The figure shows the pole–zero plots for three different causal linear time-invariant systems with real impulse responses. Indicate which of the following properties apply to each of the systems pictured: stable, IIR, FIR, minimum phase, all pass,linear phase, positive group delay at all $w$:
I am having problems specifically finding out if the system is linear phase and has a positive group delay, and all pass or not.
In the first plot: the system is stable since we have a region of convergence containing the unit circle, and FIR because it is stable, and minimum phase because we have a pair of complex conjugate pairs.
And so on with the other figures...But I still cannot quite figure out how to determine the other properties given the pole-zero mapping.
Ok, so I've gotten to this point: Plot 1 is not stable because the ROC doesn't contain the unit circle and since all zeros are in the unit circle, we have a minimum phase system. Additionally, we have a IIR system because of the 3 poles so we have feedback. Its also not an allpass filter because poles and zeros are not reciprocals of each other. I hope those are right and I understand the concept so that I can continue with the other plots. But, I'm really having difficulty trying to find out if the system is a linear phase system or not and if it has a positive group delay. I know that linear phase translates in delay in time for all frequencies by the same amount and that group delay $\tau_g = -d\theta(w)/dw$.
Thanks in advance guys.