Orthogonol signals

>Sharan123 <99077@dsprelated> wrote:
> 
>> I would like to know definition of orthogonal signals and why they are
>> significant in communication?
>
>Orthogonal means that if you multiply them, multiply by a weight
>function (which might be one) and integrate over the appropriate
>interval, the result is zero.
>
>In the case of signals, it makes them easy to separate.
>
>-- glen

Does this apply to say sine/cosine waveforms. I generated 2^20 samples
each. multiplied them and integrated but can't see the above definition
apply. What am I missing?

Kaz
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On 25.11.2015 4:52, glen herrmannsfeldt wrote:

(snip)

>
> I suspect that spherical and cylindrical coordinates come up
> in radio transmission,
>

Kinda like prolate spheroidal wave functions and such?

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=803509

--
Evgeny.

kaz <37480@DSPRelated> wrote:

> [attribution lost] wrote,

>>Orthogonal means that if you multiply them, multiply by a weight
>>function (which might be one) and integrate over the appropriate
>>interval, the result is zero.

>Does this apply to say sine/cosine waveforms. I generated 2^20 samples
>each. multiplied them and integrated but can't see the above definition
>apply. What am I missing?

Except for truncation errors you should get zero correlation
between a sine and a cosine of the same frequency.

If you get a non-zero value, compare it to the correlation of
just the sine signal with itself; the latter value should be
much much larger.

Steve

>kaz <37480@DSPRelated> wrote:
>
>> [attribution lost] wrote,
>
>>>Orthogonal means that if you multiply them, multiply by a weight
>>>function (which might be one) and integrate over the appropriate
>>>interval, the result is zero.
>
>>Does this apply to say sine/cosine waveforms. I generated 2^20 samples
>>each. multiplied them and integrated but can't see the above definition
>>apply. What am I missing?
>
>Except for truncation errors you should get zero correlation
>between a sine and a cosine of the same frequency.
>
>If you get a non-zero value, compare it to the correlation of
>just the sine signal with itself; the latter value should be
>much much larger.
>
>Steve

correlation yes gives zero mean but not multiplication as suggested in
some posts.

Kaz
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Steve Pope <spope33@speedymail.org> wrote:
> kaz <37480@DSPRelated> wrote:
 
>> [attribution lost] wrote,
 
>>>Orthogonal means that if you multiply them, multiply by a weight
>>>function (which might be one) and integrate over the appropriate
>>>interval, the result is zero.
 
>>Does this apply to say sine/cosine waveforms. I generated 2^20 samples
>>each. multiplied them and integrated but can't see the above definition
>>apply. What am I missing?
 
> Except for truncation errors you should get zero correlation
> between a sine and a cosine of the same frequency.

Well, also you need the appropriate number of periods.

If you do it as an average, dividing by the length of the integral
(or sum) then it will decrease as the length gets longer.

Mathematically, sine/cosine are orthogonal over an infinite integral,
though also for an integer number of periods. 

If you are using an integer number of periods, then there is
truncation (round off) error, as noted above.

> If you get a non-zero value, compare it to the correlation of
> just the sine signal with itself; the latter value should be
> much much larger.

-- glen

kaz <37480@DSPRelated> wrote:

>>kaz <37480@DSPRelated> wrote:

>>> [attribution lost] wrote,

>>>>Orthogonal means that if you multiply them, multiply by a weight
>>>>function (which might be one) and integrate over the appropriate
>>>>interval, the result is zero.

>>>Does this apply to say sine/cosine waveforms. I generated 2^20 samples
>>>each. multiplied them and integrated but can't see the above definition
>>>apply. What am I missing?

>>Except for truncation errors you should get zero correlation
>>between a sine and a cosine of the same frequency.

>correlation yes gives zero mean but not multiplication as suggested in
>some posts.

Correlation the same as multiplication (meaning, the inner product)
followed by summation/integration (although I suppose there are nuances 
to this).

Steve

On Wed, 25 Nov 2015 08:31:50 -0600, "kaz" <37480@DSPRelated> wrote:

>>kaz <37480@DSPRelated> wrote:
>>
>>> [attribution lost] wrote,
>>
>>>>Orthogonal means that if you multiply them, multiply by a weight
>>>>function (which might be one) and integrate over the appropriate
>>>>interval, the result is zero.
>>
>>>Does this apply to say sine/cosine waveforms. I generated 2^20 samples
>>>each. multiplied them and integrated but can't see the above definition
>>>apply. What am I missing?
>>
>>Except for truncation errors you should get zero correlation
>>between a sine and a cosine of the same frequency.
>>
>>If you get a non-zero value, compare it to the correlation of
>>just the sine signal with itself; the latter value should be
>>much much larger.
>>
>>Steve
>
>correlation yes gives zero mean but not multiplication as suggested in
>some posts.
>
>Kaz

It's multiplication in the sense of it being the dot product (or, more
generally, the inner product). 

For vectors (e.g., arrays of numbers, like are common in signal
processing),  if the dot product of two vectors (arrays) is zero they
are orthogonal.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

Thanks, everyone 

I have a couple of follow-up questions based on responses above ...

1) if we take 2 orthogonal signals and then multiply them individually
with some other signal then I assume that the resulting signals maintain
orthogonality. Is this correct?

2) From the posts above, I understand that if a transmitted signal is
composed of orthogonal signals then it is easier to recover individual
signals. Is this correct?
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On Thu, 26 Nov 2015 07:05:41 -0600, "Sharan123" <99077@DSPRelated>
wrote:

>Thanks, everyone 
>
>I have a couple of follow-up questions based on responses above ...
>
>1) if we take 2 orthogonal signals and then multiply them individually
>with some other signal then I assume that the resulting signals maintain
>orthogonality. Is this correct?

In the sense of how CDMA works, yes, but there are distortions that
can reduce the orthogonality so that there is some interference
between signals.   Also, since the other signals are still present
in-band, their energy is still there but uncorrelated with the other
"orthogonal" signals.   This appears as a reduction in SNR for each
signal.

>2) From the posts above, I understand that if a transmitted signal is
>composed of orthogonal signals then it is easier to recover individual
>signals. Is this correct?

Define "easier".   ;)

If multiple signals are trying to share the same signal space (e.g.,
frequency), then CDMA does simplify the case where they are in the
same channel.   However, an alternative approach is to maintain
orthogonality in frequency (e.g., occupy separate frequency channels),
which some would define as "easier".


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On Thu, 26 Nov 2015 07:05:41 -0600, "Sharan123" <99077@DSPRelated>

>1) if we take 2 orthogonal signals and then multiply them individually
>with some other signal then I assume that the resulting signals maintain
>orthogonality. Is this correct?

Not correct.  For example, sin(t) and cos(t) are orthogonal over
the interval from zero to 2*pi .  Consider the signal

       y(t) = 1 , 0 < t < pi/2
       y(t) = 0 otherside.

Multiplying (by means of inner product) each of sin(t) and cos(t)
by y(t) produces two signals which are strictly non-negative in
the interval 0 to pi/2, zero elsewhere, and these two signals
are not orthogonal.

>2) From the posts above, I understand that if a transmitted signal is
>composed of orthogonal signals then it is easier to recover individual
>signals. Is this correct?

Yes, usually, it is easier and also better use of your available
signal power.

Steve