Complex gain

Considering the general LTI system

${\displaystyle P(D)x=Q(D)f(t)}$

where ${\displaystyle f(t)}$ is the input and ${\displaystyle P(D),Q(D)}$ are given polynomial operators, while assuming that ${\displaystyle P(s)\neq 0}$. In case that ${\displaystyle f(r)=F_{0}\cos(\omega t)}$, a particular solution to given equation is

${\displaystyle x_{p}(t)=\operatorname {Re} {\Big (}F_{0}{\frac {Q(i\omega )}{P(i\omega )}}e^{i\omega t}{\Big )}.}$

Consider the following concepts used in physics and signal processing mainly.

${\displaystyle \bullet }$ The amplitude of the input is ${\displaystyle F_{0}}$. This has the same units as the input quantity.

${\displaystyle \bullet }$ The angular frequency of the input is ${\displaystyle \omega }$. It has units of radian/time. Often we will be casual and refer to it as frequency, even though technically frequency should have units of cycles/time.

${\displaystyle \bullet }$ The amplitude of the response is ${\displaystyle A=F_{0}|Q(i\omega )/P(i\omega )|}$. This has the same units as the response quantity.

${\displaystyle \bullet }$ The gain is ${\displaystyle g(\omega )=|Q(i\omega )/P(i\omega )|}$. The gain is the factor that the input amplitude is multiplied by to get the amplitude of the response. It has the units needed to convert

input units to output units.

${\displaystyle \bullet }$ The phase lag is ${\displaystyle \phi =-\operatorname {Arg} (Q(i\omega )/P(i\omega ))}$. The phase lag has units of radians, i.e. it’s dimensionless.

${\displaystyle \bullet }$ The time lag is ${\displaystyle \phi /\omega }$. This has units of time. It is the time that peak of the output lags behind that of the input.

${\displaystyle \bullet }$ The complex gain is ${\displaystyle Q(i\omega )/P(i\omega )}$. This is the factor that the complex input is multiplied by to get the complex output.

Suppose a circuit has an input voltage described by the equation

${\displaystyle V_{i}(t)=1\ V\cdot \sin(\omega \cdot t)}$

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

${\displaystyle V_{o}(t)=2\ V\cdot \cos(\omega \cdot t)}$

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:

${\displaystyle G={\frac {2\ V}{j\cdot 1\ V}}=-2j.}$

This (unitless) complex number incorporates both the magnitude of the change in amplitude (as the absolute value) and the phase change (as the argument).