the author should have clarified this, you are correct, as the author also makes clear he is writing for a non-technical audience.
He probably forgot to mention it because the concept is so fundamental to signal processing that he assumed it was common knowledge.
Also see page 4 here: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
edit: I looked at the first few pages of the paper but I feel none the wiser, at all.
edit2: Ah... "the poles and zeros of a transfer function may be complex, and the system dynamics may be represented graphically by plotting their locations on the complex s-plane". The transfer function (whatever that is) is a rational function of the complex variable s, i.e. (in my words) it's a fraction with complex polynomials for numerator and denominator. The zeros are the roots of the numerator and the poles are the roots of the denominator.
Ok, I still don't know what the transfer function is or means or comes from, but am much less in the dark, thank you! :-)
Some things in life leave a lasting impression[1]. :eye_roll:
It sounds like what you're looking for is an explanation of root locus analysis[2].
In the simplest control case, a transfer function is nothing more than the expression of a continuous closed-loop LTI system's output Y(s) over its input X(s) in the Laplace domain, conveniently abstracted as its forward path G(s) and negative feedback path H(s).
From there, Routh-Hurwitz method[3] can be used to determine stability of the system.
...and I will continue to use j, thanks.
[1] https://youtu.be/1rqJl7Rs6ps?t=1828
[2] https://en.wikipedia.org/wiki/Root_locus
[3] https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stabilit...
edit: Took a lil while to work out that LTI system is Linear time-invariant system.
I'd like to chime in with a more intuition-based explanation of what transfer functions are, from my recollections of college control theory classes in both electrical signals and a more general "systems engineering" application:
Basically, the transfer function is a different perspective on modelling/representing a system's output as a function of its input. Classically, when modelling and/or reasoning about a system in physics, the perspective we adopt is that of "input" being the forward advance of time (and sometimes initial conditions) and "output" being the amplitude of the physical quantity(ies) or dimension(s) of the system that interest(s) us. The transfer function, then, is when we switch perspectives to consider the "input" to be a sinusoidal signal (characterized by amplitude and phase over time), and the "output" is the new amplitude and phase of that signal [after "traversing" the system]. Of course, you're actually working with a closed-loop, but most input/output systems can be modeled as a closed-loop if you sufficiently broaden the system's boundaries.
This turns out to be useful for/in several reasons/contexts:
- many physical phenomena are sine waves (or, thanks to Fourier, a sum of sometimes many different sine waves), and often times a system's purpose (to us humans) is to control such a phenomena precisely along the lines of "do this to the amplitude, and/or adjust the phase like so" - dampening, feedback loops, more sophisticated processes like hysteresis, maintaining a steady state given incoming perturbations, etc. In these cases the transfer function ends up being the mathematical expression of that system's function in the "domain language" of that problem, so to speak.
- It turns out that often, when working with systems whose "classical" representation involve components like exponentials or sine and cosine of time (which are "just" complex exponentials of those quantities), the corresponding transfer functions are "simple" fractions of polynomials. More precisely, passing into the Langrange domain allows transforming a differential equation problem into a complex polynomial fractions problem - often much easier to crunch/solve. Furthermore, in the Lagrange domain, de-phasing a signal by pi/2 is equivalent to simply adding 1/(j * signal's frequency) to that signal (if I recall correctly). This makes much of the math more accessible to human intuition, and especially on more complex systems that have several "moving parts" the linear quality of polynomials becomes invaluable.
Personally, I remember quickly adopting, once I'd grokked it, the transfer function perspective when trying to reason about the effect of introducing a capacitor into an existing circuit - analog or DC[0] - as well as things like how the material properties of a door contribute to its behavior as a low-pass filter on sound waves. Sitting down and doing the math, the formulas that I would arrive at spoke much more clearly to me. Also, you are sort of adopting a "time-agnostic" (or perhaps time-invariant) perspective, where the system itself does not change over time. Instead, its' input is characterized by how it behaves over time, and the transfer function (especially when plotted) gives you a clear, direct sense of what the output's "behavior over time" will accordingly be. Notably, it's here that the zeroes of the OP become so meaningful.
[0]: part of what initially started making things "tick" for me was when a professor explained that an impulse on an input signal (i.e. a quasi-instant variation, then back to the preceding "steady state" value of it - i.e. a DC current "turning on"), to a transfer function, "looks like" a sine wave signal with a constant amplitude but monotonously increasing phase offset - again I forget if the rate is constant, polynomial, exponential or what.
edit: By "Lagrange" did you possibly mean to write "Laplace"? I confuse those two gentlemen too. p.s. I just learnt Lagrange was Italian! born Giuseppe Luigi Lagrangia.