Allpass filter & (locally) negative delay

[Mikko Kiviranta]

Dear colleagues,

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.

Regards,
Mikko

 

[Phil Hobbs]

Dear colleagues,

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

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.

Cheers,

Phil Hobbs

Cheers,

Phil Hobbs

 

[Tim Wescott]

Dear colleagues,

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.

Regards,
Mikko

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.

 

[John Woodgate]

I read in sci.electronics.design that Tim Wescott <[email protected]>

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.

Use AGC to combat the change in gain.

 

[Mikko Kiviranta]

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

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.

Regards,
Mikko

 

[Mikko Kiviranta]

Unfortunately you can't go backwards in time. Agreed.

All noncausal networks, i.e. those which have negative group
delay in any frequency band whatever, are unstable.

This 'any frequency band' detail sounds surprising and hence
interesting! Do you have a reference on this?

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.

I agree that getting a finite impulse response at t<0 indeed
implies poles in the right half-plane. But I don't see the connection
between the negative group delay (within a restricted frequency
range) and a non-causal impulse response. I.e. why would Fourier
transform of a complex function F(w) lead to a f(t)>0 for t<0,
just because arg(F(w)) has the property d/dw arg(F(w)) < 0 in
some w-region. It is possible that this is the case, I just don't
see it offhand.

Now it stikes me that perhaps this is related with the
faster-than-light discussion which occurred in s.p.r. some
time ago ... let me dig it out ... yes, it was this popularized article:
http://archives.cnn.com/2000/TECH/space/07/20/speed.of.light.ap/index.html
which started a fuss, briefly clarified here:
http://www.astro.ucla.edu/~wright/anomalous-dispersion.html

Ok, have to read it.

Bye for now,
Mikko

 

[PN2222A]

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).

Regards
PN2222A
"Of course I'm biased!"

 

[Mark]

Dear colleagues,

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.

Regards,
Mikko

Yes I think you can...

but you don't do it by reducing delay...you do it by increasing delay
using an all pass network...

you increase the delay for a range of frequencies, then (relativley
speaking) the other frequencies are advanced.

All pass group delay equalizers work by INCREASING the delay at those
frequencies where there is insufficent delay.

Got it? You can only add delay but you can choose the frequencies
where you do it so the result is flat or whatever shape you want...

Mark


Mark

 

[PN2222A]

Yes I think you can...

but you don't do it by reducing delay...you do it by increasing delay
using an all pass network...

you increase the delay for a range of frequencies, then (relativley
speaking) the other frequencies are advanced.

Mark

OK Mark

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.

You may freely choose the frequencies.

Regards
PN2222A
NPN (Is = 14.34f Xti = 3 Eg = 1.11 Vaf = 74.03 Bf = 255.9 Ne = 1.307 Ise =
14.34 Ikf = .2847 Xtb = 1.5 Br = 6.092 Isc = 0

Ikr = 0 Rc = 1 Cjc = 7.306p Mjc = .3416 Vjc = .75 Fc = .5 Cje = 22.01p Mje =
..377 Vje = .75 Tr = 46.91n Tf = 411.1p Itf = .6

Vtf = 1.7 Xtf = 3 Rb = 10)

 

[Mark]

OK Mark

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.

You may freely choose the frequencies.

Regards
PN2222A

A network with a couple of all pass delay poles tuned to 1 MHz such
that the total group delay is 21 uS at 1 MHz and 10 us at 2 MHz.


Mark

 

[PN2222A]

please, give an example of a linear circuit (or other system)

A network with a couple of all pass delay poles tuned to 1 MHz such
that the total group delay is 21 uS at 1 MHz and 10 us at 2 MHz.


Mark

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

Regards
PN2222A

 

[Jim Thompson]

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

(s^2 - a*s + b)/(s^2 + a*s + b)

Left half-plane complex pole mirrored by right half-plane zero ;-)

..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

Regards
PN2222A

...Jim Thompson

 

[Kiviranta, Mikko]

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

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.

Regards,
Mikko

 

[PN2222A]

(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.

Problem: You can't match the phases of the filters at the crossover points.
Thus there will be plenty of amplitude ripple
at 11, 13 and 15 MHz.

Phase and amplitude response are tightly coupled. TANSTAAFL.

Regards
PN2222A

 

[Mark]

Problem: You can't match the phases of the filters at the crossover points.
Thus there will be plenty of amplitude ripple
at 11, 13 and 15 MHz.

Phase and amplitude response are tightly coupled. TANSTAAFL.

Regards
PN2222A

read up on all pass filters and group delay equalizers, they work by
creating delay where needed

here are a few links

http://www.filter-solutions.com/delay.html

http://www.elektrorevue.cz/clanky/2004/0019/

http://www.edn.com/archives/1995/101295/21di4.htm

It's an interesting question, but there is an answer to your problem.


Mark

 

[PN2222A]

Mark replied to my assertion as follows:

read up on all pass filters and group delay equalizers, they work by
creating delay where needed

here are a few links

http://www.filter-solutions.com/delay.html

http://www.elektrorevue.cz/clanky/2004/0019/

http://www.edn.com/archives/1995/101295/21di4.htm

It's an interesting question, but there is an answer to your problem.

Mark

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.

TANSTAAFL.

PN2222A
Now lead free!

 

[Mikko Kiviranta]

Problem: You can't match the phases of the filters at the crossover points.
Thus there will be plenty of amplitude ripple
at 11, 13 and 15 MHz.

This is indeed what happens. I should calculate what the amount of
amplitude ripple would be in a (perhaps Chebysev-like) optimized filter
where, within the limited active region of frequencies, the d phi / dw
would also oscillates always remains below zero.

Phase and amplitude response are tightly coupled. TANSTAAFL.

They indeed are - Kronig Kramers again. But its not always obvious
in which manner they are coupled. I tend to agree with your
pole-zero argument, however. I.e. there is no filter with a strictly
flat magnitude response and a (locally) negative dphi / dw. Thanks
for sharing your insight.

Regards,
Mikko

 

[JosephKK]

This is indeed what happens. I should calculate what the amount of
amplitude ripple would be in a (perhaps Chebysev-like) optimized filter
where, within the limited active region of frequencies, the d phi / dw
would also oscillates always remains below zero.


They indeed are - Kronig Kramers again. But its not always obvious
in which manner they are coupled. I tend to agree with your
pole-zero argument, however. I.e. there is no filter with a strictly
flat magnitude response and a (locally) negative dphi / dw. Thanks
for sharing your insight.

Regards,
Mikko

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.