# Sketching Phase Spectra Using Group Delay and Magnitude Spectra Informations

I have only magnitude spectra and group delay information and I need to sketch phase spectra from this information. For example, group delay is given like this: $\tau_{g}(\omega) = c$ where c is a constant value.

I know the group delay is the derivative of minus phase function of the system. In this case, phase function can be written likse this: $\beta(\omega) = - c \omega + constant$, but how to determine the $constant$ here? Is it possible to find phase from these information?

• If the filter maps real signals to real signals then the phase at f=0 is an integer multiple of $\pi$. That determines your constant to be either 0 or $\pi$. Which it is can be decided by looking at the DC response of the filter. If it is negative your phase is $\pi$, if not it's 0. May 6 '15 at 17:59
• I have only magnitute spectra thus I dont know what the dc response would be, so it is hard to determine constant, I guess. Thank you Jazzmaniac.. May 6 '15 at 18:10

If the magnitude spectrum is symmetric

$$M(\omega)=M(-\omega)\tag{1}$$

(as I assume), then your system is real-valued. The phase response of a real-valued system is asymmetric:

$$\phi(\omega)=-\phi(-\omega)\quad(\mod 2\pi)\tag{2}$$

This means that there can be two cases:

1. The phase goes through zero at $\omega=0$, i.e. the phase is given by $\phi(\omega)=-c\cdot\omega$, where $c$ is the constant group delay.
2. The phase is $\pm \pi$ at $\omega=0$, which means $\phi(\omega)=-c\cdot\omega\pm\pi$.
3. The phase jumps at $\omega=0$. This is only possible if the magnitude has a zero at $\omega=0$, i.e. $M(0)=0$. The phase jumps by $\pi$, which is simply a sign change of the (bipolar) amplitude function. In this case the phase is given by $$\phi(\omega)=\begin{cases}-\pi/2-c\cdot\omega,&\quad\omega>0\\\pi/2-c\cdot\omega,&\quad\omega<0\end{cases}$$
• Thank you very much Matt L.. Well explaination as always (: May 6 '15 at 18:58