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Im reading Will Pirkles Designing audio effect plugins book and I'm not sure if I'm understanding the z transform correctly. I got up to differential equations in college, but haven't done math in over half a decade, so hopefully learning DSP isn't way, way over my head.

As I'm understanding the books explanation: the z transform changes formulas to be in terms of z, the complex sinusoid. z, when mapped from inputs of 0-pi, will output frequencies from 0hz to Nyquist. Assuming this is correct, then:

  1. Will any one of these different complex frequencies (from 0hz to Nyquist) have a different phase from any other frequency within that range?

  2. Is a z itself a function, with a specific z value basically just being a specific one of these complex frequencies (again, each with unique phase)?

  3. I assume that since it's possible to create an audio oscillator at a certain frequency and adjust it's phase however I want, that the "unique phase" thing I'm assuming is correct with the complex sinusoid doesn't limit the phase of real output, does it? (i.e. if there's an audio application where the complex sinusoid is outputting a complex frequency with 11khz with 90degree phase, it doesn't mean that an audio frequency 11khz tone coming from my speakers is required to have a specific phase, right?)

Hopefully any of that made any sense. I'm sure any correction of any part of what I wrote will be helpful. If it sounds like I have absolutely no clue what I'm talking about please ELI5. Thank you kindly to anyone that replies.

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  • $\begingroup$ "z" represents any complex number. Are you familiar with continuous time concepts including the Fourier Transform and the Laplace Transform, and how those two are related? This post may be helpful to you: dsp.stackexchange.com/questions/31830/… $\endgroup$
    – Dan Boschen
    Commented Jul 7, 2021 at 3:24
  • $\begingroup$ I should probably write "complex frequencies". The book says z transform changes things from time domain to complex frequency domain. And different values put into of z from 0 to pi will output different complex frequencies from 0hz to Nyquist. Assuming I'm understanding correctly. $\endgroup$
    – user3394004
    Commented Jul 7, 2021 at 3:36
  • $\begingroup$ Not completely. But yes the z-plane shows the correlation to the complete universe of complex frequencies that can grow or decay with time. The unit circle in the z-plane is the frequency axis you refer to-- those are all the values of z that result in phasors rotating with a constant magnitude (not decaying or growing with time). Two such phasors spinning in equal and opposite direction will result in a sinusoid along the real axis. Read my links as that should make it clearer; I really don't think there is a one paragraph answer here. $\endgroup$
    – Dan Boschen
    Commented Jul 7, 2021 at 3:39
  • $\begingroup$ Every complex sample has phase. Every time domain sample if complex has magnitude and phase, and every frequency domain sample if complex has magnitude and phase. Don't confuse phase with time delay: A fixed time delay causes a linear phase with frequency (meaning low frequencies will have a smaller added phase while higher frequencies will have a much larger added phase) I hope this is helpful- we won't be able to have the longer discussion required here as they discourage long drawn out chats in the comments, so I won't be able to detail more here, but please do read the references I gave. $\endgroup$
    – Dan Boschen
    Commented Jul 7, 2021 at 3:43

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the z transform changes formulas to be in terms of z, the complex sinusoid.

That's a narrow, and not really mathematically correct, presentation.

The z transform takes a signal in the time domain and transforms it into the z domain. It's better to think of the z domain as a frequency domain. When you work in this domain, $z$ does not represent a complex sinusoid, any more than $t$ does in the time domain. $z$ is just the frequency of a complex sinusoid (perhaps one that is decaying or growing, if you're using values of $z$ not on the unit circle).

z, when mapped from inputs of 0 to $\pi$, will output frequencies from 0hz to Nyquist.

$z = e^{j \theta}$, for $0 \le \theta \le \pi$, are the frequencies of undamped sinusoids from 0Hz to Nyquist.

Will any one of these different complex frequencies (from 0hz to Nyquist) have a different phase from any other frequency within that range?

That's kind of a meaningless question as you've stated it. A signal can have a phase, a system can induce a phase shift -- but frequencies don't have phases. A frequency is a measure of how fast phase advances.

Is a $z$ itself a function, with a specific $z$ value basically just being a specific one of these complex frequencies (again, each with unique phase)?

$z$ is a free variable, just as time is a free variable in the time domain.

I assume that since it's possible to create an audio oscillator at a certain frequency and adjust it's phase however I want, that the "unique phase" thing I'm assuming is correct with the complex sinusoid doesn't limit the phase of real output, does it? (i.e. if there's an audio application where the complex sinusoid is outputting a complex frequency with 11khz with 90degree phase, it doesn't mean that an audio frequency 11khz tone coming from my speakers is required to have a specific phase, right?)

Not sure what you're getting at. If you've built yourself a numerical oscillator with a complex output (or, more correctly a quadrature output), then you can arbitrarily choose the phase shift relative to that oscillator output of the sinusoid you want coming out of your speakers.

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    $\begingroup$ Having answered your specific questions -- in my opinion, you have a book that's giving you a technician's gloss-over of DSP, but you've got a mind that probably needs the full treatment. You may want to find those brain cells that learned differential equations five years ago, kick them out of bed and tell them to wake up. Then get a real DSP book and work through it. Khan Academy lists a Signals and Systems course; I know that MIT puts a number of courses up. Working through such a course will be work, but I suspect you'll get the level of understanding you seem to need. $\endgroup$
    – TimWescott
    Commented Jul 7, 2021 at 4:53
  • $\begingroup$ Okay. I'll review diffy q and then pick up a different course or book. That sounds like it might be necessary. $\endgroup$
    – user3394004
    Commented Jul 7, 2021 at 6:20
  • $\begingroup$ +1 Tim. For the comment. $\endgroup$
    – robert bristow-johnson
    Commented Jul 9, 2021 at 2:49
  • $\begingroup$ As this can be a point of confusion - Depending on what we call “frequencies” for example if we refer to the frequency bins in the DFT as the frequencies of a waveform (as is common to do hence my clarification) they do have phase (especially when OP refers to “these complex frequencies”) - to be more precise, phase is a rotation on the complex plane and not a delay. So the phase of “these frequencies” is the starting angle from which the time domain of “that frequency” as a complex phasor would then rotate. So each frequency component does have a magnitude and phase. $\endgroup$
    – Dan Boschen
    Commented Jul 10, 2021 at 12:14
  • $\begingroup$ Okay, so I've reviewed some math stuff and I can implement the z transform, the mathematical steps are easy, but still don't have a clear intuition for what's actually happening. It seems like your explanation in places is just restating what I wrote in a less ELI5 way. $\endgroup$
    – user3394004
    Commented Jul 13, 2021 at 0:30

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