Complex gain

In electronics, complex gain is the effect that circuitry has on the amplitude and phase of a sine wave signal. The term "complex" is used because mathematically this effect can be expressed as a complex number.
LTI systems.
Considering the general LTI system
where formula_2 is the input and formula_3 are given polynomial operators, while assuming that formula_4.
In case that formula_5, a particular solution to given equation is
Consider the following concepts used in physics and signal processing mainly.
input units to output units.
Example.
Suppose a circuit has an input voltage described by the equation
where ω equals 2π×100 Hz, i.e., the input signal is a 100 Hz sine wave with an amplitude of 1 volt.
If the circuit is such that for this frequency it doubles the signal's amplitude and causes a 90 degrees forward phase shift, then its output signal can be described by
In complex notation, these signals can be described as, for this frequency, "j"·1 V and 2 V, respectively.
The complex gain "G" of this circuit is then computed by dividing output by input:
This (unitless) complex number incorporates both the magnitude of the change in amplitude (as the absolute value) and the phase change (as the argument).