Gain and phase shift are properties of linear circuits that describe the effect a circuit has on a sinusoidal input voltage or current. The circuit's behavior that contains reactive elements will depend on the frequency of the input sinusoid. As a result, it is observed that the gain and phase shift will all be frequency functions.

Gain:

Suppose V_in is the input and V_out is the output signal to a circuit.

The input and output signals have the amplitude A and B and a phase angle of 0° and θ°, respectively.

The gain of the circuit gives the relationship between the sizes of the input and output sinusoids. In particular, the gain (K) is the ratio of the output sinusoid's amplitude to the input sinusoid's amplitude.

When the gain K is greater than unity, it implies amplification, whereas a gain less than one indicates attenuation. The gain is described using a logarithmic scale. When expressed in decibels (dB), gain provides a convenient way to represent very large and very small numbers on the same scale. The gain in decibels is calculated using the formula:

Where:

dB is the gain in decibels,

K is the gain,

log is the logarithm base 10.

Phase Shift:

Phase shift refers to the amount by which the phase of the output signal is delayed or advanced compared to the phase of the input signal. It is the angular difference between the output and input sinusoids. The input and output signal has a phase angle of 0° and θ° respectively, then the phase shift (θ_shift) is given by:

A positive θ_shift indicates that the output signal lags the input signal, while a negative means that the output signal leads the input signal.