Estimation of Phase Noise for QPSK Modulation over AWGN Channels

Florent Munier, Eric Alpman,Thomas Eriksson, Arne Svensson, and Herbert Zirath

Dept. of Signals and Systems (S2) and Microtechnology Centre at Chalmers (MC2)Chalmers University of Technology,

S-41296 Gothenburg, Swedenhttp://www.s2.chalmers.se http://www.mc2.chalmers.se/

Abstract

Every oscillator used in bandpass communication suffers from an instability of their phase (a.k.a. phase noise) that,if left unaddressed, can lead to great degradation of the system performance. In this paper, we tackle the problemof minimising the effect of oscillator phase noise on the coherent detection of a quadrature phase shift keying(QPSK) modulation operating on an Additive White Gaussian Noise (AWGN) channel. The phase noise processis modeled as a Wiener-Levy (random walk) process. Our approach uses maximum likelihood (ML) estimation ofphase noise. Thorough analysis and derivation for Decision Directed (DD), Non-Data Aided (NDA), used with andwithout symbol differential encoding, and pilot based estimators are presented. We compare these estimators withrespect to their main features and evaluate their bit error rate (BER) performances throught simulations. Resultsshow that for low signal to noise ratio (SNR) applications, the use of differential encoding along with the proposedDD or NDA estimator yields performances with an SNR penalty below the two dB imposed by the non coherentdetection methods, while pilot based estimation using wiener interpolation makes it possible to detect a QPSKmodulation with SNR penalty around two dB.

Keywords

QPSK, Phase Noise, Wiener-Levy, ML estimation,AWGN, Wiener Interpolation.

1. Introduction

In any bandpass communication system, Radio frequency (RF) hardware such as oscillators are not ideal. Thecarrier generated by this device is not ideal and experiences phase instability, or phase noise, mainly due to thepresence of thermal noise in the circuitry. Circuits designed for very high carrier frequency, such as carrier genera-tor chains used for 60GHz communication (such as in [1]), are very difficult to design with a very stable frequencysource. Moreover, these circuits typically make use of frequency multipliers to reach high carrier frequencies,increasing again the level of phase disturbance [2]. It is therefore of interest to take into account their phase noisecharacteristics when looking at system issues.

In this paper, we will address the problem of phase noise using a Wiener-Levy process [3] in order to modelphase noise. This model has been widely used and is established in the available literature (e.g [4] and [5] amongothers). The estimation methods used in this work are employing the Maximum Likelihood (ML) criterion which isdocumented in [6] and [7]. The paper is organised as follows: First we will detail the considered phase noise model(section 2.1) and present the system setup (section 2.2).Then we will present our estimations methods (section 3)and their associated results (section 4) , before concluding.

2. System Setup and Models

2.1. Phase Noise Complex Lowpass Equivalent Model

In 1966, Leeson established a power spectrum model for oscillators [8]. This model splits the spectrum into regionsof 1/fa, where a = 1, 2, 3, 4. For a properly designed frequency generation chain, the main source of problem isthe a = 2 region, cause by random walk phase modulation. In continuous time, this phase distortion is expressedby

F. Munier, E. Alpman, T. Eriksson, A. Svensson,H. ZirathEstimation of Phase Noise for QPSK Modulation over AWGN Channels

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Figure 1: A realisation of the phase noise process φn and its associated carrier power spectrum for a phase noiseprocess with a phase noise rate BT=0.01 .

φ(t) =

∫ t

0

∆(s)ds (1)

The noisy carrier in its complex lowpass equivalent model ejφ(t) now has a Lorentzian Power Spectral Densitywith a 3-dB bandwidth B controlled by the variance of the White Gaussian random variable ∆(s) [9]. For thepurpose of analysis and simulation in a digital communication system we will use a discrete time random walk,also called Wiener-Lévy process.

φn = φn−1 + ∆n (2)

∆n is refered as the stepsize of the walk and is a zero mean Gaussian random variable. Its variance sets thespeed of the process and is equal to σ2

∆ = 2πBT . The product BT is refered as the phase noise rate and expressthe relative double-sided bandwidth of the discrete time carrier ejφn with respect to the symbol period. Phase noiseis assumed to remain constant between symbols. Figure 1 shows a realisation of the process and the correspondingcarrier power spectrum.

2.2. System Description

Figure 2 shows the general block diagram of the considered system in its baseband equivalent (complex lowpass)representation. bits are modulated using a Quadrature Phase Shift Keying (QPSK) to obtain the complex symbolssn = ejθn , where θn can take values mπ

2 + π4 , m = 1, 2, 3, 4. The symbols are then multiplied by the phasor

ejφn , where φn is a random variable and accounts for the total phase noise for frequency sources in the system.The signal is then passed throught an Additive White Gaussian Noise channel, so that the received signal is

rn = snejφn + wn (3)

where wn is a zero-mean, complex Gaussian random variable with variance N0. The signal is then passed throughta phase estimator that produce an estimate φnof the phase noise event. The QPSK demodulator outputs a decisionsn of the transmitted signal based on the observation of the counter-rotated received signal rne−jφn , before thetransmitted bits are decoded.

F. Munier, E. Alpman, T. Eriksson, A. Svensson,H. ZirathEstimation of Phase Noise for QPSK Modulation over AWGN Channels

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VCO Model

Channel Model

ESTIMATOR

SOURCE

SINK

PSfrag replacements

sn

QPSK Mod.

QPSK Demod.

wn

ejφn

e−jθn

sn

rn

Figure 2: System Setup .

3. Estimators

3.1. ML estimators

This section describe the way to derive the estimators for NDA assuming that data is known. When data is notknown, we need to slightly modify the result as explained in section 3.2 for decision directed estimation and fornon data aided estimation. Prior to the estimation, we perform some transformations to the received symbol asdefined in equation 3. We rotate rn by s∗n to get rid of the data dependancy. After rotation, the received symbolbecomes

rn = ejθn + wn (4)

where wn = wns∗n1 is a rotated version of the channel noise sample, and still has the same statistical properties as

wn.ML estimators seek to find the estimate of the phasor ejφn that maximise the conditional probability density

function f(r|φn) at a given time n, where r is a vector of N observed received signal points

r =[

rn−N−1, ..., rn−2, rn

]T

(5)

from Equation 3 and the definition of the phase noise process in 2 we can express the received signal at timen − i, i = 1, ..., N as

rn−i = ej(φn+∑

i−1

u=0∆u) + wn−i (6)

Let us assume that the variable∑i−1

u=0 ∆u has a small value compare to one. Then,

ej(φn+∑

i−1

u=0∆u) ≈ ejφn(1 + j

i−1∑

u=0

∆u) (7)

Conditionning on the value φn that we seek to estimate, ˙rn−i is a function of two independant gaussian vari-ables (namely the phase noise step ∆u and the AWGN process wn−i), thus the observed vector also has a multi-variate gaussian distribution [10].

fr|φn(r|φn) =

1

(2π)N

2 detC−1exp

[

−1

2(r − mr)

HC−1(r − mr)]

(8)

The mean vector mr value at time n − i is mr(i) = E(rn−i), for all i = 0, 1, 2, ..., N − 1. Given that both∆u and wn−i are zero mean, the mean is E(rn−i) = ejφn and thus the mean vector is

mr = ejφn

[

1, ..., 1, 1]T

= ejφn1T (9)

1()∗ denotes the complex conjugate

F. Munier, E. Alpman, T. Eriksson, A. Svensson,H. ZirathEstimation of Phase Noise for QPSK Modulation over AWGN Channels

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The covariance matrix content is the correlation between points in the vector at offsets i and l, that is,

C(i, l) = E(

r∗n−i − E(rn−i))(rn−l − E(rn−l))

(10)

This is reduced as

C(i, l) = min(i, l)σ2∆ + δ(i − l)σ2

w (11)

The pdf in 8 is maximised when the log-likelihood function Λ =[

(r − mr)HC−1(r − mr)

]

is minimised. It

can be shown that this is solved for the phasor

ejφn =

N∑

u=1

αurk−u (12)

with the coefficient vectorα = 1T C−1 (13)

3.2. DD and NDA Removal of Data Dependancies

As we have seen, the estimator derived in 12 assumes that the data has been removed from the received signal. Atthis stage of the receiver, this can be done by Decision Directed (DD) methods or Non Data Aided (NDA) methods.In a DD estimator, the estimator assumes that the decisions at the receiver where correct and substitutes sn for sn

for the removal of the data dependancy to obtain equation 4. With a reasonably high SNR, mostly correct decisionsoccur and good estimates can be obtained. When employing a DD estimation method, a delay need to be introducedin the estimation. The estimation of φn is based on the observation of the past symbol rn−1, rn−N because thereis no reliable decision on the transmitted sn prior to phase estimation. The coefficient set α changes because thecovariance matrix elements of the observed signal becomes CDD(i, l) = min(i + 1, l + 1)σ2

∆ + δ(i − l)σ2w.

The NDA method for QPSK modulation raises the received signal to the power of 4 in order to remove thedata. The estimator output in this case need to be divided by four and is folded, yielding a phase ambiguity [6]. Toresolve the ambiguity, differential encoding is apply prior to transmitting the symbols (see e.g [10]).

3.3. Pilot-Based Estimation

In a pilot-based transmission, known data symbols, or pilots, are inserted into the data stream in order to recoverthe phase errors. The algorithm we propose is to use interpolation to allow tracking between pilot symbols. WienerInterpolation algorithm [7] allows to design banks of linear filter to optimally estimate phasors between pilots. TheInterpolator diagram is shown on figure 3.

The interpolator works as follow: we consider one frame of M transmitted signal as shown in 3. The Fsymbols we seek to interpolate are shown in the dashed box. Pilots symbols are inserted in a by chunks of equalsize every F symbols, and in the frame we have inserted in total N/2 Pilot symbols before and after the symbolsto interpolate as shown on figure.

We use a bank of F Wiener Filter to produce the interpolated points. Define x The vector of size M where theframe is stored. The Pilots of the frame are stored into a vector p and the position of these pilots in the frame arestored in a vector qpilots. The position of the data in the frame is stored is a vector qdata. We obtain the interpolatedphasor at time k by applying the kth linear filter with a coefficient vector ck to the vector p.

ejφn = cTk p (14)

The calculation for the coefficient vectors makes use of wiener filter theory detailed in [7]. For such a filterapplied to interpolation, the filter coefficient for the ith coefficient of the kth linear filter, are given by

c(i)k = Γ−1R(k) (15)

Where Γ is the covariance matrix of the observed pilots, and R(k) is the cross correlation between the pilots inthe frame and the kth phasor we seek to interpolate. These values are possible to calculate in advance to be storedin the receiver. Specifically the covariance matrix content is

Γ(u, v) = e−1

2|q(u)pilots−q(v)pilots|σ

2

∆ + δ(u − v)σ2w (16)

F. Munier, E. Alpman, T. Eriksson, A. Svensson,H. ZirathEstimation of Phase Noise for QPSK Modulation over AWGN Channels

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F FF

P P P PDATA DATA DATA

Length of the Frame M

Interpolation Window

N/2 pilots before interpolation window N/2 Pilots after interpolation window Filter Bank

p

Pilot Vector

c(1)

c(2)

c(F)

poin

tsIn

terp

olat

ed

Figure 3: Diagram and Frame Organisation of the interpolator. One packet of F symbols is interpolated from Npilots symbols, with half of the pilots taken from the past symbols and the other half coming from the comingsymbols (hence the need of a delay).

And the Cross correlation vector for the kth filter bank is set by

R(k)(u) = e−1

2|q(u)pilots−q(k)data|σ

2

∆ (17)

4. Results

4.1. DD and NDA Methods

Figure 4 shows an example of a counter rotated constellation for a signal to noise ratio of 10dB. Figure 5 showsthe results obtain on QPSK using differential encoding with decision directed and NDA estimation.The results areshown for five phase noise rate and compared to the theoretical probability of error for QPSK and differentiallyencoded QPSK (DQPSK). The DD algorithm designed failed to work for a QPSK modulation without differentialencoding, with DD or NDA estimators. The reason for this is the lack of reliable data symbol at such a low SNRand the propagation of estimation errors in the future estimate through the decision sn. As suggested by [6] aconstellation working at a higher SNR, such as 16QAM would be more suitable.

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Figure 4: Example of a received constellation before and after estimation of phase noise, for an SNR of 10dB. .

4.2. Interpolation

For (uncoded) QPSK, the results of interpolation shown on figure 5are satisfying. The percentage of pilot insertedwas set to keep a small estimation error variance). Another constraint was that the throughput should not dropby more than 10 percent. With these constrains, the estimator performs better than a non coherent detector whenphase noise rate does not exceeding BT = 5.10−4. The interpolator could perform better by reducing the timebetween pilots insertion, but the cost in throughput would increase.

5. Conclusions

The results show that using differential encoding, methods employing either NDA estimation or DD estimationperform well with low SNR conditions, yielding an SNR penalty of about 1.7dB for a a case of high phase noise(BT = 10.−3). Thus, the use of proposed phase estimation algorithm on coherent detection for the DQPSK

F. Munier, E. Alpman, T. Eriksson, A. Svensson,H. ZirathEstimation of Phase Noise for QPSK Modulation over AWGN Channels

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QPSK, Interpolation, F=50, P=5

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Figure 5: BER results for decision directed (DD), Non Data Aided (NDA) and pilot-based estimators. .

modulation is giving some benefit compare to non-coherent demodulation of DQPSK which typically yields a 2dBpenalty in SNR compared to uncoded QPSK. The main limitation of the DD estimator for QPSK (no differentialencoding) is that the estimator cannot work at low SNR, because of its sensibility to channel noise-induced errors,specially burst errors. Interpolation appears to be a good option but is limited to phase noise with reasonably smallphase noise rate.

6. Acknowledgements

This work has been partly supported by the PCC++ program funded by SSF.

7. References

[1] Herbert Zirath, “Afront end chipset for a 60 GHz radio receiver,” in Proc. of the GigaHertz Symposium onGigahertz Electronics. Chalmers, Mar. 2001.

[2] Christian Fager, MMIC FET Frequency Doublers and FMCW radar Transceivers, Licenciate thesis,Chalmers University of Technology, Goteborg, Sweden, Mar. 2001.

[3] A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd Edition, McGraw-Hill, 1991.

[4] G.J. Foschini and G Vannucci, “Characterizing filtered light waves corrupted by phase noise,” in IEEETransactions on Information Theory. IEEE, nov 1988, vol. 34, pp. 1437–1448.

[5] L. Tomba, “On the effect of wiener phase noise in ofdm systems,” in IEEE Transactions on Communications.IEEE, May 1998, vol. 46, pp. 580–583.

[6] M. Moeneclaey H. Meyr and S.A. Fechtel, Digital Communication Receivers, John Wiley and sons, 1998.

[7] S.M Kay, Fundamentals of Statistical Signal Processing, Prentice Hall International, 1993.

[8] D.B Leeson, “A simple model of feedback oscillator noise spectrum,” in Proceedings of the IEEE. IEEE,1966, vol. 54, pp. 329–330.

[9] J. Roychowdhury A. Demir, A. Mehrotra, “Phase noise in oscillators: a unifying theory and numericalmethods for characterization,” in IEEE Transactions on Circuits and Systems. IEEE, May 2000, vol. 47, pp.655–674.

[10] J. G. Proakis, Digital Communications, McGraw Hill, 1995.

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