Mathematical advantages of the ZT, DTFT and DT?

Question

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.

Answer 1

My friend, i hope you can gain a proper command of those techniques for achieving an excelent result at your exam. Of course the question is broad and more discussionfull, but the purpose of this is "clarify" not to turn this into a "closary" of questions.

Indeed. there are a lot of variants for the transforms, but only a few of them have a clear direct meaning, and the others are generalizations seldom used.

At first, the standard, classical FT, is the main entry for your analysis. $$\mathbf{F}=X(f)=\int_{-\infty}^{\infty}x(t)e^{-i2 \pi ft}dt$$ $$\mathbf{F}^{-1}=x(t)=\int_{-\infty}^{\infty}X(f)e^{i2 \pi ft}df$$ That is the base, plain and simple, sweet, the FT and the IFT are the same (of course exchanging the signs and variables :) ) and with the easiest expressions $\mathbf{F}\{e^{-\pi t^2}\}=e^{-\pi f^2}$, $\mathbf{F}\{1\}=\delta(f)$, $\mathbf{F}\{\cos(\pi t)\}=\frac{1}{2}\delta(f-1)+\frac{1}{2}\delta(f+1)$, $\mathbf{F}\{\Delta(t)\}=\Pi(f)$, $\mathbf{F}\{\Pi(t)\}=\Delta(f)$, $\mathbf{F}\{|||(t)\}=|||(f)$ (where $\Delta$, $\Pi$ and $|||$ are the triangle, step and comb functions:)), and so on. You use this on the handwritting, papers, reports, analysis, etc.

We can safely ommit the DTFT and DFFT, and the tricky $i$ and $k$ symbols for discrete time and frequency and keep $t$ and $f$, discretized, with the sampling time $dt$ and the frequency periodic spacing $df$ (again with the same notation as the integral). Thus for the DTFT direct transform is: $$\mathbf{F}_{DT}=\sum_{t=-\infty}^{\infty}x(t)e^{-i2\pi ft}dt$$ and the DFFT inverse is: $$\mathbf{F}^{-1}_{DF}=\sum_{f=-\infty}^{\infty}X(f)e^{i2\pi ft}df$$

because the $|||(t)$ (and $|||(f)$) comb function: $$|||(t)=\sum_{i=-\infty}^{\infty}\delta(t-i)$$

is all what we need for recovering the related expressions and cumbersome hard to learn theorems regarding sampling (DT) and perioditizicing (DF). And you recover all your symmetries just by multiplying by your combs:

$$\mathbf{F}_{DT}(x(t))=\mathbf{F}(x(t)\frac{1}{dt}|||(\frac{t}{dt}))=X(f)\frac{1}{dt}\frac{1}{F}|||(\frac{f}{F}), F=\frac{1}{dt}$$

Hence, at my second place, you have your well forgotten ZT, unilateral, again considering the $t$ as a discretized variable under a sampling time $dt$: $$X(z)=\sum_{t=0}^\infty x(t)z^{-t}$$

You have a DAQ with a lot of sensors and controllers, and you have some fixed sampling time and you need to know the continuous parameters of the system based on the discrete behaviour, or tuning some controllers, or identifying a system. Soon or later you need to pass onto the Z domain and check if your IIR filters are stable or nor, or if your FIR smoothers are too short on delay, and comparing your ARMA models with your differential equations.