When we apply a periodic input to a linear, time-invariant system, the output is periodic and has Fourier series coefficients equal to the product of the system's frequency response and the input's Fourier coefficients (Filtering Periodic Signals). The way we derived the spectrum of non-periodic signal from periodic ones makes it clear that the same kind of result works when the input is not periodic: If serves as the input to a linear, time-invariant system having frequency response , the spectrum of the output is .
Let's use this frequency-domain input-output relationship for linear, time-invariant systems to find a formula for the -circuit's response to a pulse input. We have expressions for the input's spectrum and the system's frequency response.
Derive the filter's output by considering the terms in [link] in the order given. Integrate last rather than first. You should get the same answer.
The inverse transform of the frequency response is . Multiplying the frequency response by means subtract from the original signal its time-delayed version. Delaying the frequency response's time-domain version by results in . Subtracting from the undelayed signal yields . Now we integrate this sum. Because the integral of a sum equals the sum of the component integrals (integration is linear), we can consider each separately. Because integration and signal-delay are linear, the integral of a delayed signal equals the delayed version of the integral. The integral is provided in the example.
In this example, we used the table extensively to find the inverse Fourier transform, relying mostly on what multiplication by certain factors, like and , meant. We essentially treated multiplication by these factors as if they were transfer functions of some fictitious circuit. The transfer function corresponded to a circuit that integrated, and to one that delayed. We even implicitly interpreted the circuit's transfer function as the input's spectrum! This approach to finding inverse transforms -- breaking down a complicated expression into products and sums of simple components -- is the engineer's way of breaking down the problem into several subproblems that are much easier to solve and then gluing the results together. Along the way we may make the system serve as the input, but in the rule , which term is the input and which is the transfer function is merely a notational matter (we labeled one factor with an and the other with an ).
The notion of a transfer function applies well beyond linear circuits. Although we don't have all we need to demonstrate the result as yet, all linear, time-invariant systems have a frequency-domain input-output relation given by the product of the input's Fourier transform and the system's transfer function. Thus, linear circuits are a special case of linear, time-invariant systems. As we tackle more sophisticated problems in transmitting, manipulating, and receiving information, we will assume linear systems having certain properties (transfer functions) without worrying about what circuit has the desired property. At this point, you may be concerned that this approach is glib, and rightly so. Later we'll show that by involving software that we really don't need to be concerned about constructing a transfer function from circuit elements and op-amps.
Another interesting notion arises from the commutative property of multiplication (exploited in an example above): We can rather arbitrarily choose an order in which to apply each product. Consider a cascade of two linear, time-invariant systems. Because the Fourier transform of the first system's output is and it serves as the second system's input, the cascade's output spectrum is . Because this product also equals , the cascade having the linear systems in the opposite order yields the same result. Furthermore, the cascade acts like a single linear system, having transfer function . This result applies to other configurations of linear, time-invariant systems as well; see this Frequency Domain Problem. Engineers exploit this property by determining what transfer function they want, then breaking it down into components arranged according to standard configurations. Using the fact that op-amp circuits can be connected in cascade with the transfer function equaling the product of its component's transfer function (see this analog signal processing problem), we find a ready way of realizing designs. We now understand why op-amp implementations of transfer functions are so important.