# Orthogonal signals

#### mentaaal

Joined Oct 17, 2005
451
Hey guys, we are learning in telecommunications class that orthogonal signals do not interfere. My iderstanding of this is that for 2 signals to be orthogonal, they have to have a phase shift of 90 degrees. I do not understand how these 2 signals do not interfere. They both have instantaneous amplitude (be it voltage or current or whatever) so how can these amplitudes not interfere with eachother?

Thanks

#### studiot

Joined Nov 9, 2007
4,998
A phase shift of 90 degrees is not the property that makes two functions orthogonal, the phase shift just happens in certain cases.

For example sinx and cosx are orthogonal and are phase shifted by 90 degrees.

The property of interest uses the fact that any vector can have no influence at 90 deg to its line of action.

If you draw (plot) the instantaneous vectors (phasors) at any instant for sinwt and coswt you will find they are at right angles to each other for each and every instant (value).

If you want a slightly more mathematical version

Two vectors are orthogonal if their dot product is zero

So take the dot product of each instantaneous vector pair in my example. They all equal zero.

Try a few values for yourself.

#### mentaaal

Joined Oct 17, 2005
451
Thanks for the reply studiot, I understand what you mean about cos(x) and sin(x) being 90 degrees out of phase and I can imagine clearly how this would be the case for any value for x but, perhaps its just a lack of understanding on my part but i still dont see how they dont interfere. Say for example the orthogonal sinx and cosx. If a point received sinx + cosx (i.e. two signals) wont the amplitudes of the two signals still interfere? (at any given instant in x)
I have included a plot to illustrate my confusion.

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#### marwanm

Joined Sep 16, 2008
5
The orthognality in signals means that the receiver can detect any one of them seperatly, and that is the meaning of non interference.
As you said, their instantanious values interfere, but over the time, the receiver will be able to recover every signal on its own, without interference from the other.

#### studiot

Joined Nov 9, 2007
4,998
OK lets go back and start at the beginning.

Two single individual vectors are orthogonal if their dot product is zero.

Thus i.j = j.k = i.k = 0

Now you are talking about orthogonal functions.

This term only applies to vector valued functions.

A vector valued function is like a regular function except that the numbers y and x in

y = f(x)

are replaced by vectors eg

V = f(U)

Where the function f assigns a vector V to any vector U.

Now consider two different functions of U, say f(U) and g(U).

the condition for orthogonality is fulfilled if for every pair of vectors, one from each function, f(U) and g(U) the product

f(U).g(U) = modulus f(U) times modulus g(U) times cos angle between them equals zero.

Now look again at your graphs. The phase angle is the angle between them. You can see this more clearly on a phasor diagram drawn as a rotating arm

Cos90 = 0, so whatever the modulus of the vector (= amplitude in this case) their product with cos 90 = zero

#### mentaaal

Joined Oct 17, 2005
451
Thanks again for the replies. Ok so what i get from your replies is that, although the two signals are adding here, because they are always 90 degrees apart their signals arent interfering with each other as such?

Ok say for example, a receiver were to pick up two signals like this, their phases being 90 degrees apart, how would the receiver separate the two and not have them add like they did in my plot?

#### Mark44

Joined Nov 26, 2007
628
The dot product that studiot mentioned is one example of an inner product. Assuming that a function space has an inner product, a typical one is a definite integral. (A function space has properties similar to a vector space, except that it is inhabited by functions instead of vectors.)

For the two functions f and g in a function space, an inner product <f, g> could be defined in this way:

<f, g> = $$\int f(t) g(t) dt$$

where the limits of integration are 0 and 2$$\pi$$.

It's fairly to easy to show that <sin(t), cos(t)> = 0, which should convince you that these functions are orthogonal.

A tip: Whenever you hear the word orthogonal, you should be thinking dot product or inner product.

Mark

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#### mentaaal

Joined Oct 17, 2005
451
Thanks mark, ok after much thought, i got these ideas in order in my head but i am still not sure how someone would separate two signals phase shifted by 90 degrees from each other? I understand now after readiing parseval's theorem that the total energy is sum of the energy of individual signals but my question is how does the receiver know that the signals are phase shifted and how to separate them?

#### Mark44

Joined Nov 26, 2007
628
Thanks mark, ok after much thought, i got these ideas in order in my head but i am still not sure how someone would separate two signals phase shifted by 90 degrees from each other? I understand now after readiing parseval's theorem that the total energy is sum of the energy of individual signals but my question is how does the receiver know that the signals are phase shifted and how to separate them?
I don't know. My expertise is more on the mathematics side, and not so much on the applied side that you're asking about.

#### mentaaal

Joined Oct 17, 2005
451
No problem, you cant be my own personal Jesus for everything i guess ;-)
But yeah i have a much better understanding of orthogonality now than before i posted this thread!

#### Dave

Joined Nov 17, 2003
6,969
Thanks mark, ok after much thought, i got these ideas in order in my head but i am still not sure how someone would separate two signals phase shifted by 90 degrees from each other? I understand now after readiing parseval's theorem that the total energy is sum of the energy of individual signals but my question is how does the receiver know that the signals are phase shifted and how to separate them?
Can I suggest you look up something call Phase Sensitive Demodulation. I started looking into this a few years back and, although information is scarce, it is out there.

Dave

#### mentaaal

Joined Oct 17, 2005
451
I certainly shall, thanks dave

#### markm

Joined Nov 11, 2008
16
Ok say for example, a receiver were to pick up two signals like this, their phases being 90 degrees apart, how would the receiver separate the two and not have them add like they did in my plot?

The receiver has to have a time reference so it knows when the peaks of the sin and cos waves arrive. In practice, that means that both waves are being transmitted from the same source, with the data stream(s) interrupted periodically to send a sync code. One way to proceed from there is to have a local oscillator synchronized to the sync codes producing a local copy of the sine wave carrier (that is, constant amplitude but in phase with the modulated sine wave). A phase shift circuit gives a local cos wave. Then you mix the received signal with these reference signals in two circuits that respond depending on the correlation between the reference and the received signals. I've forgotten the circuit details; nowadays, you usually just buy an IC that does it all.

Or if the signal isn't too high a frequency, you might just digitize it and program an FPGA to do fast Fourier transforms...

#### KL7AJ

Joined Nov 4, 2008
2,229
It's one thing to show how orthogonal signals work from a pure math standpoint, but it's quite another to visualize the physical implementation.
Coming from an R.F. background, it's natural for me to think of orthogonal signals as the equivalent of cross-polarized antennas. (Interestingly, but not surprisingly, the math for orthogonal antennas in the SPATIAL domain, translates precisely to orthogonal signals in the TIME domain).

At any rate, it's quite simple to see how a cross-polarized wave...either in light or R.F. or even wobbling rubber bands....can totally isolate energy in one plane from energy in the other plane. This is one of the wonderful symmetries of nature.

#### mentaaal

Joined Oct 17, 2005
451
Thanks for the replies guys, i actually hadnt noticed this thread in a while, i have so many questions on the go on this forum Yeah from a mathematical point of view its MUCH easier to understand than trying to grapple with conecept by looking at a plot!

Thanks guys!

#### berry-o

Joined Dec 1, 2008
1
can anybody please give me a concept of orthogonal signals with an example

#### mentaaal

Joined Oct 17, 2005
451
Sine(t) and cos(t) are orthogonal signals. i.e. they are 90° apart and as such they contain no components of each other.

#### hayden

Joined Jul 2, 2009
2
i am trying to understand the basis of orthogonal signals (from stratch) and i am tring to make the link between the following equations can anyone shed some clarity on it please

the first equation is a simple fourier series expansion which they used to represent a signal with and the other is some Nth dimensional expansion
x= $$\sum$$ $$\alpha$$$$\nu$$