Kalman Filter Carrier Synchronization Algorithm

Andrei Câmpeanu Communications Department

Faculty of Electronics and Telecommunications Timisoara, Romania

andrei.campeanu@etc.upt.ro

János Gál Communications Department

Faculty of Electronics and Telecommunications Timisoara, Romania janos.gal@etc.upt.ro

Abstract— A new Kalman carrier synchronization algorithm was developed for high-order QAM modulated signals which showed fast acquisition times and good frequency tracking performances. The proposed solution relies on a Decision Directed Kalman filter combined with a lock detector which determines the status of the synchronization process, modifying consequently the filter parameters. Simulation results show that, compared to the performance of a basic Kalman synchronization system, a tenfold improvement or better is achieved in terms of acquisition speed, while keeping a low residual phase noise.

Keywords-component; Acquisition Mode and Tracking Mode, Decision-Directed Carrier Synchronization, Extended Kalman Filter Algorithm, Quadrature Amplitude Modulation, Random Walk.

I. INTRODUCTION High-Order Quadrature Amplitude Modulation (QAM) is a

high-bandwidth-efficient modulation scheme employed especially in the field of cable communications. For this type of modulation, the carrier recovery is a critical component of the coherent communication system [1]. To extract correctly the transmitted symbols as the carrier frequency offset between the transmitter and receiver is inevitable, the receiver must to provide blind fast carrier frequency synchronization.

In order to track the frequency of the received signal, the receiver uses commonly a phase-locked-loop (PLL) technique [2-4]. The use of a PLL carrier recovery loop to synchronize a QAM receiver requires fast convergence rates and small steady-state phase tracking errors, two opposing requirements. Simple decision-directed phase error estimators used in this PLL loops provide limited performances for frequency acquisition, since the phase error range is very low for high-power symbols, e.g. ±3.7° for corner symbols in 256 QAM [2]. In practice, the problems arise also from the process of changing the PLL loop bandwidth in a variety of parameter setup stages in the convergence process [2, 4].

An alternative to PLL carrier recovery is the use of Kalman filter algorithms [5, 6]. The Kalman filter is known as the optimally recursive linear filter in the minimum mean squared sense. Whenever the system environment varies, the Kalman filter can adaptively cope with the system environment changes better than a traditional PLL can do. Linear Kalman filters or extended Kalman filters (EKF) based on a carrier phase dynamic model [7, 8], were successfully used in carrier

synchronization of QAM signals. In all these applications, the system model incorporates as variables, the phase and frequency of the incoming signal. Since the relationship between these variables and the observations is nonlinear, EKF filters are used to solve the problems.

There are several objectives of this paper. In the first row, it intends to develop a basic decision-directed EKF carrier synchronization algorithm (DD-EKF) for QAM modulation based on a Gaussian random walk model of the phase offset between the received signal and the reconstructed carrier [6]. Then, in order to improve the performances of the basic EKF algorithm, we adopt the architecture proposed by Matsuo and Namiki [2, 9] which rely on a coarse frequency acquisition mode and a fine phase tracking mode, selected by a lock detection module. Filter parameters are modified when system status changes from acquisition to tracking mode, thereby achieving a faster acquisition and a lower phase noise in tracking mode.

A drawback of EKF algorithms are the important number of divergence cases that arises even at large S/N ratios. To minimize these cases and to speed up acquisition, the adaptive carrier synchronization EKF algorithm uses different values of parameters in the acquisition mode as compared to tracking mode. Also, in order to avoid relatively large frequency offset and false synchronization on image frequencies, the algorithm uses saturation to keep the estimated frequency in the region of interest.

Several simulations show good acquisition and frequency tracking performances of the proposed adaptive decision-directed EKF algorithm.

II. BASIC SYSTEM MODEL Assuming perfect timing synchronization and proper gain

control, the receiver in Fig. 1 [8] observes the discrete time down-converted signal

( ) ( ) ( ) ( ) ( ) ( ) ( )j n j nr n m n e w n m n e w nϕ θΩ += + = + (1)

where ( ) ( ) ( )m n a n jb n= + is the nth transmitted complex QAM symbol, φ and Ω are the carrier residual phase and frequency offset and ( )w n is a zero-mean complex Gaussian

noise with distribution ( )20, wN σ .

978-1-4673-1176-2/12/$31.00 ©2012 IEEE

The decision-directed EKF filter estimates the total phase of the incoming signal carrier and brings to the input of hard-decision QAM detector the signal:

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )ˆj n n j n

z zz n m n e w n m n e w nθ θ θ− Δ= + = + (2)

where is a phase error and is

complex Gaussian noise with the same power as . The

carrier synchronization system aim is to cancel .

Finally, a hard decision is made on the output to

generate the detected complex symbol , part of QAM constellation. The detected signal along with the received signal are used to estimate the total phase of the

incoming signal carrier by the EKF filter.

III. CARRIER SYNCHRONIZATION WITH A DECISION-DIRECTED EKF FILTER

A. The State Space Model In order to estimate the phase and frequency of the carrier

of a M-QAM modulated signal by Kalman filtering, it is necessary to model it in the state space. The parameters of the model are the phase ( )nθ and frequency ( )nΩ , with:

( ) ( ) ( )1n n nθ θΩ = − − (3) To synchronize the model with the received signal, the

evolution of ( )nΩ is driven by the random walk model:

( ) ( ) ( )1n n v nΩ = Ω − + (4)

where ( )v n is a sequence of i.i.d. random scalars with distri-

bution ( )20,N γ . Thus the rate of evolution of the signal frequency estimation is controlled by γ. By combining (3) and (4), the state evolution equation of Kalman filter is obtained:

( )( )

( )( ) ( )1

1

1 1 0with and

0 1 1

n nv n

n nθ θ⎡ ⎤ ⎡ − ⎤

= +⎢ ⎥ ⎢ ⎥Ω Ω −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

F G

F G

(5)

The In-phase and Quadrature components of the received

signal are the elements of the observation vector ( )nr :

[ ] ( )( ) ( )( )Re ImT

n r n r n⎡ ⎤= ⎣ ⎦r (6)

In this conditions, eq. (1) gives the observation vector as a nonlinear function of state vector ( ) ( ) ( ) T

n n nθ= Ω⎡ ⎤⎣ ⎦x :

( ) ( )( ) ( )n n n= +r h x w (7)

with

( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )( )( )( )

cos sinsin cos

Reand

Im

a n n b n nn

a n n b n n

w nn

w n

θ θθ θ

⎡ − ⎤= ⎢ ⎥+⎣ ⎦

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

h x

w

(8)

where the correlation matrix of the noise vector ( )nw is

2 1 0

0 12wσ ⎡ ⎤

= ⎢ ⎥⎣ ⎦

Q (9)

In order to use EKF, we apply the first order linearization procedure to ( )( )nh x in (8) around the estimation of the state

vector :

( )( ) ( )( )( )

( ) ( )( )ˆ 1

ˆ ˆ1 1n n

n n n n n nδδ = −

= − + − −x x

hh x h x x xx

(10)

with

( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

ˆ 1

ˆ ˆsin 1 cos 1 0ˆ ˆsin 1 cos 1 0

n n

n

a n n n b n n n

b n n n a n n n

δδ

θ θ

θ θ

= −

=

⎡ ⎤− − − −⎢ ⎥=⎢ ⎥− − + −⎣ ⎦

x x

hHx

(11)

The calculation of ( )nH requires the values of transmitted

symbol ( ) ( ) ( )m n a n jb n= + which are unknown at the receiver. If the receiver works with a sufficiently low error rate, then the output of the Hard-Decision device

( ) ( ) ( )ˆˆ ˆm n a n jb n= + can be used in place of transmitted symbol. Therefore, in the decision-directed EKF synchronization algorithm ( )nH is replaced by:

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

ˆˆ ˆˆ sin 1 cos 1 0ˆˆ ˆ ˆˆsin 1 cos 1 0

a n n n b n n nn

b n n n a n n n

θ θ

θ θ

⎡ ⎤− − − −⎢ ⎥=⎢ ⎥− − + −⎣ ⎦

H (12)

Similarly, the nonlinear function ( )( )ˆ ˆ 1n n −h x is calculated as:

( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

ˆˆ ˆˆ cos 1 sin 1ˆ ˆ 1ˆˆ ˆˆ sin 1 cos 1

a n n n b n n nn n

a n n n b n n n

θ θ

θ θ

⎡ ⎤− − −⎢ ⎥− =⎢ ⎥− + −⎣ ⎦

h x (13)

B. The EKF Algorithm The procedure known as Extended Kalman Filter (EKF)

algorithm [10] uses state space equations(5) and (7) as well as the linearization of the observation function around the current

( ) ( ) ( )ˆ ˆ ˆn n n nθ ϕ= Ω +

( ) ( ) ( )ˆn n nθ θ θΔ = − ( )zw n

( )w n

( )nθΔ

( )z n

( )m n

( )r n

( )ˆ nθ

( )ˆ 1n n −x

Fig. 1 Basic decision-directed EKF filter carrier synchronization system.

vector estimate (12). Assume that the initial state [ ]1x , the

observation noise [ ]nw and the state noise [ ]v n are jointly

Gaussian and mutually independent. Let ˆ 1n n⎡ − ⎤⎣ ⎦x and

1n n⎡ − ⎤⎣ ⎦R be the conditional mean and the conditional

variance of [ ]ˆ nx given the observations [ ] [ ]1 , , 1n −r r… and

let ˆ n n⎡ ⎤⎣ ⎦x and n n⎡ ⎤⎣ ⎦R be the conditional mean and conditio-

nal variance of [ ]ˆ nx given the observations [ ] [ ]1 , , nr r… . Then, according [10] and using (12) and(13), the EKF algorithm is:

1. Measurement Update Equations

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 1ˆ ˆ ˆ1 1T Tn n n n n n n n n−

= − − +K R H H R H Q (14)

( ) ( ) ( ) ( ) ( )( )( )ˆˆ ˆ ˆ1 1n n n n n n n n= − + − −x x K r h x (15)

( ) ( ) ( ) ( ) ( )1 1n n n n n n n n= − − −R R K H R (16)

2. Time Update Equations ( ) ( )ˆ ˆ1n n n n+ =x Fx (17)

( ) ( ) 21 T Tn n n n γ+ = +R FR F GG (18)

where ( )nK is the Kalman gain matrix at moment n . Viewing the four-quadrant symmetry of M-QAM signals, the action of the algorithm is confined to a carrier frequency offset between 8S−Ω and 8SΩ , where SΩ is the symbol rate.

For an initial frequency offset near to 8SΩ the algorithm can converge to 4S±Ω + Ω instead of Ω , the real value. This

“false-lock” situation can be avoided by imposing to ( )ˆ nΩ , the estimated frequency, and the following saturation condi-tion:

( )ˆ8 8S Sn−Ω ≤ Ω ≤ Ω (19)

IV. ADAPTIVE SYSTEM MODEL Carrier synchronization process consists of two distinct

stages: acquisition and tracking. Both stages involve different modes of action of the synchronization system in order to

achieve both fast carrier synchronization in the acquisition stage and low phase error in the tracking stage. To meet these antagonistic requirements firstly, it is necessary to detect in which of two stages of the synchronization process is the system and then, to make the EKF filter parameters dependent on the stage where the synchronization process is found. The model in Fig. 2 fulfills these requirements, using in addition to the adaptive EKF filter, a lock detector [2] to estimate the status of the synchronization process.

The lock detector monitors the derotated signal ( )z n and

the output of the Hard-Decision detector ( )m n to determine whether or not the output constellation is phase-locked. The lock detector generates the binary signal

( ) ( ) ( )ˆ1, if

0, otherwise

z n m ny n

λ⎧ − <⎪= ⎨⎪⎩

(20)

In presence of a frequency offset between Ω and ( )ˆ nΩ ,

the received signal points in ( )m n will not distribute closely around the expected points of constellation [1]. Hence, when the system is phase-locked, ( )y n will be equal to 1 in majority. Every 1N symbol, the lock detector compares the average value of ( )y n taken on 2N symbols with a threshold value β to determine if the system has acquired or not synchronization. By choosing adequate values for

1 2, , and N N λ β , the probability of false erroneous lock-detection is practically zero.

The PLL carrier recovery systems use two different time constants in their loop filters. These filters have a larger band-width in acquisition mode to speed up synchronization and a lower bandwidth in tracking mode to prevent loss of synchro-nization. The DD-EKF synchronization system implements this solution through the averaging parameters of lock detec-tor, 1N and 2N . The lock detector uses two sets of values: a lower value in acquisition mode to allow a faster synchroniza-tion and a higher value in tracking mode to fight against the accidental loss of synchronization.

Lock detector output is used to control adaptively the performances of EKF filter. This control is exercised through the variance of Gaussian random walk model of estimated frequency, γ2. An intuitive explanation of the action of random walk step size γ on the performance of the EKF algorithm in a decision-directed carrier recovery system observes on one hand that large values of γ give fast convergence rates for the algorithm in the acquisition mode. On the other hand, lower values of variance γ2 are useful when the synchronization system is in tracking mode because it determines lower levels of phase noise of the estimated frequency ( )ˆ nΩ . Moreover, the use of a large γ in tracking mode determines a rapid loss of synchronization, while with a low γ in the acquisition mode, the synchronization no longer occurs. Consequently, the use of a single value for the variance of estimated carrier frequency as in [8], cannot provide both a fast convergence rate in acquisition mode and a low frequency error in tracking mode.

Fig 2. Adaptive decision-directed EKF filter carrier synchronization system.

11 2

2

, in acquisition mode,

, in tracking modeγ

γ γ γγ⎧

= ⎨⎩

(21)

V. SIMULATION RESULTS The goal of performed simulations was to compare the

performance of DD-EKF algorithms presented in the paper. The basic DD-EKF algorithm is used in the structure of Fig. 1 and the adaptive DD-EKF algorithm uses the system described by Fig. 2. Each simulation trial consisted of 50,000 symbols with DVB-C 64-QAM and 256-QAM constellations and symbol rate SΩ . The modulated signal propagates through a communication channel with additive white and Gaussian noise (AWGN) of zero-mean and variance 2

wσ . The received signal, applied at the input of decision-directed synchronization systems, has a carrier frequency offset Ω , one sample per symbol, perfect timing synchronization and proper gain control.

The single controlling parameter of basic DD-EKF algo-rithm is the variance 2γ . In this first case, γ takes a single value, experimentally established, both in acquisition and tracking mode. The value used was 35 10 Sγ −= ⋅ Ω .

The second version examined, the adaptive DD-EKF algorithm uses a lock detector to determine when the synchronization system switches between acquisition and tracking mode. Based on results reported in [2] and our simulations, the threshold values are set as follow: the lock detection threshold λ is set to 0.7 and the threshold for the averaging operation β is set to 0.6. As was stated in the previous Section, the averaging parameters of lock detector, 1Nand 2N change when the system switches between the two modes. They are lower in acquisition mode:

and higher in tracking mode: 1 64 N = and 2 256N = . The false lock-detection probability is easily found using binomial coefficients. For instance, using and , the false lock-detection probability is less than 10-10.

Besides the averaging constants, in the case of adaptive DD-EKF algorithm, when the system switches, the variance

of EKF filter is also modified from 21 2 10 Sγ −= ⋅ Ω in

acquisition mode to 52 5 10 Sγ −= ⋅ Ω in tracking mode for a 64-

QAM modulated signal. These values were determined experimentally.

A typical description of the way the carrier synchronization process works for the two versions of DD-EKF algorithm is made for a 64-QAM signal constellation by Fig. 3. The

simulation used the following parameters: SNR 25 dB= and 0.02 SΩ = Ω , where Ω is the carrier frequency offset. A first

conclusion, taken from Fig. 3, concerns the utility of the lock detector to determine precisely the moment of transition from acquisition to tracking. It can be seen also, that under same conditions, the adaptive DD-EKF algorithm takes much less time to acquire a residual frequency offset than the basic algorithm.

In the case of 256-QAM signals the values used by the adaptive DD-EKF algorithm for the variance of EKF filter were 2

1 1.25 10 Sγ −= ⋅ Ω and 62 3.125 10 Sγ −= ⋅ Ω . Fig. 4 presents

the synchronization process of the adaptive algorithm for a signal with SNR 35 dB= and 0.04 SΩ = Ω .

1 216 and 64N N= =

0.6β =

2 256N =

2γ

2γ

Fig. 3 Description of the synchronization process for a 64-QAM signal in the two DD-EKF algorithms: (a) Output of Lock Detector, (b)

and (c) In-Phase input signals to Hard-Decision detectors.

Fig. 4 The synchronization of the adaptive EKF filter for a 256-QAM modulated signal

Fig. 5 shows that random walk is the way used by the

estimate ( )ˆ nΩ to converge to the actual carrier frequency Ω. The simulation uses SNR 15dB= and 0.05 SΩ = − Ω . It also reveals that the use of saturation avoids the risks of false-lock cases.

Each curve shown on Figs. 6-8 is obtained by averaging the results of 100-run simulations made on 64-QAM signals. Fig. 6 presents the acquisition time performances for various initial carrier frequency offsets with a signal-to-noise ratio SNR 30 dB= . Lock is considered achieved when the rms value of the phase error ( ) ( )ˆn nθ θ− over the last 256 symbols is less than 1° rms. The simulations were carried out on basic and adaptive DD-EKF algorithms and compared with the results reported by Gagnon et al. [2] for a very fast PLL carrier synchronization loop in terms of acquisition speed. The adaptive DD-EKF algorithm is at least 10 times faster than the basic algorithm and has better performances than those reported by Gagnon et al.

Considering as a case of divergence, the simulation where synchronization is not reached after 50,000 symbols, a comparison between the two DD-EKF algorithms was made for this aspect over 100 simulations. Fig. 7 compares the rate of convergence of DD-EKF algorithms at SNR 30 dB= for various initial carrier frequency offsets. The adaptive DD-EKF algorithm has much higher performances than its basic version.

In order to characterize the performance of the adaptive DD-EKF algorithm in tracking mode, the bit error rate (BER) of binary signal at the output of Hard-Decision device was compared with the theoretical curve. Fig. 8 reveals that for moderate carrier frequency offsets, the difference between our results and the theoretical curve is lower than 1 dB.

Fig. 5 A typical evolution of estimated frequency during the synchro-nization process for DD-EKF algorithms which use or not

saturation.

Fig. 6 Acquisition performances of the carrier synchronization systems for 64-QAM. For the PLL carrier synchronization loop,

the results are taken directly from Gagnon et al. [2].

Fig. 7 Rate of convergence vs. carrier frequency offset at SNR 30 dB= comparison for DD-EKF algorithms for 64-

QAM modulated signal.

Fig. 8 The dependence of BER on SNR at various initial carrier frequency offsets for the adaptive DD-EKF algorithm for a 64-

QAM signal.

VI. CONCLUSIONS

This paper introduces a carrier synchronization system for M-QAM modulated signals, based on the new adaptive DD-EKF algorithm. The algorithm uses a very efficient Lock Detector to assess whether or not the system attained synchronization. The Kalman filter modifies its parameters when system status changes from acquisition to tracking mode, thereby permitting a faster acquisition time and a lower phase noise in tracking mode. Significant improvements in terms of algorithm convergence were obtained by limiting the range of variation of Kalman filter variables. Simulations results show that, in terms of acquisition time, the adaptive DD-EKF algorithm is at least 10 times faster than the basic algorithm and has better performances than PLL synchronization loops.

REFERENCES [1] Kim, K.-Y., Choi H.-J., Design of carrier recovery algorithm for high-

order QAM with large frequency acquisition range. In Proc. IEEE International Conference on Communications (ICC’01). June 2001.

[2] Gagnon, G., et al., A simple and fast carrier recovery algorithm for high-order QAM. IEEE Trans. Commun. Lett., Oct. 2005. 49(4): p. 918-920.

[3] Best, R.E., Phase-Locked Loops: Design, Simulation, and Applications. 5 ed., 2003, New York: McGraw-Hill.

[4] 4. Ouyang, Y., Wang C.-L., A new carrier recovery loop for high-order quadrature amplitude modulation. In Proc. Of IEEE Global Telecommunications Conference (GLOBECOM’02). Nov. 2002.

[5] Aghamohammadi, A., Meyr H., Ascheid G., Adaptive synchronization and channel parameter estimation using an extended Kalman filter. IEEE Trans. on Communications, Nov. 1989. Vol. 37(1212-1219 ).

[6] Gál, J., Câmpeanu A., Naforniţă I., Kalman Noncoherent Detection of CPFSK Signal. In Proc. of 8th International Conference on Communications "COMM 2010”. June 10-12 2010. Bucharest.

[7] Patapoutian, A., Application of Kalman filters with a loop delay in synchronization. IEEE Trans. on Communications, May 2002. Vol. 50(No. 5): p. 703-706.

[8] Lin, W.-T., Chang D.-C., Adaptive carrier synchronization using Decision-Aided Kalman filtering algorithms. IEEE Trans.on Consumer Electronics, Nov. 2007 53(4): p. 1260-1267.

[9] Matsuo, Y., Namiki J., Carrier recovery systems for arbitrarily mapped APK signals IEEE Trans. on Communications, 1982. 30(10): p. 2385 - 2390

[10] Haykin, S., Adaptive Filter Theory. 3rd ed1996, New Jersey, U.S.A.: Prentice-Hall, Englewood Cliffs