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A New Algorithm for 16QAM Carrier Phase Estimation Using QPSK Partitioning
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 Phase Estimation Using QPSK Partitioning Feng Rice,∗ Mark Rice,† and Bill Cowley‡ ∗ Cooperative
Research Centre for Sensor Signal and Information Processing, Technology Park, South Australia, Australia; † DSpace Pty. Ltd., Technology Park, South Australia, Australia, and Institute for Telecommunications Research, University of South Australia, Mawson Lakes, South Australia 5095, Australia; and ‡ Cooperative Research Centre for Satellite System, Institute for Telecommunications Research, University of South Australia, Mawson Lakes, South Australia 5095, Australia E-mail: [email protected], [email protected], [email protected] Rice, F., Rice, M., and Cowley, B., A New Algorithm for 16QAM Carrier Phase Estimation Using QPSK Partitioning, Digital Signal Processing 12 (2002) 77–86. A new algorithm for 16QAM carrier phase estimation using QPSK partitioning is presented in this paper. The estimator performance is close to the Cramér–Rao lower bound (CRLB). In particular the performance of the new algorithm is compared to the CRLB for very short packets and shows that the equivalent loss in SNR is less than 0.5 dB for 20 symbol packets at 16 dB. The estimator is very suitable for fast acquisition performance in radio transmission systems, which may be required for burst transmission systems or conditions where the phase can change rapidly, e.g., multipath fading or local oscillator phase noise. The algorithm can be readily implemented in real time using digital signal processing techniques. 2002 Elsevier Science (USA) Key Words: phase estimation; synchronization; quadrature amplitude modulation; Cramér–Rao lower bound.
1. INTRODUCTION Quadrature amplitude modulation (QAM) is an important technique in digital 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 simpler modulation schemes such as binary phase shift keying. Sometimes a pilot signal can be used to assist with carrier synchronization; however, this reduces the power and/or bandwidth efficiency of the transmission. Alternatively, the 77 1051-2004/02 $35.00 2002 Elsevier Science (USA) All rights reserved.
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modulated carrier itself can be processed using blind phase recovery techniques. Many suboptimum algorithms have been developed and implemented for QAM schemes. Examples for square 16QAM include Sari and Moridi [2], Horikawa et al. [3], Chen et al. [4], and Georghiades [5]. Recently, a two-stage approach was described in [6] utilizing a combination of a non-data-aided algorithm and a decision-directed algorithm. For all these cases, it has been difficult to judge the effectiveness of the methods due to the absence of an appropriate theoretical result. Recently the authors have derived the phase estimate Cramér–Rao lower bound (CRLB) for 16QAM (to be presented in [7]), and this is included here for comparison. These methods are not very effective at low SNR and require a relatively large number of symbols to produce a reliable phase estimate. In this paper, a new 16QAM algorithm is presented which is also a two-stage approach, applying the Viterbi and Viterbi Phase Estimate (VVPE) [8] first on the inner and outer signal constellation rings and second on the middle ring partitioned into two sets of quaternary phase shift keyed (QPSK) signals. Its advantages are that it uses ail the received symbols, approaches the CRLB performance, and works effectively with small observation intervals. Certain practical implementation issues are also discussed.
2. QPSK PARTITIONING SCHEME It is assumed that the received square 16QAM signal has a fixed phase offset φ, and the signal has been ideally filtered and sampled at the optimum sampling instant. In this case, the received samples are xk = ak ej φ0 + 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 real and imaginary parts are independent zero-mean Gaussian random variables, each with variance σ 2 and the wk s are mutually independent. The symbol SNR is 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 of QPSK signal points. The set of points close to the inner and outer rings is called S1 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 works by first separating the received signals into two groups of signals based on their amplitude. 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
FIG. 1.
79
16QAM square constellation.
where T1 and T2 are thresholds for S1 and So or Sx . The thresholds can be set to optimum values which may depend on the operating conditions. In the simulations this was done by experiment to obtain a single fixed pair of thresholds. In a real 16QAM demodulator, automatic gain control (AGC) is an essential function in order to distinguish between signal points with amplitude variation and so the threshold can be easily set relative to the normalized signal amplitude levels. In practice this is very simple using digital signal processing techniques. The signals which are likely to have been transmitted as S1 constellation points are processed using a non-data-aided technique to produce an initial phase estimate. This is possible because the S1 points are equivalent to high and low energy QPSK signal points, for which the VVPE is known to be an effective technique. The coarse phase estimate φˆ 1 is x4 |x k|3 x ∈ S 1 k 1 k φˆ 1 = tan−1 xk4 . 4 x k ∈S1
|xk
(2)
|3
As with standard QPSK, there exists a four-fold phase ambiguity due to the square shape. This is typically resolved in a practical receiver by the use of a known sequence of symbols, i.e., a unique word. However, the phase ambiguity does not affect the phase estimator operation and so is ignored in this paper. The phase is corrected for ail received signals xk using φˆ 1 . ˆ
xk = xk e−j (φ1 −π/4) .
(3)
Hard decisions are made to classify the So and Sx by demodulating the phase corrected middle ring signals and choosing corresponding estimated transmitted symbols based on decision regions. Transmitted points may be erroneously assigned to So and Sx due to the effect of Gaussian noise and the residual phase offset.
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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 by e∓j θ respectively to match the S1 QPSK signal. ySx = xk ej θ ,
(4)
ySo = xk e−j θ ,
(5)
where xk ∈ Sx . And xk
where ∈ So . Then VVPE is applied to xk ∈ S1 , ySx and ySo to yield an estimated residual phase φˆ 2 . As previously noted, there is a four-fold phase ambiguity. This is not of concern as the mapping to Sx and So is relative to S1 and is arbitrary in terms of the original transmit signal mapping. Finally, the estimated phase offset is φˆ 0 = φˆ 1 + φˆ 2 . The implementation complexity is largely set by the use of the VVPE, which is commonly used in digital phase, shift keyed modems. Although there are two VVPE stages, the first one only operates on half the symbols (on average, assuming equiprobable symbols are transmitted). Therefore the complexity is between 1.5 and 2 times the standard VVPE complexity. The phase estimate can be further refined by iterating the approach to generate φ as illustrated in Fig. 2 by the dashed line. The iteration gives a small
FIG. 2. New phase estimation scheme.
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improvement and works better for the smaller obseration intervals. For the case of a signal that bas been coded using forward error control techniques, it is likely that the probability of symbol error can be significantly reduced by decoding the signal. This would improve the overall phase estimator performance and system error rate at an increased cost in terms of complexity.
3. SIMULATION RESULTS In order to determine performance the phase estimator was simulated using a Monte Carlo technique. The square 16QAM signal is generated from a pseudo random data source. The signal is then shifted by multiplying ej φ0 , where φ0 is a static phase offset taken from a random uniform distribution. Ideal AGC is assumed. The received signals are partitioned into S1 and So ∪ Sx using the received signal amplitude and comparing with the thresholds. Calibrated zero mean, 2 is added to the signal to generate x . complex Gaussian noise with variance k √ 2σn √ √ √ 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 found empirically to be near optimum across a wide range of SNR values. It was noted that there was not great sensitivity to those values, and only minor improvements are possible by further optimization. In a practical receiver the thresholds would be fixed. The Monte Carlo simulation results are shown in Figs. 3–7. In all simulations the minimum number of runs at the each SNR point was 5000 to ensure convergence 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.
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FIG. 6. Performance of the estimator versus observation interval in symbols.
phase CRLB of square 16QAM as derived in [7] and the unmodulated carrier wave CRLB given by σ 2 /N [9]. Figures 3–5 show the performance of the new phase estimator, the coarse phase estimator (generated using S1 points only by VVPE), and the phase CRLB for 16QAM. The vertical scale indicates phase error standard deviation in radians and the horizontal scale shows the signal-to-noise ratio. The solid line in 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 the estimators in the range 6 dB ≤ SNR ≤ 16 dB and the estimator approaches the CRLB for SNR ≥ 16 dB. The relationship between observation interval and phase estimator is shown in Fig. 6 for a range of SNRs. The solid lines are for the performance of the estimators and the dashed lines for the square 16QAM CRLBs at different SNRs. The performance of the estimator closely follows the corresponding CRLB. From the square 16QAM CRLB derivation [7] there is a “hump” at SNR ≈ 10 dB. Figure 5 shows this effect when the estimator is closer to the CRLB at SNR ≈ 4 dB compared with SNR ≈ 10 dB. In general the standard deviation of the estimator decreases as the SNR increases and the observation intervals increase. The estimator converges to the square 16QAM CRLB for SNR > 20 dB. In order to determine the effect on bit error performance a Monte Carlo simulation was performed. The probability of bit error is shown in Fig. 7 for SNR between 12 and 22 dB for a block size of 20 symbols. Also shown is the theoretical 16QAM error curve. The results show that for a probability of bit error of 10−6 , the loss due to phase error is approximately 1.6 dB for the coarse estimator, reducing to only 0.8 dB for the fine estimator. Note that the result assumes 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 partitioning is presented. The performance of the new estimator closely follows the CRLB of square 16QAM. At high SNR, the CRLB is virtually achieved, and at low SNR the performance deviates from the CRLB for small observation intervals. Figures 3–5 quantify the performance loss for N = 20, 200, and 1000 symbols, respectively. The accuracy of the phase estimator is an important factor in receiver error performance, and also for other derived measurements such as Doppler frequency. The two-stage estimator can be slightly improved by further iterations. Figure 7 shows that the probability of bit error shows significant improvement 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. This makes the estimator very suitable for fast acquisition performance, which may be required for burst transmission systems or conditions where the phase can change rapidly, e.g., multipath fading or local oscillator phase noise. Under fading conditions, the requirement for accurate AGC is more difficult to meet and often it is necessary to utilize pilot signals to assist with rapid tracking. This was not considered in this paper and is a subject for further work. The staged coarse/fine principle of operation used for this algorithm and in [6] can be applied to other high-order modulation schemes including nonsquare constellation and is also a subject for further work.
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ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Cooperative Research Center (CRC) for Sensor Signal and Information Processing and the CRC for Satellite Systems in undertaking this work.
REFERENCES 1. Webb, W. T. and Hanzo, L., Modern Quadrature Amplitude Modulation Principles and Applications 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 QAM systems. IEEE Trans. Commun. 9 (1988), 1035–1043. 3. Horikawa, I., Murase, T., and Saito, Y., Design and performances of a 200 Mbit/s 16QAM digital radio system. IEEE Trans. Commun. COM-27 (1979), 1953–1958. 4. Chen, L., Kusaka, H., and Kominami, M., Blind phase recovery in QAM communication systems using 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 carrier recovery 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 and frequency estimation. IEEE Trans. Commun., to appear. 8. Viterbi, A. J. and Viterbi, A. M., Nonlinear estimation of PSK-modulated carrier phase with application 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 observations. 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 engineering from City University, London. In 1993 she worked in the area of biomedical engineering at Flinders University of South Australia. From 1995 to 1996 she worked as a research engineer at the Institute for Telecommunications Research, University of South Australia. Since 1996 she has been consulting in the area of simulation and modeling of telecommunications systems for a number of organizations. Currently she is completing her Ph.D. studies with the Cooperative Research Centre for Sensor Signal and Information Processing, University of South Autralia. Her current research interests are performance bounds and algorithms for frequency and phase estimation of digitally modulated signals. MARK RICE was born in Southport, United Kindom. He received his B.Sc. and Ph.D. in electrical 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 from 1984 to 1988, working in the fields of error control coding and future satellite communication systems. From 1989 to 1996 he worked as a research fellow/senior research fellow at the Institute for Telecommunications Research, University of South Australia. He led research activities to develop new coding and modulation schemes for mobile satellite systems, including the specifications of the world’s first national L-band digital voice service and the first commercial application of turbo coding to high speed data transmission. In 1995 Dr. Rice was awarded a Telecommunications Advancement Organisation Fellowship from the Japanese Government, sponsored by the Communications Research Laboratory (CRL) of Japan. Since 1996, Dr. Rice has been the technical director of DSpace, leading the company’s efforts in advanced physical layer design for wireless systems. Dr. Rice is an adjunct associate professor with the University of South Australia, where he participates in turbo coding, digital modem, and satellite systems research. He is the 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 Defence Research Center in Salisbury, South Australia. In 1985 he joined the Digital Communications Group at SAIT, which later became the Institute for Telecommunications Research (ITR) at the University of South Australia. He is currently Director of ITR and the leader of the Satellite Communications Program in the Australian Cooperative Research Center for Satellite Systems. His technical interests include modem signal processing, low-rate speech coding, radar signal processing, spectral estimation techniques, and real-time signal processing implementation methods.
A New Algorithm for 16QAM Carrier Phase Estimation Using QPSK Partitioning Feng Rice,∗ Mark Rice,† and Bill Cowley‡ ∗ Cooperative
Research Centre for Sensor Signal and Information Processing, Technology Park, South Australia, Australia; † DSpace Pty. Ltd., Technology Park, South Australia, Australia, and Institute for Telecommunications Research, University of South Australia, Mawson Lakes, South Australia 5095, Australia; and ‡ Cooperative Research Centre for Satellite System, Institute for Telecommunications Research, University of South Australia, Mawson Lakes, South Australia 5095, Australia E-mail: [email protected], [email protected], [email protected] Rice, F., Rice, M., and Cowley, B., A New Algorithm for 16QAM Carrier Phase Estimation Using QPSK Partitioning, Digital Signal Processing 12 (2002) 77–86. A new algorithm for 16QAM carrier phase estimation using QPSK partitioning is presented in this paper. The estimator performance is close to the Cramér–Rao lower bound (CRLB). In particular the performance of the new algorithm is compared to the CRLB for very short packets and shows that the equivalent loss in SNR is less than 0.5 dB for 20 symbol packets at 16 dB. The estimator is very suitable for fast acquisition performance in radio transmission systems, which may be required for burst transmission systems or conditions where the phase can change rapidly, e.g., multipath fading or local oscillator phase noise. The algorithm can be readily implemented in real time using digital signal processing techniques. 2002 Elsevier Science (USA) Key Words: phase estimation; synchronization; quadrature amplitude modulation; Cramér–Rao lower bound.
1. INTRODUCTION Quadrature amplitude modulation (QAM) is an important technique in digital 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 simpler modulation schemes such as binary phase shift keying. Sometimes a pilot signal can be used to assist with carrier synchronization; however, this reduces the power and/or bandwidth efficiency of the transmission. Alternatively, the 77 1051-2004/02 $35.00 2002 Elsevier Science (USA) All rights reserved.
Digital Signal Processing Vol. 12, No. 1, January 2002
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modulated carrier itself can be processed using blind phase recovery techniques. Many suboptimum algorithms have been developed and implemented for QAM schemes. Examples for square 16QAM include Sari and Moridi [2], Horikawa et al. [3], Chen et al. [4], and Georghiades [5]. Recently, a two-stage approach was described in [6] utilizing a combination of a non-data-aided algorithm and a decision-directed algorithm. For all these cases, it has been difficult to judge the effectiveness of the methods due to the absence of an appropriate theoretical result. Recently the authors have derived the phase estimate Cramér–Rao lower bound (CRLB) for 16QAM (to be presented in [7]), and this is included here for comparison. These methods are not very effective at low SNR and require a relatively large number of symbols to produce a reliable phase estimate. In this paper, a new 16QAM algorithm is presented which is also a two-stage approach, applying the Viterbi and Viterbi Phase Estimate (VVPE) [8] first on the inner and outer signal constellation rings and second on the middle ring partitioned into two sets of quaternary phase shift keyed (QPSK) signals. Its advantages are that it uses ail the received symbols, approaches the CRLB performance, and works effectively with small observation intervals. Certain practical implementation issues are also discussed.
2. QPSK PARTITIONING SCHEME It is assumed that the received square 16QAM signal has a fixed phase offset φ, and the signal has been ideally filtered and sampled at the optimum sampling instant. In this case, the received samples are xk = ak ej φ0 + 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 real and imaginary parts are independent zero-mean Gaussian random variables, each with variance σ 2 and the wk s are mutually independent. The symbol SNR is 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 of QPSK signal points. The set of points close to the inner and outer rings is called S1 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 works by first separating the received signals into two groups of signals based on their amplitude. 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
FIG. 1.
79
16QAM square constellation.
where T1 and T2 are thresholds for S1 and So or Sx . The thresholds can be set to optimum values which may depend on the operating conditions. In the simulations this was done by experiment to obtain a single fixed pair of thresholds. In a real 16QAM demodulator, automatic gain control (AGC) is an essential function in order to distinguish between signal points with amplitude variation and so the threshold can be easily set relative to the normalized signal amplitude levels. In practice this is very simple using digital signal processing techniques. The signals which are likely to have been transmitted as S1 constellation points are processed using a non-data-aided technique to produce an initial phase estimate. This is possible because the S1 points are equivalent to high and low energy QPSK signal points, for which the VVPE is known to be an effective technique. The coarse phase estimate φˆ 1 is x4 |x k|3 x ∈ S 1 k 1 k φˆ 1 = tan−1 xk4 . 4 x k ∈S1
|xk
(2)
|3
As with standard QPSK, there exists a four-fold phase ambiguity due to the square shape. This is typically resolved in a practical receiver by the use of a known sequence of symbols, i.e., a unique word. However, the phase ambiguity does not affect the phase estimator operation and so is ignored in this paper. The phase is corrected for ail received signals xk using φˆ 1 . ˆ
xk = xk e−j (φ1 −π/4) .
(3)
Hard decisions are made to classify the So and Sx by demodulating the phase corrected middle ring signals and choosing corresponding estimated transmitted symbols based on decision regions. Transmitted points may be erroneously assigned to So and Sx due to the effect of Gaussian noise and the residual phase offset.
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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 by e∓j θ respectively to match the S1 QPSK signal. ySx = xk ej θ ,
(4)
ySo = xk e−j θ ,
(5)
where xk ∈ Sx . And xk
where ∈ So . Then VVPE is applied to xk ∈ S1 , ySx and ySo to yield an estimated residual phase φˆ 2 . As previously noted, there is a four-fold phase ambiguity. This is not of concern as the mapping to Sx and So is relative to S1 and is arbitrary in terms of the original transmit signal mapping. Finally, the estimated phase offset is φˆ 0 = φˆ 1 + φˆ 2 . The implementation complexity is largely set by the use of the VVPE, which is commonly used in digital phase, shift keyed modems. Although there are two VVPE stages, the first one only operates on half the symbols (on average, assuming equiprobable symbols are transmitted). Therefore the complexity is between 1.5 and 2 times the standard VVPE complexity. The phase estimate can be further refined by iterating the approach to generate φ as illustrated in Fig. 2 by the dashed line. The iteration gives a small
FIG. 2. New phase estimation scheme.
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improvement and works better for the smaller obseration intervals. For the case of a signal that bas been coded using forward error control techniques, it is likely that the probability of symbol error can be significantly reduced by decoding the signal. This would improve the overall phase estimator performance and system error rate at an increased cost in terms of complexity.
3. SIMULATION RESULTS In order to determine performance the phase estimator was simulated using a Monte Carlo technique. The square 16QAM signal is generated from a pseudo random data source. The signal is then shifted by multiplying ej φ0 , where φ0 is a static phase offset taken from a random uniform distribution. Ideal AGC is assumed. The received signals are partitioned into S1 and So ∪ Sx using the received signal amplitude and comparing with the thresholds. Calibrated zero mean, 2 is added to the signal to generate x . complex Gaussian noise with variance k √ 2σn √ √ √ 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 found empirically to be near optimum across a wide range of SNR values. It was noted that there was not great sensitivity to those values, and only minor improvements are possible by further optimization. In a practical receiver the thresholds would be fixed. The Monte Carlo simulation results are shown in Figs. 3–7. In all simulations the minimum number of runs at the each SNR point was 5000 to ensure convergence 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.
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FIG. 6. Performance of the estimator versus observation interval in symbols.
phase CRLB of square 16QAM as derived in [7] and the unmodulated carrier wave CRLB given by σ 2 /N [9]. Figures 3–5 show the performance of the new phase estimator, the coarse phase estimator (generated using S1 points only by VVPE), and the phase CRLB for 16QAM. The vertical scale indicates phase error standard deviation in radians and the horizontal scale shows the signal-to-noise ratio. The solid line in 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 the estimators in the range 6 dB ≤ SNR ≤ 16 dB and the estimator approaches the CRLB for SNR ≥ 16 dB. The relationship between observation interval and phase estimator is shown in Fig. 6 for a range of SNRs. The solid lines are for the performance of the estimators and the dashed lines for the square 16QAM CRLBs at different SNRs. The performance of the estimator closely follows the corresponding CRLB. From the square 16QAM CRLB derivation [7] there is a “hump” at SNR ≈ 10 dB. Figure 5 shows this effect when the estimator is closer to the CRLB at SNR ≈ 4 dB compared with SNR ≈ 10 dB. In general the standard deviation of the estimator decreases as the SNR increases and the observation intervals increase. The estimator converges to the square 16QAM CRLB for SNR > 20 dB. In order to determine the effect on bit error performance a Monte Carlo simulation was performed. The probability of bit error is shown in Fig. 7 for SNR between 12 and 22 dB for a block size of 20 symbols. Also shown is the theoretical 16QAM error curve. The results show that for a probability of bit error of 10−6 , the loss due to phase error is approximately 1.6 dB for the coarse estimator, reducing to only 0.8 dB for the fine estimator. Note that the result assumes 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 partitioning is presented. The performance of the new estimator closely follows the CRLB of square 16QAM. At high SNR, the CRLB is virtually achieved, and at low SNR the performance deviates from the CRLB for small observation intervals. Figures 3–5 quantify the performance loss for N = 20, 200, and 1000 symbols, respectively. The accuracy of the phase estimator is an important factor in receiver error performance, and also for other derived measurements such as Doppler frequency. The two-stage estimator can be slightly improved by further iterations. Figure 7 shows that the probability of bit error shows significant improvement 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. This makes the estimator very suitable for fast acquisition performance, which may be required for burst transmission systems or conditions where the phase can change rapidly, e.g., multipath fading or local oscillator phase noise. Under fading conditions, the requirement for accurate AGC is more difficult to meet and often it is necessary to utilize pilot signals to assist with rapid tracking. This was not considered in this paper and is a subject for further work. The staged coarse/fine principle of operation used for this algorithm and in [6] can be applied to other high-order modulation schemes including nonsquare constellation and is also a subject for further work.
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ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Cooperative Research Center (CRC) for Sensor Signal and Information Processing and the CRC for Satellite Systems in undertaking this work.
REFERENCES 1. Webb, W. T. and Hanzo, L., Modern Quadrature Amplitude Modulation Principles and Applications 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 QAM systems. IEEE Trans. Commun. 9 (1988), 1035–1043. 3. Horikawa, I., Murase, T., and Saito, Y., Design and performances of a 200 Mbit/s 16QAM digital radio system. IEEE Trans. Commun. COM-27 (1979), 1953–1958. 4. Chen, L., Kusaka, H., and Kominami, M., Blind phase recovery in QAM communication systems using 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 carrier recovery 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 and frequency estimation. IEEE Trans. Commun., to appear. 8. Viterbi, A. J. and Viterbi, A. M., Nonlinear estimation of PSK-modulated carrier phase with application 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 observations. 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 engineering from City University, London. In 1993 she worked in the area of biomedical engineering at Flinders University of South Australia. From 1995 to 1996 she worked as a research engineer at the Institute for Telecommunications Research, University of South Australia. Since 1996 she has been consulting in the area of simulation and modeling of telecommunications systems for a number of organizations. Currently she is completing her Ph.D. studies with the Cooperative Research Centre for Sensor Signal and Information Processing, University of South Autralia. Her current research interests are performance bounds and algorithms for frequency and phase estimation of digitally modulated signals. MARK RICE was born in Southport, United Kindom. He received his B.Sc. and Ph.D. in electrical 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 from 1984 to 1988, working in the fields of error control coding and future satellite communication systems. From 1989 to 1996 he worked as a research fellow/senior research fellow at the Institute for Telecommunications Research, University of South Australia. He led research activities to develop new coding and modulation schemes for mobile satellite systems, including the specifications of the world’s first national L-band digital voice service and the first commercial application of turbo coding to high speed data transmission. In 1995 Dr. Rice was awarded a Telecommunications Advancement Organisation Fellowship from the Japanese Government, sponsored by the Communications Research Laboratory (CRL) of Japan. Since 1996, Dr. Rice has been the technical director of DSpace, leading the company’s efforts in advanced physical layer design for wireless systems. Dr. Rice is an adjunct associate professor with the University of South Australia, where he participates in turbo coding, digital modem, and satellite systems research. He is the 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 Defence Research Center in Salisbury, South Australia. In 1985 he joined the Digital Communications Group at SAIT, which later became the Institute for Telecommunications Research (ITR) at the University of South Australia. He is currently Director of ITR and the leader of the Satellite Communications Program in the Australian Cooperative Research Center for Satellite Systems. His technical interests include modem signal processing, low-rate speech coding, radar signal processing, spectral estimation techniques, and real-time signal processing implementation methods.