t .

Derivation II: The Long Approach

This derivation is really not too different from the one above. It is, however, a little more rigorous and a little longer. Hopefully, if you think you "kind of" get the derivation above, this will help you gain a more complete understanding of convolution.

The first step in this derivation is to define a particular realization of the unit impulse function. For this, we will use δ Δ t=1Δif-Δ2<t<Δ20otherwise δ Δ t 1 Δ Δ 2 t Δ 2 0

Figure 5: The realization of the unit impulse function that we will use for this derivation.

After defining our realization of the unit impulse response, we can derive our convolution integral from the following steps found in the table below. Note that the left column represents the input and the right column is the system's output given that input.

Implementation of Convolution

Taking a closer look at the convolution integral, we find that we are multiplying the input signal by the time-reversed impulse response and integrating. This will give us the value of the output at one given value of tt. If we then shift the time-reversed impulse response by a small amount, we get the output for another value of tt. Repeating this for every possible value of tt, yields the total output function. While we would never actually do this computation by hand in this fashion, it does provide us with some insight into what is actually happening. We find that we are essentially reversing the impulse response function and sliding it across the input function, integrating as we go. This method, referred to as the graphical method, provides us with a much simpler way to solve for the output for simple (contrived) signals, while improving our intuition for the more complex cases where we rely on computers. In fact Texas Instruments develops Digital Signal Processors which have special instruction sets for computations such as convolution.

Basic Example

Let us look at a basic continuous-time convolution example to help express some of the ideas mentioned above through a short example. We will convolve together two unit pulses, xt xt and ht ht .

Figure 6: Here are the two basic signals that we will convolve together.

(a) (b)

Reflect and Shift

Now we will take one of the functions and reflect it around the y-axis. Then we must shift the function, such that the origin, the point of the function that was originally on the origin, is labeled as point τ τ. This step is shown in the figure below, ht−τ h t τ . Since convolution is commutative it will never matter which function is reflected and shifted; however, as the functions become more complicated reflecting and shifting the "right one" will often make the problem much easier.

Figure 7: The reflected and shifted unit pulse.

Convolution Results

Thus, we have the following results for our four regions:

yt= 0ift<0tif0≤t<12−tif1≤t<20ift≥2 y t 0 t 0 t 0 t 1 2 t 1 t 2 0 t 2 (6)

Now that we have found the resulting function for each of the four regions, we can combine them together and graph the convolution of xt*ht x t h t .

Figure 8: Shows the system's response to the input, xt x t .