What is Orthogonal Variable Spreading Factors? (A note about OVSF Codes)

With the advancement in the cellular technology and convergence of wireless technologies, now it is the need to combine two messages having different data rates in an orthogonal manner. Take an example, the date rate of user 1 is r₁ and of user 2 is r₂ and we have to spread the user 1 message by spreading factor s₁ and that of user 2 by s₂, so that we can produce and overall chip rate of ρ. we can use Walsh Hadamard sequences if the spreading factors are powers of 2. The result so obtained is referred to as Orthogonal Variable Spreading factor and is showed in figure 1.

 

Orthogonal Variable Spreading Factor - An Illustration

 

For OVSF the orthogonality requirement can be stated mathematically as (for the given figure)

 

ovsf orthogonality requirement

 

 

This is, code 1, of duration T₁ is orthogonal to all subsequences of code 2, of the same length, and offset by a multiple of T₁, the length of code 1.

Now, as we know, OVSF is important in terms of WCDMA and LTE, the question is how we will generate OVSF codes? We can generate OVSF codes by following algorithm.

Suppose we want to generate OVSF codes of length n₁ and n₂, where n₁ < n₂, then:

1. Construct H_n1 by usual Walsh-Hadamard algorithm.
   
2. Choose one row of matrix H_n1 as the code length of 2ⁿ¹. Let H_n1‘ represent the Hadamard matrix with the selected row removed.
   
3. Continue the Walsh- Hadamard algorithm with H_n1‘: that is
   

 

Modified Matrix for OVSF

 

 

Then continue until desired H_n2‘ is constructed.

1. Choose any row of H_n2‘ as the spreading code of length 2n2
   
2. If a third code is needed, then continue in a similar manner.
   

A matlab function can be downloaded from the given below links. The usage is given below.

Usage

% x = ovsf(y,z)

%x, the output cell array having required OVSF codes

%y, number of codes required

%z, array having length of each code to the base 2 i.e. 2.^2 = 4

%example y = 3 z = [4 2 3]

%this will give a cell array with [1x4] [1x8] [1x16]

%length = 2.^n , enter n in length of array

%The code is not optimized yet, user is free to optimize

%do not forget to leave a comment

 

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Author : Ashwini Patankar

A note about Walsh Code !

In IS 95, 64 Walsh Functions are used to identify the forward (down) link channels. Walsh Codes or Walsh functions are generated from a matrix called Walsh Hadamard Matrix and also known as Walsh Hadamard Sequences. We can construct two orthogonal sequences of length 2 easily and are given as the rows of the below matrix

 

Walsh Matrix

 

 

We can construct 2ⁿ orthogonal sequences of length 2ⁿ from sequences of length 2^n-1 by equation below.

 

Walsh Matrix 2

 

 

By using “walshcode.m” we can generate any number of walsh sequences. The usage is given below:

Var = walshcode(n);

The function will return walsh matrix, you can also get a randomly picked sequence from the matrix, for this you will have to modify the file little bit (you will have to cut and paste ‘%’ from the beginning of a line to another). The matlab script can be downloaded from here

the file is also available on Matlab File Exchange

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Author: Ashwini Patankar

A note about PN Sequence

PN sequences or Pseudo Noise sequence is a periodic binary code which is random in nature generated by the use of shift registers, but generated with taking into considerations some generator polynomials. For sequence to be a pseudo noise or pseudo random it should follow the following basic rules. The rules mentioned below are simply mentioned in brief:

1. The relative frequency of 0′s and 1′s are each ½
2. The run lengths of 0′s and 1′s are: ½ of all run lengths are of length 1; ¼ are of length 2; 1/8 are of length 3; and so on.
3. If a PN sequence is shifted by any non zero number of elements, the resulting sequence will have an equal number of agreements and disagreements with respect to the original sequence.

These properties are known also known as balance property, run property, and correlation property respectively. This code is orthogonal in nature. PN sequence is also known as Maximal Length Sequences.

PN sequences are used in spread spectrum systems, like CDMA, WCDMA, and Radar etc. In CDMA IS 95, 64 ling PN sequence codes are used for the identification if the reverse link channels.

The matlab script for generating pn sequence is given below, and also can be downloaded from this link.

refer figure as a companion for the matlab file

 

block diagram of pn sequence generator

 

 

function [op_seq] = pnseq (a, b, c)

% a : no of fliflops; b = tapp _ function starting frm highest order; c = initial stae

%tapp functions or genrator polynomial

%e.g. no of flip flops 4 ==> a = 4

%generator polynomial x4+x+1 ==> b = [1 0 0 1]

%initial state [1 0 0 0] ==> c = [1 0 0 0]

%refere figure to set a relation between tap function and initial state

%

%

%

%

% <

% ———–X____________________________________

% | ___ | _____ ____ _____ |

% |> | | | | | | | | | ^|

% —–| |——–| |—| |——-| |————->o/p

% —– —– —– —-

% x1 + x2 + x3 + x4

%initial state

% 1 0 0 1

%

%take care of the reverse order of genrator polynomial and intiial states

%

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

close all;

clc;

x = a;

tap_ff =b;

int_stat= c;

for i = 1:1: length(int_stat)

old_stat(i) = int_stat(i);

gen_pol(i) = tap_ff(i);

end

len = (2 ^x)-1;

gen_pol(i+1)= 1;

gen_l = length(gen_pol);

old_l = length(old_stat);

for i1 = 1: 1:len

% feed back input genration

t = 1;

for i2 = 1:old_l

if gen_pol(i2)==1

stat_str(t) = old_stat(gen_l – i2);

i2 = i2+1;

t = t+1;

else

i2 = i2+1;

end

end

stat_l = length(stat_str);

feed_ip = stat_str(1);

for i3 = 1: stat_l-1

feed_ip = bitxor(feed_ip,stat_str(i3 + 1));

feed_ipmag(i1) = feed_ip;

i3 = i3+1;

end

% shifting elements

new_stat = feed_ip;

for i4 = 1:1:old_l

new_stat(i4+1) = old_stat(i4);

old_stat(i4)= new_stat(i4);

end

op_seq(i1) = new_stat(old_l +1);

end

%op_seq;

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Author: Ashwini Patankar