A definite integral is written and are commonly used to find the area between the graph of a function and the x-axis.
are constants, is a variable and , are functions.
Differentiation is the symmetric operation of integrating:
The convolution of two functions (over the same variable, e.g. and ) is an operation that produces a separate third function that describes how the first function modifies the second one. Conversely, the resulting function can also be used to describe how the second function modifies the fist function.
The convolution of two functions is denoted with the operator and written as:
where is a dummy variable.
Here is an example of the convolution of two square pulses and :
resulting in .
where is the unit impulse.
Most electrical circuits are designed to be linear, time-invariant systems (
LTI) meaning that the magnitude of a circuit's output signal is a scaled version of the input signal's magnitude. Furthermore, and
LTI system that is given two independent signal sources will output the sum of the scaled versions of each signal. Given a function that causes an
LTI system to output , then:
where is a multiplicative constant. Given a number of independent signal sources, this gives rise to the concept of signal superposition which allows us to say:
A time-invariant system means it does not matter when the input signal is applied - a specific input signal will always result in the same output signal for a given
LTI system. The time-invariance can be expressed as:
where can be viewed as a time delay when dealing with signals in time (i.e. "time-domain signals"). Indirectly, this concept also signifies than an output signal cannot contain frequency components not inherent in the input signal (causality).
The vast majority of circuits are
LTI systems, each with a specific impulse response. The "impulse response" of a system is a system's output when its input is fed with an impulse signal - a signal of a brief duration. An example of an "impulse signal" would be a lighting bolt or any form of
ESD (electro-static discharge). Any voltage or current that spikes in magnitude for a relatively short period of time may be viewed as an impulse signal. The impulse response of a circuit will always be a time-domain signal and exists because no signal can propagate through a circuit will always be a time-domain signal, and exists because no signal can propagate through a circuit in zero time; each individual electron involved can only move so quickly through each component. Real-world electronic
LTI systems exhibit an impulse response that consists of an initial spike in magnitude, followed by an everlasting and ever-decreasing exponential relationship in signal magnitude.
The integral of the impulse response and the input square wave stepped through time can be observed in the image above. The convolution equation is seen as the operation is done with respect to . In reality, we are taking an input signal, flipping it vertically through the origin and determining whether the what the integral is for each value of which represents in this case a delay through time. Since the output of any
LTI system is non-casual (it cannot exist until the signal that excites the output has seen applied), we must step through time to see how each impulse signal of the input affects the
LTI system's impulse response - again, achieved by stepping through - the "time-delay" dummy variable.
LTI system with an impulse response of , calculate the output given the input of .
Using the convolution integral, we have to solve:
It only makes sense to integrate between and because at the signal is applied and both and will have zero magnitude at any time .
Now we can calculate:
Since for all , we can write the output as:
which describes the output function for an
LTI system with an initial impulse response when fed the input signal .