Finding phase from group delay

Question

I know that group delay is defined as :

$$\begin{align} \tau(\omega) = grd [H(e^{j\omega})] &=-\frac{d[\arg[H(e^{j\omega})]]}{d\omega}\end{align}$$

But if what is available is group delay itself, is it possible to recover the phase?

According to Discrete-Time Signal Processing 3rd edition by Alan Oppenheim equation 5.86 the expression would look like this:

$$\begin{align} \arg[H(e^{j\omega})] = -\int\limits_{0}^{\omega} \end{align} grd [ H(e^{j\phi})]\, d \phi + \arg[H(e^{j0})] $$

and my doubt is:

Where did the second term $$\arg[H(e^{j0}) ]$$ came from?

Answer 1

Note that if $F(x)$ is an antiderivative of $f(x)$, i.e., if $F'(x)=f(x)$, then from the fundamental theorem of calculus we have

$$\int_a^bf(x)dx=F(b)-F(a)\tag{1}$$

Since the phase $\phi(\omega)$ is an antiderivative of the negative group delay $-\tau(\omega)$ we get from $(1)$

$$-\int_a^b\tau(\Omega)d\Omega=\phi(b)-\phi(a)\tag{2}$$

With $b=\omega$ and $a=0$ we get from $(2)$

$$-\int_0^\omega\tau(\Omega)d\Omega=\phi(\omega)-\phi(0)\tag{3}$$

By adding $\phi(0)$ on both sides of $(3)$ we obtain the given formula:

$$\phi(\omega)=-\int_0^\omega\tau(\Omega)d\Omega+\phi(0)\tag{4}$$

Answer 2

When integrating any expression, there is a constant term that is added. This represents an offset that may exist. For example, if I take the derivative of the function $y=x^2+3$. The derivative $\frac{dy}{dx} = 2x$. If I integrate $\frac{dy}{dx}$, then I end up with the expression $y=x^2+c$, where $c$ is an unknown constant. In your question, $arg[H(e^{j0})]$ is the phase offset at DC. This is the missing constant offset when you integrate the group delay. Without it you, you can only know the phase at some frequency $\omega1$, relative to some other frequency $\omega2$.

I'll see if I can clarify (hopefully not confuse) and answer the comment below. Let's look at a different problem. Let's say I want to integrate velocity $v$ over time to find out my position $p$ on the highway. I started at time 0 and drove for $\omega$ seconds. My position relative to my starting position is $p=\int\limits_{0}^\omega vdt$. The key here is relative to my starting position. If I don't know my position at time 0, I do not know my absolute position. I can tell you I travelled 10 miles, but I don't know where from and therefore where to.

The same is true for integrating the group delay to produce a phase. If we wanted phase offset from DC, the $arg[H(e^{j0})]$ wouldn't be necessary. If we want absolute phase, we must add it.