and phase shift:
phi(omega) = arg(Y) - arg(X) = arg( H(j omega)).
The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:
au_{phi}(omega) = -frac{phi(omega)}{omega}.
The group delay (i.e., the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency omega ,