Convolution has many forms. It can be applied to continuous functions and discrete functions. The scope is restricted here to continuous functions. In particular, we consider functions that have time as its independent variable. Convolution integral can be applied to functions that have frequency as its independent variable, but that is outside of the scope of what is presented here.
Convolution is a mathematical operation on two functions that results in a third function. The resulting third function tends to be some type of combination of the two input functions. Let the two input functions be f(t) and g(t), and let the result be called y(t). Then y(t) is an integral that indicates the variation in the amount of overlap as one of the input functions is shifted over the other function.
The relevance of convolution integral to circuit analysis is expressed by the block diagram in Fig. 1.
When the impulse response of a system is known, we can apply convolution integral to determine response y(t), to some other input f(t). The application of convolution integral to causal systems is described in this page and the pages to follow.
Let the input to the linear, time-invariant, continuous system be an impulse input. The output or the response y(t) of this system is then the impulse response, usually called h(t).
If the impulse input is shifted in time, the impulse response of the system is also shifted in time, with the delay in response equaling the delay in the input function, since the system is a time-invariant system.
Since we are dealing with a linear system, the response varies linearly with the input.
Now let the input be the integral of f(τ).δ(t - τ). Since we are dealing with a LTIC system, the response is the integral of f(τ).h(t - τ), and hence we can express the convolution integral as shown by equation (1.1).
By the use of sampling property of an impulse function, the input function can be defined as shown in equation (1.1). Equation (1.1) states that the response y(t) obtained is the integral of impulse responses over time, with the magnitude of impulse at each instant being defined by the amplitude of the input function at that instant. Understanding of equation (1.1) leads to a graphical technique for evaluating the convolution integral. To illustrate the graphical operations associated with the convolution integral, an example is presented.
To understand equation (1.1), we can visualize the input as a series of closely spaced impulse functions, where the magnitude of impulse function at an instant equals the value of input function f(t) at that instant. The response then is the sum of responses due to the series of impulse functions of varying magnitude. In the limit, the summation turns out to be an integral, as shown by equation (1.1).
Obtain the response y(to) due to the three input impulse functions, shown in Fig. 2.
SOLUTION:
The response y(t) is the sum of three responses, with each impulse input yielding a response. The response yo(t), due to δ(t) is shown in Fig. 3.
From yo(t), value of yo(to) can be obtained as shown in Fig. 3. Let yo(to) = A = h(to). Next the response y1(t), due to δ(t - 1) is obtained and it is shown in Fig. 4.
The impulse input δ(t - 1) is applied one second after t = 0 and the response to this impulse is delayed or time-shifted by one second. The plot of h(t - 1) is shown in Fig. 4 and it is called y1(t). From y1(t), value of y1(to) can be obtained as shown in Fig. 4. Let y1(to) = B =h(to- 1) .
Next the response y2(t), due to δ(t - 2) is obtained and it is shown in Fig. 5. The impulse input δ(t - 2) is applied two seconds after t = 0 and the response to this impulse is delayed or time-shifted by two seconds. The plot of h(t - 2) is shown in Fig. 5 and it is called y2(t). From y2(t), value of y2(to) can be obtained as shown in Fig. 5. Let y2(to) = C = h(to- 2).
The response y(t) is the sum of yo(t), y1(t), and y2(t). Also y(to) = yo(to) + y1(to) + y2(to). That is, y(to) = A + B + C. The plots in Fig. 6 show how we can obtain the same value in a different way. Since yo(to) = A = h(to), y1(to) = B = h(to- 1), and y2(to) = C = h(to- 2), we obtain the plots in Fig. 6. The original input sequence of impulse functions, starting from t = 0, can be plotted in the reverse order starting at t = to and the value of h(t) at the instant at which the impulse is applied yields its component of y(to).
The reverse sequence of input impulse functions is obtained by reflecting the sequence at t = 0, and then shifting the input impulse functions. The same result can be obtained by shifting the train of input impulse functions by to seconds and then reflecting them at the vertical axis located at t = to.
The process of reflecting and shifting the train of input impulse functions is illustrated in Fig. 7.
Given two continuous functions, f(t) and g(t), the convolution of these functions over a range [0 , t], is expressed by equation (1.2) shown below.
As shown in equation (1.2), the symbol for convolution is '*'. The convolution can also taken over an infinite range, as shown below.
Equation (1.3) illustrates the commutative property of convolution integral. The convolution integral has other algebraic properties, such as the associative and the distributive properties.
We can obtain the Laplace transform of convolution integral as follows. By definition, we get the Laplace transform of convolution integral by taking the Laplace transform of both sides of equation (1.3). Since the system is a causal system, the lower limit is set to be 0-.
Interchanging the order of integration, we have that
Let p = t - τ for the second integral.
The inner integral equals F(s). The next step is as follows.
We can see that the convolution integral provides the basis for using Laplace transforms to obtain the system response.
The convolution integral is also known as the Faltung integral, with the word Faltung meaning “folding” or “plaiting.” The convolution integral involves four operations, which are reflection, shifting, multiplication and integration. Let us start with equation (1.8).
The four operations contained in a convolution integral are presented above. The process of shifting and reflection by an example in the next page.
Some aspects of a convolution integral have been presented in this page. Examples to evaluate convolution integral have been presented in the pages to follow.