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Continuous-Time Convolution

Module by: Melissa Selik, Richard Baraniuk

Summary: Defines convolution and derives the Convolution Integral.

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
yt=∫-∞∞xτht-τdτ y t τ x τ h t τ (1)
Convolution is such an important tool that it is represented by the symbol * , and can be written as
yt=xt*ht y t x t h t (2)
By making a simple change of variables into the convolution integral, τ=t-τ τ t τ , we can easily show that convolution is commutative:
xt*ht=ht*xt x t h t h t x t (3)
For more information on the characteristics of the convolution integral, read about the Properties of Convolution.
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.
    Brief Overview of Derivation Steps:
  1. An impulse input leads to an impulse response output.
  2. A shifted impulse input leads to a shifted impulse response output. This is due to the time-invariance of the system.
  3. We now scale the impulse input to get a scaled impulse output. This is using the scalar multiplication property of linearity.
  4. We can now "sum up" an infinite number of these scaled impulses to get a sum of an infinite number of scaled impulse responses. This is using the additivity attribute of linearity.
  5. Now we recognize that this infinite sum is nothing more than an integral, so we convert both sides into integrals.
  6. Recognizing that the input is the function ft f t , we also recognize that the output is exactly the convolution integral.
ctconvir1.png
Figure 1: We begin with a system defined by its impulse response, ht h t .
ctconvir2.png
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.
ctconvir4.png
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.
ctconvir3.png
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
unitimpulsee.png
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.
Derivation II of Convolution Integral
Input   Output
limΔ→0 δ Δ t Δ 0 δ Δ t → h →→ h → limΔ→0ht Δ 0 h t
limΔ→0 δ Δ t-nΔ Δ 0 δ Δ t n Δ → h →→ h → limΔ→0ht-nΔ Δ 0 h t n Δ
limΔ→0fnΔ δ Δ t-nΔΔ Δ 0 f n Δ δ Δ t n Δ Δ → h →→ h → limΔ→0fnΔht-nΔΔ Δ 0 f n Δ h t n Δ Δ
limΔ→0∑nfnΔ δ Δ t-nΔΔ Δ 0 n f n Δ δ Δ t n Δ Δ → h →→ h → limΔ→0∑nfnΔht-nΔΔ Δ 0 n f n Δ h t n Δ Δ
∫-∞∞fτδt-τdτ τ f τ δ t τ → h →→ h → ∫-∞∞fτht-τdτ τ f τ h t τ
ft f t → h →→ h → yt=∫-∞∞fτht-τdτ y t τ f τ h t τ

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.
Example 1 
This demonstration illustrates the graphical method for convolution. See here for instructions on how to use the demo.

Download LabVIEW Source

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 .
xte.pnghte.png
Subfigure 6.1
Subfigure 6.2
Figure 6: Here are the two basic signals that we will convolve together.

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.
refshiftpulsee.png
Figure 7: The reflected and shifted unit pulse.

Regions of Integration

Next, we want to look at the functions and divide the span of the functions into different limits of integration. These different regions can be understood by thinking about how we slide ht-τ h t τ over the other function. These limits come from the different regions of overlap that occur between the two functions. If the function were more complex, then we would need to have more limits so that that overlapping parts of both function could be expressed in a single, linear integral. For this problem we will have the following four regions. Compare these limits of integration to the sketches of ht-τ h t τ and xt x t to see if you can understand why we have the four regions. Note that the t t in the limits of integration refers to the right-hand side of ht-τ h t τ 's function, labeled as t t between zero and one on the plot.
    Four Limits of Integration
  1. t<0 t 0
  2. 0≤t<1 0 t 1
  3. 1≤t<2 1 t 2
  4. t≥2 t 2

Using the Convolution Integral

Finally we are ready for a little math. Using the convolution integral, let us integrate the product of xtht-τ x t h t τ . For our first and fourth region this will be trivial as it will always be 00. The second region, 0≤t<1 0 t 1 , will require the following math:
yt=∫0t 1 dτ=t y t τ 0 t 1 t (4)
The third region, 1≤t<2 1 t 2 , is solved in much the same manner. Take note of the changes in our integration though. As we move ht-τ h t τ across our other function, the left-hand edge of the function, t-1 t 1 , becomes our lowlimit for the integral. This is shown through our convolution integral as
yt=∫t-11 1 dτ=1-t-1=2-t y t τ t 1 1 1 1 t 1 2 t (5)
The above formulas show the method for calculating convolution; however, do not let the simplicity of this example confuse you when you work on other problems. The method will be the same, you will just have to deal with more math in more complicated integrals.

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

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