The circuit shown is a simple example
of a linear time invariant system. Clearly the input voltage is the sum of the iR drop across the resistor and
the voltage across the capacitor, which constitutes the output. This leads to the input output relation
This relationship is thus expressed by a linear differential equation with time independent coefficients, the general feature of this type of system. Setting the time constant RC = t and multiplying by the integrating factor et/t leads to
In proceeding further it is important to take care in distinguishing between the variable of integration t' and the time at which the solution is sought t. The general solution may be written
In this solution the effect of a boundary value condition is implicitly included with the physically acceptable assumption that in the distant past the output was zero. An explicit initial condition can be introduced by making the lower limit the initial time, usually but not necessarily zero and adding the known output at that time to the integral. However the approach above is more suitable for general discussion.
The second form of the solution is invaluable in that it provides insight for a physical interpretation also leading to both important concepts and generalization. Suppose the input is an extremely narrow spike occurring at time t1, represented mathematically by v(t1)d(t1-t). Then it follows that
For v(t1) = 1, the input is referred to as the unit impulse, and in general the output of the system for a unit impulse input is referred to as the impulse response. The time t1 is usually taken as zero in which case the impulse response is simply
In the above figure is shown the situation when the input consists of three spikes each of a different strength. The output is the linear superposition of scaled and translated impulse responses. Each response is scaled by the magnitude of the spike and translated according to the time the spike occurs. Each response can occur only after the impulse which evoked it, a fact embodied in the principle of causality. While this is obvious on physical grounds care must be taken to ensure the principle is incorporated into the mathematical description. To do so the impulse response function must be completely defined so that it vanishes when its argument is negative. A description of the output in terms of the actual exponentials involved is difficult because of the above nature of the response function and would require separate equations for the regions t < t1, t between t1 and t2, between t2 and t3, and after t3. By using the general form of the impulse response, which incorporates the principle of causality, the mathematical representation of the output is simplified.
Thus if the input is written as
then the output becomes
In the description of the input the symmetry of the delta function is used to make it apparent that the output is constructed from the input using the transformation
expressing the fact that the impulse at ti evokes the corresponding response at this time. However, unlike the impulse, the response exists continuously for times following ti. Thus at an arbitrary time t the output consists of contributions from all those impulse inputs that occurred before t.
Finally, this approach can be generalized to a continuous signal on fundamental grounds. Any input function satisfies
This can be interpreted as a representation of the continuous input by an infinite number of impulses of strength vin(t')d't. Then each impulse generates the corresponding output vin(t') h(t-t')dt' giving by linear superposition
The second line results from causality. Since for all t' > t the argument of the response function is negative the integrand vanishes so the infinite upper limit of integration produces the same result as an upper limit of t. This result is entirely equivalent to that obtained from integration of the differential equation in Eqn(3) with the substitution from Eqn(5). However, the result as presented in Eqn(10) can now be considered generally applicable to any linear time invariant system. The detailed properties of the system will be contained in the characteristic impulse response function, which may be quite different from the simple exponential for the example with which this discussion began. The input output relation is a special type of integral transform. A general integral transform relating two functions can be written
where K(t,t') is referred to as the kernel. The class of transforms describing linear time invariant systems have a kernel depending only on the difference between the two variables, so that the transform is given the specific name of convolution. This special form of the kernel reflects the fact that the nature of the system does not change with time so that all impulse responses are related by a linear translation in time. Thus the properties of a linear time invariant system may be summarized by the fact that the output is the convolution of the input with the impulse response function, written in short form as
The description of the behaviour of linear time invariant systems given above has been entirely in the time domain. An important alternative description arises from using the frequency domain representation. Let
with a similar relation for the output. Note that differentiation with respect to time and integration with respect to frequency are independent operations which commute. The differential equation governing the system can be written for each frequency component as
One advantage of the frequency domain representation is that the differential equation describing the system is transformed into an algebraic equation. The solution is straightforward and can be written in the general form
where
in this case. The quantity H(w) is referred to as the system transfer function and its form will depend upon the properties of the system.
It should be noted that H(w) could be calculated by consideration of the impedance of the network. The input voltage is across the impedance R+i/wC while the output is the fraction across i/wC.
The relationship between the frequency domain and time domain description can be made by again considering an impulse input. If vin(t) = d(t) then the frequency domain description becomes
Thus the frequency domain representation of the system response to a unit impulse is the transfer function H(w). Since the time domain representation of this same output is the impulse response function h(t), it follows that the transfer function and impulse response functions are Fourier transform pairs. A very important fact which follows from this identification and a comparison of the general input output relations of Eqn(12) and (15) is the convolution theorem. This states that the frequency domain description (Fourier transform) of a function which results from a convolution in the time domain (vout(t) in this case) is the product of the Fourier transforms of the two functions being convolved, vin(t) and h(t).
The reciprocal relation is also true. That is if two functions are multiplied in the time domain the Fourier transform of the resultant is the convolution of the frequency domain representations of the two functions.
The general concepts discussed here are equally applicable to quite different systems for which a linear invariant model is adequate. For example, the approach is used to describe the image object relation of an optical system. In this example the image of a point object plays the role of the impulse response function. The description must be extended to two dimensions of course. Moreover the causality constraint no longer applies.