Reality check -- What constitutes a "real signal" ?

I have some conceptual problems properly relating "physical real" and 
"mathematical real".

I don't have problems with equations such as

sin(wt) = (e^jwt - e^-jwt)/2  [ did I confuse sin with cos ? ;]
That's a mathematical expression following mathematical rules &/or 
conventions.

I don't have any real { poor word? ;} problem taking the FFT of a 
physically realizable signal ( eg lab square wave generator ). The FFT 
result is complex. But I sort of grok* that its complex nature provides 
me with phase information.

What confuses me is "reality"/"physical significance" of a complex input 
to an FFT. I can accept that as a mathematical operator an FFT can 
operate on real, imaginary, or complex data and produce a result.

But just
  a. where do you physically get said complex input?
  b. what is a complex input?
  c. what question should I be asking?  ;]

Jerry objects to me claiming "newbie" status ;}
I'll just claim "confused" status ;!

Richard Owlett wrote:

> I have some conceptual problems properly relating "physical real" and 
> "mathematical real".
> 
> I don't have problems with equations such as
> 
> sin(wt) = (e^jwt - e^-jwt)/2  [ did I confuse sin with cos ? ;]
> That's a mathematical expression following mathematical rules &/or 
> conventions.

You got half & half -- your expression is for -j sin(wt) -- multiply by 
j for the correct answer.
> 
> I don't have any real { poor word? ;} problem taking the FFT of a 
> physically realizable signal ( eg lab square wave generator ). The FFT 
> result is complex. But I sort of grok* that its complex nature provides 
> me with phase information.

Actually it's complex nature is constrained such that if you know the 
input was real-valued you only need half of the samples to predict the 
other half.
> 
> What confuses me is "reality"/"physical significance" of a complex input 
> to an FFT. I can accept that as a mathematical operator an FFT can 
> operate on real, imaginary, or complex data and produce a result.
> 
> But just
>  a. where do you physically get said complex input?

The place that I know of to get such an input is by demodulating a band 
limited signal with a complex sinusoid.  For example if I want to build 
a SSB radio and I have a signal that occupies 1900 - 1903kHz I can 
multiply it by e^jwt, with w = 2 * pi * 1900000.  I'll get a signal with 
  energy from 0 to 3kHz and energy from 3800 to 3803kHz.  I can easily 
discard the stuff at 3.8kHz and save the stuff around DC.

As to how to do this I have two mixers:  my inphase mixer multiplies by 
cos(wt)and my quadrature mixer multiplies by sin(wt).  I sample the 
inphase and quadrature channels, and internally in my software I declare 
the quadrature channel to be imaginary.

>  b. what is a complex input?

See above.

>  c. what question should I be asking?  ;]

Gee, that's a tough one -- it's so open ended.  "Why am I here"?  "Why 
am I Republican (Democrat, Nazi, Communist, Labor, Christian Democrat, 
Likud, etc.)"?  "Why didn't I go into prostitution when I was good looking"?

> 
> Jerry objects to me claiming "newbie" status ;}
> I'll just claim "confused" status ;!
> 


-- 

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

Richard Owlett wrote:

> I have some conceptual problems properly relating "physical real" and 
> "mathematical real".

> I don't have problems with equations such as

> sin(wt) = (e^jwt - e^-jwt)/2  [ did I confuse sin with cos ? ;]
> That's a mathematical expression following mathematical rules &/or 
> conventions.

Is should be divided by 2i, but pretty close.

> I don't have any real { poor word? ;} problem taking the FFT of a 
> physically realizable signal ( eg lab square wave generator ). The FFT 
> result is complex. But I sort of grok* that its complex nature provides 
> me with phase information.

If you really don't like complex numbers there is the Hartley 
transform.  Just as valid physically, it isn't as nice 
mathematically.

> What confuses me is "reality"/"physical significance" of a complex input 
> to an FFT. I can accept that as a mathematical operator an FFT can 
> operate on real, imaginary, or complex data and produce a result.
> 
> But just
>  a. where do you physically get said complex input?
>  b. what is a complex input?
>  c. what question should I be asking?  ;]

In most cases, signals will be real.  There are some physical 
quantities that are best described as complex, but not usually 
the ones you Fourier transform.   Index of refraction, and the
related dielectric constant, where the imaginary part is related 
to absorption (attenuation).   It works pretty much the same as 
complex impedance.

It could also be used to describe quadrature modulation, such as 
the color subcarrier in TV signals.  While it is really 
amplitude and phase it is sometimes easier to understand as real 
and imaginary parts of a complex value, though with no odd real 
or even imaginary part.

-- glen

in article 10uqp7od5u07579@corp.supernews.com, Richard Owlett at
rowlett@atlascomm.net wrote on 01/18/2005 14:35:

> a. where do you physically get said complex input?

complex signals are "constructed" in the mind of the computer or DSP.  they
are a pair of real signals where one of those real signals is called the
"real part" or the "in-phase signal" and the other real signal is called the
"imaginary part" or the "quadrature signal".  then the pair are always
processed together with the rules of doing mathematics to complex numbers
(how they're added, multiplied, divided, exponentiated, etc.).

> b. what is a complex input?

an input consisting of two part (both real signals), one representing a real
part, the other representing the imaginary part.

> c. what question should I be asking?  ;]

i dunno.

there have been some philosophical discussions here regarding if complex
quantities really exist in nature.  i'm of the belief that complex numbers
are a very useful mathematical abstraction and that you will not be hooking
up a voltmeter to some voltage and read a complex value (without some
interpretation, at least).

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

in article BE12DE92.3E10%rbj@audioimagination.com, robert bristow-johnson at
rbj@audioimagination.com wrote on 01/18/2005 15:41:

> there have been some philosophical discussions here regarding if complex
> quantities really exist in nature.  i'm of the belief that complex numbers
> are a very useful mathematical abstraction and that you will not be hooking
> up a voltmeter to some voltage and read a complex value (without some
> interpretation, at least).

more specifically, i meant to say that *imaginary* numbers are a useful
mathematical abstraction (and then so also are complex numbers).  there are
folks that say the terms "real" and "imaginary" are sorta misnomers for the
two parts of a complex number, but i think they are very appropriate terms.
imaginary numbers truly exist only in our imagination.  all mathematics done
at the root physical level are real, IMHO.  even, ultimately, quantum
mechanics, although doing it with complex numbers sure saves you a headache.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

"robert bristow-johnson" <rbj@audioimagination.com> wrote in message 
news:BE12DE92.3E10%rbj@audioimagination.com...
> in article 10uqp7od5u07579@corp.supernews.com, Richard Owlett at
> rowlett@atlascomm.net wrote on 01/18/2005 14:35:
>
>> a. where do you physically get said complex input?
>
> complex signals are "constructed" in the mind of the computer or DSP.  they
> are a pair of real signals where one of those real signals is called the
> "real part" or the "in-phase signal" and the other real signal is called the
> "imaginary part" or the "quadrature signal".  then the pair are always
> processed together with the rules of doing mathematics to complex numbers
> (how they're added, multiplied, divided, exponentiated, etc.).

That meshes with the way I think about it too.  I was once told that sometimes 
we make real signals into complex ones because the analysis becomes easier. 
Then when you are done with the math, you often throw away part of the data to 
get a real answer (e.g. the magnitude of a particular FFT frequency bin). 
Another simple example is phasor analysis in elementary circuits.  A lot of 
complex numbers are thrown in to deal with e.g. both voltage and current 
simultaneously. 

Richard Owlett wrote:
> I have some conceptual problems properly relating "physical real" and

> "mathematical real".

Some people have accused me (and probably rightly so) of being too
concerned about linguistic details. I prefer to use the terms
"real-world signal" for an actually measured (as opposed to
simulated) signal, and the terms "real-valued data" for measured
or simulated data that are represented as real (as opposed to
complex) numbers. Some words have just too many possible meanings
to take any chances wih.

> I don't have problems with equations such as
>
> sin(wt) = (e^jwt - e^-jwt)/2  [ did I confuse sin with cos ? ;]
> That's a mathematical expression following mathematical rules &/or
> conventions.

You didn't confuse it with cos, you forgut the "i" in the denominator.
Now, what you need to contemplate with the above equation, is that
the real-valued sin(wt) is *represented* by two complex numbers.
These two *representations* are equally valid since the imaginary
parts of the complex-valued exp terms cancel.

I am sure you remember calculus in school, where one often
encountered tricks like adding one and subtracting one to an
equation. The overall value of the equation is unchanged, but
the algebraic representation has changed so that it is easier
to get a step or two further.

> I don't have any real { poor word? ;} problem taking the FFT of a
> physically realizable signal ( eg lab square wave generator ). The
FFT
> result is complex. But I sort of grok* that its complex nature
provides
> me with phase information.

Well, that's the difference between the real-valued and complex-
valued representations. Remember that the DFT of a realvalued signal
is ymmetrical, i.e. F(-w) = conj(F(w)), so you need to take the
whole spectrum into account to get a mathematically correct
representation. Usually, we prefer to view only the half spectrum
(the band 0 - Fs/2) since we basically only nead that to understand
the essentials of the spectrum.

> What confuses me is "reality"/"physical significance" of a complex
input
> to an FFT. I can accept that as a mathematical operator an FFT can
> operate on real, imaginary, or complex data and produce a result.
>
> But just
>   a. where do you physically get said complex input?

Generally, you don't. There *might* be situations where one measures
two real-valued signals and combine them as a real and imaginary
part of a complex number prior to further processing. If that can be
dne and if anything useful come out of it, it would be purely
coincidential. You generally have to take one or more real-valued
inputs, do some voodoo on them, and produce complex numbers as
a result of manipulations (e.g. quadrature sampling, Hilbert
transforms, other weird stuff).

>   b. what is a complex input?

A complex-valued input is an input to a routine, that consist of
complex numbers.

Just to give you a hint of how useful the distinction is, I exploited
the difference between complex-valued and real-valued inputs in the
techniques I developed in my thesis (you can find a PDF copy at
http://www.fysel.ntnu.no/~allnor/thesis/thesis.pdf but be quick,
I'm clearing out my office now so the link will disappear in a
couple of days).

What we did was to use an array of M sensors to record time series.
What you end up with, is a set of M timeseries, each containing N
samples. This is arranged in an N x M matrix, where all coefficients
are real-valued. The analysis we did was based on inspecting the 2D
spectrum of these 2D (t,x) data. The naive way of doing that is to
compute a 2D DFT of the NxM matrix.

Now, let's just take a closer look at how one actually does that.
Step 1 is to compute the 1D DFT (using a standard FFT routine) along
the t axis, the columns. After doing that, you still have a NxM
matrix, but the coefficients are no longer real-valued, they are
complex-valued. There is a symmetry in that the complex conjugated
complex numbers that merge to form real-valued time series are found
in different rows. But there is no connection between the numbers
found in each row.

The last step of computing the 2D DFT is to compute the 1D DFT of the
rows. These are complex valued, and are therefore complex-valued
inputs to the 1D FFT routine.

>   c. what question should I be asking?  ;]

I think you are (and have been, for a long time now) asking the
correct questions. These questions are good indicators that you
actually have been contemplating the material you ask about.
Which makes it more inspiring to try to help out.

> Jerry objects to me claiming "newbie" status ;}

And right he is! As I said above, these types of questions are
founded on a certain type of contemplation that separate the
"experienced student" from the complete newbie.

> I'll just claim "confused" status ;!

Heh, that makes at least two of us. As they say, "once you've
become one of us, there is no way back." Once confused, always
confused.

Seriously, over the years I have found myself more and more often
confused about seemingly "obvious" technical details, like the
questions you ask. Interestingly, I find that the keys to solving
problems are often found in those tiny details. For instance,
in my PhD work I found that since the Nyquist sampling theorem
only applies to real-valued data, I could achieve my target
results with only half as long arrays as everybody expected.

Nah, you are asking just the kind of questions that can bring
you a long way forward. Contemplating "trivial" questions
and being aware of "curious" answers, are the key.

Rune

Rune Allnor wrote:
(big snip)

> Well, that's the difference between the real-valued and complex-
> valued representations. Remember that the DFT of a realvalued signal
> is ymmetrical, i.e. F(-w) = conj(F(w)), so you need to take the
> whole spectrum into account to get a mathematically correct
> representation. Usually, we prefer to view only the half spectrum
> (the band 0 - Fs/2) since we basically only nead that to understand
> the essentials of the spectrum.

Well, if it really is symmetric, then you only need half.
On the other hand, you don't know that the original function
was real valued unless you see the symmetry in the transform
result.

Consider the Fourier transform symmetries of transform pairs:

real even <--> real even
real odd  <--> imaginary odd
imaginary even <--> imaginary even

If you consider the Hartley transform instead, where real
and imaginary aren't mixed, but instead

real <--> real
imaginary <--> imaginary
odd <--> odd
even <--> even

It seems much simpler to understand.  But then if you
want to see the important parts of the result, you
have to separate the result into even and odd parts.

>>What confuses me is "reality"/"physical significance" of 
 >> a complex input to an FFT. I can accept that as a
 >> mathematical operator an FFT can operate on real,
 >> imaginary, or complex data and produce a result.

>>But just
>>  a. where do you physically get said complex input?

One place is from the result of Fourier transforms.

-- glen

glen herrmannsfeldt wrote:
> Rune Allnor wrote:
> (big snip)
>
> > Well, that's the difference between the real-valued and complex-
> > valued representations. Remember that the DFT of a realvalued
signal
> > is ymmetrical, i.e. F(-w) = conj(F(w)), so you need to take the
> > whole spectrum into account to get a mathematically correct
> > representation. Usually, we prefer to view only the half spectrum
> > (the band 0 - Fs/2) since we basically only nead that to understand
> > the essentials of the spectrum.
>
> Well, if it really is symmetric, then you only need half.
> On the other hand, you don't know that the original function
> was real valued unless you see the symmetry in the transform
> result.

Good point. Most of the time, one knows that from the statement
of the problem. In most applications one deals with real-valued
signals, and know from the scope of the problem that half the
spectrum suffices. In fact, this assumption is so common that
one could think that it is a fundamental property of DSP,
based on most, if not all, entry-level and mid-level texts on
DSP. Actually, off the top of my head I can't think of a single
book on DSP that actually discusses complex-valued signals in
their own right.

Rune

Richard Owlett wrote:
> I have some conceptual problems properly relating "physical real" and

> "mathematical real".
>
> I don't have problems with equations such as
>
> sin(wt) = (e^jwt - e^-jwt)/2  [ did I confuse sin with cos ? ;]
> That's a mathematical expression following mathematical rules &/or
> conventions.
>
> I don't have any real { poor word? ;} problem taking the FFT of a
> physically realizable signal ( eg lab square wave generator ). The
FFT
> result is complex. But I sort of grok* that its complex nature
provides
> me with phase information.
>
> What confuses me is "reality"/"physical significance" of a complex
input
> to an FFT. I can accept that as a mathematical operator an FFT can
> operate on real, imaginary, or complex data and produce a result.
>
> But just
>   a. where do you physically get said complex input?
>   b. what is a complex input?

I am newbie to posting here. So pls. bear with my mistakes.

First of all I agree with all the previous replies. I would like to add
some of my thoughts here.

IMHO, one way to make sense out of complex numbers is, as special cases
of two dimensional vector spaces.

A 2D vector space is spanned by the two orthogonal(not unique) basis
vectors. In the case of complex numbers we call one of them as real and
the other imaginary. IMHO, What makes their case special is that, in
addition to all the mathematical properties of vector spaces, complex
numbers gives the ability to describe the operation of rotating a
vector by 90 degrees, by the operation of multiplying by j.

Rotation is useful in describing many physical phenomena where a system
changes the phase of an input signal and the phase information of the
output is important. Then one may be able to use the complex number
operations to model parts of that system. One example is phase
modulations in communications.

The concept of angle(phase) is very fundamental to vector spaces.
Specifically a vector space defines the vector inner product. From that
we can calculcate the angle between the two multiplicand vectors.

angle b/w vectors x and y = arccos(<x,y>/(mag(x)*mag(y))

But vector spaces do not need to define a rotation operation. (I am not
sure if it a byproduct of any of the vector space properties.)

Now lets take an example where complex numbers are used, the fourier
transform. In the case of fourier transforms, each frequency represents
a 2D vector space spanned by the basis vectors cos(wt) and sin(wt)
which are orthogonal bases. Note that we can obtain the second basis
vector sin(wt) by rotating the first basis vector cos(wt) by 90
degrees( = pi/2 rad). Physically this rotation operation is achieved by
delaying the cost(wt) signal by td = pi/2/w.  cos(w(t - td)) = cos(wt -
pi/2) = cos(pi/2 - wt) = sin(wt).

If we take the rotated signal sin(wt) and again rotate it by the same
delay we get -cos(wt). i.e. sin(w(t - td)) = sin(wt - pi/2) = -sin(pi/2
- wt) = -cos(wt). I.e, two rotations by 90 degrees each in the same
direction result in multiplication by -1. So intuitively, rotation by
90 degrees is multiplication by sqrt(-1). (Also think about the polar
coordinate representations, where j = exp(j*pi/2) )

Now lets see how some of the vector space quantities are defined in the
case of complex numbers. Vector spaces require that a vector inner
product be defined *uniquely* on a vector space between any two
non-zero vectors. In the case of complex numbers this is
<z1,z2> = real(z1 * Z2') where * is the complex multiplication and ' is
the conjugation operation. Vector inner products are by definition
scalars. So the output of real(Z1*Z2) is interpreted as a scalar.

Interestingly complex multiplication also provides the vector outer
product(cross product) as the imaginary part of the complex
product(i.e. z1*z2'). But by definition the vector outerproduct is a
vector and is perpendicular to the direction of both the multiplicands
vectors. IMHO, this is a place where the complex numbers differ from
the usual vector spaces, that here the vector outer product *SEEMS to*
lie within the 2D complex plane itself. But usually this does not
create any confusion because there are only two possible directions
that the vector outerproduct can have *IF* the two multiplicands are
always lie in a certain 2D plane. So we can interpret the outerproduct
as a real number either positive or negative depending on the two
directions.

I have not listed the other vector space properties, but one can easily
show that complex numbers do provide all of them as definitions of
complex number operations. So IMHO complex numbers represent a 2D
vector space. To describe physical phenomena/systems where one is
interested in the relative phases of the input/output signals, complex
numbers is a very useful tool.

Going back to the question of fourier transforms. We use complex
multiplications and not just either vector inner/outer products in the
case of fourier transfoms. And there are physical phenomena to back up
these mathematical operations. E.g. Fraunhauffer doffraction pattern is
the 2D fourier transform of the aperture of the light source. (Note,
here each of the dimension in the "time domain" actually has units of
length. And so frequencies are spatial frequencies with units
1/length.)

>   c. what question should I be asking?  ;]

Here are some suggestions.

1. What is the definition of a vector space? (Also think about inner
products, norms, angle, ...)

2. Give a proof that complex plane is a 2D vector space defined over
the field of real numbers.

3. How do we interpret vectors in the physical world? E.g. we know
force is a vector, but we measure force as a scalar quantity. The
direction is implied. Can we apply a similar approach to complex
numbers where we do not measure the j part, but we infer it from the
context?

4. How to interpret negative frequencies and complex frequencies?

There are many more interesting questions. There was a thread on a very
similar topic on this newsgroup, perhaps a few yrs ago. You can google
it i think. Also I found a short thread "why do complex numbers work in
physics? " on sci.physics.research. Very interesting discussion. Half
of it was greek and latin to me though!

I am interested to hear some examples and counter examples from
Physicist on this "complex" problem.

>
> Jerry objects to me claiming "newbie" status ;}
> I'll just claim "confused" status ;!

I have seen some other threads on comp.dsp where it was asked whether
one can measure a complex signal? I think, it is possible to build a
device that measures the magnitude and also phase (w.r.to a reference)
of a signal. And IMHO, that is measuring a complex signal. A watt meter
is close to such a device 'coz, it measures real(<v(t),i(t)>) and if we
know the magnitude of the v(t) and i(t) signals we can calculate the
angle between v(t) and i(t) and we have the relative phase between v
and i(arc cosine of power factor).
Thanks for putting up with me until here :)

-Kal