Here's my thoughts (maybe there's some additional insights) and sorry if there's overlap as we were apparently writing these at the same time.
Kraku wrote:1. If the filter is designed in the S-domain, it will eventually need to be transferred into Z-domain anyway? (BLT was mentioned) If this is the case, I might equally well place the poles/zeros in Z-domain to begin with.
Only if you want to implement it as a digital effect. You can always build the analog filter too.
As far as placing poles/zeroes goes, see the comments (eg. mine below) about analog s-plane being much easier geometry to work in.
2. Quickly browsing Vadim's book based on his comments, am I right assuming that the following two trapezoidal integrators can actually be used equally well for ZDF purposes with equivalent (or even identical) results? The integrators in questions can be seen in figures 3.9 and 3.12. There seems to be some disagreement about it in this thread.
I assume you mean 3.11 since 3.12 is a complete one-pole low-pass.
The forms shown in 3.9 and 3.10 are equivalent. However, the substitution of the "embedded cutoff gain" wc*T/2 in place of 1/2 that is done from 3.10 to 3.11 should not be done in the same way (as a simple substitution) for 3.9, otherwise one gets the problems we've been discussing (however, there is no such "unsafe" substitution pictured in the book). For 3.9, one should first move the 1/2 multiplier from the output to the input (ie. same side as 3.10) at which point the substitution should be fine again (or one can add wc*T in front and keep the 1/2 afterwards; the point is you don't want changing coefficients at the output).
3. Regardless of how I come up with the transfer function, in the end I always need to choose one topology / filter structure which I'll use to implement the filter. Just like with traditional non-ZDF filters which you design for Z-plane? So there's no "you just magically integrate the signal into discrete samples".
There is no verified scientific evidence of any magic.
Pretty much the only alternative to choosing an implementation structure is to sample the impulse response (from the transfer function) directly until it decays into a noise and then use the resulting IR for FIR filtering like any other... but generally speaking it's easier to just pick some implementation structure. For 2nd order sections (or filters that you can reasonably decompose into 2nd order sections) it's usually safe to choose SVF (the ZDF transformed analog SVF, not the classic digital Chamberlin thing) unless you have some specific reason to use something else.
edit: I'd also like to add that your favourite math package can help you with numerical root-finding to decompose arbitrary filters into 2nd order sections... although I'd recommend working with something more straight-forward first
4. S-plane and Z-plane filter structures are identical, differing only by the integrator/delay block? This is based on the comments that BLT would need to be used for integrators.
On the continuous time S-plane (based on integrators 1/s) the filter response can be found on the imaginary axis, while on the discrete time Z-plane (based on unit-delays 1/z or z^-1 as it's usually written) it's found on the unit circle. This differences make the actual coefficients behave in very different ways (with the "linear" S-plane being much more straight-forward than the trigonometric Z-plane) and since the filter structures are based on different constructs (integrators vs. delays) they behave quite differently as well.
That said, when doing TPT style transforms, what we do is approximate the analog integrator with a "complete" digital filter (ie. the BLT integrator) that is only "internally" based on delays, which then allows using the continuous-time (analog) structures.
5. The actual integrator implementation can be completely ignored when deriving the implicit functions so that the result isn't used in the right hand side of the formula? So basically the resulting function/formula is completely separate from the integrator block implementation? The integrator blocks could have a built in Pacman game in them and it wouldn't change the filter function/formula at all?
I'm not sure if I really understand the question, but the idea is that you can build an "analog" design in continuous time (without worrying much about digital) and then "drop in" the "digital wannabe integrate" to arrive at the final digital filter (at least in the abstract sense; you then need to solve the resulting equations so you can implement them).
6. Any existing filter can be made into ZDF if it uses TDF2 or DF1?
Not really. Any existing filter (in the transfer function sense, analog or digital) can be implemented with a ZDF-style filter (at least in principle), but in the implementation structure sense you can build a ZDF filter out of any analog (well, continuous time) filter structure. An existing digital structure doesn't really do anything, you essentially have to start from scratch (although some of them are similar to related analog structures that one can transform).