A Novel QAM Joint Frequency-Phase Carrier Recovery Method Qijia Liu1 , Zhixing Yang, Jian Song, and Changyong Pan Department of Electronic Engineering Tsinghua University Beijing, P.R.China 100084 Email: 1
[email protected]
Abstract— In this paper, we propose a novel structure of the joint frequency-phase carrier recovery loop for all-digital QAM satellite receivers. Unlike the existing methods, the initial phase offset is recovered through the utilization of a frequency recovery loop. By introducing a non-zero-mean input to the Stop-and-Go controller, the steady-state estimation variance is greatly reduced. This method is especially useful in the low SNR region. Keywords— Carrier Recovery, Stop and Quadrature-Amplitude-Modulation (QAM).
Go
(S&G),
1. Introduction
2. Mathematical Model Consider an M -state QAM signal transmitted over an additive white Gaussian noise (AWGN) channel. The received complex signal rc (t) can be represented as the sum of the transmitted signal sc (t) and noise nc (t) rc (t)
The Quadrature-Amplitude-Modulation (QAM) plays an important role in the broadband satellite communications among which most of current digital satellite systems use QPSK(i.e.4QAM). And along with the increase of transmission rates, high-order QAM is promising to be the main modulation scheme in future satellite systems owing to its high spectral efficiency[1]-[3]. A good-performance carrier recovery is an essential part of any coherent QAM receiver which requires very small steady-state phase jitter and it must be achieved blindly in satellite broadcasting systems. Conventionally, the blind carrier recovery is accomplished by various phase-locked loop (PLL) algorithms[4]-[6]. Unfortunately, PLL methods require residual frequency errors much less than the symbol rate; otherwise, frequency-acquisition capabilities should be resorted to, such as the FPD method as well as its variations in [7]-[9] and frequency estimation methods in [10], both of which require great extra cost of complexity. On the other hand, small estimation variance is critical to QAM modems. As PLL methods are originally designed for loworder QAM, the performance will be significantly degraded for high-order QAM. A reduced-constellation PLL has been proposed in [11], the performance, however, is not satisfactory when the occurrence probability of the reduced-constellation is small. Therefore, new QAM carrier recovery schemes without significant complexity increase are in need. In this paper, a frequency detector (FD) for QAM signals is introduced at first. Based on our presented QAM FD, we propose an all-digital joint frequency-phase carrier recovery method. It can accomplish both the phase recovery and the frequency recovery simultaneously with low complexity, and
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can greatly reduce the estimation variance. A variation of the reduced-constellation method is then introduced to further improve the performance for high-order QAM receivers.
= sc (t, d, ε, θc , ωc ) + nc (t) (1) X j(ωc ·t+θc ) = dk g(t − kT − εT )e + nc (t), (2) k
where g(t) is the transmit filter impulse response, T is the symbol interval and nc (t) is a narrow-band Gaussian noise with double sided power spectral density N0 /2. The fractional symbol time shift ε, the transmitter carrier frequency ωc and the initial phase θc are assumed unknown at the receiver and will be determined by the clock synchronizer as well as the carrier recovery. Moreover, in rc (t), d = {dk } is the sequence of transmitted QAM symbols that dk = ak + jbk = Ak · [cos (arg dk ) + j sin (arg dk )],
(3)
where {ak } and {bk } are in the alphabet sets determined by the modulation schemes and Ak is the modulus of the transmitted symbol. In all-digital zero-intermediate-frequency (i.e. baseband) receivers, the received signal rc (t) is first demodulated to the baseband signal of r(t) with the local oscillator frequency ωo and the initial phase θo . Denoting the carrier frequency offset and initial phase offset between transmitter and receiver as ∆ω = ωc − ωo and ∆θ = θc − θo , we have r(t)
= rc (t) × e−j(ωo ·t+θo ) (4) ( ) X = dk g(t − kT − εT ) · ej(∆w·t+∆θ) + No (t), k
where No (t) = nc (t) · e−j(ωo ·t+θo ) is a zero-mean noise component. The carrier recovery aims at estimating and compensating both ∆ω and ∆θ on the baseband samples. To address the carrier recovery, assume the clock synchronization is perfect that the samples are taken at the optimum
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sampling instant nT + εT . Therefore, the demodulated signal samples will be denoted as rn = r(nT + εT ). Finally, for notational simplicity, denote the phase offset between the signal sample rk and the transmitted symbol dk as ϕk = ∆ω · (kT + εT ) + ∆θ.
(5)
3. A QAM Frequency Detector Here we propose a decision-directed (DD) frequency detector for the QAM signals. The FD obtains the frequency error signal by two consecutive baseband samples and their tentative slicer decisions denoted as dˆk . The intuitive explanation will be first presented and the algorithmic description will follow. When the frequency offset ∆ω > 0, the phase offset ϕk is expected to be larger than that at the previous instant, ϕk−1 ; otherwise, if ∆ω < 0, we have ϕk − ϕk−1 < 0. Therefore, the phase difference between two consecutive phase offsets can represent the frequency error. Mathematically, derived from Equation (5), consecutive phase offsets ϕk and ϕk−1 have the relation ϕk − ϕk−1 = ∆ω · T . Because the symbol interval T is constant, the difference is proportional to the frequency offset. Thus, for a feedback loop, the frequency error signal at instant k could be ϕk − ϕk−1 . Unfortunately, in practice, the transmitted symbols {dk } and their corresponding phase offsets {ϕk } are both unknown. Then the tentative slicer decisions dˆk should be used to estimate the phase offset ϕˆk . Therefore, it results in the decision-directed frequency error signal ek = ϕˆk − ϕˆk−1 .
where sgn(x) is the signum function for variable x. Denote φk = arg rk and ρk = arg dˆk . Given that ϕˆk is not too large, the phase offset estimation can be trigonometrically derived as follows ≈ sin ϕˆk = sin (φk − ρk ) ≈ sin ϕk = sin φk cos ρk − cos φk sin ρk 1 [Im(rk )Re(dˆk ) − Re(rk )Im(dˆk )]. ≈ A2k
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Furthermore, for the different QAM schemes, the FD will hold the following different forms: 1) QPSK: The symbol alphabets in QPSK have ak , bk√∈ {−1, 1}. Specifically, QPSK signals’ modulus is Ak = 2 for all k, shown in Figure 1(a). The coefficient A12 can be k regarded as a constant gain in the loop. Hence, the estimation of the phase offset (8) could be simplified for QPSK signals ϕˆk = Im(rk )Re(dˆk ) − Re(rk )Im(dˆk ).
(8)
(9)
2) High-order QAM: Unlike QPSK, higher QAM symbols have varying modulus. Although Ak is not a constant and unknown at the receiver, its estimation Aˆk can be determined by the tentative decisions. Subsequently, the phase offset estimation becomes
(6)
In the non-steady state, due to the residual frequency offset, the phase offset is changing. A tentative decision dˆk different from the transmitted symbol dk will come forth when the varying phase offset ϕk crosses the decision boundary. In this case, it can be easily demonstrated that the phase offset estimation has a different sign from that at the previous instant and the consequent difference between them will lead to a wrong loop error signal. Therefore, a modification should be introduced to alleviate this problem: when two consecutive phase offset estimations have opposite signs, the frequency detector outputs the error signal of the previous instant instead of calculating a new one, just like the well-known stop-andgo (S&G) method[12]. The frequency detector algorithm thus becomes ϕˆk − ϕˆk−1 sgn(ϕˆk ) = sgn(ϕˆk−1 ) ek = , (7) ek−1 sgn(ϕˆk ) 6= sgn(ϕˆk−1 )
ϕˆk
Figure 1. FD construction for carrier recovery: (a)QPSK; (b)16QAM.
ϕˆk =
1 [Im(rk )Re(dˆk ) − Re(rk )Im(dˆk )]. Aˆ2
(10)
k
Together, (7) and (10) define the complete FD algorithm. This QAM FD’s characteristic curves are periodic at the normalized frequency offset 14 , providing a large acquisition range. For high-order QAM, another modification can be adopted to lower the estimation variance even further. With different coefficients Aˆ12 , the phase offset estimation (10) suffers k different interferences from the same noise level. Namely, smaller modulus symbols are more sensitive to the noise. Therefore, the strategy is to use the symbols with large estimated modulus for the phase offset estimation and the frequency error signal generation while keeping the estimation unchanged and resetting the error signal when the estimated modulus is small. By doing so, the estimation variance is expected to be reduced. This is a variation of the reducedconstellation method in [11] but the number of constellation points used in our method could be larger. As a consequence, the acquisition performance is improved. To distinguish this modification from the original one, we call it the Partial FD. In conclusion, the proposed QAM FD is composed of a decision-directed phase error detector, an S&G controller and a differentiator. Because the phase recovery is achievable by the same phase detector, it is straightforward to combine the frequency and the phase recoveries, to simultaneously estimate the frequency offset and compensate the initial phase offset.
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signal in a similar form to that of the above FD: α ˆk − α ˆ k−1 sgn(ˆ αk ) = sgn(ˆ αk−1 ) Ek = (12) Ek−1 sgn(ˆ αk ) 6= sgn(ˆ αk−1 ) ek + C · ϕˆk sgn(ˆ αk ) = sgn(ˆ αk−1 ) = . (13) Ek−1 sgn(ˆ αk ) 6= sgn(ˆ αk−1 )
Figure 2. Joint frequency-phase carrier recovery loop structure
The complexity is amazingly low since only one detector and one phase rotator are needed.
4. Joint Frequency-Phase Carrier Recovery Based on the given QAM FD, a joint frequency-phase carrier recovery loop is proposed in this section. It can compensate not only the frequency offset but also the initial phase offset at low expense of the complexity. The concept of completing the initial phase recovery by a frequency recovery loop is what’s special here. At the same time, the new loop structure is delicately designed to further reduce the estimation variance and enlarge its applicable signal-to-noise ratio (SNR) range. To compensate the initial phase offset by the frequency recovery loop, we introduce the phase offset estimation into the frequency error signal. The error signal thus becomes Ek = ek + C · ϕˆk ,
(11)
where C is a positive constant which influences the acquisition time and the steady-state estimation variance. In the steady state, because E{ek } = 0 and E{ϕˆk } = 0, it yields E{Ek } = 0. And in the non-steady state, the non-zero phase offset estimation ϕˆk existing in (11) will bring the frequency estimation to change. Specifically, when ϕˆk > 0, the frequency estimation will increase in order to ’chase’ to compensate the residual phase offset; otherwise, the estimation is deduced to ’wait for’ the negative phase offset to grow to zero. This procedure will continue until both the residual frequency offset and the phase offset converge to zero. Through analysis, however, we found that (11) suffers a large estimation variance. Because the phase offset is zeromean in the steady state, existing noise will cause the sign of its instantaneous estimation to change frequently. Particularly in low SNR region, the S&G method introduced in Section 3 will change the probability distribution of the noise and yield a relatively large fluctuation on the steady-state frequency estimation, as shown in Figure 5(a). Even worse, it can make the steady state unstable. To address this problem, we introduce the phase offset estimation into the error signal in another way. Replacing the Pkinput to the S&G controller ϕˆk in (7) by α ˆ k = ϕˆk + C · i=1 ϕˆi , we construct a new error
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The corresponding joint frequency-phase carrier recovery loop structure is illustrated in Figure 2. The phase compensation part can calculate and send the phase offset estimation into the original frequency recovery loop. Evidently, the additional hardware is just two adders. It is clear that (11) and (13) share the similar form. So the new loop error signal Ek in (12) maintains the feature of recovering both the frequency and the phase offsets at the same time. However, by replacing ϕˆk with α ˆ k , the performances at the steady state are quite different. This is because unlike from ϕˆk , E{ˆ αk } 6= 0. Therefore, the joint frequency-phase carrier recovery loop’s error signal α ˆ k rarely changes its sign in the presence of noise. Consequently, the steady-state frequency estimation is equivalently smoothed so that its variance is greatly reduced, especially in low SNR region. In summary, by skillfully introducing the phase offset estimation into the loop error signal, the joint recovery method combines the frequency recovery and the phase recovery together with low complexity of just one phase detector and one rotator, and it also avoids the large estimation’s fluctuation that the original frequency recovery loop has.
5. Simulation Results In this section, we present the simulation results of our proposed frequency detector and joint frequency-phase carrier recovery method. The simulations are performed using SPW and MATLAB. Unless specially indicated, the transmit filter g(t) is assumed to be a square-root raised cosine one with roll-off factor α = 0.35 and the constant C = 0.01. The FD’s characteristic curves are plotted in Figure 3 for QPSK and 16QAM. They vary with the SNR and the slope at the origin which determines the normalized loop bandwidth BL T becomes smaller with decreasing SNR. Consequently, the matched filter has to be utilized before the frequency detector to achieve a sufficiently large SNR at the FD input, as our loop in Figure 2 does. Given that the characteristic curve is periodic, the tolerable frequency offset is restricted within one period around the origin. The features of the joint frequency-phase carrier recovery loop are illustrated in Figure 4-6. On one hand, both the frequency offset estimation and the initial phase offset compensation are accomplished by the loop, as shown in the 16QAM constellations in Figure 4. On the other hand, by comparing the frequency estimation results of (a)frequency recovery and (b)joint recovery with the same parameters in figure 5, it is obvious that the proposed joint recovery method greatly reduces the steady-state estimation fluctuation. Figure 6 provides the estimation variance curves changing with SNR for QPSK. Clearly, the estimation variance at low SNR is
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Figure 5. Comparison of the frequency estimation for (a)frequency recovery and (b)joint frequency-phase carrier recovery. (SNR=10dB and ∆ωT = 0.16rad)
Figure 3. FD characteristics: (a)QPSK; (b)16QAM.
Figure 4. 16QAM constellations: (a)Before carrier recovery; (b)After frequency recovery; (c)After joint frequency-phase carrier recovery. (SNR=20dB and ∆ωT = 0.32rad)
Figure 6. Frequency estimation variance versus SNR for QPSK signals. (∆ωT = 0.16rad and BL T ≈ 0.05)
ulations should be α-independent, to which Figure 9 conforms. greatly reduced by the joint recovery loop. Correspondingly, we can apply the proposed method in an enlarged SNR range. Moreover, with the same normalized loop bandwidth, our method is shown to achieve the similar variance performance to the analytical result of the NDA Dε frequency control loop presented in Chapter 8 [10], with the advantages of very low complexity and easy implementation. Finally, some results for 16QAM scheme are presented. The frequency estimation variance curves changing with SNR, the normalized frequency offset and the roll-off factor are presented in Figure 7, 8 √ and 9 respectively. As expected, the √ 10, 3 2 ) significantly improves Partial FD (using Aˆk ∈ the performance of the estimation variance, especially in low SNR region. Moreover, since the carrier recovery is at the symbol rate with the perfect timing synchronization, the sim-
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6. Conclusion A low-complexity joint frequency-phase carrier recovery loop is proposed in this paper for the QAM signals. It accomplishes both the frequency recovery and the phase recovery simultaneously. The steady-state frequency estimation variance is significantly reduced owing to the delicate use of a non-zero-mean input to the S&G controller. This method is especially useful in the low SNR region. Moreover, a Partial FD method is introduced to further improve the estimation variance performance for high-order QAM. This joint carrier recovery method is designed to be used in future broadband satellite receivers. More research effort will focus on the impact of the I/Q unbalance on this method as well as the
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optimization of the constant C.
R EFERENCES
Figure 7. Frequency estimation variance versus SNR for 16QAM signals. (∆ωT = 0.16rad)
[1] H. Jiang and P. A. Wilfred, ”A Hierarchical Modulation for Upgrading Digital Broadcast Systems”, IEEE Trans.Broadcast., vol.51, no.2, pp.223-229, Jun.2005. [2] J. C. Chen, L. Cooper, D. Taggart et al., ”Performance of MPSK and 16QAM in the Satellite Communication Environment”, in Proc. 2004 IEEE Aerospace Conference, vol.2, pp.1383-1391, Mar.2004. [3] ETSI EN 300 744 Ver 1.3.1, Digital Video Broadcasting (DVB); Framing Structure, Channel Coding and Modulation for Digital Terrestrial Television, Aug.2000. [4] F. Classen, H. Meyr and P. Schier, ”Maximum Likelihood Open Loop Carrier Synchronizer for Digital Radio”, in Proc. ICC1993, Geneva, pp.493-497, May 1993. [5] R. L. Cupo and R. D. Gitlin, ”Adaptive Carrier Recovery Systems for Digital Data Communications Receivers”, IEEE J.Select.Areas Commun., vol.7, no.9, pp.1328-1339, Dec.1989. [6] K. Yamanaka et al., ”A Multilevel QAM Demodulator VLSI with Wideband Carrier Recovery and Dual Equalizing Mode”, IEEE J.Solidstate Circuits, vol.32, no.7, pp.1101-1107, Jul.1997. [7] H. Sari and S. Moridi, ”New Phase and Frequency Detectors for Carrier Recovery in PSK and QAM Systems”, IEEE Trans.Commun., vol.36, no.9, pp.1035-1043, Sep.1988. [8] Y. Ouyang and C. L. Wang, ”A New Carrier Recovery Loop for HighOrder Quadrature Amplitude Modulation”, in Proc. IEEE GLOBECOM2002, Taipei, vol.1, pp.478-482, Nov.2002. [9] A.Mouaki Benani and F. Gagnon, ”Comparison of Carrier Recovery Techniques in M-QAM Digital Communication Systems”, in Proc. 2000 Canadian Conference on Electrical and Computer Engineering, vol.1, pp.73-77, Mar.2000. [10] H. Meyr, M. Moeneclaey and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, John Wiley & Sons, New York, 1998. [11] N. K. Jablon, ”Joint Blind Equalization, Carrier Recovery and Timing Recovery for High-Order QAM Signal Constellations”, IEEE Trans.Signal Processing, vol.40, pp.1383-1398, Jun.1992. [12] G. Picchi and G. Prati, ”Blind Equalization and Carrier Recovery Using a ’Stop-and-Go’ Decision-Directed Algorithm”, IEEE Trans.Commun., vol.35, no.9, pp.877-887, Sep.1987.
Figure 8. Frequency estimation variance versus normalized frequency offset for 16QAM signals. (SNR=10dB)
Figure 9. Frequency estimation variance versus roll-off factor for 16QAM signals. (SNR=10dB)
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