Orthogonal Codes in CDMA

8/10/2019 Orthogonal Codes in CDMA

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Orthogonal Codes in CDMA:

Generation and Simulation

Introduction

CDMA (Code Division Multiple Access) is a communication technique that allowsmultiple users to communicate simultaneously over one frequency. This is achieved

through the use ofspreading codes, whereby a single data bit is "spread" over a longer

sequence of transmitted bits. These codes, known as chip sequences, must be carefully

chosen so that the data may be correctly "despread" at the receiver. Such codes areknown as orthogonalcodes.

This document introduces some fundamentals of CDMA, necessary for understanding of

the subsequent material. It discusses some of the issues involved, then describes theappended C programs as guides to generating orthogonal codes, and simulating the

coding/decoding procedure.

CDMA Basics

Each user is assigned a chip sequence. This is simply a sequence of bits (chips). The

spreading process is shown below, using an example:

Chip Sequence: 1 0 1 1 0 0

Spreading Sequence: 1 -1 1 1 -1 -1

Transmitted bits, data = 1: 1 -1 1 1 -1 -1

Transmitted bits, data = 0: -1 1 -1 -1 1 1

No transmission: 0 0 0 0 0 0

In its simplest form, decoding involves calculating the dot product of the received bitstream with one users spreading sequence. Note that when signals arrive at the receiver

from different users, they linearly add. See the example below.

Length of chip sequences = n= 4


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StationSpreading

SequenceData

Transmitted

Sequence

Dot

ProductDP / n

A -1 +1 -1 +1 Low +1 -1 +1 -1 -4 1

B +1 +1 -1 0 High +1 +1 -1 -1 +4 -1

C -1 -1 1 0 N/T 0 0 0 0 0 0

Received: 2 0 0 -2

Thus all transmitted data has been recovered.

Investigation

The above method makes two key assumptions: code synchronization and chip

synchronization. Code synch indicates that the transmitted sequences from each user

begin at the same time. Chip synch implies that all users transmit a chip (bit) at the sametime. In a realistic CDMA system, neither assumption is valid.

Definition of Practical Orthogonality

Ultimately, the chip sequences must be orthogonal for correct transmission/reception.

What is orthogonality? Mathematically speaking, if we have nusers and n-bit chip

sequences, then a set of vectors in n-space are orthogonal if any point in n-space may be

expressed as only one linear combination of these vectors. In a more practical sense,vectors are orthogonal if they allow correct decoding of transmitted data. This definition

may sound contrived at first, and slower to calculate, but its usefulness will become

apparent later as we investigate chip synchronization issues.

Here is aC programthat tests for practical orthogonality. Near the top of the file, thenumber of users, code lengths, and actual codes are entered. Then compile the programand run it to see if the codes are orthogonal. When testing, it uses the idea that each user

has three transmission possibilities: logic high, logic low, and no transmission. All

possible combinations are tried.
http://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_simple.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_simple.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_simple.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_simple.c


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Code Synchronization

In this section, we will investigate chip sequences that are orthogonal regardless of code

asynchrony. The key idea is to use codes that are longer than nbits (for nusers). This"wasted" bandwidth will solve code synchronization problems, thus negating the need for

other solutions (such as a high-power synch carrier).

The test for orthogonality despite code synchronization issues is very similar to the basic

orthogonality test. However, for every transmission combination, each code must be

rotated in all possibilities relative to each other. So for 4-user, 8-bit codes, this programwill take 8

4=4096 times as long as the previous program. However, the run time is still

acceptable.

The steps for configuringthis programare identical to those above: set the number of

users and code length, then the chip sequences.

Finding Suitable Codes

Another programfinds a set of chip sequences that solve the code synchronization

problem above. To configure, simply choose the desired number of users. Uponinitialization, the program begins to try all possible codes, determining orthogonality by

the same criteria as above. It begins with codes of length n(the number of users), and if

necessary, increments the code length until a solution is found. If one solution is found

for a particular code length, all solutions of that code length are generated and output

before the program finishes.

Unfortunately, due to the number of possible codes, this program is too slow for actualuse. Significant optimization (i.e. identifying and pruning combinations that cannot work,

rather than trying all possibilities) would be necessary for this program to be useful.

However, it was written, and is presented here should anyone find a use for it. Note thatthis program is written in C++ (almost pure C, but variables are defined anywhere, not

just at the top of functions). This was done to make the program easier to understand.

Chip Synchronization

We will not be developing a program to solve chip synchronization issues. Unlike the

code asynchrony problem, which had a discrete (and finite) number of possibilities, the

number of ways in which bits may be unsynchronized is infinite. Therefore, this programwill be a simulation.
http://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_rotate.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_rotate.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_rotate.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/find_orth.cchttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/find_orth.cchttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/find_orth.cchttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/orth_rotate.c


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To solve the chip synchronization issue, supersampling is used. That is, rather than

sample the received waveform once per bit period, multiple samples are taken during the

same period. A supersampling value of 4, for example, should provide enough extraresolution to handle all cases (i.e. sampling on a bit transition). The purpose of the

simulation is to determine how accurately the transmitted signals can be decoded.

This programsimulates the CDMA transmission and decoding process. It requires an

input file that specifies the transmission sequence. Here is asample input file.See the

comments at the top of the program file for more details. This program creates a set offiles for every station being simulated. These results are intended to be viewed with

gnuplot, a freeware plotting program. If running in a unix environment, with the sample

input file given above, use the following command line: simul1 simul1.in appThiswill produce the following files:

app0.comm app0.data

app1.comm app1.dataapp2.comm app2.data

app3.comm app3.data

To start gnuplot, simple type gnuplot. To view the results of the first station from within

gnuplot, type load "app0.comm"(this command, as well as all that follow, can easily beextended for any particular station). This will graph the results of the simulation,considering the decoded data for the first chip sequence. The following three series are

produced:

1.

"User X Data" - indicates the correct value that should be received for this user. It

stays at this value for the first four samples (i.e. throughout the first chip in thesequence) and drops to -50 elsewhere to indicate where the sequence should NOT

be despread correctly at these times. The received value, (-1, 0, or +1), is

multiplied by 32 for easy comparison with other series.

2. "User X Dot Prod" - is the dot product with taking one sample during each over-sampling period (this is the method actually used for this project). So the dot

product vectors will be as wide as the chip sequences. This result is multiplied but

the over-sampling rate (4) for easy comparison.

3. "User X DP Quad Sum" - this is the result of adding four adjacent dot products,each calculated as above for the second series. In effect, this averages the dot

product values. This series is just for comparison.

The ideal case shows all three plots having the same value when the data graph is not -50

(this is when the incoming samples should be decoded). In other words, correlation is

highest at these points and is lower elsewhere. A sample plot is shown below: Clickherefor the original Postscript file.
http://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.chttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.inhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.inhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.inhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.commhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.commhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.datahttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.datahttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.pshttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.pshttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.pshttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.pshttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.datahttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/app0.commhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.inhttp://www.ece.ualberta.ca/~elliott/ee552/studentAppNotes/2000f/misc/CDMA/simul1.c


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To generate a postscript file for a particular plot, use the following sequence of gnuplot

commands:

set terminal postscript

set output "app0.ps"

load "app0.comm"

Now instead of displaying the graph on the current display, it will be saved to disk usingthe given filename.

Stephen Somogyi

CDMA Group

December 4, 2000

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Orthogonal Codes in CDMA