Question about causal system group delay in DSP

[chingkui]

I have been doing some reading from the book Discrete Time Signal Processing by Oppenheim & Schafer (2nd Edition). In the book, I come across an IIR system (example 5.8 on p265) with transfer function:
H(z)=1/[(1-r*e^(j*theta)*z^(-1))*(1-r*e^(-j*theta)*z^(-1))]
The causal impulse response is:
h[n]=(r^n)*sin[(n+1)*theta]*u[n]/sin(theta)

A plot of the group delay is given for r=0.9 and theta=pi/4 (figure 5.16c). From the plot, there is an extensive region where the group delay is negative (somewhere between -1 and 0). From my understanding, group delay represents the number of time steps delay of the output response to an applied input. For a causal system, I would expect the group delay is positive at all frequency (which means input precede output). However, in this case, a negative group delay is seen for some frequency (which would mean output comes before input!) How can this happen? Can anyone please point to me where my logic went wrong? Thank you.

 

[doodle]

For a causal system, I would expect the group delay is positive at all frequency (which means input precede output).

How did you arrive at this conclusion?

 

[piercas]

Thank you for your question. Group delay is a measure of the delay in a system's output response to an applied input at a specific frequency. The negative group delay observed in this IIR system is not a violation of causality, but rather a result of the system's frequency response.

In the frequency domain, the transfer function H(z) has a pole at z = r*e^(j*theta) and a zero at z = r*e^(-j*theta). When r is close to 1 and theta is small, these two points are close to each other and the frequency response has a sharp peak. This peak corresponds to the negative group delay observed in the plot.

To understand this further, we can look at the formula for group delay:
GD(omega) = -d(phi(omega))/d(omega)
where phi(omega) is the phase response of the system. In this case, the phase response is:
phi(omega) = -theta + (n+1)*omega
where n is the sample index.

When we take the derivative of this phase response, we get:
d(phi(omega))/d(omega) = (n+1) - theta
At the peak of the frequency response, theta is small and n is large, resulting in a negative value for the group delay.

In summary, the negative group delay observed in this IIR system is a result of its frequency response and does not indicate a violation of causality. I hope this explanation helps clarify your understanding.