Algebraic minimum-phase variant of sinc function

[Aleksey Vaneev]

Eh... Unfortunately, that 'sinc-filtered dirac spike' after resampling becomes unstable... and beside that it has zeros on the unit circle (it's a low-pass filter). so, it can't be used for deconvolution.

On the other hand, it can be converted into min-phase filter, but this will induce an additional phase shift.. and so, the final response will be definitely longer. maybe there's another way available...

[Christian Schüler]

from this equation you can't say [a0 a1 a2] = [b0 b1 b2], but you should say:
[a0 a1 a2] * H(z) = [b0 b1 b2]

You can't throw H(z) from the equation.

Read my post! The output filtered by the b coeffs equals the input filtered by the A coeffs:

Y * B = X * A

[Aleksey Vaneev]

Y * B = X * A

This equation does not make sense, sorry. (makes sense if Y=H(z) and X = 1, or vice versa. )

[Aleksey Vaneev]

Seems like I've found a way to overcome that 'sinc-filtered dirac spike' unstability. From my previous deductions, filter is unstable in IIR topology if it contains zeros on the unit cycle (zeros in the frequency response) AND discontinuities in phase response. It is obvious that the steep sinc low-pass filter has a discontinuity in its phase response at cutoff point.

So, to solve this problem (and make that 'sinc-filtered dirac spike' usable in the IIR topology) the filter should be converted to a more relaxed one: for example, it should attenuate stop-band by some given value (e.g. 100 dB), and the transition band should be as wide as possible while still providing a good bandwidth and minimum amount of aliasing after resampling.

[Christian Schüler]

This equation does not make sense, sorry.

I'd say it makes all sense in the world, since H(z) = Y/X = A/B.


Anyway, since there is only so much a biquad filter can do, resampling its coefficients and then truncating to biquad again is probably not optimal. An approximation algorithm (least squares, Chebychef, Remez) could give you a better choice of coeffs to match a filter curve.

Cheers

[MackTuesday]

This equation does not make sense, sorry.

I'd say it makes all sense in the world, since H(z) = Y/X = A/B.

I agree.

[Aleksey Vaneev]

This equation does not make sense, sorry.

I'd say it makes all sense in the world, since H(z) = Y/X = A/B.

I agree.

Of course, I agree as well. But what the sense in equating Y to A, and X to B? There's no such entity as "input" or "output" in transfer function - it is a function that should be THEN multiplied by signal... That's why I'm talking about 'no sense'.

[MackTuesday]

This equation does not make sense, sorry.

I'd say it makes all sense in the world, since H(z) = Y/X = A/B.

I agree.

Of course, I agree as well. But what the sense in equating Y to A, and X to B? There's no such entity as "input" or "output" in transfer function - it is a function that should be THEN multiplied by signal... That's why I'm talking about 'no sense'.

But Y is not being equated to A, nor is X to B. The input signal is already denoted in this equation by the variable 'X'. Y refers to the output signal. The roots of A and B are the zeros and poles of the transfer function, respectively.

[Aleksey Vaneev]

But Y is not being equated to A, nor is X to B. The input signal is already denoted in this equation by the variable 'X'. Y refers to the output signal. The roots of A and B are the zeros and poles of the transfer function, respectively.

Allright. Probably I was thinking about a different thing...

Anyway, this is the same as:

Y = X * (A / B), where A/B = H(z)

which I cannot argue, of course.

(but I do not see a reason to think in these terms, because Y in practice is a result of multiplication of X by H(z). You can't obtain Y other way).

[Christian Schüler]

For the theory, the filter does not need to be causal.

[Aleksey Vaneev]

For the theory, the filter does not need to be causal.

I'm not talking about causality. I just wanted to say that you can't put Y anywhere, and be practical. You can first obtain Y, and then manipulate it, in equations that follow... I mean, Y does not exist until you apply H(z) to X.

So, it is exactly because of IIR filter theory. Basic math operations mangling won't work here.

[Luce & Gae]

The sinc impulse, is a sine, decaying at 1/T.
By superficial comparisons of the regular sinc impulse, and the minimal phase version, a different decay rate would seem sufficient at first guess. Have you tried this?

[Aleksey Vaneev]

Luce & Gae, min-phase has a bit different response it seems - it is not just a sine, or it is sine with some given phase function (i.e. it changes non-linearly, at least initially). It's an idea anyway, thanks.