In a causal system there is a relationship between the amplitude variation and the phase delay -- you can get some phase lead with a differentiator, but only at the cost of an increasing gain vs. frequency characteristic. This of course causes _other_ problems, but you find a lot of differentiators in control loops just for this predictive capability.

Unfortunately you can't go backwards in time. All noncausal networks, i.e. those which have negative group delay in any frequency band whatever, are unstable. The reason is that when you compute the impulse response from the transfer function, you have to close the contour at

-infinity for t < 0 and +infinity for t > 0. (Modulo your choice of Fourier transform sign convention.) Getting a response before t=0 means that there have to be poles in the unstable half plane.

If I'm not mistaken, the standard all-pass filter sections, the first-order T(s)=(s - a0) / (s + a0) and the second-order T(s)=(s^2 - wr/Q s + wr^2)/(s^2 + wr/Q s + wr^2) both yield a monotonously increasing phase shift as a function of frequency.

Does an all-pass transfer function exist, which would result in a decreasing phase shift within a limited range of frequencies? This would correspond to a negative group delay, but I don't immediately see why such a phenomenon would be forbidden by the Kramers-Kronig relation (a.k.a the Bode relation, or causality). Provided that outside of the (limited) frequency range the group delay would be positive, of course.

Such an effect takes place in a notch filter, when one moves from the low-pass slope to the hi-pass slope with the associated switch from a lagging phase shift into a leading phase shift. I wonder if it would be possible to construct a filter function with (locally) similar phase behaviour, but a constant magnitude.

Yes, this is exactly the (Bode version of) the Kronig-Kramers relation. Its rule-of-thumb version says that on a log-log plot, at each frequency the phase shift is proportional to the slope of the gain, and the constant of proportionality is 90 degrees for a 20 dB/decade slope. Lagging phase shift for a downhill slope and leading for an uphill slope.

The more careful version, however, says that the phase shift is not proportional to the slope at the same frequency point, but to the weighted average of the slope at surrounding frequencies. The weighing function log{coth(u/2)) has 90% of its weight within the f/10...10*f range. So there is some room to fiddle with contributions which cancel out within this frequency range. The ordinary allpass filters succeed in generating the phase shift without any gain slope, after all.

This line of thinking makes me wonder why would it be impossible to obtain a brief region of negative dphi/df in the transfer function.

I been thinkin bout this. The first and second all-pass sections you originally described are, I think, the only ones available. (and combinations of them...)

Draw the complex plane and plot those two as pole-zero pairs (quads for the second order). Poles always on left half plane, zeros always on right half. (pole-zero pairs always symmetric about the axis, too.)

No way to get phase advance from these in any combination.

The negative sign on deltaphase/deltafrequency that's achievable with bandlimiting filters must always come from one or more zeros in the left half plane. (causal filters only -- poles in right half plane are a Bad Idea).

please, give an example of a linear circuit (or other system) with frequency response flat over the frequency range of interest, in which there are two frequencies, F0 and F1, F1>F0 and for which the phase shift of the transfer function at F1 is less than the phase shift at F0.

Sorry, I didn't get enough detail from your post to discover how to actually build this circuit.

(Note: 10 uS at 2 MHz is 628 radians -- perhaps you had a different delay TIME in mind?)

Would you kindly post one of: ..a more detailed explanation of your "all pass delay poles" ..an s-plane sketch showing pole and zero locations ..the laplace-domain polynomial describing the filter ..a schematic diagram or netlist ..a photograph of your breadboard with 'scope photos or another bit of information that would 'allow one normally skilled in the state of the art' to duplicate the design

Sounds like Mark is groping for a circuit resembling the one I considered before posting here in s.e.d.

(1) Take a bank of bandpass filters, with center frequencies at, say,

10MHz, 12MHz, 14MHz, 16MHz ... (2) Couple lengths of cable at outputs of the filters, giving delays of, say, 100ns, 80ns, 60ns, 40ns ... (3) Sum the output from the cables to get the phase shift of 360deg,
346deg, 302deg, 230deg...

The idea is that the signal at an increasing frequency would get coupled to shorter and shorter cables. If passbands of the filters are separated from each other, the phase oscillates wildly between -90deg and +90deg as alternating uphill and downhill slopes of the bandpass filters kick in. When the passbands get closer to each other, the behaviour gets nicer.

This was a clumsy way to construct such a filter, and I wondered that perhaps there is a body of theory (going beyond elementary filter books which I know of) on how to construct a minimal complexity filter given the constraints.

You have shown a number of websites where the all-pass function (pole-zero pairs symmetric about the Y-Axis, zeros in RHP, poles in RHP) is described and used. I'm choosing not to consider the Discrete-time versions.

All of these articles and all of the functions and circuits are topologically the same. All of the functions have a monotonically increasing phase as function of frequency. And there are no others, except the case of poles in RHP, zeros in LHP which I claim are not of practical use.

Mikko proposes a decreasing phase-with-frequency implementation by splitting the signal and delaying the low-frequency section. In the simplest case,

V1 = Vin * (s/s+1), V2 = (1/s+1). Delay V1 with, say, a long piece of coax. Recombine the signals with an adder. At the cutoff frequencies, there is an ugly phase shift characteristic ADDED BY THE COAX. So the two signals recombine with amplitude ripples whose number is proportional to the amount of delay.

So, says Mark. We'll throw an all-pass function into V2, to equalize the phase as needed. Indeed, this can be done. BY ADDING DELAY TO V2. The all pass function to remove the ripple in the cutoff band duplicates the function of the coax.

Net result, there is still no example given of a linear circuit (or other system)with frequency response flat over the frequency range of interest, in which there are two frequencies, F0 and F1, F1>F0 and for which the phase shift of the transfer function at F1 is less than the phase shift at F0.

Sounds to me that what you want is an inverse Chebysev, which places the amplitude ripple and phase irregularities in the stop band. Notably twichier in design and implementation though.

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