# Power Code System

Effect of using different codes for Power reduction

Abstract

This paper presents the investigative study of power reduction technique using different coding schemes while keeping the low probability of bit error. Hamming, extended Golay and BCH (Bos-chdhuri-Hocquenghem) are selected to illustrate the purpose of power reduction. For the simulation purpose different rates along with different coding techniques are selected. The results show that an efficient and rightful selection of code can improve the performance of any communication system with lower Eb/No (Ratio of the signal energy per bit to noise power density) and bit error values.

1. Introduction

Recently there is a tremendous growth in digital communications sector especially in the fields of wireless and computer communication. In these communication systems, the information is represented as a sequence of binary bits. These binary bits are transformed into analog wave forms using different modulation techniques. The communication channel introduces noise and interference that corrupts the transmitted signal. So at the time of reception corrupted signal is received. Bit errors may result due to the transmission and the number of bit errors depends on the amount of noise and interference in the communication channel. Channel coding is often used in digital communication systems to protect the digital information from noise and interference and reduce the number of bit errors.

Use of channel coding as shown in figure 1 to design low bit error rate communication system has recently remained an active research area [1][2][3]. The ability of different codes to detect and correct data at receiving side improves the quality of communication and also minimizes the chances of re-transmissions. The importance of error rate is realized in [1] [2] [4] [5].

Apart from the ability of using codes to minimize the error rate the possibility of reducing peak power by using specific codes is now under consideration [1][2][3]. Use of codes to reduce power will help in building robust and stable systems with better quality of voice and data.

With the preference of wireless devices for the communication purpose the focus of research has now mainly shifted to wireless communication system design and also takes the channel coding aspect with it. The battery power is a big constraint for the long duration communication and adjustment of power during calls cause a lot of power usage. This problem gives a clear motivation for investigating such codes that can reduce the power transmission while keeping the same or low bit error rate [1].

This paper presents an investigative study of using popular codes to minimize the transmission power. Effect of using codes on transmitted power is investigated in detail and observations are noted about the effect on bit error rate, which remains the main quality standard.

This paper is divided into 5 sections. Section 2 provides details about the well know codes like Hamming, Golay and BCH that are used in simulations. Section 3 presents details about simulation setup. Section 4 provides results and discussion on them. Finally section 5 gives conclusion and future directives.

2. Types of Codes

Channel coding is mostly used in digital communication systems to protect the digital information from noise and interference and reduce the number of bit errors. Channel coding is mostly accomplished by selectively introducing redundant bits into the transmitted information stream. Addition of these bits will allow detection and correction of bit errors in the received data bit stream, and provide more reliable information transmission. There are two main types of channel codes, Block codes & Convolution codes.

Block codes are based rigorously on finite field arithmetic and abstract algebra. They can be used to either detect or correct errors. Block codes accept a block of k information bits and produce a block of n coded bits. By predetermined rules, n-k redundant bits are added to the k information bits to form the_{} coded bits. Commonly, these codes are referred to as_{} block codes. Some of the commonly used block codes are Hamming codes, Golay codes, Extended Golay, BCH codes, and Reed Solomon codes [6]. This research work has utilized three of the well know block code namely Hamming, Extended Golay and BCH. Brief description is given as below.

2.1 Hamming

Extension of hamming codes that can correct one and detect more than one error is widely used in different applications. The main principal behind working of Hamming codes is parity. Parity is used to detect and correct errors. These parity bits are the resultant of applying parity check on different combination of data bits. Structural representation of Hamming codes can be given as [6]

(1)

Where_{} .. For hamming codes Syndrome decoding is well suited. It is possible to use syndrome to act as a binary pointer to identify location of error.

If hard decision decoding is assumed then the probability of bit error_{}can be given as

(2)

Where p is the channel symbol error probability [6]. An identical equation can be written as [6].

(3)

2.2 Extended Golay

The extended Golay code uses 12 bits of data and coded it in 24-bit word. This (24,12) extended Golay is derived by adding parity bit to (23, 12) Golay code. This added parity bit increases the minimum distance_{}from 7 to 8 and produces a rate code, which is easier to implement than the rate_{} that is original Golay code [6].

Though the advantages of using extended Golay is much more to that of ordinary Golay but at the same time the complexity of decoder increases and with the increase in code word size the bandwidth is also utilized more. Extended Golay is also considered more reliable and powerful as compared to Hamming code. If probability of bit error is given by Pb and dmin is 8 with the assumption of hard decision then error probability is given by [6]

(4)

2.3 BCH Codes

BCH belongs to powerful class of cyclic codes. BCH codes are powerful enough to detect multiple errors. The most commonly used BCH codes employs a binary alphabet and a codeword block length of n= 2^m-1, where m= 3, 4,...... [6].

3. Simulation

In this simulation we used fixed data rate for e.g 9600 with different code rates of hamming, golay and BCH.We have also assumed the modulation to be BPSK (binary phase shift keying). Matlab is used as a model simulating tool. Eb/No is taken as a power comparison parameter for coded and uncoded signal. The_{} parameter is calculated using formula [6]

(5)

Where R is the data rate in bits per second. Pr/No is ratio the received power to the noise.

Apart from_{}comparison, the value of bit error rate is also compaired for coded and uncoded bit stream which is calculated by the formula given in eq 6 & eq respectively [6]

(6)

(7)

Where Q(x) is called complementary error function or co-error function, it is commonly used symbol for probability under the tail of Gaussian pdf. Where Pu is probability of error in un-coded bit sequence and Pc is the probability of error in coded bit sequence. Ec/No is ratio of energy per bit to noise spectrum density of coded bit sequence.

Finally the most important parameter that shows the edge of using codes with the data is calculated which is probability of bit detected correctly for coded and uncoded bit sequence which is given by equations [6]

(8)

(9)

_{}and _{}are probability of un-coded message block received in error and probability of coded block received in error respectively.

4. Simulation results

This section presents graphs that are obtained through simulation. The parameter of investigation are _{}of coded and uncoded sequences, Correlation Coefficient (coded power, code rate and error) and bit error propabiliy at receiving end.

Graph of figure 2 represents ratio of energy per bit to noise power density_{}on X-axis and error probability on the Y-axis. Graph shows that (error right curve) using code we have aquired reduction in power from 4 to 2 dB with same low probability of error i.e 3x10^{-2. }. Thus with the help of codes it is shown that more reliable transmission with the reduced power is posible. Thus the power can be efficiently used and will help to improve the up time for the mobile devices with better quality of data transaction.

Graph of figure 3 shows correlation coefficient (Coded Power & Coded rate). This graph is based of three attributes of signal. One is Coded power, second is coded rate and third is Error. Interresting part to be noted about the graph is that from 12 dB(coded power) onwards we have almost '0' probability of error.

Finally graph of Figure 4 represents the most important comparision of different codes performances. This graph is obtained by keeping the same data rate and varying code rate and code types. For example as decsribed in ealier section three well know block code like Hamming, Extended Golay and BCH are taken for study. For hamming three different rate that is (7,4), (15,11) and (31,26) are taken. For Extended Golay (24,12) and for BCH (127,64) and (127,36) rates are used. The graph show the performance of all the codes with Eb/No on X-axis and Error probability on the Y-axis. It can be eassily infered by the given graph that Golay and BCH shows better performance and these codes give the optimal power of 3.6 dB.

5. Conclusions

In this research we inferred that using codes we have low probaility of error with reduced power. Golay and BCH gave optimized power. For future work, codes may be used to implement on multi-carrier systems such as CDMA and OFDM.

References

[1]. *Kenneth G. Paterson, Member, IEEE On Codes with Low Peak-to-Average Power Ratio for Multi-Code CDMA IEEE Transactions on Information Theory, VOL 50 (3), 2004, 550-559 *

[2]. *Tony Ottosson Precoding in Multicode DS-CDMA Systems 1997 IEEE*

[3]. *R. Neil Braithwaite Using Walsh Code Selection to Reduce the Power Variance of Band-Limited Forward-Link CDMA Waveforms IEEE Journal on Selected Areas on Communications, VOL. 18, NO. 11 NOVEMBER 2000.*

[4]. *Samuel C. Yang: CDMA RF SYSTEM ENGINEERING, Artech House, INC, 685 Canton Street Norwood, MA 02062, 1998*

[5].* Henrik Schulze and Christen Luders Theory and application of OFDM and CDMA Wideband Wireless Communication, John Wiley and Sons, Ltd, 2005*

[6].* Bernard Sklar, Digital Communications: Fundamentals and Applications, 2nd Edition, Pearson Education, 2006*

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