# Is there a $z$-transform like for variable sampling rate signals?

I'm working with signals with variable sampling rate (the time space between samples is not constant). I know the delay between samples but I don't wont to interpolate the signal.

Is there a way to design a filter in $$z$$ domain and filter a signal with variable sampling rate without interpolation?

• when you say variable sample rate, do you mean multiple data rates within a system, sample rates that change on the fly, or something else? – Stanley Pawlukiewicz Oct 29 '17 at 22:10
• I mean that the space between samples is not constant – Andrea Oct 29 '17 at 23:18
• A continuous time State Space system doesn’t necessarily require constant sample intervals but you have to solve a set matrix differential equations at each time step, which can be more involved than interpolation – Stanley Pawlukiewicz Oct 29 '17 at 23:36
• Have you documentation about it? – Andrea Oct 30 '17 at 1:02
• If you happen to have chapter 7 of BP Lathi's , Signal, Systems, and Control, 1974, there is a good discussion. I'll look around around if I can find something more available. There is also something called the Advanced Z transform en.wikipedia.org/wiki/Advanced_Z-transform that is peripherally relevant, – Stanley Pawlukiewicz Oct 30 '17 at 1:52

Since the powers in the $$z$$-transform relate to delays or lags, in the philosophy of Laurent series, I would like to mention Puiseux series, a generalization of power series with negative and fractional exponents, or the Hahn (or Hahn–Mal'cev–Neumann) series, with even more flexibility in the exponents. However, their use in signal processing remains quite shallow to me yet.
I suspect than rational exponents could be still tractable. Indeed, they are used on multiband ($$M$$-band) filterbanks, where you can spot weirdities like $$X(z^{1/2})$$ or $$X(z^{1/M})$$, but most of them are based only on rationals with $$M^p$$ denominators. More generic powers (irrational) could be way more involved. The only mentions of Puiseux series I have found on IEEE Xlore are in the domain of control and automation (definitely not my cup of tea).
• Good source. I however wonder whether this can be called "$z$-transform like, as it seems to remains "very Fourier". As there is a literature on "fllter-design" with non-uniform sampling, I may have focused on the "$z$-transform term, for which I haven't seen DSP mention of Puiseux or Hahn – Laurent Duval Apr 25 at 19:41