Estimation of Phase Noise for QPSK Modulation over AWGN Channels

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1 Florent Munier, Eric Alpman, Thomas Eriksson, Arne Svensson, and Herbert Zirath Dept. of Signals and Systems (S) and Microtechnology Centre at Chalmers (MC) Chalmers University of Technology, S-496 Gothenburg, Sweden 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 problem of minimising the effect of oscillator phase noise on the coherent detection of a quadrature phase shift keying (QSK) modulation operating on an Additive White Gaussian Noise (AWGN) channel. The phase noise process is modeled as a Wiener-Levy (random walk) process. Our approach uses maximum likelihood (ML) estimation of phase noise. Thorough analysis and derivation for Decision Directed (DD), Non-Data Aided (NDA), used with and without symbol differential encoding, and pilot based estimators are presented. We compare these estimators with respect to their main features and evaluate their bit error rate (BER) performances throught simulations. Results show that for low signal to noise ratio (SNR) applications, the use of differential encoding along with the proposed DD or NDA estimator yields performances with an SNR penalty below the two db imposed by the non coherent detection methods, while pilot based estimation using wiener interpolation makes it possible to detect a QSK modulation with SNR penalty around two db. Keywords QSK, hase Noise, Wiener-Levy, ML estimation,awgn, Wiener Interpolation.. Introduction In any bandpass communication system, Radio frequency (RF) hardware such as oscillators are not ideal. The carrier generated by this device is not ideal and experiences phase instability, or phase noise, mainly due to the presence of thermal noise in the circuitry. Circuits designed for very high carrier frequency, such as carrier generator chains used for 6GHz communication (such as in []), are very difficult to design with a very stable frequency source. Moreover, these circuits typically make use of frequency multipliers to reach high carrier frequencies, increasing again the level of phase disturbance []. It is therefore of interest to take into account their phase noise characteristics 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 model phase noise. This model has been widely used and is established in the available literature (e.g [4] and [5] among others). The estimation methods used in this work are employing the Maximum Likelihood (ML) criterion which is documented in [6] and [7]. The paper is organised as follows: First we will detail the considered phase noise model (section.) and present the system setup (section.).then we will present our estimations methods (section 3) and their associated results (section 4), before concluding.. System Setup and Models.. hase Noise Complex Lowpass Equivalent Model In 966, Leeson established a power spectrum model for oscillators [8]. This model splits the spectrum into regions of /f a, where a =,, 3, 4. For a properly designed frequency generation chain, the main source of problem is the a = region, cause by random walk phase modulation. In continuous time, this phase distortion is expressed by

2 ower [db] Normalised Frequency ft Figure : A realisation of the phase noise process φ n and its associated carrier power spectrum for a phase noise process with a phase noise rate BT=.. φ(t) = t (s)ds () The noisy carrier in its complex lowpass equivalent model e jφ(t) now has a Lorentzian ower Spectral Density with a 3-dB bandwidth B controlled by the variance of the White Gaussian random variable (s) [9]. For the purpose of analysis and simulation in a digital communication system we will use a discrete time random walk, also called Wiener-Lévy process. φ n = φ n + n () n is refered as the stepsize of the walk and is a zero mean Gaussian random variable. Its variance sets the speed of the process and is equal to σ = πbt. The product BT is refered as the phase noise rate and express the relative double-sided bandwidth of the discrete time carrier e jφn with respect to the symbol period. hase noise is assumed to remain constant between symbols. Figure shows a realisation of the process and the corresponding carrier power spectrum... System Description Figure shows the general block diagram of the considered system in its baseband equivalent (complex lowpass) representation. bits are modulated using a Quadrature hase Shift Keying (QSK) to obtain the complex symbols s n = e jθn, where θ n can take values m π + π 4, m =,, 3, 4. The symbols are then multiplied by the phasor e jφn, where φ n 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 r n = s n e jφn + w n (3) where w n is a zero-mean, complex Gaussian random variable with variance N. The signal is then passed throught a phase estimator that produce an estimate ˆφ n of the phase noise event. The QSK demodulator outputs a decision s n of the transmitted signal based on the observation of the counter-rotated received signal r n e j ˆφ n, before the transmitted bits are decoded.

3 SOURCE sn QSK Mod. e jφn VCO Model wn Sfrag replacements Channel Model rn SINK sn QSK Demod. j ˆθn e ESTIMATOR Figure : System Setup. 3.. ML estimators 3. Estimators This section describe the way to derive the estimators for NDA assuming that data is known. When data is not known, we need to slightly modify the result as explained in section 3. for decision directed estimation and for non data aided estimation. rior to the estimation, we perform some transformations to the received symbol as defined in equation 3. We rotate r n by s n to get rid of the data dependancy. After rotation, the received symbol becomes r n = e jθn + w n (4) where w n = w n s n is a rotated version of the channel noise sample, and still has the same statistical properties as w n. ML estimators seek to find the estimate of the phasor e jφn that maximise the conditional probability density function f(r φ n ) at a given time n, where r is a vector of N observed received signal points r = [ ṙ n N,..., ṙ n, ṙ n ] T (5) from Equation 3 and the definition of the phase noise process in we can express the received signal at time n i, i =,..., N as ṙ n i = e j(φn+ i u= u) + ẇ n i (6) Let us assume that the variable i u= u has a small value compare to one. Then, e j(φn+ i i u= u) e jφn ( + j u ) (7) Conditionning on the value φ n that we seek to estimate, r n i is a function of two independant gaussian variables (namely the phase noise step u and the AWGN process ẇ n i ), thus the observed vector also has a multivariate gaussian distribution []. f r φn (r φ n ) = u= [ (π) N det C exp ] (r m r) H C (r m r ) The mean vector m r value at time n i is m r (i) = E(ṙ n i ), for all i =,,,..., N. Given that both u and ẇ n i are zero mean, the mean is E(ṙ n i ) = e jφn and thus the mean vector is () denotes the complex conjugate m r = e jφn [,...,, ] T = e jφ n T (9) (8) 3

4 The covariance matrix content is the correlation between points in the vector at offsets i and l, that is, ( ) C(i, l) = E ṙn i E(ṙ n i ))(ṙ n l E(ṙ n l ) () This is reduced as C(i, l) = min(i, l)σ + δ(i l)σw () [ ] The pdf in 8 is maximised when the log-likelihood function Λ = (r m r ) H C (r m r ) is minimised. It can be shown that this is solved for the phasor with the coefficient vector N e j ˆφ n = α u ṙ k u () u= α = T C (3) 3.. DD and NDA Removal of Data Dependancies As we have seen, the estimator derived in assumes that the data has been removed from the received signal. At this 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 s n for s n for the removal of the data dependancy to obtain equation 4. With a reasonably high SNR, mostly correct decisions occur and good estimates can be obtained. When employing a DD estimation method, a delay need to be introduced in the estimation. The estimation of φ n is based on the observation of the past symbol r n, r n N because there is no reliable decision on the transmitted s n prior to phase estimation. The coefficient set α changes because the covariance matrix elements of the observed signal becomes C DD (i, l) = min(i +, l + )σ + δ(i l)σ w. The NDA method for QSK modulation raises the received signal to the power of 4 in order to remove the data. The estimator output in this case need to be divided by four and is folded, yielding a phase ambiguity [6]. To resolve the ambiguity, differential encoding is apply prior to transmitting the symbols (see e.g []) ilot-based Estimation In a pilot-based transmission, known data symbols, or pilots, are inserted into the data stream in order to recover the phase errors. The algorithm we propose is to use interpolation to allow tracking between pilot symbols. Wiener Interpolation algorithm [7] allows to design banks of linear filter to optimally estimate phasors between pilots. The Interpolator diagram is shown on figure 3. The interpolator works as follow: we consider one frame of M transmitted signal as shown in 3. The F symbols we seek to interpolate are shown in the dashed box. ilots symbols are inserted in a by chunks of equal size every F symbols, and in the frame we have inserted in total N/ ilot symbols before and after the symbols to 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 the frame is stored. The ilots of the frame are stored into a vector p and the position of these pilots in the frame are stored in a vector q pilots. The position of the data in the frame is stored is a vector q data. We obtain the interpolated phasor at time k by applying the kth linear filter with a coefficient vector c k to the vector p. e j ˆφ n = c T k p (4) The calculation for the coefficient vectors makes use of wiener filter theory detailed in [7]. For such a filter applied to interpolation, the filter coefficient for the ith coefficient of the kth linear filter, are given by c(i) k = Γ R(k) (5) Where Γ is the covariance matrix of the observed pilots, and R(k) is the cross correlation between the pilots in the frame and the kth phasor we seek to interpolate. These values are possible to calculate in advance to be stored in the receiver. Specifically the covariance matrix content is Γ(u, v) = e q(u)pilots q(v)pilots σ + δ(u v)σ w (6) 4

5 c () Interpolated p Interpolation Window ilot Vector DATA DATA F DATA F c F N/ pilots before interpolation window points c () Length of the Frame M N/ ilots after interpolation window (F) Filter Bank Figure 3: Diagram and Frame Organisation of the interpolator. One packet of F symbols is interpolated from N pilots symbols, with half of the pilots taken from the past symbols and the other half coming from the coming symbols (hence the need of a delay). And the Cross correlation vector for the kth filter bank is set by R(k) (u) = e q(u)pilots q(k)data σ (7) 4. Results 4.. DD and NDA Methods Figure 4 shows an example of a counter rotated constellation for a signal to noise ratio of db. Figure 5 shows the results obtain on QSK using differential encoding with decision directed and NDA estimation.the results are shown for five phase noise rate and compared to the theoretical probability of error for QSK and differentially encoded QSK (DQSK). The DD algorithm designed failed to work for a QSK modulation without differential encoding, with DD or NDA estimators. The reason for this is the lack of reliable data symbol at such a low SNR and the propagation of estimation errors in the future estimate through the decision s n. As suggested by [6] a constellation working at a higher SNR, such as 6QAM would be more suitable Figure 4: Example of a received constellation before and after estimation of phase noise, for an SNR of db Interpolation For (uncoded) QSK, the results of interpolation shown on figure 5are satisfying. The percentage of pilot inserted was set to keep a small estimation error variance). Another constraint was that the throughput should not drop by more than percent. With these constrains, the estimator performs better than a non coherent detector when phase noise rate does not exceeding BT = The interpolator could perform better by reducing the time between 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 estimation perform well with low SNR conditions, yielding an SNR penalty of about.7db for a a case of high phase noise (BT =. 3 ). Thus, the use of proposed phase estimation algorithm on coherent detection for the DQSK 5

6 DQSK, DD, WA, N= DQSK, NDA, WA, N= QSK, Interpolation, F=5, =5 BER 3 BT=e 5 4 BT=5e 5 BT=. BT=.5 5 BT=. DQSK non coherent DQSK coherent QSK SNR [db] BER 3 BT=e 5 4 BT=5e 5 BT=. BT=.5 5 BT=. DQSK non coherent DQSK coherent QSK SNR [db] BER 3 QSK DQSK, non coherent 4 DQSK, coherent BT=e 5 BT=5e 5 5 BT=. BT= SNR [db] 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 DQSK which typically yields a db penalty in SNR compared to uncoded QSK. The main limitation of the DD estimator for QSK (no differential encoding) 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 small phase noise rate. 6. Acknowledgements This work has been partly supported by the CC++ program funded by SSF. 7. References [] Herbert Zirath, Afront end chipset for a 6 GHz radio receiver, in roc. of the GigaHertz Symposium on Gigahertz Electronics. Chalmers, Mar.. [] Christian Fager, MMIC FET Frequency Doublers and FMCW radar Transceivers, Licenciate thesis, Chalmers University of Technology, Goteborg, Sweden, Mar.. [3] A. apoulis, robability, Random Variables and Stochastic rocesses, 3rd Edition, McGraw-Hill, 99. [4] G.J. Foschini and G Vannucci, Characterizing filtered light waves corrupted by phase noise, in IEEE Transactions on Information Theory. IEEE, nov 988, vol. 34, pp [5] L. Tomba, On the effect of wiener phase noise in ofdm systems, in IEEE Transactions on Communications. IEEE, May 998, vol. 46, pp [6] M. Moeneclaey H. Meyr and S.A. Fechtel, Digital Communication Receivers, John Wiley and sons, 998. [7] S.M Kay, Fundamentals of Statistical Signal rocessing, rentice Hall International, 993. [8] D.B Leeson, A simple model of feedback oscillator noise spectrum, in roceedings of the IEEE. IEEE, 966, vol. 54, pp [9] J. Roychowdhury A. Demir, A. Mehrotra, hase noise in oscillators: a unifying theory and numerical methods for characterization, in IEEE Transactions on Circuits and Systems. IEEE, May, vol. 47, pp [] J. G. roakis, Digital Communications, McGraw Hill,