Linear Time-Invariant Systems: Impulse response and Convolution

If a system is linear and time-invariant, then it is possible to establish a simple relationship between its input and output. Given such an input-output relationship we can calculate the response of the Linear Time-Invariant (LTI) system to any input.

The key to establishing the input-output relationship is the idea that any signal can be broken up into a set primitive elements. For convenience we shall use narrow strips for such a primitive element. For example, consider the signal x(t) shown below which is broken into a set of narrow strips, s₀, s₁, s₂,

If the strips are of the same width, then the only difference in the strips is the amplitude and time location (time shift). Using the notation from module 1 for time shift, we can write:

All the strips are similar to the first strip, s₀, and the only difference between the strips is the amplitude the time shift. If T is the width of each strip, and the amplitude of the k^th strip is represented by x_k, then we can write:

The impulse function

In the above discussion the signal x(t) was broken into thin strips of duration T. By comparing the two figures above, it is clear that the thinner the strips are the better will be the approximation to the original x(t). This leads to the notion of an impulse function which has infinitesimal duration. As we make the strip very narrow, the energy contained in it becomes zero. Therefore, the impulse function is defined to have unit area and is defined as:

Since the Greek letter delta is used to represent the impulse function, it is also called the delta function. As the duration of the impulse function goes to zero, its amplitude goes to infinity; therefore, the impulse function is denoted graphically by a vertical arrow.

Now we can describe any signal, x(t), in terms of impulse functions:

Impulse Response

Now that we can describe any signal in terms of scaled, time shifted impulses, we can say that if the response of a Linear Time-Invariant system to an impulse is known then we can determine its response to any signal. The following picture shows an LTI system whose impulse response is h(t):

Therefore, if h(t) is the impulse response of a system the response, y(t), to an arbitrary signal is given by the convolution integral:

Two ways of looking at convolution:

A. The simplest way to think about convolution is to regard the input signal x(t) as a sequence of scaled+shifted impulse functions, and the corresponding output as the accumulation of scaled+shifted impulse response:

A linear system will respond in a proportional manner to changes in the amplitude of the input; and moreover it responds identically to repeated application of the same input. The accompanying figure shows a system with a triangular response upon input of a brief rectangular pulse. When another signal is input to this system, its output can now be determined. The new signal is broken up into a sequence of rectangular pulses of the same fixed brevity, but amplitude depending on the new signals amplitude at various points in time. The output can now be calculated as the combination of a sequence of primitive response scaled suitably. The set of responses shown in the lower right side must now be added at every point in time to obtain the shape of the actual response of the system.

In actual practice convolution can be calculated accurately and precisely by mathematical means. The primitive impulse that is used to describe other signals has the principle property of being extremely brief in time. Thus, any signal including those with rapid variations can be broken up in terms of this primitive signal. The primitive signal is the Impulse function. In order to characterize any system, we would like to obtain its response to such an impulse. Once the impulse response of the system is known, its response to any arbitrary signal can be calculated using convolution.

B. The second way to think about convolution is the use of the integral. This involves obtaining the product of x(T)h(T-t) and then integrating with respect to the dummy variable T. Since, the variable is T, note that h(T-t) is a time-reversed and shifted version of h(t). (Note: Since, convolution is commutative, x(t)*h(t) = h(t)*x(t) and so we can time-reverse either x(t) or h(t) depending on calculation convenience).

Use the following program to understand how the above two views of convolution are equivalent.

1. There are four graphical panels. The top panels use method A, and the lower panels calculate the convolution integral (method B). The left panels show the calculation method and the right panels show the output.
2. Method A uses superposition of the scaled impulse responses for each time step. The size of the scaled impulse response will depend on the size of the strip of the input, i.e., determined by the time step.
3. Method B uses the integration of the overlapped h(T) and x(t-T). The area of integration is shown in pink.
4. Select signals x(t) and h(t) from the sets on the top left
5. Select the time step for the calculations using the buttons on the top (right side).
6. Run the calculations through time using the Run button.
7. If you want to stop and inspect the procedure, use the Stop button.

Interactive example: Convolution of signals

In addition to simple first order low pass and high pass systems, second order systems can also be simulated in this program. Many common transducers can be described as second order systems. They are described in more detail below.

© Suresh Devasahayam