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What are orthogonal binary sequences?

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Orthogonal

short answer whose cross corelation is ideally zero and autocorelation is equal to the half the length of the sequence (if even)

the number of ones and zeros (-1) are the same

e.g
Walsh codes, Gold codes

hope that helps you out. for more confusion do let me know.

Gauiver
 

Re: Orthogonal

salam (in farsi = Hello)
example from walsh code

W(0,4) = 1, 1, 1, 1
W(1,4) = 1,-1, 1,-1
W(2,4) = 1, 1,-1,-1
W(3,4) = 1,-1,-1, 1
 

Re: Orthogonal

What about PN Sequence? It is also "orthogonal", isn't it?

Added after 20 minutes:

Walsh Codes are binary vectors or vectors with ±1?
 

Re: Orthogonal

In a Code Division Multiplexing system, a spreading sequence is used to spread the message signal in frequency domain.
I'm a starter in CDMA system, I've been refering to several article and books. Some show the message signal multiplied with the spreading sequence while some show XOR operations (See the figure). I am confused. Which operation is used between message signal and spread spectrum?

Regards
M
 

Orthogonal

no pn-codes are not orthognal codes. they are codes with low auto-correlation. Walsh codes or gold codes are orthognal codes. about 0 or -1 it doesnot matter. they are replaced just to check the auto-corelation and cross corelation property.
hope that helps, do correct me if wrong.
 

Orthogonal

So although PN codes are not orthogonal codes, due to their low auto correlations, they can still be used in CDMA system, would that be correct?
 

Orthogonal

Here are two sequences I obtained using LFSR with 4 register and taps at [4,3]
Seq1 = [111100010011010]
Seq2 = [011110001001101]

In order to prove orthogonality of the two sequences, I multipled the sequences bitwise (i.e AND operation) and added the resulting sequence bitwise (i.e XOR) to obtain zero.

[111100010011010]
[011110001001101]
------------------
[011100000001000]

Since there are four 1's and eleven 0's, after adding individual bits, result is zero.
Am I correct in this proof?
If not how do i do it?

M
 

Re: Orthogonal

Hi
For proving orthogonal consider w(1,4) = 1, -1, 1, -1 and W(3,4)=1, -1, -1, 1

Multiply bit wise and add...
result= (1x1)+(-1x-1)+(1x-1)+(-1x1)=0
Thus the two sequences w(1,4) and w(3,4) are orthogonal

In terms of XOR...
Consider the 1,0 representation of w(1,4)=1 0 1 0 and w(3,4)=1 0 0 1
Bitwise XOR will give 0 0 1 1 and then add bitwise 0+0+1+1=0
Hence orthogonal....
I hope this helps you
 

Orthogonal

Thanks xischaune, i agree with the first part of ur explanation, but regarding "In terms of XOR..." how does it show orthogonality of two sequence?
For orthogonality, integration of product of two sequence should be zero...isn't it so?
 

Re: Orthogonal

Classical Product integration is more easily understood if the sequences are analog in nature...
For digital (discrete) case...product integration is equivalent to bitwise multiplication and then adding...as ive shown earlier...
Hope this is clear...
 

Orthogonal

But, you have used bitwise XOR operation instead of bitwise multiplying? XOR is modulo-2 addition operation isn't it?
 

Re: Orthogonal

magnetra said:
What are orthogonal binary sequences?
Click to expand...
if two binary sequence of the same length have the inner product equal to zero the it is said that these two sequence are orthogonal. inner product in every spave is defined
in a particular way, usually the cross correlation is defined as inner product in communication
 

Re: Orthogonal

magnetra said:
But, you have used bitwise XOR operation instead of bitwise multiplying? XOR is modulo-2 addition operation isn't it?
Click to expand...

yes definitely it is. but actually thats what is happening in binary addition.
secondly its better to check the orthognality or auto corelation property or cross corelation property its better to check as said by xischaune

xischaune said:
But, you have used bitwise XOR operation instead of bitwise multiplying? XOR is modulo-2 addition operation isn't it?
Click to expand...

thirdly in the upper part sorry guys for mistake. gold codes are also not orthognal rather codes with low auto / cross correlation property like pn codes.

for more reference check
CDMA system Engineering
digital communication systems by sklar

hope that it helps
rgds
Gauiver
 

Re: Orthogonal

Hello,
I think so that the others provide you with good explenation about the codes orthogonality, but however, Gold code is not an orthogonal code, because the crosscorrelation value between two gold codes is 1/N, where N is the code length, walsh codes are fully orthogonal codes so the cross correlation is zero, there are also m-sequence, Kasami ...etc
try to find this manual, it is very good for PN code generation

A. Lam and S. Tantaratana, Theory and Applications of Spread-Spectrum Systems, IEEE, Inc, USA, 1994.
I prepared few explenations about the PN codes, maybe you can understand the idea in a good way by refering to the attached doc.
Best Wishes
Nidhal
 

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