```robert bristow-johnson wrote:
> in article 1151077967_12507@sp6iad.superfeed.net, jim at
> "sjedgingN0sp"@m@mwt.net wrote on 06/23/2006 12:00:
>
>> Andor wrote:
>>
>>> Any other phase shift than +/- 90&#2013266096; however does include DC.It's just
>>> that a 90&#2013266096; phase shifted DC happens to be zero.
>> I haven't been following this discussion but there must be something
>> mis-typed in the above 2 sentences. What does this DC with phase of zero
>> look like? And if all DC has a phase of zero when shifted 90&#2013266096; then
>> doesn't that mean all DC has a phase of +/-90&#2013266096; to begin with. So it
>> seems that the only phase shift allowed for DC is 180&#2013266096;, Right? But then
>> what does the first sentence "Any other phase shift than +/- 90&#2013266096; however
>> does include DC" mean?
>
> even though i took a cursory look at the thread, i'm still clueless.  what
> is the issue debated in the simplest physical, mathematical, and/or
> philosophical language?  (is it the physical reality of complex signals or
> quantities?)  except for the simple thing they tell us in school about phase
> and the meaning/consequences about phase, i don't understand another
> definition of it.  if it's something like that, can someone boil it down for
> this bown-hed enjenear?

I wish I could. Friends who shall remain nameless (unless they choose to
claim credit themselves) have been trying to tell me that phase shifting
the power source of of a flashlight is mathematically meaningful. :-)

I just wanted to know if the 180-degree phase shift supposedly produced
by an inverter is lead or lag. I expected the question to provoke
discussion -- call it a troll if you like -- but not this much!

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```in article 1151077967_12507@sp6iad.superfeed.net, jim at
"sjedgingN0sp"@m@mwt.net wrote on 06/23/2006 12:00:

> Andor wrote:
>
>>
>> Any other phase shift than +/- 90&#2013266096; however does include DC.It's just
>> that a 90&#2013266096; phase shifted DC happens to be zero.
>
> I haven't been following this discussion but there must be something
> mis-typed in the above 2 sentences. What does this DC with phase of zero
> look like? And if all DC has a phase of zero when shifted 90&#2013266096; then
> doesn't that mean all DC has a phase of +/-90&#2013266096; to begin with. So it
> seems that the only phase shift allowed for DC is 180&#2013266096;, Right? But then
> what does the first sentence "Any other phase shift than +/- 90&#2013266096; however
> does include DC" mean?

even though i took a cursory look at the thread, i'm still clueless.  what
is the issue debated in the simplest physical, mathematical, and/or
philosophical language?  (is it the physical reality of complex signals or
quantities?)  except for the simple thing they tell us in school about phase
and the meaning/consequences about phase, i don't understand another
definition of it.  if it's something like that, can someone boil it down for
this bown-hed enjenear?

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```Andor wrote:

> Jani Huhtanen wrote:
>> Andor wrote:
>>
>> > Ray Andraka wrote:
>> >> Jerry Avins wrote:
>> >>
>> >> >
>> >> > Show me two simultaneous DC signals which are out of phase one with
>> >> > the other, and I'll accept the math.
>> >> >
>> >> > Jerry
>> >>
>> >>
>> >> Well, Jerry, if they are complex DC signals (.707 + .707j  and  0-1j,
>> >> for example) you've got two DC signals that are out of phase.  It has
>> >> to
>> >>   be complex though in order to be sensible.
>> >
>> > I see you belong to Wilson's school of phase shifting (ie. multiplying
>> > with a complex unit length number). Note that Andor's / Oli's school of
>> > phase shifting says that 1 and 1/sqrt(2) are two out-of-phase DC
>> > signals (the second is shifted by 45&#2013266096; w.r.t. to the first), while
>> > Jerry's school of phase shift cannot shift DC signals. If we keep
>> > going, I'm sure more schools pop out of Usenet like mushrooms out of
>> > dung heeps.
>>
>> Your school, Andor, has some serious issues with ambiguity between
>> amplitude and phase. You have to know the amplitude in order to measure
>> the phase and vice versa.
>>
>> Wilson's school, however, does not share this ambiguity. At least, as
>> long as amplitude A is defined in (0,+inf) and for real DC there is only
>> two phases 0 and 180. In his scool, Hilbert transform for DC is well
>> defined and it causes a 90' phase shift:
>>
>>         A*cos(p0) + i*A*sin(p0) = A*cos(p0)
>
> Not quite sure what you mean with that. This equation holds for all p0
> = k pi (where sin(p0) = 0), where k is an integer, and in those cases
> it reduces to the triviality A = A.
>

I meant that if p0 can only be 0 or pi (or any multiple of pi if you like),
then the result of a HT of a real DC can be interpreted to be shifted by 90
degrees (this is similar to your school). However, this interpretation is
only valid when the signal is viewed as a part of complex valued signal,
thus the representation as analytical signal (which contains the same
information as the original, but is merely redundant).

In "my" school (I don't know if this is Wilson's school anymore ;)), a phase
shift of a real signal is "just" a projection of the "true" complex valued
phase shifted signal:

x_p(t) = R{ xc(t) * exp(i*p) } = R{ (x(t) + i*HT{ x(t) }) * exp(i*p) }  (1)

After the projection it is impossible to determine the value of phi for the
DC... unless you have side information (such as the amplitude). In
projection one loses some sample wise information (namely amplitude and
phase). In case of real signals (excluding DC and Nyquist), this
information is stored in the time structure and can be measured against
some reference sinusoid. For complex signals the same information can be

However, equally x(t)*exp(i*p) is a phase shift, just not the same one
explained above. Inversion can be achieved with both when p = pi;

>>
>> where the imaginary part is the HT, real part is the original signal, and
>> p0 is phase in {0,pi}.
>
> If you are under the impression that you can get a Hilbert transform
> from the imaginary part of Wilson's 90&#2013266096; phase shift, then you are
> wrong. Wilson's 90&#2013266096; shift just puts an "i" in front the signal, ie.
> s(t) -> i s(t).

I'm not under that impression.

> "My" school of p radian phase shift has the following impulse response:
>
> h_p(n) = cos(p + n pi /2) sin(n pi /2) / (n pi / 2),   n = .., -2, -1,
> 0, 1, 2, ...
>
> (for n = 0 take the obvious h_p(0) = cos(p)), which reduces to the well
> known impulse response of the Hilbert transformer for p = - pi/2. Due
> to the antisymmetry of the impulse response about 0, the response to DC
> is simply cos(p) (which is also the response at Nyquist, by the way).

Yes, I think (1) does pretty much the same thing, unless I made a mistake.
However, this does not solve the ambiguity of the phase and amplitude in
your school. Please tell me how you determine from a two arbitrary, but
positive, constant signals (i.e., pure DC) the phase difference and
amplitude?

My answer would be: They have 0' phase difference and the amplitudes are
just the values of the constants.

Whether or not this was true for the complex valued signal from which the
real valued signal has been projected is irrelevant for the information
can't be recovered. Note that when I speak of projecting a real signal from
the "true" complex valued signal, it's only a mind game. The complex valued
signal may not have ever existed but it could have. You can view the answer
as least squares approximation if you will ;)

If the other signal would have been negative, I would have answered that the
phase diff is 180'.

--
Jani Huhtanen
Tampere University of Technology, Pori
```
```Andor wrote:

> Jani Huhtanen wrote:
>> Andor wrote:
>>
>> > Ray Andraka wrote:
>> >> Jerry Avins wrote:
>> >>
>> >> >
>> >> > Show me two simultaneous DC signals which are out of phase one with
>> >> > the other, and I'll accept the math.
>> >> >
>> >> > Jerry
>> >>
>> >>
>> >> Well, Jerry, if they are complex DC signals (.707 + .707j  and  0-1j,
>> >> for example) you've got two DC signals that are out of phase.  It has
>> >> to
>> >>   be complex though in order to be sensible.
>> >
>> > I see you belong to Wilson's school of phase shifting (ie. multiplying
>> > with a complex unit length number). Note that Andor's / Oli's school of
>> > phase shifting says that 1 and 1/sqrt(2) are two out-of-phase DC
>> > signals (the second is shifted by 45&#2013266096; w.r.t. to the first), while
>> > Jerry's school of phase shift cannot shift DC signals. If we keep
>> > going, I'm sure more schools pop out of Usenet like mushrooms out of
>> > dung heeps.
>>
>> Your school, Andor, has some serious issues with ambiguity between
>> amplitude and phase. You have to know the amplitude in order to measure
>> the phase and vice versa.
>>
>> Wilson's school, however, does not share this ambiguity. At least, as
>> long as amplitude A is defined in (0,+inf) and for real DC there is only
>> two phases 0 and 180. In his scool, Hilbert transform for DC is well
>> defined and it causes a 90' phase shift:
>>
>>         A*cos(p0) + i*A*sin(p0) = A*cos(p0)
>
> Not quite sure what you mean with that. This equation holds for all p0
> = k pi (where sin(p0) = 0), where k is an integer, and in those cases
> it reduces to the triviality A = A.
>

I meant that if p0 can only be 0 or pi (or any multiple of pi if you like),
then the result of a HT of a real DC can be interpreted to be shifted by 90
degrees (this is similar to your school). However, this interpretation is
only valid when the signal is viewed as a part of complex valued signal,
thus the representation as analytical signal (which contains the same
information as the original, but is merely redundant).

In "my" school (I don't know if this is Wilson's school anymore ;)), a phase
shift of a real signal is "just" a projection of the "true" complex valued
phase shifted signal:

x_p(t) = R{ xc(t) * exp(i*p) } = R{ (x(t) + i*HT{ x(t) }) * exp(i*p) }  (1)

After the projection it is impossible to determine the value of phi for the
DC... unless you have side information (such as the amplitude). In
projection one loses some sample wise information (namely amplitude and
phase). In case of real signals (excluding DC and Nyquist), this
information is stored in the time structure and can be measured against
some reference sinusoid. For complex signals the same information can be

However, equally x(t)*exp(i*p) is a phase shift, just not the same one
explained above. Inversion can be achieved with both when p = pi;

>>
>> where the imaginary part is the HT, real part is the original signal, and
>> p0 is phase in {0,pi}.
>
> If you are under the impression that you can get a Hilbert transform
> from the imaginary part of Wilson's 90&#2013266096; phase shift, then you are
> wrong. Wilson's 90&#2013266096; shift just puts an "i" in front the signal, ie.
> s(t) -> i s(t).

I'm not under that impression.

> "My" school of p radian phase shift has the following impulse response:
>
> h_p(n) = cos(p + n pi /2) sin(n pi /2) / (n pi / 2),   n = .., -2, -1,
> 0, 1, 2, ...
>
> (for n = 0 take the obvious h_p(0) = cos(p)), which reduces to the well
> known impulse response of the Hilbert transformer for p = - pi/2. Due
> to the antisymmetry of the impulse response about 0, the response to DC
> is simply cos(p) (which is also the response at Nyquist, by the way).

Yes, I think (1) does pretty much the same thing, unless I made a mistake.
However, this does not solve the ambiguity of the phase and amplitude in
your school. Please tell me how you determine from a two arbitrary, but
positive, constant signals (i.e., pure DC) the phase difference and
amplitude?

My answer would be: They have 0' phase difference and the amplitudes are
just the values of the constants.

Whether or not this was true for the complex valued signal from which the
real valued signal has been projected is irrelevant for the information
can't be recovered. Note that when I speak of projecting a real signal from
the "true" complex valued signal, it's only a mind game. The complex valued
signal may not have ever existed but it could have. You can view the answer
as least squares approximation if you will ;)

If the other signal would have been negative, I would have answered that the
phase diff is 180'.

--
Jani Huhtanen
Tampere University of Technology, Pori
```
```Ray Andraka wrote:
> Jerry Avins wrote:
>
> >
> > Show me two simultaneous DC signals which are out of phase one with the
> > other, and I'll accept the math.
> >
> > Jerry
>
>
> Well, Jerry, if they are complex DC signals (.707 + .707j  and  0-1j,
> for example) you've got two DC signals that are out of phase.  It has to
>   be complex though in order to be sensible.

I see you belong to Wilson's school of phase shifting (ie. multiplying
with a complex unit length number). Note that Andor's / Oli's school of
phase shifting says that 1 and 1/sqrt(2) are two out-of-phase DC
signals (the second is shifted by 45=B0 w.r.t. to the first), while
Jerry's school of phase shift cannot shift DC signals. If we keep
going, I'm sure more schools pop out of Usenet like mushrooms out of
dung heeps.

```
```Andor wrote:
> Jerry Avins wrote:
>
> ...
>>>>> Such a definition can be via specification of the response for one
>>>>> single sinusoid, and requiring linearity and time-invariance. As soon
>>>>> as you do that, the response at DC and Nyquist follows, and we can stop
>>>>> quibbering and just look at the facts.
>> You find a reasonable mathematical way to express an intuitively simple
>> concept in a restricted "region of applicability" and declare without
>> justification that it is valid everywhere. How is that "fact"?
>
> I was going to wait with the justification until you laid the basics
> for a discussion:  a simple definition of the phase shift. The response
> of the phase shift at DC can be derived from this definition - that's
> fact.
>
> How do you seriously want to define a system by
>
> "If you prefer, Phase shift -- or difference -- is the relative phase
> of two signals at the same frequency"
>
> ?
>
> That's a tautology, not a definition. Until you specify what phase
> shift is, it is pointless to discuss its properties.

You snipped another formulation: "That fraction of a cycle, however
expressed, that one waveform needs to be shifted along its time axis in
order to coincide with another." There is an unstated assumption that
either amplitude is not germane, or it has been adjusted. The signals
must, of course, have the same repetition rate.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Andor wrote:
> Jerry Avins wrote:
>
> ...
>>>>> Such a definition can be via specification of the response for one
>>>>> single sinusoid, and requiring linearity and time-invariance. As soon
>>>>> as you do that, the response at DC and Nyquist follows, and we can stop
>>>>> quibbering and just look at the facts.
>> You find a reasonable mathematical way to express an intuitively simple
>> concept in a restricted "region of applicability" and declare without
>> justification that it is valid everywhere. How is that "fact"?
>
> I was going to wait with the justification until you laid the basics
> for a discussion:  a simple definition of the phase shift. The response
> of the phase shift at DC can be derived from this definition - that's
> fact.
>
> How do you seriously want to define a system by
>
> "If you prefer, Phase shift -- or difference -- is the relative phase
> of two signals at the same frequency"
>
> ?
>
> That's a tautology, not a definition. Until you specify what phase
> shift is, it is pointless to discuss its properties.

You snipped another formulation: "That fraction of a cycle, however
expressed, that one waveform needs to be shifted along its time axis in
order to coincide with another." There is an unstated assumption that
either amplitude is not germane, or it has been adjusted. The signals
must, of course, have the same repetition rate.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Andor wrote:
> Jerry Avins wrote:

...

>> Phase shift -- or difference -- is the
>> relative phase of two signals.
>
> This is not a definition, Jerry.

between signals of the same frequency. If you prefer, Phase shift -- or
difference -- is the relative phase of two signals at the same
frequency. You can generalize this to two signals of the same frequency
and shape. Phase is readily understood with square, sawtooth, and
triangle waves. (The Hilbert transformation of a square wave is not square.)

> I (and others) defined the "phase shift of d degrees" as a LTI system,
> and specified its frequency response (I'll also accept an impulse
> response as a proper definition). You don't need, and in fact can't use
> "the relative phase of two signals" (what is relative phase of two
> arbitrary signals?)  to define phase shift. If you feel that the phase
> shift is not an LTI system (and therefore not definable via impulse or
> frequency response), specify some other sensible definition from which
> it follows that inversion =/= 180&#2013266096; phase shift.

That is a reasonable mathematical encapsulation of the concept for
special cases, but is is neither entirely general nor universally true.
(There are cases it erroneously includes and excludes) Square waves in
quadrature is what one expects out of an incremental encoder. If you
have another classification of that output, I'd like to see it. It will
certainly be more cumbersome than "quadrature square waves".

>>> Insisting on a definition on your part is not nit picking - we have
>>> seen two incompatible defintions of the "phase shift of d degrees", and
>>> perhaps you come up with a third?

That fraction of a cycle, however expressed, that one waveform needs to
be shifted along its time axis in order to coincide with another.

>>> Such a definition can be via specification of the response for one
>>> single sinusoid, and requiring linearity and time-invariance. As soon
>>> as you do that, the response at DC and Nyquist follows, and we can stop
>>> quibbering and just look at the facts.

You find a reasonable mathematical way to express an intuitively simple
concept in a restricted "region of applicability" and declare without
justification that it is valid everywhere. How is that "fact"?

>> You seem to be happy with special cases, so I define phase for a special
>> case, the phase difference between voltage and current in a single wire
>> carrying a single frequency.  Power = V(RMS)*I(RMS)*cos(phi). This
>> defines phi, with only an ambiguity as to sign. I can resolve the
>> ambiguity if you will tell me whether inversion is lead or lag. Phase,
>> defined this way or any other, is not a property of DC.
>
> A "phase shift of d degrees" is an LTI system with well defined
> response at DC. The same thing goes for the equality of inversion and
> 180&#2013266096; phase shift - it is just a property of that system.

Call it a leg if you like, but it nevertheless wags the dog.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Andor wrote:
> Jerry Avins wrote:

...

>> Phase shift -- or difference -- is the
>> relative phase of two signals.
>
> This is not a definition, Jerry.

between signals of the same frequency. If you prefer, Phase shift -- or
difference -- is the relative phase of two signals at the same
frequency. You can generalize this to two signals of the same frequency
and shape. Phase is readily understood with square, sawtooth, and
triangle waves. (The Hilbert transformation of a square wave is not square.)

> I (and others) defined the "phase shift of d degrees" as a LTI system,
> and specified its frequency response (I'll also accept an impulse
> response as a proper definition). You don't need, and in fact can't use
> "the relative phase of two signals" (what is relative phase of two
> arbitrary signals?)  to define phase shift. If you feel that the phase
> shift is not an LTI system (and therefore not definable via impulse or
> frequency response), specify some other sensible definition from which
> it follows that inversion =/= 180&#2013266096; phase shift.

That is a reasonable mathematical encapsulation of the concept for
special cases, but is is neither entirely general nor universally true.
(There are cases it erroneously includes and excludes) Square waves in
quadrature is what one expects out of an incremental encoder. If you
have another classification of that output, I'd like to see it. It will
certainly be more cumbersome than "quadrature square waves".

>>> Insisting on a definition on your part is not nit picking - we have
>>> seen two incompatible defintions of the "phase shift of d degrees", and
>>> perhaps you come up with a third?

That fraction of a cycle, however expressed, that one waveform needs to
be shifted along its time axis in order to coincide with another.

>>> Such a definition can be via specification of the response for one
>>> single sinusoid, and requiring linearity and time-invariance. As soon
>>> as you do that, the response at DC and Nyquist follows, and we can stop
>>> quibbering and just look at the facts.

You find a reasonable mathematical way to express an intuitively simple
concept in a restricted "region of applicability" and declare without
justification that it is valid everywhere. How is that "fact"?

>> You seem to be happy with special cases, so I define phase for a special
>> case, the phase difference between voltage and current in a single wire
>> carrying a single frequency.  Power = V(RMS)*I(RMS)*cos(phi). This
>> defines phi, with only an ambiguity as to sign. I can resolve the
>> ambiguity if you will tell me whether inversion is lead or lag. Phase,
>> defined this way or any other, is not a property of DC.
>
> A "phase shift of d degrees" is an LTI system with well defined
> response at DC. The same thing goes for the equality of inversion and
> 180&#2013266096; phase shift - it is just a property of that system.

Call it a leg if you like, but it nevertheless wags the dog.

Jerry
--
Engineering is the art of making what you want from things you can get.
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```Andor wrote:
> Jerry Avins wrote:

...

>> Phase shift -- or difference -- is the
>> relative phase of two signals.
>
> This is not a definition, Jerry.

between signals of the same frequency. If you prefer, Phase shift -- or
difference -- is the relative phase of two signals at the same
frequency. You can generalize this to two signals of the same frequency
and shape. Phase is readily understood with square, sawtooth, and
triangle waves. (The Hilbert transformation of a square wave is not square.)

> I (and others) defined the "phase shift of d degrees" as a LTI system,
> and specified its frequency response (I'll also accept an impulse
> response as a proper definition). You don't need, and in fact can't use
> "the relative phase of two signals" (what is relative phase of two
> arbitrary signals?)  to define phase shift. If you feel that the phase
> shift is not an LTI system (and therefore not definable via impulse or
> frequency response), specify some other sensible definition from which
> it follows that inversion =/= 180&#2013266096; phase shift.

That is a reasonable mathematical encapsulation of the concept for
special cases, but is is neither entirely general nor universally true.
(There are cases it erroneously includes and excludes) Square waves in
quadrature is what one expects out of an incremental encoder. If you
have another classification of that output, I'd like to see it. It will
certainly be more cumbersome than "quadrature square waves".

>>> Insisting on a definition on your part is not nit picking - we have
>>> seen two incompatible defintions of the "phase shift of d degrees", and
>>> perhaps you come up with a third?

That fraction of a cycle, however expressed, that one waveform needs to
be shifted along its time axis in order to coincide with another.

>>> Such a definition can be via specification of the response for one
>>> single sinusoid, and requiring linearity and time-invariance. As soon
>>> as you do that, the response at DC and Nyquist follows, and we can stop
>>> quibbering and just look at the facts.

You find a reasonable mathematical way to express an intuitively simple
concept in a restricted "region of applicability" and declare without
justification that it is valid everywhere. How is that "fact"?

>> You seem to be happy with special cases, so I define phase for a special
>> case, the phase difference between voltage and current in a single wire
>> carrying a single frequency.  Power = V(RMS)*I(RMS)*cos(phi). This
>> defines phi, with only an ambiguity as to sign. I can resolve the
>> ambiguity if you will tell me whether inversion is lead or lag. Phase,
>> defined this way or any other, is not a property of DC.
>
> A "phase shift of d degrees" is an LTI system with well defined
> response at DC. The same thing goes for the equality of inversion and
> 180&#2013266096; phase shift - it is just a property of that system.

Call it a leg if you like, but it nevertheless wags the dog.

Jerry
--
Engineering is the art of making what you want from things you can get.
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```