The imaginary axis is not phase... or is it?

I recently caught one of my textbooks discussing the Fourier Transform 
and referring to the imaginary results as the 'phase' of the frequency 
component(s).  That struck me as wrong, but I let it pass.

Recently on this newsgroup a number of people have been using 'j' and 
the imaginary axis interchangeably with 'phase,' so it seems appropriate 
to raise a question here.

My understanding is that the Real and Imaginary components of the 
complex result of an FFT represent the rectangular coordinates of the 
frequency domain values.  I also understand that you must convert these 
values from rectangular to polar coordinates before you have the 
magnitude and phase.

My point is that the phase involves both Real and Imaginary axes, just 
as the magnitude does, so it seems rather misleading to be so cavalier 
about a fundamental concept.


No offense, but comp.dsp seems to involve an inordinate amount of 
nit-picking when it comes to being very precise about terminology.  In 
that light, I would hope that posters would be more careful about mixing 
up 'j' and 'phase' as if they're the same thing.  If I am wrong about 
this, then please set me straight, otherwise please try to use the terms 
correctly.

Trying my hand at the math; it would seem that with the term e^(jwt), 
't' would have more to do with phase than 'j' ... or perhaps (wt) would 
incorporate phase more completely than 't' alone.  Literally, 'j' is a 
constant, so how can it represent phase all on its own?  Also, if 
e^(-jwt) is the negative frequency mirror of e^(jwt), then again I don't 
see how 'j' can be phase (rather it would look like frequency).  Sure, 
there is a phase difference between sine and cosine, and 'j' might be a 
convenient symbol to recognize this fixed phase relationship, but that's 
hardly the indication of the phase component of an arbitrary input signal.

Anyway, I probably should have stopped with my fourth paragraph.  I have 
a feeling that my subsequent choice of words will probably cause way 
more trouble than my primary question, but I am trying to be 
conversational...

Brian Willoughby
Sound Consulting

On Apr 1, 6:14=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:
> I recently caught one of my textbooks discussing the Fourier Transform
> and referring to the imaginary results as the 'phase' of the frequency
> component(s). =A0That struck me as wrong, but I let it pass.
>
> Recently on this newsgroup a number of people have been using 'j' and
> the imaginary axis interchangeably with 'phase,' so it seems appropriate
> to raise a question here.
>
> My understanding is that the Real and Imaginary components of the
> complex result of an FFT represent the rectangular coordinates of the
> frequency domain values. =A0I also understand that you must convert these
> values from rectangular to polar coordinates before you have the
> magnitude and phase.

You are correct.

> Trying my hand at the math; it would seem that with the term e^(jwt),
> 't' would have more to do with phase than 'j' ... or perhaps (wt) would
> incorporate phase more completely than 't' alone.

Again, you're correct. exp(j*w*t) is a complex sinusoid whose phase is
a function of time. At any given time "t", the phase of the sinusoid
is w*t.

Where you may have seen people use "phase" and "j" in a tangled way is
in an example like this one. "e" raised to an imaginary power yields a
complex number whose phase is equal to the scale factor applied to
"j":

exp(j*0) =3D 1 (phase =3D 0)
exp(j*pi/2) =3D j (phase =3D pi/2)

And so on.

Jason

Jason

On Apr 1, 6:14=A0am, Brian Willoughby
<Sound_Consulting-...@Sounds.wa.com> wrote:
> I recently caught one of my textbooks discussing the Fourier Transform
> and referring to the imaginary results as the 'phase' of the frequency
> component(s). =A0That struck me as wrong, but I let it pass.
>
> Recently on this newsgroup a number of people have been using 'j' and
> the imaginary axis interchangeably with 'phase,' so it seems appropriate
> to raise a question here.
>
> My understanding is that the Real and Imaginary components of the
> complex result of an FFT represent the rectangular coordinates of the
> frequency domain values. =A0I also understand that you must convert these
> values from rectangular to polar coordinates before you have the
> magnitude and phase.
>
> My point is that the phase involves both Real and Imaginary axes, just
> as the magnitude does, so it seems rather misleading to be so cavalier
> about a fundamental concept.
>
> No offense, but comp.dsp seems to involve an inordinate amount of
> nit-picking when it comes to being very precise about terminology. =A0In
> that light, I would hope that posters would be more careful about mixing
> up 'j' and 'phase' as if they're the same thing. =A0If I am wrong about
> this, then please set me straight, otherwise please try to use the terms
> correctly.
>
> Trying my hand at the math; it would seem that with the term e^(jwt),
> 't' would have more to do with phase than 'j' ... or perhaps (wt) would
> incorporate phase more completely than 't' alone. =A0Literally, 'j' is a
> constant, so how can it represent phase all on its own? =A0Also, if
> e^(-jwt) is the negative frequency mirror of e^(jwt), then again I don't
> see how 'j' can be phase (rather it would look like frequency). =A0Sure,
> there is a phase difference between sine and cosine, and 'j' might be a
> convenient symbol to recognize this fixed phase relationship, but that's
> hardly the indication of the phase component of an arbitrary input signal=
.
>
> Anyway, I probably should have stopped with my fourth paragraph. =A0I hav=
e
> a feeling that my subsequent choice of words will probably cause way
> more trouble than my primary question, but I am trying to be
> conversational...
>
> Brian Willoughby
> Sound Consulting

On my website I have four interactive flash programs that may give you
some more insight into complex numbers:


http://www.fourier-series.com/fourierseries2/complex_tutorial.html

First, an apology. I started this topic with what I hopes would be a throw-=
away comment, and embroiled us in what is probably a fruitless exercise in =
omphaloskepsis. The trouble with contemplating one's navel is that one ofte=
n discovers an excess of lint.

Complex numbers are useful tools for many kinds of calculations. The meanin=
gs assigned to the axes of graphs relating to those calculations are govern=
ed by the information being displayed. On what we call the s plane, the rea=
l axis depicts gain and the imaginary axis depicts frequency. When plotting=
 impedance, the real axis depicts resistance and the imaginary axis depicts=
 reactance.=20

It makes no more sense to say that an axis is phase than it does to say tha=
t another is time. It depends.

Jerry
--=20
Engineering is the art of making what you want from things you can get.

On 04/01/2011 06:35 AM, Jerry Avins wrote:
> First, an apology. I started this topic with what I hopes would be a throw-away
 > comment, and embroiled us in what is probably a fruitless exercise in
 > omphaloskepsis. The trouble with contemplating one's navel is that 
one often
 > discovers an excess of lint.
>
> Complex numbers are useful tools for many kinds of calculations. The meanings
 > assigned to the axes of graphs relating to those calculations are 
governed
 > by the information being displayed. On what we call the s plane, the real
 > axis depicts gain and the imaginary axis depicts frequency.

 >> snip <<

Uh -- nuh uh.  The entire s plane is in frequency*.  The real axis is 
Napiers/sec, and the imaginary axis is radians/sec, but it's all in 
units of (some name for a dimensionless thing)/sec.

* As is the entire z plane, in a twisted-up sort of way, and normalized 
to the sampling rate.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

On Friday, April 1, 2011 12:08:26 PM UTC-4, Tim Wescott wrote:

  ...

> Uh -- nuh uh.  The entire s plane is in frequency*.  The real axis is 
> Napiers/sec, and the imaginary axis is radians/sec, but it's all in 
> units of (some name for a dimensionless thing)/sec.

Oops! Yes, of course.

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

On Friday, April 1, 2011 9:05:34 AM UTC-4, Jason wrote:


>     ... exp(j*w*t) is a complex sinusoid whose phase is
> a function of time. At any given time "t", the phase of the sinusoid
> is w*t.

It's not a complex sinusoid. It's a complex exponential whose real and imag=
inary parts are real sinusoids. Not keeping things like that straight is wh=
at haunts beginners. After a while, we get to create our own mythologies th=
at seem to make it hang together until it matters.=20

Jerry
--=20
Engineering is the art of making what you want from things you can get.

Years ago, I had a colleague who was always angry about something, and he h=
ad the silly idea that any mistake he made demeaned him to the extent that =
he had to find a way to defend it. I once pointed out a mistake he had made=
 (dividing instead if multiplying, I think) and he insisted that he was cor=
rect. I showed that the dimensional analysis came out all wrong his way, an=
d he said angrily, "You're making me look like a damn fool!" I answered, "T=
hat is an office one only perform for oneself." A few days later, he died -=
- presumably in his sleep -- of a massive heart attack that I remain convin=
ced was brought on largely by chronic anger. The thought came to me as I to=
ssed a shovelful of earth onto his coffin, "This is an office one can not p=
erform for oneself."

The whole incident came to mind yesterday at the funeral of a friend.

Jerry

On Apr 1, 6:20&#4294967295;pm, Jerry Avins <j...@ieee.org> wrote:
>
> The whole incident came to mind yesterday at the funeral of a friend.
>

Jerry, i meant to send you condolences regarding your friend Ruth.
was this a person i met when i was at your house?

i hope that Ruth wasn't chronically angry like your other acquaintance
(thus connecting the two incidents).

L8r,

r b-j

Yes, You met her. Although her last few months were spent in great pain from spinal stenosis, she died peacefully while asleep. Members of both of the choirs she sang in sang at the service.

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