2008 International Conference on Emerging TechnologiesIEEE-ICET 2008Rawalpindi, Pakistan, 18-19 October, 2008

GOLAY CODE BASED CARRIER PHASE RECOVERY

Aamir HussainDepartment of Electrical Engineering

College of Electrical & Mechanical EnggNational university of Sciences and Technology, NUST

Rawalpindi, Pakistanaamirhussain7@hotmail.com

Mohammad Bilal MalikDepartment of Electrical Engineering,

College of Electrical & Mechanical Engg;National university of Sciences and Technology, NUST

Rawalpindi, Pakistanmbmalik@ieee.org

Abstract- In this paper we present a Golay code basedcarrier phase recovery scheme. Golay code is a pair ofcomplementary codes. Golay codes have been used forchannel coding in communications. We have used theside lobe suppression property of Golay Code inrecovery of the phase of the carrier. We begin with thediscussion of side lobes suppression property of theGolay code. Then we present the pass band scheme i.e.the QPSK scheme for the transmission of Golay code,and the algorithm for the recovery of the phase of GolayCode based QPSK signal. We have adopted the serialsearch, parallel phase search and maximum likelihoodestimation (MLE) techniques for carrier phase recovery.Carrier phase recovery algorithm using Golay code is anew addition in the carrier phase recovery methods.This new scheme exhibits excellent carrier phaserecovery performance. With its sound mathematicalbasis, the Golay code based carrier phase recoveryalgorithm is expected to become an important tool incommunications and carrier phase recoveryapplications.

Key words: Phase recovery, QPSK, Correlation

are presented to demonstrate the Golay code based carrierphase recovery algorithm using these techniques. Finallythere are comments on the performance of the three carrierphase recovery techniques. This concludes the paper.

II. GOLAY CODE BASED QPSK TRANSMISSION

For transmitting both the complementary Golay codes, saycode-c and code-k, simultaneously we use the QPSKscheme. Code-c is used to modulate the in-phase carriercos(men) and code-k is used to modulate the quadrature

carrier sin(men). These two BPSK modulated signals are

added which gives the Golay code based QPSK signal to betransmitted. The transmitted QPSK signal is [1]

s[n] =c[n]cos(men) + k[n]sin(mcn) (2.1)

___~~__c_o~:(Wf)Icode-c ..0

Figure. 1 Golay Code based QPSK transmitter

III. PHASE RECOVERY

A. Framework

Transmitted QPSKsignal s[n]Icode-k

The transmitted QPSK signal reaches with some delay

I. INTRODUCTION

Carrier phase recovery using Golay Code has beenpresented in this paper. This is a useful idea for recoveringthe phase of the received Golay code based QPSK signal.Golay codes are used in channel coding in communications([2]-[4]). A Golay code is a pair of complementary codes,

each having length 2N , where N is positive integer [5].Golay Code have a most remarkable property thatautocorrelation of one Golay code when added to theautocorrelation of the complementary part, gives zero sidelobe level ([6], [7]). This side lobe suppression property hasbeen used for recovery of the phase of the Golay code basedQPSK received signal in this paper. We present a pass bandscheme i.e. QPSK Scheme followed by an algorithm for therecovery of the phase of the carrier using Golay code. Webegin with the serial search strategy for the recovery of thephase followed by the parallel phase search and maximumlikelihood estimation (MLE) techniques. Then simulations

978-1-4244-2211-1/08/$25.00 ©2008 IEEE

no at the receiver. We assume that some phase distortion ¢

is introduced in the received signal due to the channel. Thesignal received at the receiver is [1]

r[n] =e[n - no]cos(men + ¢J) + k[n - no]sin(men + ¢J)

(2.2)

¢ is a function of the delay no. The receiver for the Golay

code based QPSK transmitted signal is represented inFigure 2.

rQ[no]= [c[n - no][sin(~- ¢) + sin(2OJen + ¢J+ ¢)] +(2.6)

k[n - no][cos(¢-¢) + cos(2liJcn + ¢+¢)]]

The high frequency terms in the two demodulated signalsare filtered out and we get

r/[no] =c[n - no]cos(¢-¢)+k[n - no]sin(¢-¢) (2.7)

rQ[no] =c[n -no]sin (¢-¢)+k[n - no]cos(¢-¢) (2.8)Correlation of the in-pfiase component (equation 2.7) iscarried out with code-c at base band which gives

The in-phase component of the complex demodulation is

Here r(cn ' kn-no

), r(kn , Cn-no ) are the two cross correlation

terms of 2.9 and 2.10 respectively.

no+M-l no+M-l

L k[n]c[n-no]sin(~-¢)+ L k[n]k[n -no]cos(~-~)n=no n=no

(2.10)The result (2.10) is the addition of autocorrelation terms ofcode-k with itself and the cross-correlation of code-k withcode-C.

Our algorithm for side lobe suppression works asfollows. When autocorrelation of code-c is added with theautocorrelation of code-k, side lobes are completelysuppressed ([6],[7]). Getting rid of the two cross-correlationterms in the correlation result (2.9) of the in-phasedemodulated signal (2.7) and the correlation result (2.10) ofthe quadrature phase demodulated signal (2.8) is achallenging task. We proceed as follows. We add the twocorrelation results (2.9 and 2.10) and get

no+M-l no+M-l

L c[n]c[n - no] cos(¢ - ¢) + L c[n]k[n - no] sin (¢ - ¢)n=no n=no

(2.9)The result (2.9) is the addition of autocorrelation of

code-c and the cross-correlation of code-c with code-k.Correlation of the Q-Component of demodulation (2.8) iscarried out with code- k which results in

Correlation andaddition result

k- correlator

c- correlator

The received signal r[n] is demodulated by the in-phasecarrier cos(OJen) , and the quadrature-phase carriers

sin(mcn) [1]. To demodulate the received signal we

introduce a phase offset ~ in the two demodulating

carriers. ¢ is varied as 0 ~ ~ ~ 21l. The reason for

introducing this phase offset into the demodulatingcarriers will become apparent in foregoing discussion.

Figure 2. Demodulation, correlation and addition(DCA) block at the receiver

B. Carrier phase recovery using serial search strategy

cos(men + t/J)Received ISignal r[n] +

L:i A

sin(men +t/J)

r/ [Ii0] = 2 [ c [n - no] cos ( OJen + ~) +

k[n - no]sin(men + ~)]cos(men + ¢)(2.3)

r/[no] = [c[n - noHcos(¢-¢) + cos(2liJcn + ¢+ ¢)] +

+k[n - noHsin(¢- ¢) + sin (2liJcn+¢+ ¢)]J

(2.4)And the quadrature component of the complexdemodulation is

rQ[no]=2[e[n - no] [cos(men + ¢J)] +

k[n - no] [sin(OJen +¢J)]]sin(OJen +¢) (2.5)

In (2.11) we get a delta function 8[n - no] having amplitude

2N cai//J-¢;) and the addition o! the cross-correlation terms

scaled with sir(¢r-¢;). When ~ =~ the cross-correlation terms

become zero and we get a perfect delta function at the

output of the receiver at the delay no ([6],[7],[9]). The phase

of the received QPSK signal is thus recovered using the sidelobe suppression property of the Golay code. It is the valueof the self introduced phase ~ at the demodulating carriers

at which delta function of correlation is achieved afterfollowing the correlation and addition algorithm. Thisscheme is demonstrated with the help of computersimulations in section IV.

c. Phase bank technique

Serial search strategy for finding the phase of thereceived Golay code based QPSK signal is a very timetaking process as we have to try many values of the phaseone by one. Instead of the serial search strategy we can usethe parallel phase bank technique. In this technique we

place the demodulation, correlation and addition block(DCA-block of Figure 2) in a number of times in parallel.The incoming signal is introduced to each of the DCAblock simultaneously, as shown in Figurel0.

We have assumed that the phase offset introduced inthe signal is t/J =1!/4, and we vary the self introduced phase

¢ at the demodulating carriers from 0 to 2lZ" i.e 0 ~¢~ 21r

in equal increments of 1! / 16 each. Phase introduced into

the demodulated carriers of first DCA block is ~ =0

radians, and the phase offset to the demodulating carriers of

the last DCA block is 4=21! radians. The phase offset is

incremented in equal increments of Jr / 16 in each of theconsecutive DCA blocks.

ReceivedSignal r[n]

cos(men + tPt) "~=O

Correlation andaddition result

~

~

offset equal to the phase of the received signal, gives adelta function of correlation with suppressed side lobes atits output as shown in Figure 2. This method is a very fastone as compared to the serial search strategy as only oneiteration is required for the completion of the algorithm.

If the received signal is corrupted by noise, then side"lobes may appear in the final correlation result even when¢ = ¢ . The algorithm for finding the phase of the receivedGolay code based QPSK signal in this case will be as under:

Phase of the incoming Golay codes based QPSK signal isthe one out of (~ ,~2 , ••••••• •~n ) that gives maximum value of

( main -lobe - height) at the output of the corresponding DCAL:abs(y)

block. Here Yi is the height of the individual side lobe.

D. Maximum Likelihood Estimation (MLE) ofthe phaseofthe signal

In this approach first we find the maximum likelihoodestimate (MLE) of the phase of the received signal. Thenwe introduce the MLE of the phase of the signal ~ at the

demodulating carriers of the DCA block (Figure 2) to getthe delta function of correlation.

i) Finding the MLE ofphase ofthe received signal:

After demodulating the received signal r[n] with the in-

cos(men)Received

~Signal r[n]~rI[nO]

C ~ ["]

sintmcn)

rq no

and

phase and quadrature phase carriers followed by correlationwith code-c and code-k respectively, the maximum value ofthe correlation and its index is picked. The maximum

likelihood estimate (MLE) of the carrier phase is found bythe following relation [8] .

MLE ofphase =¢" = -arc tan(maxrq[~o]J (2.12)max r/[no]

Whereno+M-l no+M-l

r/[no]= L c[n]c[n-no]cos(~-¢)+ L c[n]k[n-no]sin(~-¢)n=no n=no

Figure 4. Demodulation at the receiver

DCA Blockn

DCA Block 4

sin(men + tP2) I" I

cos(men + tP4) " It/J4 =4Jr/16

sin(men + tP4)

cos(men + tPn) ~ = nlZ" /16 = 2JZ"• 'rn

sinemen +¢1)cos(men + tP2 )

Figure 3. Phase bank technique for finding the phase of the receivedGolay code based QPSK Signal

The received signal is introduced to all the DCA blockssimultaneously. In each DCA block, demodulation,correlation and addition processes are taking place. TheDCA block, whose demodulating carriers have phase

no+M -1 no+M-l

rQ[fio]= L k[n]c[n-no]sin(¢-¢)+ L k[n]k[n-no]cos(¢-¢)n=no n=no

In phase component of the correlation

100

ii) Introducing a phase offset equal to the MLE ofthe signalphase in the receiver demodulating carriers 80

60

Correlation (2.10) of Q-component of demodulated signal~2.8) with code-k is given in figure 4

Quadrature phase component of the correlation

600500200 300 400number of samples

100

40

20

100

Figure 6. Correlation of I-component of demodulation with code-c

k- correlator

c- correlator

MLE of the phase

After finding the MLE of the phase, the demodulatingcarri~rs at the leceiver are given phase offset by an amountof t/J. As t/J = t/J the delta function of correlation is achievedfrom the relation (equation 2.11) at the output of thereceiver. There may be side lobes in the final result due tothe slight mismatch between t/J and J.

"cos(mcn + ¢) ¢ =

:i~~~;~n][: LPF

SiD(Jn+J)

Figure 7. Correlation of Q-component of demodulation with code-k

And the addition of the two correlation results (2.9 and2.10) gives (2.11) as shown in Figure 5.

Addition of the two correlation results

600500200 300 400number of samples

100

80

Q):::>

~ 60c:0

al

I 40

20

IV. SIMULATION RESULTS

We demonstrate the serial search technique for therecovery of the phase of the Golay code based receivedQPSK signal with the help of computer simulations. Codelength used for Golay code is 256 each for code-c and code­k. Delay introduced in the received signal is n = 8 samples.We have assumed that the phase offset introduced in the

signal is t/J =1r/4, and we vary the self introduced phase ¢at the demodulating carriers from 0 to 2JZ' i.e 0 ~¢~ 21r in

equal increments of 1f /16 each.The simulation results are as under

Figure 5. Introducing the MLE of the signal phase into thedemodulating carriers of the receiver

250..-------,----....----------.-------;---------;---~

Here ¢ = Phase distortion in the signal due to the

channel200

Phase introduced into the demodulating

carriers

150

100

i) Case 1 t/J =1l / 4, t/J =1l / 1650

600500200 300 400number of samples

100_50'------.-----l----'---------'-------L-------'------'

oCorrelation (2.9) of in phase component of demodulatedsignal (2.7) with code-c is given in Figure 3.

Figure. 8. Addition of the two correlation results

ii) Case 2 fjJ = ;r 14, fjJ =2;r 116Addition of the two correlation results

300~-_r_-_r____-____r_--r-----_r_-______,

The addition of the two correlation results (2.11) for thiscase is given in Figure 6.

250

200

150

Addition of the tv.<> correlation results250......------------,--------.-------,--------,----,----..,

100

200 50

150~~

.~ 100

I_50L..-_---J.-__L--_-----'----__-'----_-----'----_------J

o 100 200 300 400 500 600number of samples

Figure 11 Addition of the two correlation results

50

Here it is clearly seen that as t/J = ~, the side lobes arecompletely suppressed. Hence we have found that

the phase of the received signal is ¢ = ¢ = 1l / 4 ._50L--_-----L__-----I....-__--L-__..L.-_-----'__-----'

o 100 200 300 400 500 600number of samples

Figure 9 Addition of the two correlation results

iii) Case 3 fjJ =;r 14,fjJ =3;r116

v) Case 5 fjJ=;r14,fjJ=5;r/16The addition of the two correlation results is given in

Figure 9Addition of the two correlation results

The addition of the two correlation results for this case isgiven in Figure 7.

250

200Addition of the tv.<> correlation results

250

~~ 150c:oiI 100

50

600500100 200 300 400number of samples

-5O'-------'------"-----~-------'------"-------'

o

Figure 12. Addition of the two correlation results

~~ 150c:o~§100

200

50

_50L--_-----L__--l.-__--'-----_----J__------l.-__--J

o 100 200 300 400 500 600number of samples

It is clearly visible that as the difference between t/J and Jincreases, the level of side lobes in the final result increases.

Figure 10. Addition of the two correlation results

It can be observed that as the difference between

¢ and ¢ decreases the level of side lobes reduces in

the final correlation result.

iv) Case 4 fjJ =;r14,fjJ =4;r 116The addition of the two correlation results is given inFigure8

V. COMMENTS ON THE PHASE ESTIMATION

TECHNIQUES

A. The serial search strategy

This technique for finding the phase of the Golay codebased QPSK signal is a very time consuming technique.For finding the phase of the received signal we have tochange the phase offset at the demodulating carriers inequal increments of J= tr /16 each. This technique

requires 2pi/(pi/16) = 32 iterations, with this incrementalvalue of phase selected, for finding the true phase of theGolay code based QPSK received signal betweenO~J~2tr.

B. The phase bank technique

This is a much faster method than the serial searchstrategy. Here we have introduced different phase offsetsto the demodulating carriers in a number of parallel DCAblocks; with increasing the phase in equal increments inthe consecutive blocks. The received signal is introducedsimultaneously to all the DCA blocks. The algorithm(equation 2.11) is used to find the DCA block with whichthe phase of the incoming signal matches. In this methodthe time required for the phase estimation is considerablyreduced. This method requires a lot of hard ware andresources as we need (2n)/(n/16)=32 DCA blocks to

find the phase of the received signal between 0 ~¢~ 21l

with this selected incremental value n / 16 of the phase;but the desired result is achieved in one iteration only.

c. The maximum likelihood estimation (MLE) technique

This is also an efficient method for finding the phaseof the received signal. Two iterations are required to findthe phase of the received signal. In the first iteration theMLE of the phase is found , and in the second iterationthis MLE is introduced into the demodulating carriers toget delta function of correlation after following thecorrelation and addition algorithm.

VI. CONCLUSION

We have seen that the development of QPSK Scheme andthe carrier phase recovery algorithm using Golay Code(2.11) is a novel idea for finding the phase of the receivedGolay code based QPSK signal. We have exploited the sidelobe suppression property of the Golay code in the recoveryof the phase of the Golay code based QPSK received signal.To recover the phase distortion introduced by the channel,we have adopted three techniques namely serial searchstrategy, parallel search and the maximum likelihoodestimation (MLE) technique. We have discussed all thethree techniques with sufficient details. In the end there arecomments on the number of iterations required for eachtechnique to recover the phase of the received signal. Ourwork furnishes sufficient details so as to make the basictheory of the carrier phase recovery algorithm using Golaycode clear and ready to be used by a designer.

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