On Sat, 06 Sep 2014 17:15:48 -0500, Tim Wescott
<seemywebsite@myfooter.really> wrote:
>On Sat, 06 Sep 2014 13:20:25 -0700, Rick Lyons wrote:
>
>> On Sat, 06 Sep 2014 12:01:54 -0500, Tim Wescott
>> <seemywebsite@myfooter.really> wrote:
>>
>>>On Sat, 06 Sep 2014 09:36:12 -0700, Rick Lyons wrote:
>>>
>>>> On Fri, 05 Sep 2014 14:35:45 -0500, tim <tim@seemywebsite.com> wrote:
>>>>
>>>>>On Fri, 05 Sep 2014 05:11:07 -0700, Rick Lyons wrote:
>>>>>
>>>>>> On Wed, 03 Sep 2014 17:54:38 -0500, Tim Wescott
>>>>>> <seemywebsite@myfooter.really> wrote:
>>>>>>
>>>>>> [Snipped by Lyons]
>>>>>>
>>>>>>>I only just noticed this part of your question.
>>>>>>>
>>>>>>>e^jwt and e^(jwt + pi/2) are not orthogonal, because no signal is
>>>>>>>orthogonal with itself, and e^(jwt + pi/2) is just e^jwt rotated by
>>>>>>>90 degrees.
>>>>>>>
>>>>>>>In a sense e^jwt BY ITSELF already gives you a carrier for
>>>>>>>transmitting two independent real signals -- asking for more is just
>>>>>>>greedy.
>>>>>>
>>>>>> Hi Tim,
>>>>>> miladsp described two complex signals:
>>>>>>
>>>>>> p = e^(j*w_c*t) and q1 = e^(j*w_c*t + pi/2).
>>>>>>
>>>>>> His q1 is equal to:
>>>>>>
>>>>>> q1 = e^(j*w_c*t)*e^(pi/2)
>>>>>>
>>>>>> where e^(pi/2) = 4.81. His q1 has a magnitude of 4.81. And his p
>>>>>> and q1 = 4.81*p are orthogonal.
>>>>>>
>>>>>>
>>>>>> What I think he MEANT to describe was:
>>>>>>
>>>>>> p = e^(j*w_c*t) and q2 = e^(j*(w_c*t + pi/2)).
>>>>>>
>>>>>> (Notice how q2 does not equal q1. q2 has a magnitude of one.)
>>>>>>
>>>>>> And as far as I can tell, p and q2 are orthogonal.
>>>>
>>>> Hi,
>>>>
>>>>>Over one cycle of p the integral of p and p* is a real, positive
>>>>>number.
>>>>>Hence, p is not orthogonal to itself.
>>>>
>>>> If by p* you mean the conjugate of p, then I agree. But we were not
>>>> talkin' about the orthogonality of p and the conjugate of p.
>>>>
>>>>>Over one cycle of p (and q2), the integral of q2 * p* is a purely
>>>>>imaginary number of absolute value distinctly greater than zero.
>>>>>hence,
>>>>>p is not orthogonal to q2.
>>>>
>>>> You have written p* again. The value p* was not part of our
>>>> discussion.
>>>>
>>>> I claim that p = e^(j*w_c*t) and q2 = e^(j*(w_c*t + pi/2))
>>>> are orthogonal. That is, over one cycle the integral of the product
>>>> of p times q2 is zero.
>>>>
>>>> Here's my thinking:
>>>> Over one cycle we can evaluate the integral of p times q2 as:
>>>>
>>>> 2pi
>>>> --
>>>> /
>>>> / e^(jx)e^(jx + pi/2) dx
>>>> /
>>>> --
>>>> 0
>>>>
>>>> 2pi
>>>> --
>>>> /
>>>> = / e^(jx)e^(jx)e^(pi/2) dx
>>>> /
>>>> --
>>>> 0
>>>>
>>>> 2pi
>>>> --
>>>> /
>>>> = / je^(jx)e^(jx) dx
>>>> /
>>>> --
>>>> 0
>>>>
>>>> 2pi - - 2pi
>>>> -- | |
>>>> / | e^(j2x) |
>>>> = / je^(j2x) dx = j |----------|
>>>> / | j2 |
>>>> -- | | 0 -
>>>> - 0
>>>>
>>>> e^(j4pi) e^(0)
>>>> = ---------- - -------
>>>> 2 2
>>>>
>>>> 1 1
>>>> = --- - --- = 0.
>>>> 2 2
>>>>
>>>>
>>>> If I've screwed up here, please let me know.
>>>>
>>>> [-Rick-]
>>>
>>>I think the problem is your definition of orthogonality. If you start
>>>by saying that a signal with nonzero energy absolutely positively cannot
>>>be orthogonal with itself, then your definition of orthogonality leads
>>>to a contradiction, because the integral over N cycles of p * p is zero.
>>>
>>>So, I'm doing the usual engineer's stretch, and claiming that
>>>orthogonality must be calculated using signal1 times the conjugate of
>>>signal2. When used with p, this comes up with the sensible result that
>>>the time average of p * p* is the power level of p.
>>>
>>>If you really insist that your definition of orthogonality is correct
>>>then I think we both need to hit the books.
>>
>> Hi,
>> Wow. I don't know why you and I are not
>> communicating too well.
>>
>> I did not say a signal is not orthogonal with itself.
>
>By your definition of orthogonality, p is orthogonal with itself, because
>if you substitute p for q2 in your equation above, the result of the
>integration is zero.
>
>> I said p was orthogonal to q2.
>
>Yes, and I checked your math by substituting p in place of q2, and found
>it math wanting.
>
>> As for "hitting the books", my "Engineer's Guide to DSP" book by Steven
>> Smith mentions that a sine wave is orthogonal with a cosine wave. But
>> he doesn't give an integral equation definition for orthogonality.
>> There is no entry in the Index of my Opp & Schafer DSP book for the word
>> orthogonal.
>>
>> I'm basing my definition of orthogonality on the definition given at:
>>
>> http://en.wikipedia.org/wiki/Orthogonality
>>
>> The same definition is also provided at:
>>
>> http://mathworld.wolfram.com/OrthogonalFunctions.html .
>> Tim, I'll bet our disagreement is a problem of not agreeing upon the
>> definition of orthogonality.
>> Resolving such controversies is educational, at least they are for me.
>
>I think the difficulty comes about either with a definition of
>orthogonality for complex signals, or with the validity of using the
>concept of orthogonality for complex signals. I tried to resolve the
>problem with suitable definition of "orthogonal".
The definition I've always seen and used is that the inner product is
zero. A zero-magnitude vector is magic, kind of like the number zero
is magic, in that it is orthogonal to anything. That stretches the
concept of "orthogonality", though, just like the goofy proofs
discussed here lately where zero or division by zero is used to
"prove" something that isn't true.
Likewise using a zero-magnitude vector to test for orthogonality or
the validity of a statement isn't very useful.
>The original question was, if you have two complex carriers, e^jwt and
>e^j(wt + pi/2), can you transmit two independent signals?
Clearly you can. There are many, many ways to do it. There's time
orthogonality, there's frequency orthogonality, and if you want to
transmit them at the same time on the same frequency you can spread
them with orthogonal codes or use MIMO in many cases (if the channels
are faded independently).
>The answer to THAT question depends, I guess, on whether you mean two
>independent _complex_ signals, or two independent _real_ signals. Because
>regardless of how you may define orthogonality, the answer to "can you use
>these two carriers to transmit two independent complex signals over one
>channel" is HELL NO, while the answer if the independent signals are real
>is HELL YES.
As I just mentioned, you can do it in either case.
>I think you can see why -- the result of
>
>A * e^jwt + B * e^j(wt + pi/2) is (if I'm doing my math right)
>
>((Re(A) - Im(B)) + j(Im(A) + Re(B)) * e^jwt
>
>It should be obvious by inspection why an arbitrary A and B, with both
>numbers complex, cannot be sent using the two cited carriers.
Not always, but very often, you can.
You can extend what you've written to an even more complex case:
s1(t) = A(t)e^jwt and s2(t) = B(t)e^(jwt+pi/2). A(t) and B(t) could
be two independent NRZ streams of symbols from the same bit clock, and
then s1(t) and s2(t) can be interpreted as two BPSK signals, both at
frequency = w. If the phase relationship between them created by the
pi/2 offset in s2 is strictly maintained, then
s1(t) + s2(t) = s3(t)
just creates a QPSK signal, s3(t). This is done routinely.
So, yes, as far as I can tell there's no reason for me to believe that
your two example signals are not orthogonal and cannot be transmitted
at the same time on the same channel and properly recovered.
There are LOTS of things that could happen to make them not orthogonal
and no longer separable, but I haven't seen any of those conditions
specified. Maybe I missed it.
Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com