Is S21 equal to the gain of ac response

As the title, I am confused ...
is the s parameter s21 always equal to the gain of ac response or not ?
thanks!

Depends on exactly what you mean by "gain".

If you are driving the unit with a signal generator of Z0 characteristic impedance, and you are measuring output power into a detector of that same Z0 characteristic impedance, then yes, gain equals S21.

For example, 0 dBm in, and you measure 22 dBm out, so the power gain of the unit is 22 dB, so 22 = 20 Log (S21)

Z0 is comonly 50 ohms, but can be other values, such as 75 ohms, etc.

S21 is NOT valid when you talk about voltage or current gains, since the boundary conditions are different. Also, the magnitude of S21 would measure to be different if you changed Z0 (the characteristic impedance of the measuring system).

if zg=zl=z0. Gt=abs(s21)^2
Gt(transducer power gain)