On Wiener Phase Noise Channels at High Signal-to-Noise Ratio

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1 On Wiener Phase Noise Channels at High Signal-to-Noise Ratio Hassan Ghozlan Department of Electrical Engineering University of Southern California Los Angeles, CA 989 USA Gerhard Kramer Institute for Communications Engineering Technische Universität München 8333 Munich, Germany arxiv:3.693v [cs.it 9 Jan 3 Abstract Consider a waveform channel where the transmitted signal is corrupted by Wiener phase noise additive white Gaussian noise AWGN. A discrete-time channel model that takes into account the effect of filtering on the phase noise is developed. The model is based on a multi-sample receiver which, at high Signal-to-Noise Ratio SNR, achieves a rate that grows logarithmically with the SNR if the number of samples per symbol grows with the square-root of the SNR. Moreover, the pre-log factor is at least / in this case. I. INTRODUCTION Phase noise is an impairment that often arises in coherent communication systems. Different models are adopted for the phase noise process depending on the application. In [, Katz Shamai studied a discrete-time model of a phase noise channel partially coherent channel in which the phase noise is independent identically distributed i.i.d. with a Tikhonov distribution. This model is reasonable for the residual phase error of a phase-tracking scheme, such as a Phase-Locked Loop PLL. In [, the authors investigate white Gaussian phase noise for which they observed a spectral loss phenomenon. The white phase noise approximates the nonlinear effect of cross-phase modulation XPM in a Wavelength-Division Multiplexing WDM optical communication system. Lapidoth studied in [3 a discrete-time phase noise channel Y k X k e jθ k +N k at high SNR, where {Y k } is the output, {X k } is the input, {Θ k } is the phase noise process {N k } is the additive noise. He considered both memoryless phase noise phase noise with memory. He showed that the capacity grows logarithmically with the SNR with a pre-log factor /, where the pre-log is due to amplitude modulation only. The phase modulation contributes a bounded number of bits only. In this paper, we study a communication system in which the transmitted waveform is corrupted by Wiener phase noise AWGN. The model is rt xt expjθt+nt, for t R where xt rt are the transmitted received signals, respectively, while nt θt are the additive phase noise, respectively. A detailed description of the model is given in Sec. II. One application for such a channel model is optical communication under linear propagation, in which the laser phase noise is a continuous-time Wiener process see [4 references therein. Since the sampling of a continuous-time Wiener process yields a discrete-time Wiener process Gaussian rom walk, it is tempting to use the model with {Θ} as a discrete-time Wiener process, but this ignores the effect of filtering prior to sampling. It was pointed out in [4 that even coherent systems relying on amplitude modulation phase noise is obviously a problem in systems employing phase modulation will suffer some degradation due to the presence of phase noise. This is because the filtering converts phase fluctuations to amplitude variations. It is worth mentioning that filtering is necessary before sampling to it the variance of the noise samples. The model thus does not fit the channel it is not obvious whether a pre-log / is achievable. The model that takes the effect of matched filtering into account is Y k X k H k +N k 3 where {H k } is a fading process. The model 3 falls in the class of non-coherent fading channels, i.e., the transmitter receiver have knowledge of the distribution of the fading process {H k }, but have no knowledge of its realization. For such channels, Lapidoth Moser showed in [5 that, at high SNR, the capacity grows double-logarithmically with the SNR, when the process {H k } is stationary, ergodic, regular. Rather than using a matched filter sampling its output at the symbol rate, we use a multi-sample receiver, i.e., a filter whose output is sampled many times per symbol. We show that this receiver achieves a rate that grows logarithmically with the SNR if the number of samples per symbol grows with the square-root of the SNR. Furthermore, we show that a pre-log of / is achievable through amplitude modulation. In this paper, we study only rectangular pulses but we believe that the results hold qualitatively for other pulses. The paper is organized as follows. The continuous-time model is described in Sec. II the discretization is described in Sec. III. We derive a lower bound on the capacity in Sec. IV discuss our result in Sec. V. Finally, we conclude the paper with Sec. VI.

2 II. CONTINUOUS-TIME MODEL We use the following notation: j, denotes the complex conjugate, δ D is the Dirac delta function, is the ceiling operator, R[ is the real part of a complex number, log is the natural logarithm we use X k to denote the k-tuple X,X,...,X k. Suppose the transmit-waveform is xt the receiver observes rt xt expjθt + nt 4 where nt is a realization of a white circularly-symmetric complex Gaussian process Nt with E[Nt E[Nt N t σ N δ Dt t. 5 The phase θt is a realization of a Wiener process Θt: Θt Θ+ t Wτdτ 6 whereθ is uniform on[ π,π Wt is a real Gaussian process with E[Wt 7 E[Wt Wt πβ δ D t t. 8 The processes Nt Θt are independent of each other independent of the input as well. N σn is the single-sided power spectral density of the additive noise. The parameter β is called the full-width at half-maximum FWHM, because the power spectral density of e jθt has a Lorentzian shape, for which β is the full-width at half the maximum. The transmitted waveforms must satisfy the power constraint [ T E Xt dt P 9 T where T is the transmission interval. III. DISCRETE-TIME MODEL Let x,x,...,x n be the codeword sent by the transmitter. Suppose the transmitter uses a unit-energy rectangular pulse, i.e., the waveform sent by the transmitter is xt x m gt m T symbol m where T symbol is the symbol interval { /Tsymbol, t < T gt symbol,, otherwise. Let L be the number of samples per symbol L define the sample interval as T symbol L. The received waveform rt is filtered using an integrator over a sample interval to give the output signal yt t t rτ dτ. 3 whereyt is a realization ofyt. The outputyt is sampled every seconds which yields the discrete-time model: Y k X k/l e jθ k F k +N k 4 for k,...,nl, where Y k Yk, Θ k Θk, F k N k k k k k e jθτ Θ k dτ 5 Nτ dτ. 6 The process {N k } is an i.i.d. circularly-symmetric complex Gaussian process with mean E[ N k σ N while the process {Θ k } is the discrete-time Wiener process: Θ k Θ k +W k 7 where Θ is uniform on [ π,π {W k } is an i.i.d. real Gaussian process with mean E[ W k πβ. The process {F k } is an i.i.d. process. Moreover, {F k } {W k } are independent of {N k } but not independent of each other. Equations 9 imply the power constraint n E[ X m P PT symbol. 8 m IV. LOWER BOUND For the kth input symbol X k we have L outputs, so it is convenient to group the L samples per symbol in one vector define Y k Y k L+,Y k L+,...,Y k L+L. We further define X A X X Φ X. We decompose the mutual information using the chain rule into two parts: IX n ; Yn IXn A, ; Yn +IXn Φ, ; Yn Xn A,. 9 The first term represents the contribution of the amplitude modulation while the second term represents the contribution of the phase modulation. We focus on the amplitude contribution use IXΦ, n ; Yn Xn A, to obtain the lower bound IX n ; Yn IXn A, ; Yn. Suppose that XA, n is i.i.d. Hence, we have IXA, n ; Yn a IX A,k ; Y n Xk A, b c d k k HX A,k HX A,k Y n X k A, IX A,k ; Y k k IX A,k ;V k k

3 where V k Y k L+l. l Step a follows from the chain rule of mutual information, b follows from the independence of X A,,X A,,...,X A,n, c holds because conditioning does not increase entropy, d follows from the data processing inequality. Since XA, n is identically distributed, then V n is also identically distributed we have, for k, IX A,k ;V k IX A, ;V. 3 In the rest of this section, we consider only one symbol k drop the time index. Moreover, we assume that T symbol for simplicity. By combining 4, we have V X A F l +X A R[e jφx e jθ l F l Nl + N l l X A G+X A Z +Z 4 where G, Z Z are defined as G F l 5 L Z Z l R[e jφx e jθ l F l Nl 6 l N l. 7 l The second-order statistics of Z Z are E[Z Var[Z E[GσN / E[Z σn Var[Z σn 4 E[Z Z E[Z. 8 By using the Auxiliary-Channel Lower Bound Theorem in [6, Sec. VI, we have IX A ;V E[ logq V V+E[logQ V XA V X A 9 where Q V XA v x A is an arbitrary auxiliary channel Q V v P XA x A Q V XA v x A dx A 3 where P XA is the true input distribution, i.e., Q V is the output distribution obtained by connecting the true input source to the auxiliary channel. E[ is the expectation according to the true distribution. We choose the auxiliary channel Q V XA v x A exp 4πx A σn v x A σ N 4x A σ N It follows that [ V X E[ logq V XA V X A E A σn 4XA σn. 3 +log + log4πσ N + E[logX A. 3 By using 4, we have V X A σ N X A G +X A Z +Z σ N X 4 A G +4X A Z +Z σ N +4X 3 A G Z +X A G Z σ N +4X A Z Z σ N 33 hence, using the second-order statistics 8, we have [ V X E A σn 4XA σn P E [ G + [ E[G+ σ N 34 where we also used E[G Z. 35 Substituting 34 into 3 using E[G yield E[ logq V XA V X A log + log4πσ N + E[logX A + P E [ G + [ + σ N. 36 It is convenient to define X P XA. We choose the input distribution { P XP x P exp xp Pmin, xp P min 37, otherwise where < P min < P P P min, so that E[X P E[X A P. 38 It follows from 3 37 that Q V v P min exp x P P min Q V XP v x P dx P expp min / F V v 39 where F V v Q V XP v x P Q V XA v x P 4 exp x P Q V XP v x P dx P. 4 The inequality 39 follows from the non-negativity of the integr. By combining 3, 4, 4 making the change of variables x x P, we have F V v e x/ 4πx σ N exp +4 σ N exp 4 σ N [v σ N v σn v x σ N 4x σ N + 4 σ N dx 4

4 where we used equation 4 in Appendix A of [7: a exp x exp u x dx a πbx bx [ exp u u + b. 43 aa+b b a Therefore, we have E[ logf V V log + [E[V σn σ N E[ V σn + 4σ N a log + σn E[V σ N [ + 4σ N b log 44 wherea holds because the logarithmic function is monotonic E[ E[, b holds because E[V σ N E[X A E[G+E[X A E[Z +E[Z σ N P E[G. 45 The monotonicity of the logarithmic function 39 yield [ E[ logq V V E log e Pmin/ F V V log +log P min 46 where the last inequality follows from 44. It follows from 9, that IX A ;V log P min log4πσ N E[logX A P E [ G [ σ N. 47 If P min P/, then P P min P/ we have E[logX P [ E X P a log+ P min P e x / logxdx e u logudu 48 b log+ 49 where a follows by the change of variables u x/, b holds because logu u for all u >. Substituting into 47, we obtain IX A ;V log SNR log8π SNR 4 SNR E[ G 5 where SNR P/σN. Suppose L grows with SNR such that L β SNR. 5 Since /L, then we have SNR SNR which implies SNR SNR β 5 IX A;V SNR log SNR π log8π because see Appendix E[G πβ By combining,, 3 53, we have SNR n IXn ; Yn log SNR π log8π This shows that the information rate grows logarithmically at high SNR with a pre-log factor of /. V. DISCUSSION There is a wide literature on the design of receivers for the channel model with a discrete-time Wiener phase noise, e.g., see [8, [9, [ references therein. One may want to make use of these designs, which raises the following question: when is it justified to approximate the non-coherent fading model 3 with the discrete-time phase noise model? Our result suggests that this approximation may be justified when the phase variation is small over one symbol interval i.e., when the phase noise linewidth is small compared to the symbol rate also the SNR is low to moderate. It must be noted that the SNR at which the high-snr asymptotics start to manifest themselves depends on the application. We remark that the authors of [ treated on-off keying transmission in the presence of Wiener phase noise by using a double-filtering receiver, which is composed of an intermediate frequency IF filter, followed by an envelope detector squarelaw device then a post-detection filter. They showed that by optimizing the IF receiver bwidth the double-filtering receiver outperforms the single-filtering matched filter receiver. Furthermore, they showed via computer simulation that the optimum IF bwidth increases with the SNR. This is similar to our result in the sense that we require the number of samples per symbol to increase with the SNR in order to achieve a rate that grows logarithmically with the SNR. Finally, we remark that we have not computed the contribution of phase modulation to the information rate. We believe that using the multi-sample receiver it is possible to achieve an overall pre-log that is larger than /. This matter is currently under investigation.

5 VI. CONCLUSION We studied a communication system impaired by Wiener phase noise AWGN. A discrete-time channel model based on filtering oversampling is considered. The model accounts for the filtering effects on the phase noise. It is shown that at high SNR the multi-sample receiver achieves rates that grow logarithmically with at least a / pre-log factor if the number of samples per symbol grows with the square-root of the SNR. ACKNOWLEDGMENT H. Ghozlan was supported by a USC Annenberg Fellowship NSF Grant CCF G. Kramer was supported by an Alexer von Humboldt Professorship endowed by the German Federal Ministry of Education Research. APPENDIX We discuss the it in 54. We express E[G as E[G VarG+E[G L Var F + E[ F 56 where the last equality follows from the definition of G in 5 because {F k } is i.i.d. Next, we outline the steps for computing E[ F 4 E[ F. Let M be a positive integer, c c,...,c M T be a constant vector, t t,...,t M T be a non-negative real vector Θt Θt Θ,...,Θt M Θ T where Θt is defined in 6. We have E [ M a M b c expjc T Θtdt E [ expjc T Θt dt M exp ct Σtc dt exp ct Σtc du 57 where dt dt M...dt Σt is the covariance matrix of Θt whose entries are given by We also have, using M 4 c,,, T in 57, E[ F a+a4 +54loga+4aloga+44loga 8loga 4. Computing the integrals is tedious but straightforward. Finally, it follows from 56, 59 6 that E[G πβ E[G a loga REFERENCES πβ. 9 6 [ M. Katz S. Shamai. On the capacity-achieving distribution of the discrete-time noncoherent partially coherent AWGN channels. IEEE Trans. Inf. Theory, 5:57 7, Oct. 4. [ B. Goebel, R. Essiambre, G. Kramer, P.J. Winzer, N. Hanik. Calculation of mutual information for partially coherent gaussian channels with applications to fiber optics. IEEE Trans. Inf. Theory, 579: , Sep.. [3 A. Lapidoth. Capacity bounds via duality: A phase noise example. In Proc. nd Asian-Euro. Workshop on Inf. Theory, pages 58 6,. [4 G.J. Foschini G. Vannucci. Characterizing filtered light waves corrupted by phase noise. IEEE Trans. Inf. Theory, 346: , Nov [5 A. Lapidoth S.M. Moser. Capacity bounds via duality with applications to multiple-antenna systems on flat-fading channels. IEEE Trans. Inf. Theory, 49:46 467, Oct. 3. [6 D.M. Arnold, H.-A. Loeliger, P.O. Vontobel, A. Kavcic, Wei Zeng. Simulation-based computation of information rates for channels with memory. IEEE Trans. Inf. Theory, 58: , Aug. 6. [7 S.M. Moser. Capacity results of an optical intensity channel with inputdependent gaussian noise. IEEE Trans. Inf. Theory, 58:7 3, Jan.. [8 A. Barbieri, G. Colavolpe, G. Caire. Joint iterative detection decoding in the presence of phase noise frequency offset. IEEE Trans. Commun., 55:7 79, Jan. 7. [9 A. Spalvieri L. Barletta. Pilot-aided carrier recovery in the presence of phase noise. IEEE Trans. Commun., 597: , July. [ A. Barbieri G. Colavolpe. On the information rate repeataccumulate code design for phase noise channels. IEEE Trans. Commun., 59:33 38, Dec.. [ G.J. Foschini, L.J. Greenstein, G. Vannucci. Noncoherent detection of coherent lightwave signals corrupted by phase noise. IEEE Trans. Commun., 363:36 34, Mar Σ ij t πβmin{t i,t j }, for i,j,...,m. 58 Step a follows from the linearity of expectation, b follows by using the characteristic function of a Gaussian rom vector, c follows from the transformation of variables t u. We define a e πβ 59 use M c, T in 57 to compute E[ F a loga loga. 6