# Optimizing Peak To Average Power Ratio Biology Essay

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Abstract--This paper present the high PAPR in MC-CDMA and investigates the fact that Reed Muller Codes plays a distinct role in PAPR optimization. Simulation results also proved that Reed Muller code, RM (2, 6) gives the optimal PAPR, which helps in providing high variable data rates as per application demands.

I. Introduction

Today requirements for applications such as multi-media, Internet, video conferencing needs high data rates. Certainly there are some solutions available to meet these needs. One of them is a Multi-Carrier Communication system for example 4th. generation OFDM, which provide high data rates with excellent performance. Similarly, MC-CDMA is an evolving 3rd. generation Multiple Access Technique, its merits are several: - 1) It is a derivative of Code Division Multiple Access (CDMA), 2) it provides high variable data rates, 3) It is orthogonal (Cross correlation between rows and columns is zero). The idea behind MC-CDMA is to assign multiple numbers of channels to the user, who wants to transmit at higher data rates. These multiple channels appear at the Base Station (BS) as multiple users. Each channel is multiplied by the given data rates of CDMA, which give rise to the data rates of MC-CDMA [1, 3, chapter 1, p.1].

MC-CDMA is based on summation of signals for transmission, from an orthogonal set, which are called discrete time Walsh-Hadamard sequences. This summation of signals in the form of high peaks present at the forward link of transmitter (BS) becomes hurdle for it and degrades the performance of the Power Amplifier (PA) [7, 8, 9], which works best in its linear region. In order to maintain best efficiency of PA, its Input Back off (IBO) should be high. In other words average input power of the signal should be lower to the maximum input power of signal.

(1)

The said hurdle PAPR, can be resolved by encoding the message [1,3]or other techniques but our concentration will be on use of codes , which yields optimal PAPR, good code rate and good minimum hamming distance (PAPR, Rc, dmin). We will address this issue in the subsequent sections. This paper is

organized as follows: In Section II we will discuss MC-CDMA Communication Model, which includes PAPR, Coding Solution, and Reed Muller Code, Section III gives Simulation results and Section IV provides Conclusion with open issues.

II. MC-CDMA Communication Model

## Message

## (m1,m2â€¦mn-1)

Fig 1: Transmitter End (Partial)

As shown in Fig.1, our communication model consists of Message (m), Message Encoder (ENCm), and Walsh Hadamard Orthogonal Transform (WHOT). However, this is a basic model but, it describes our key objective, which is to achieve optimal PAPR. The message consists of a binary vector m = (m0, m1, m2â€¦mn-1) is input to the (ENCm), which encodes the message using code depending on the length of the message. The encoded message (Cm) is then feed to WHOT, which made the Cm orthogonal for transmission.

Each data bit of the message mi is modified from 0 and 1 to 1 and -1 respectively and then multiplied by the Walsh Hadamard code and the result is the sum of the 'n' modulated function, so the transmitted signal Scm(t) [1], which is 'n' times the basic rate of CDMA is given as follows:

(2)

Where,

'n' is the size or length of matrix and WH(2k)

(3)

is a Sylvester type 2k x 2k Walsh Hadamard matrix, whose cross correlation of columns and rows is zero. Note that Scm(t) is a discrete signal and transform is discrete.

A. Peak-to-Average Power Ratio

The instantaneous Power Pm (t) [1] of a signal at time't' for the message 'm' is given by

(4)

Now, we define PAPR as a measuring technique for high Peaks or Amplitude of a signal strength [8,9]. It is the ratio of the maximum Peak to the Average Power of a message 'm' and mathematically can be expressed as follows:

(5)

Please note that the maximum PAPR will be equal to 'n'.

Example 1.

Let n=16 and the message m1 = 01011111100001112 and message m2 = 11111111111111112. The calculated PAPR of MC-CDMA signal corresponding to the said messages are 4 and16 respectively and further note that max PAPR = 16 is equal to n as shown in Fig. 2.

## Fig 2: PAPR of MC-CDMA Signal

B. Coding Solution

As stated earlier PAPR is a problem for both OFDM and MC-CDMA [1, 3]. Now the question arises, what method should we adopt to cure this problem? There are ways to reduce it, like Clipping and filtering, where high order harmonics are filtered using low pass filtering, which results in re-production of peak power. This method is simple, but suffers with degradation in bit error performance. Another method is Partial Transmit Sequences (PTS). This method gives about 4dB reduction in PAPR for 256 sub-carrier with low redundancy and without distortion [5]. There are some more methods, but regretfully, it is not possible to discuss each of them, instead a nice consolidated comparison table 1 [11, Chapter 8, p.248] is given below:-

Table 1

Comparison of methods

## Method

## PMEPR

## Distortion

## Rate

## Hit

## Side

## Inform

## Complexity

Coding

H

N

H

N

H

Clipping

H

Y

L

N

L

SLM

M

N

L-H

Y

L-H

Balancing

M

N

H

N

L

Codes of

Strength (CS)

M

N

L

Y

H

Trellis

Shaping(TS)

M

N

L-H

N

H

Tone Injection (TI)

M

N

H

N

L-H

ACE

L

N

L

N

H

Constellation

Shaping

H

N

H

N

H

PTS

M

N

L-H

Y

L-H

Reduction

Carrier(PRC)

L

N

L-H

N

H

## L, low; M, moderate; H, high; Y, yes; N, no.

Our focus is on coding solution. Here theidea is to select a code Cm {0, 1} n from the existing one or construct a new one, which when multiplied by the message yields optimal PAPR. The message encoder (ENCm) is inserted between message (m) and WHOT [1], already shown in Figure 1. This gives optimize PAPR and reliable data, but at the expense of higher bandwidth. At the receiver end inverse transform and decoding is performed for the recovery of original data.

Before we elaborate the above paragraph, there are several things to consider:

Achieve high code rate of coded message Rm

Minimum distance of coded message dcm should be high for good error correction capabilities

Low PAPR(m) for optimal power and economical communication

Redundancy Rd is introduced in the encoded message, but at the expense of Bandwidth.

Theorem 1. The minimum distance of a linear code is the minimum weight of any non-zero codeword. [10]

Theorem 2. A code with minimum distance d can correct [1/2(d-1) errors*. If d is even, the code can simultaneously correct 1/2 (d-2) errors and detect d/2 errors. [10]

*[x] denotes the greatest integer less than or equal to x. E.g. [3.5] =3, [-1.5] = -2

The following example will reinforce the concept of Rm, dcm and Rd. We will use the notation (k, m), where 'k' is the total length of message and 'm' is the message.

Example 2.

This example demonstrate the use of simple Code (4, 2)

Table 2

Generator Matrix (4, 2)

## Messages Code words

00 0000

01 1001

10 1010

11 0011

Rm = m/k = 1/2 (each code bit contains Â½ bit of information)

dcm = 2 (It can correct simultaneously zero error and detect one error)

Rd = (k-m)/m = 100% (redundancy will require double the bandwidth)

The sum of any two code words generates a codeword, which is member of the subspace as shown in the above-mentioned example. This example is simple to implement, because we have 16 message vectors and 4 code words, but what about if we have a code (64,32), which will require 1.8447e+019 message vectors and 4.2950e+009 code words so this will require an exhaustive search to find code, which should meet closure property requirement. However, it is possible to reduce the exhaustive search by generating the required codeword as needed [13, Chapter 6, p.355] with low PAPR. There are existing codes available, which can give us good minimum distance, error correction, and code rate like BCH, and Reed Muller code. Regarding new code designing, which results in low PAPR,[ 2,4] has a great contribution.

C. Reed Muller Codes

Reed Muller (or RM) codes are some of the oldest best error correcting and easy to understood families of codes. Their application is useful in Telecommunication. Reed Muller codes were invented in 1954 by D.E. Muller and I.S.Reed.

However, except for first order RM codes and codes of modest block length, their minimum length is lower than BCH codes. But its great merit is easy decoding using majority-logic circuits [10 Chapter, 13, p.370]

We define binary Reed Muller code RM(r, m) of order r, r = 0, 1 â€¦.m [11, Chapter 3, p.57] has the triplets (n, k, d) as follows:-

(6)

Where n denotes the total length of the message, k is used to determine the number of rows in RM code, and d is the minimum distance. Following example is used to understand the generation matrix of Reed Muller code:

Example 3.

RM (2, 4) is of order 2, has length 16, dimension 4 and minimum distance is also 4.

As per theorem 2 it can correct 1 error and detect 2 errors.

Table 3

Generator Matrix RM (2, 4)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

x1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

x2 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0

x3 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0

x4 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

x1x2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

x1x3 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0

x1x4 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0

x2x3 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0

x2x4 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0

x3x4 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0

The following steps (1 - 4) are extracted from [6] in constructing the matrix of the above example:

The vector associated with monomial 1 is simply a vector of length 2m, where every entry of the vector is 1. For example 24 , the vector associated with 1 is (1111111111111111).

Vector associated with monomial x1 is 2m-1 ones followed by 2m-1 zeros.

Similarly, vector associated with monomial x2 is 2m-2 ones followed by 2m-2 zeros, then another 2m-2 ones followed by another 2m-2 zeros.

In general, the vector associated with a monomial xi is a pattern of 2m-i ones followed by 2m-i zeros until 2m or length of the code is achieved.

After putting the monomial in reduced form, then perform the logical operation AND with the vectors associated with each monomial xi in the reduced form. For example the value of x3x4 is obtained by ANDed the operation between the vectors associated with x3 and x4. This process goes on until we reached the maximum number of rows i.e. 'k'.

III. Simulation results

Table 4

Simulation Parameters

## S.no

## Assumptions

## 1

## Multiple Access Technique: CDMA

## 2

## n users and 64 channels per cell

## 3

## data rate:- Rate set 2 (14.4kbps)

## 4

## Modulation: BPSK

## 5

## Reed Muller code = (2,6)

## "Channel Coding techniques are

## commonly applied in OFDM and

## CDMA" [12]

## 6

## frequency reuse factor 100%

## 7

## Area 2 Sq.km

## 8

## Two ray ground model

We have used MatLab to run the simulation. For coding solution RM (2, 6) has been used, which has 7 error corrections and 8 error detection capabilities. Using the notation (n, k, d) to represent 2nd. order RM (2, 6), we have generated a matrix of 22 x 64 of RM code, where n=64, k=22 and d=8. This triplet (64, 22, 8) has code rate of 0.340 and redundancy about 3, and PAPR is 98% optimal values, which are 1, 4 and 2% (16) as shown in Fig.3. Variable data rates up to approximately 900kbps are also possible, because of this low PAPR, but again at the expense of double bandwidth.

Reason for taking 64 bit length of RM is that, because CDMA has 64 Walsh Codes. Out of which, W0 is used as pilot channel, W1 to W7 only one for paging and W32 for sync. Remaining 61 Walsh codes may be used for traffic channels.

We have also run the simulation for different message length for e.g... 21,20,19,18 etcâ€¦ but PAPR is same i.e... 1 & 4. However, the efficient code rate is obtained at the message length of 22.

## Fig 3: Optimal PAPR

IV. Conclusion

From the section of Coding Solution and Reed Muller codes, we have seen that for an optimal PAPR, there are two important parameters to be consider: - which are good code rate Rc, and large minimum distance dm.

These characteristics are available in RM code, which helped us in achieving our target i.e.â€¦optimal PAPR

and high variable data rates. Beside the said features of RM code its matrix generation and encoding are easy too.

There are some following open issues:-

We have achieved optimal but variable PAPR (99% low, 1% high). There should be a way to have constant amplitude PAPR.

"QPSK modulation in place of BPSK may be used in Multi-code CDMA" [1]

"Are there mappings from binary (or quaternary) to Octary or, more generally, to q-ary constant amplitude codes?" [3]