Digital Signal Processing 12, 77–86 (2002)doi:10.1006/dspr.2001.0400, available online at http://www.idealibrary.com on

A New Algorithm for 16QAM Carrier PhaseEstimation Using QPSK PartitioningFeng Rice,∗ Mark Rice,† and Bill Cowley‡

∗Cooperative Research Centre for Sensor Signal and Information Processing,Technology Park, South Australia, Australia; †DSpace Pty. Ltd., TechnologyPark, South Australia, Australia, and Institute for TelecommunicationsResearch, University of South Australia, Mawson Lakes, South Australia 5095,Australia; and ‡Cooperative Research Centre for Satellite System, Institute forTelecommunications Research, University of South Australia, Mawson Lakes,South Australia 5095, Australia

E-mail: feng@cssip.edu.au, mark@dspace.com.au, bill.cowley@unisa.edu.au

Rice, F., Rice, M., and Cowley, B., A New Algorithm for 16QAM CarrierPhase Estimation Using QPSK Partitioning, Digital Signal Processing 12(2002) 77–86.

A new algorithm for 16QAM carrier phase estimation using QPSKpartitioning is presented in this paper. The estimator performance is closeto the Cramér–Rao lower bound (CRLB). In particular the performanceof the new algorithm is compared to the CRLB for very short packetsand shows that the equivalent loss in SNR is less than 0.5 dB for 20symbol packets at 16 dB. The estimator is very suitable for fast acquisitionperformance in radio transmission systems, which may be required forburst transmission systems or conditions where the phase can changerapidly, e.g., multipath fading or local oscillator phase noise. The algorithmcan be readily implemented in real time using digital signal processingtechniques. 2002 Elsevier Science (USA)

Key Words: phase estimation; synchronization; quadrature amplitudemodulation; Cramér–Rao lower bound.

1. INTRODUCTION

Quadrature amplitude modulation (QAM) is an important technique indigital communications systems to achieve high bandwidth efficiency (refer,for example, [1]). Usually, coherent detection of QAM signals is required;however, carrier synchronization is not as straightforward as for simplermodulation schemes such as binary phase shift keying. Sometimes a pilotsignal can be used to assist with carrier synchronization; however, this reducesthe power and/or bandwidth efficiency of the transmission. Alternatively, the

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78 Digital Signal Processing Vol. 12, No. 1, January 2002

modulated carrier itself can be processed using blind phase recovery techniques.Many suboptimum algorithms have been developed and implemented for QAMschemes. Examples for square 16QAM include Sari and Moridi [2], Horikawaet al. [3], Chen et al. [4], and Georghiades [5]. Recently, a two-stage approachwas described in [6] utilizing a combination of a non-data-aided algorithm anda decision-directed algorithm. For all these cases, it has been difficult to judgethe effectiveness of the methods due to the absence of an appropriate theoreticalresult. Recently the authors have derived the phase estimate Cramér–Rao lowerbound (CRLB) for 16QAM (to be presented in [7]), and this is included herefor comparison. These methods are not very effective at low SNR and requirea relatively large number of symbols to produce a reliable phase estimate. Inthis paper, a new 16QAM algorithm is presented which is also a two-stageapproach, applying the Viterbi and Viterbi Phase Estimate (VVPE) [8] first onthe inner and outer signal constellation rings and second on the middle ringpartitioned into two sets of quaternary phase shift keyed (QPSK) signals. Itsadvantages are that it uses ail the received symbols, approaches the CRLBperformance, and works effectively with small observation intervals. Certainpractical implementation issues are also discussed.

2. QPSK PARTITIONING SCHEME

It is assumed that the received square 16QAM signal has a fixed phaseoffset φ, and the signal has been ideally filtered and sampled at the optimumsampling instant. In this case, the received samples are

xk = akejφ0 + wk, k = l,2, . . . ,N, (1)

where ak are the transmitted symbols of 16QAM square constellation C ∈{±1 ± j, ±3 ± 3j, ±3 ± j, ±1 ± 3j } and wk is the kth noise sample whose realand imaginary parts are independent zero-mean Gaussian random variables,each with variance σ 2 and the wks are mutually independent. The symbol SNRis given by the energy per symbol to noise spectral density Es/No = P/2σ 2 [1],where P is the average power of 16QAM square constellation points.

As illustrated in Fig. 1, the constellation points are divided into three sets ofQPSK signal points. The set of points close to the inner and outer rings is calledS1 and the others points close to the middle ring belong to two sets, called So

and Sx, as illustrated in Fig. 1 by the “o” and “x” points. The algorithm worksby first separating the received signals into two groups of signals based on theiramplitude.

S1 = {xk, k = 1,2, . . . ,N, s.t. |xk| ≤ T1 or |xk| ≥ T2}and the remaining samples are either in So or Sx; i.e.,

So ∪ Sx = {xk, k = 1,2, . . . ,N, s.t. T1 < |xk| < T2},

Rice, Rice, and Cowley: 16QAM Carrier Phase Estimation 79

FIG. 1. 16QAM square constellation.

where T1 and T2 are thresholds for S1 and So or Sx. The thresholds can beset to optimum values which may depend on the operating conditions. Inthe simulations this was done by experiment to obtain a single fixed pair ofthresholds. In a real 16QAM demodulator, automatic gain control (AGC) is anessential function in order to distinguish between signal points with amplitudevariation and so the threshold can be easily set relative to the normalized signalamplitude levels. In practice this is very simple using digital signal processingtechniques.

The signals which are likely to have been transmitted as S1 constellationpoints are processed using a non-data-aided technique to produce an initialphase estimate. This is possible because the S1 points are equivalent to highand low energy QPSK signal points, for which the VVPE is known to be aneffective technique. The coarse phase estimate φ̂1 is

φ̂1 = 14

tan−1

∑xk∈S1

�( x4k

|xk |3)

∑xk∈S1

�( x4k

|xk |3) . (2)

As with standard QPSK, there exists a four-fold phase ambiguity due to thesquare shape. This is typically resolved in a practical receiver by the use of aknown sequence of symbols, i.e., a unique word. However, the phase ambiguitydoes not affect the phase estimator operation and so is ignored in this paper.

The phase is corrected for ail received signals xk using φ̂1.

x ′k = xke

−j (φ̂1−π/4). (3)

Hard decisions are made to classify the So and Sx by demodulating the phasecorrected middle ring signals and choosing corresponding estimated transmittedsymbols based on decision regions. Transmitted points may be erroneouslyassigned to So and Sx due to the effect of Gaussian noise and the residual phaseoffset.

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Sx and So correspond to two sets of QPSK signals with phase offsets of ±θ

with respect to S1 points, as shown in Fig. 1b. The two sets are rotated back bye∓jθ respectively to match the S1 QPSK signal.

ySx = x ′ke

jθ , (4)

where x ′k ∈ Sx. And

ySo = x ′ke

−jθ , (5)

where x ′k ∈ So.

Then VVPE is applied to xk ∈ S1, ySx and ySo to yield an estimated residualphase φ̂2. As previously noted, there is a four-fold phase ambiguity. This is notof concern as the mapping to Sx and So is relative to S1 and is arbitrary in termsof the original transmit signal mapping.

Finally, the estimated phase offset is

φ̂0 = φ̂1 + φ̂2.

The implementation complexity is largely set by the use of the VVPE, whichis commonly used in digital phase, shift keyed modems. Although there aretwo VVPE stages, the first one only operates on half the symbols (on average,assuming equiprobable symbols are transmitted). Therefore the complexity isbetween 1.5 and 2 times the standard VVPE complexity.

The phase estimate can be further refined by iterating the approach togenerate φ as illustrated in Fig. 2 by the dashed line. The iteration gives a small

FIG. 2. New phase estimation scheme.

Rice, Rice, and Cowley: 16QAM Carrier Phase Estimation 81

improvement and works better for the smaller obseration intervals. For the caseof a signal that bas been coded using forward error control techniques, it is likelythat the probability of symbol error can be significantly reduced by decoding thesignal. This would improve the overall phase estimator performance and systemerror rate at an increased cost in terms of complexity.

3. SIMULATION RESULTS

In order to determine performance the phase estimator was simulated usinga Monte Carlo technique. The square 16QAM signal is generated from a pseudorandom data source. The signal is then shifted by multiplying ejφ0 , where φ0 isa static phase offset taken from a random uniform distribution. Ideal AGC isassumed.

The received signals are partitioned into S1 and So ∪ Sx using the receivedsignal amplitude and comparing with the thresholds. Calibrated zero mean,complex Gaussian noise with variance 2σ 2

n is added to the signal to generate xk.The thresholds area is set to T1 = √

2 + √2/2 and T2 = (

√10 + √

18)/2, i.e.,the mean radii between two sets of constellation points, which was foundempirically to be near optimum across a wide range of SNR values. It wasnoted that there was not great sensitivity to those values, and only minorimprovements are possible by further optimization. In a practical receiver thethresholds would be fixed.

The Monte Carlo simulation results are shown in Figs. 3–7. In all simulationsthe minimum number of runs at the each SNR point was 5000 to ensureconvergence of results. The performance of the estimator is compared to the

FIG. 3. Performance of the phase estimators with 20 symbols.

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FIG. 4. Performance of the phase estimators with 200 symbols.

FIG. 5. Performance of the phase estimators with 1000 symbols.

Rice, Rice, and Cowley: 16QAM Carrier Phase Estimation 83

FIG. 6. Performance of the estimator versus observation interval in symbols.

phase CRLB of square 16QAM as derived in [7] and the unmodulated carrierwave CRLB given by σ 2/N [9].

Figures 3–5 show the performance of the new phase estimator, the coarsephase estimator (generated using S1 points only by VVPE), and the phaseCRLB for 16QAM. The vertical scale indicates phase error standard deviation inradians and the horizontal scale shows the signal-to-noise ratio. The solid linein Figs. 3–5 is the performance of the new estimator. For 200 symbol packets,it shows that the new estimator follows the square 16QAM CRLB curve in the

FIG. 7. Probability of bit error (PBE) with 20-symbol observation interval.

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range of Es/No ≥ 6 dB. There is 1-dB difference between the CRLB and theestimators in the range 6 dB ≤ SNR ≤ 16 dB and the estimator approachesthe CRLB for SNR ≥ 16 dB.

The relationship between observation interval and phase estimator is shownin Fig. 6 for a range of SNRs. The solid lines are for the performance of theestimators and the dashed lines for the square 16QAM CRLBs at differentSNRs. The performance of the estimator closely follows the correspondingCRLB. From the square 16QAM CRLB derivation [7] there is a “hump” atSNR ≈ 10 dB. Figure 5 shows this effect when the estimator is closer to theCRLB at SNR ≈ 4 dB compared with SNR ≈ 10 dB. In general the standarddeviation of the estimator decreases as the SNR increases and the observationintervals increase. The estimator converges to the square 16QAM CRLB forSNR > 20 dB.

In order to determine the effect on bit error performance a Monte Carlosimulation was performed. The probability of bit error is shown in Fig. 7 forSNR between 12 and 22 dB for a block size of 20 symbols. Also shown is thetheoretical 16QAM error curve. The results show that for a probability of biterror of 10−6, the loss due to phase error is approximately 1.6 dB for the coarseestimator, reducing to only 0.8 dB for the fine estimator. Note that the resultassumes that the four-fold phase ambiguity bas been resolved by other means,e.g., a unique word.

4. CONCLUSION

A new algorithm for square 16QAM phase estimation using QPSK parti-tioning is presented. The performance of the new estimator closely follows theCRLB of square 16QAM. At high SNR, the CRLB is virtually achieved, and atlow SNR the performance deviates from the CRLB for small observation inter-vals. Figures 3–5 quantify the performance loss for N = 20, 200, and 1000 sym-bols, respectively. The accuracy of the phase estimator is an important factor inreceiver error performance, and also for other derived measurements such asDoppler frequency. The two-stage estimator can be slightly improved by furtheriterations. Figure 7 shows that the probability of bit error shows significant im-provement between coarse and fine estimators for a block size of 20 symbols.The loss due to phase estimation is less than 0.5 dB at Es/No = 16 dB. Thismakes the estimator very suitable for fast acquisition performance, which maybe required for burst transmission systems or conditions where the phase canchange rapidly, e.g., multipath fading or local oscillator phase noise. Under fad-ing conditions, the requirement for accurate AGC is more difficult to meet andoften it is necessary to utilize pilot signals to assist with rapid tracking. Thiswas not considered in this paper and is a subject for further work. The stagedcoarse/fine principle of operation used for this algorithm and in [6] can be ap-plied to other high-order modulation schemes including nonsquare constellationand is also a subject for further work.

Rice, Rice, and Cowley: 16QAM Carrier Phase Estimation 85

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of the Cooperative Research Center (CRC) forSensor Signal and Information Processing and the CRC for Satellite Systems in undertaking thiswork.

REFERENCES

1. Webb, W. T. and Hanzo, L., Modern Quadrature Amplitude Modulation Principles and Appli-cations for Fixed and Wireless Communications. IEEE Press, New York, and Pentech, London,1994.

2. Sary, H. and Moridi, S., New phase and frequency detectors for carrier recovery in PSK and QAMsystems. IEEE Trans. Commun. 9 (1988), 1035–1043.

3. Horikawa, I., Murase, T., and Saito, Y., Design and performances of a 200 Mbit/s 16QAM digitalradio system. IEEE Trans. Commun. COM-27 (1979), 1953–1958.

4. Chen, L., Kusaka, H., and Kominami, M., Blind phase recovery in QAM communication systemsusing higher order statistics. IEEE Signal Process. Lett. 3 (1996), 147.

5. Georghiades, C., Blind carrier phase acquisition for QAM constellations. IEEE Trans. Commun.45 (1997), 1477–1486.

6. Morelli, M., D’Andrea, A., and Mengali, U., Feedforward estimation techniques for carrierrecovery in 16-QAM modulation. In Broadband Wireless Communications (Luise, M. and Pupolin,S., Eds.). Springer-Verlag, London, 1998, pp. 35–45.

7. Rice, F., Cowley, B., Moran, B., and Rice, M., Crámer–Rao lower bounds for QAM phase andfrequency estimation. IEEE Trans. Commun., to appear.

8. Viterbi, A. J. and Viterbi, A. M., Nonlinear estimation of PSK-modulated carrier phase withapplication to burst digital transmission. IEEE Trans. Inform. Theory IT-29 (1983), 543–551.

9. Rife, D. C. and Boorstyn, R. R., Signal-tone parameter estimation from discrete-time observa-tions. IEEE Trans. Inform. Theory IT-20 (1974), 591–598.

FENG RICE graduated from Shandong University of Technology, China, with a B.Eng. (Hons)in electronic engineering. In 1992 she received the master of philosophy in information engineeringfrom City University, London. In 1993 she worked in the area of biomedical engineering at FlindersUniversity of South Australia. From 1995 to 1996 she worked as a research engineer at theInstitute for Telecommunications Research, University of South Australia. Since 1996 she has beenconsulting in the area of simulation and modeling of telecommunications systems for a number oforganizations. Currently she is completing her Ph.D. studies with the Cooperative Research Centrefor Sensor Signal and Information Processing, University of South Autralia. Her current researchinterests are performance bounds and algorithms for frequency and phase estimation of digitallymodulated signals.

MARK RICE was born in Southport, United Kindom. He received his B.Sc. and Ph.D. inelectrical and electronic engineering from the University of Manchester in 1984 and 1989,respectively. He held the position of research associate with the University of Manchester from1984 to 1988, working in the fields of error control coding and future satellite communicationsystems. From 1989 to 1996 he worked as a research fellow/senior research fellow at theInstitute for Telecommunications Research, University of South Australia. He led research activitiesto develop new coding and modulation schemes for mobile satellite systems, including thespecifications of the world’s first national L-band digital voice service and the first commercialapplication of turbo coding to high speed data transmission. In 1995 Dr. Rice was awardeda Telecommunications Advancement Organisation Fellowship from the Japanese Government,sponsored by the Communications Research Laboratory (CRL) of Japan. Since 1996, Dr. Rice hasbeen the technical director of DSpace, leading the company’s efforts in advanced physical layerdesign for wireless systems. Dr. Rice is an adjunct associate professor with the University of SouthAustralia, where he participates in turbo coding, digital modem, and satellite systems research. He isthe co-inventor of two patents in turbo coding and adaptive rate transmission schemes, respectively.

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BILL COWLEY received the B.Sc., B.E., and Ph.D from the University of Adelaide in 1975, 1976,and 1985, respectively. Between 1979 and 1983 he worked on radar signal processing at the DefenceResearch Center in Salisbury, South Australia. In 1985 he joined the Digital Communications Groupat SAIT, which later became the Institute for Telecommunications Research (ITR) at the Universityof South Australia. He is currently Director of ITR and the leader of the Satellite CommunicationsProgram in the Australian Cooperative Research Center for Satellite Systems. His technicalinterests include modem signal processing, low-rate speech coding, radar signal processing, spectralestimation techniques, and real-time signal processing implementation methods.