I apologize if this question is too general to answer concretely, but I was hoping more to perhaps be pointed towards some resources that could help more extensively. Essentially, I have a Discrete-Time Signal Processing Final exam coming up soon, and although I understand and am familiar with the mathematical descriptions of the ZT, DTFT and DFT, I was hoping to gain some intuition about which one to use in which scenario.
To me it seems that the DFT is a discretely sampled version of the DTFT, and the DTFT is the ZT specified on the unit circle. In this way, they are all essentially the same thing with increasing generality towards the ZT.
However, there are mathematical subtleties associated with each one (can Parseval's only be applied for DTFT and DFT?), even between the bilateral and unilateral ZT, for example with respect to the time shift property; when you shift a sample to the left in the time domain, then you must subtract the value of the ZT at n = 0 in the unilateral case, but no such subtraction (or addition) is required in the bilateral case. So why use the unilateral ZT, when it is simply the same as the bilateral ZT for causal sequences?
More generally, I was hoping for some insight into when each of these 4 transforms is most useful. For practical purposes I suppose the final transform, the DFT, is useful because computers process discrete information so it is useful to have samples that are discrete in both the time and the frequency domain, but this kind of answer is not what I am looking for (although I would appreciate more information about this as well). In my final I will be given a system in the form of perhaps a difference equation or a block diagram and asked to find the impulse response sequence, or perhaps given the impulse response sequence and asked to find a description of the system. Sometimes I will be asked to use a specific transform, but other times it is up to me to decide which is the quickest/easiest path to what I am looking for.
I understand that part of this kind of analysis is an art and the intuition comes with time, but maybe some of you know of some resources which can accelerate that process.
Thanks in advance for any responses.