Numerical Performance Evaluation for OFDM Systems affected by Phase Noise and Channel Estimation Errors

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1 Numerical Performance Evaluation for OFDM Systems affected by Phase Noise and Channel Estimation Errors Marco Krondorf, Steffen Bittner and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universität Dresden, D-162 Dresden, Germany Abstract In this paper we present a numerical approach to evaluate the bit error rate BER) and mutual information of OFDM links subject to phase noise, channel estimation error and frequency selective fading channels. Based on the analytical modeling of the correlation between channel estimates and received signals affected by phase noise, the statistical properties of the received signal can be numerically evaluated by means of a probability density function. The results illustrate that our analysis can approximate the simulative performance very accurately if the power delay profile of the fading channels and the phase noise properties are known. Hence, we present a useful numerical tool for OFDM performance analysis under the presence of phase noise that can be used for planning and design of mobile device architectures, without running extensive simulations. I. INTRODUCTION Orthogonal Frequency Division Multiplexing OFDM) is a widely applied technique for wireless communications, which enables simple one-tap equalization in contrast to single carrier systems. However, one main drawback of OFDM systems is their sensitivity to RF impairments such as phase noise. Phase noise describes a multiplicative phase distortion caused by RF imperfections, e.g. imperfect oscillators, used for upand downconversion, leading to a common phase rotation and additionally to intercarrier interference ICI). In the literature the influence of phase noise in OFDM has been extensively studied. Furthermore, possible phase noise mitigation algorithms are also discussed [8], [1]. However, in previous contributions the remaining ICI noise has implicitly be assumed to been Gaussian distributed, which is in general not the case. In [6] a first approach was presented indicating that the intercarrier interference caused by phase noise is not Gaussian distributed. The authors also came up with an analytical expression of the ICI, which, however, is only valid for small phase noise based on a free running oscillator. Extending the work done in [4], we come up with a new semi-analytic approach to express the overall received signal statistics of an OFDM receiver chain including channel estimation errors, for any given oscillator type and phase noise strengths. With the help of the PDF of the decision variable, one is now able to compute relevant performance metrics such Part of this work has been performed in the framework of the IST project IST ORACLE, which is partly funded by the European Union. as symbol error rate, bit error rate and ergodic link capacity, without running extensive Monte-Carlo simulations. The remainder of this paper is organized as follows. In Sec. II we introduce the OFDM system model including mobile channel characteristics, the channel estimation error model. We also present a short overview of different oscillator types together with their phase noise characteristics. Sec. III deals with the probability density function analysis where we explain the general work flow of deriving the conditional PDF. Finally, Sec. IV and Sec. V present some numerical examples before we conclude in Sec.VI. II. OFDM SYSTEM MODEL In this paper we consider a N-point FFT OFDM system. The data on each subcarrier is modulated by a QAM modulator and transformed to a time domain signal by performing an IFFT operation. Subsequently, a cyclic prefix is added which is chosen to be longer than the channel impulse response CIR). In this paper, we assume that one OFDM symbol is used as pilot symbol for channel estimation. Afterwards, data is transmitted in the following OFDM symbols leading to the well-known OFDM block structure. Furthermore, we assume static channel characteristics during one OFDM burst. After direct down conversion the time domain OFDM signal in the presence of receiver phase noise φn) can be expresses as: yn) = sn) hn))e jφn) wn), 1) where stands for convolution and wn) represents the additive white Gaussian noise with variance σ 2 n. Details about generating phase noise samples are given in Sec. II-C. The result after removing the cyclic prefix and FFT operation at the receiver can be obtained by the following reasoning: Phase noise affects the signal as an angular multiplicative distortion. Multiplication of two signals in the time domain is equivalent to convolving the spectra of the corresponding signals in the frequency domain. The complex received signal Y l on subcarrier l can be written as Y l = X l H l I) }{{} CPE N 2 1 X kh k Il k) W l, 2) k= N 2 }, {{} ICI

2 where X l represents the transmitted complex M-QAM data symbol on subcarrier l and W l is a complex Gaussian noise sample. In the following, we will abbreviated the summation N 2 1 as k= N 2,. The coefficient H l denotes the frequency domain channel transfer function on subcarrier l, which is discrete Fourier transform of the CIR hτ) having L taps, i.e. L 1 H l = hτ)e j2πlτ/n. 3) τ= Phase noise affects the signal in two ways. Firstly it causes a common phase error I) CPE), which is a common phase rotation on all subcarriers. Secondly, an intercarrier interference ICI) term is introduced. Note, that the common phase shift is robustly estimated and compensated by continuous pilots inserted in the OFDM data symbols. In this work, we consider a preamble-based channel estimation where one entire OFDM symbol is used for pilot transmission. Hence, the phase noise coefficients during the pilot symbol are marked with index P - Pilot and are ordered subcarrier-wise in the vector I P,l. The least squares frequency domain channel estimation is used to obtain the channel state information on subcarrier l: Ĥ l = Y P,l = I P )H l X P,kH k I P k l)) W l X P,l X P,l 4) where X P,l and Y P,l denote the transmitted and received pilot symbol on subcarrier l. Further assumptions are that the Gaussian noise of the preamble part W l has the same variance as W l of the data part σw 2 = σ 2 l W l ). After channel estimation, Ĥ l is used for frequency domain zero-forcing equalization of the OFDM data symbol on subcarrier l Z l = Y l Ĥ l, 5) where Z l denotes the decision variable that is fed in the OFDM receivers detector/decoder stage. Note that the conditional probability density function PDF) p Zl z X l ) of those decision variables are of integral importance for the computation of relevant performance metrics such as bit error rate or ergodic link capacity. It is important to know that the phase noise coefficients are different for each OFDM symbol. Thus, we use index D - Data to subdivide phase noise coefficients during data detection from the ones during pilot symbol transmission. Such phase noise coefficients per subcarrier l are orderd into a vector I D,l. The power of preamble signals and the average power of transmitted data signals is normalized to one X P,l 2 = σ 2 X = 1) on all subcarriers. Hence, we define the average subcarrier SNR γ = σ2 X σ2 H σ 2 W = σ2 H σ 2 W which we use for performance evaluation in Sec.V. A. Mobile channel characteristics To obtain precise performance analysis results in the case of subcarrier crosstalk induced by phase noise, it is desirable to use exact expressions of the subcarrier channel cross-correlation properties. The cross-correlation between frequency domain channel coefficients is mainly determined by the CIR power delay profile and the CIR tap crosscorrelation properties. Furthermore, the discrete nature of the sampled CIR is modeled as tapped delay line having L channel taps. Although our analysis is not limited to a specific type of frequency selective channel, we consider mobile channels having an exponential power delay profile PDP) in our numerical examples: σ 2 τ = 1 C e Dτ/L, τ =, 1,...,L 1. Here, στ 2 = E{ hτ) 2 } and the factor C = L 1 τ= e Dτ/L is chosen to normalize the PDP as L 1 τ= σ2 τ = σh 2 = E{ H l 2 }. The channel taps hτ) are assumed to be complex zeromean Gaussian random variable RV) with uncorrelated real and imaginary parts. Hence, after FFT according to Eq.3), the channel coefficients are zero-mean complex Gaussian RV. Additionally, the CIR length L is assumed to be shorter/equal than the cyclic prefix. The cross-correlation coefficient of the channel transfer function on subcarriers k and l in the case of frequency selective fading is defined as r k,l = E{H kh l } σ 2 H 1, k l, 6), where σh 2 is equivalent for all subcarriers. Assuming mutually uncorrelated CIR taps and applying Eq.3), one obtains E{H k H l } στ 2 e j2πk l)τ/n. 7) = L 1 τ= Finally, the cross correlation of the complex Gaussian channel coefficients can be formulated as H k = r k,l H l V k,l, 8) where V k is a complex zero-mean Gaussian RV with variance σ 2 V k,l = σ 2 H 1 r k,l 2 ) and E{V k,l H l } =. B. Channel estimation error model Substituting the cross-correlation properties of the channel coefficients into Eq.4), the channel estimate can be written as Ĥ l = I P )H l 1 X ) P,kI P k l)r k,l ν l, 9) I P )X P,l }{{} α l where the effective noise term of the channel estimate is given by ν l = 1 X P,k I P k l)v k W l 1) X P,l

3 having a variance σ 2 ν l = σ 2 W X P,k XP,mI P k l)ipm l) m l r k,m r k,l r m,lσ 2 H). 11) The factor α l given in Eq.9) α l = 1 X ) P,kI P k l)r k,l I P )X P,l represents the cross-correlation coefficient between H l and its estimate Ĥl. In impairment-free OFDM i.e. no phase noise and no carrier crosstalk) we get α l = 1. Conversely, α l deviates from 1 in the case of carrier crosstalk due to phase noise. C. Phase Noise modeling In this part we provide a short summary of three phase noise models of practical implementation of the local oscillator. As introduced in [2], we will characterize the phase noise imperfections by a random time shift κt), resulting in an oscillator output of x s t) = x t κt)) where x t) is the noiseless periodic steady state response of an oscillator. The phase shift itself is given by: φt) = 2πf c κt). Using a discrete time basis, the n-th sample is related to the previous one as φn) = φn 1) φ, where φ is defined by the phase noise model. The phase noise vector I = I N 2,, I l, I N 2 1)T introduced in the previous section, represents the FFT coefficients of the realization of the phase process e jφn) in the current OFDM symbol and is given as: I l = 1 N N 1 n= e j2πln/n e jφn). 12) 1) Free Running VCO: In the case of a free running voltage controlled oscillator VCO) the random time shift κt) is given as: κt) = t c vco ξt )dt 13) } {{ } Wt) where Wt) is a standart Wiener or Brownian motion process and ξt ) indicates a differential Wiener process. The constant c vco describes the oscillator quality. Its value is in practice not directly available and is defined via the 3-dB frequency f 3dB of the power spectral density Fig. 2). The relation between c vco and f 3dB is given by c vco = f 3dB /πf 2 c [2]. 2) Second Order PLL: A PLL oscillator consists of an interconnection with an integrated low quality VCO and a stable reference oscillator as shown in Fig. 1. The advantage of such a setup is to combine the very good phase noise properties of the reference oscillator with the large frequency range of the VCO. Furthermore, a low pass filter is introduced, guaranteing that the spectrum follows the VCO for high frequencies. Even though the reference signal is very stable it still suffers from a time fluctuation κ in t), resulting in a Wiener process. The time deviation of the VCO is denoted as κ vco t) and of the Single Side PSD dbc/hz) Crystal Phase Detector Fig. 1. Low Pass Filter 1/N PLL block diagram VCO Sec. Order PLL Ref. Oscillator VCO Charge Pump PLL Offset Frequency Fig. 2. Single Side PN spectrum; f c = 5.25GHz c vco = s, c in = s, f Gpll = 3kHz, f LP = f CP = 5kHz overall PLL as κ pll t). Using the results presented in [5] the output of the PLL can be written as: κ pll t) = κ in t) βt) 14) where βt) is a one dimensional Ornstein-Uhlenbeck process. The low pass loop filter in Fig. 1 is given by the differential equation: γt) = 2πf LP γt) βt)), with f LP as 3dB cut off frequency of the filter. Following the derivation in [5] the stochastic differential equation describing the PLL is given as: [ ] [ ] [ ] βt) 2πf = Gpll βt) γt) 2πf LP 2πf LP γt) [ cvco c in ] [ ξvco t) ξ n t) ]. 15) The PSD around the first harmonic for the given values is depicted in Fig. 2. Note that the spectrum follows the reference signal for low frequencies and follows the VCO for high frequencies. 3) Charge Pump PLL: The presented second order PLL has the disadvantage that the phase difference between the VCO and the reference signal cannot be compensated completely, resulting in a so called steady state error [5]. One can achieve the zero phase error using an integrator after the linear phase detector knowing as Charge Pump. However, the integrator degrades the stability of the system. This stability can be regained by introducing an additional zero in the transfer function, which can be realized by a series combination of a resistor and a capacitor. The linear differential equation is given by: γt) = βt) 2πf CP βt)), with f CP as the 3dB cut off frequency of the charge pump phase detector. The overall PLL differential equation can now be written as: [ ] [ ] [ ] βt) f = 2π Gpll βt) γt) f CP f Gpll γt) [ cvco ][ c in ξvco t) c vco cin ξ n t) ]. 16)

4 The PSD is shown in Fig. 2. In the case of the charge pump PLL, the spectrum follows even longer the reference signal, given a better phase noise properties in the low frequency range. III. PROBABILITY DENSITY FUNCTIONS ANALYSIS The PDF of the decision variables among the OFDM data carriers is of integral importance for performance analysis such as bit error rate or capacity calculation. Hence, in this section we illustrate how to numerically compute this PDF for any dedicated data carrier without time-consumptive simulation of the entire OFDM receiver chain. A. General Workflow Based on the results of [7], in [4] we derived the conditional PDF p Zl z X l, I P,l, I D,l ) of decision variable Z l on subcarrier l for deterministic carrier crosstalk vectors I P,l and I D,l. The PDF analysis was used in [4] to compute symbol error rates for OFDM systems under carrier frequency offset and channel estimation errors. Later, we extended the findings of [4] toward time selective channels and I/Q imbalance in [3]. Since there is a formal equivalence between the OFDM system model used in Sec.II and the one presented in [4], we can use p Zl z X l, I P,l, I D,l ) for phase noise analysis here in this paper. While under CFO the vectors I P,l and I D,l are deterministic among all OFDM symbols, they are getting stochastic under phase noise since phase noise samples randomly changes from symbol to symbol. Therefore, we have to compute the marginal PDF p Zl z X l ) that does not depend on the carrier crosstalk vectors anymore: p Zl z X l ) = E IP,l,I D,l {p Zl z X l, I P,l, I D,l )} 17) Here, E IP,l,I D,l {.} denotes the expectatoin operation w.r.t. the random vectors I P,l and I D,l. The computation of p Zl z X l, I P,l, I D,l ) given in [4] is shortly reviewed in the next section III-B. Generally, the analytical solution of Eq.17) requires the joint PDF of the random vectors I P,l and I D,l which is hard to derive in closed form for any oscillator type. Hence, we decided to solve Eq.17) by using a semi analytical approach. In each run we randomly generate appropriate phase noise trajectories according to Sec.II-C and compute the instantaneous PDF p Zl z X l, I P,l, I D,l ). Finally, the instantaneous PDFs are averaged to obtain p Zl z X l ). B. PDF calculation Based on the results of [4], the conditional PDF of the complex decision variable Z l in the case of transmit symbol X l is given by the following equation: p Zl z X l, I P,l, I D,l ) = a 2 X l, I P,l, I D,l ) π z bx l, I P,l ) 2 a 2 X l, I P,l, I D,l )) 2. 18) The PDF mainly depends on two parameters a 2 X l, I P,l, I D,l ) and bx l, I P,l ) that are detailed in [4] and that can be calculated as follows: a 2 X l, I P,l, I D,l ) = X l 2 σ 2 ν l I P ) 2 σ 2 H αl 2 I P ) 2 σ 2 H σ2 ν l ) 2 σ 2 w α l 2 I P ) 2 σh 2. 19) σ2 ν l Here, σ w 2 denotes the variance of the effective noise term that consists of the AWGN and the ICI part as defined in [4]: σ w 2 = σ2 W σ2 H I D k l) 2 1 r k,l 2 ). 2) The complex parameter bx l, I P,l ) can be determined by solving α l bx l, I P,l ) = X I P) 2 σ 2 ) H l α l 2 I P ) 2 σh 2. 21) σ2 ν l IV. OFDM PERFORMANCE METRICS The marginal PDF of the decision variables can be used to compute relevant performance metrics such as symbol error rate SER), bit error rate BER) and ergodic link capacity on certain subcarriers. For SER computation for instance we have to solve the following equation to yield the corresponding average probability of error on subcarrier l: P e l) = 1 1 M M m=1 R m p Zl z r jz i X m ) dz r dz i 22) where M is the number of constellation points and R m is the decision region of point X m. If the bit mapping is taken into account, bit error rates can be calculated in an equivalent manner of solving the double integral w.r.t. appropriate decision regions and averaging over all OFDM data carriers. For information theoretic concerns, ergodic OFDM link capacity in terms of mutual information I l X; Z) on subcarrier l between the discrete complex input alphabet X mostly the M-QAM modulation alphabet) and the continuous complex alphabet Z of the decision variables, is of greater importance than BER or SER. Hence, it is well known that I l X; Z) can be computed as: I l X; Z) = PX m ) p Zl z r jz i X m ) 23) m Z i log 2 Z r p Zl z r jz i X m ) n p Z l z r jz i X n )PX n ) V. NUMERICAL EXAMPLES ) dz r dz i. In numerical examples, we consider a OFDM system with 64-point FFT. The data is 16-QAM and 64-QAM modulated to different subcarriers, then transformed to a time domain signal by IFFT operation and prepended by a 16 samples cyclic prefix. The data is randomly generated and one OFDM symbol preamble was used for channel estimation. The used BPSK

5 training symbols of the preamble in frequency domain is given by X P,l = 1) l for subcarrier index l = [ 32 : 1 : 31] We consider frequency selective Rayleigh fading channels having an exponential power delay profile according our channel model of Sec. II-A: σ 2 τ = 1 C e Dτ/L, τ =, 1,...,L 1 where is set to be D=7 and the factor C = L 1 τ= e Dτ/L is chosen to normalize σh 2 for a given average subcarrier SNR γ = σ2 H σ 2. Additionally, the CIR length is set to be L = 8. For W the numerical integration of p Zl z X l ) for BER calculation see Eq.23)) we used the Matlab 7..1 build-in numerical integration function. The system bandwidth is chosen to be 2 MHz. In all numerical examples, the free running VCO phase noise model is used. Fig.3 illustrates the calculated and simulated bit error rates vs. SNR under different c vco for 16-QAM and 64-QAM. We set the carrier frequency f c to be 5.2 GHz. The results illustrate that our analysis can approximate the simulative performance accurately, if the channel power delay profile and phase noise settings are known. Fig.4 depicts the ergodic link capacity according to Eq.23) BER = 2e 17 = 2e 17 = 8e 18 = 8e 18 = 3e 18 = 3e 18 Channel estimation error only 64 QAM 16 QAM Average Subcarrier SNR γ in db Fig. 3. Comparison of calculated and simulated BER vs. SNR under different c vco in frequency selective Rayleigh fading channel for 64-QAM modulation averaged over all subcarriers. The results are plotted over SNR for different carrier frequencies and a worse c vco of s. The results illustrate how critical phase noise can get especially at higher carrier frequencies. Mutual Information in Bit/Channel Use GHz 7GHz 1GHz Channel estimation error only Average Subcarrier SNR γ in db Fig. 4. Comparison of 64-QAM mutual information vs. SNR under different carrier frequencies for c vco = s relevant oscillator types are reviewed as well. Furthermore, our results show that uncoded bit error rate can be calculated exactly when the channel power delay profile and phase noise settings are known. Finally, the tool is able to support system engineers to find a proper OFDM receiver design. REFERENCES [1] S. Bittner, W. Rave, and G. Fettweis. Joint Iterative Transmitter and Receiver Phase Noise Correction using Soft Information. In IEEE International Conference on Communications, 27. [2] A. Demir, A. Mehrotra, and J. Roychowdhury. Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterisation. IEEE Trans. Circuits Syst. I, 475): , 2. [3] M. Krondorf and G. Fettweis. OFDM Link Performance Analysis under Various Receiver Impairments. In EURASIP Journal on Wireless Communications and Networking, online available under September 27. [4] M. Krondorf, T. J. Liang, and G. Fettweis. Symbol Error Rate of OFDM Systems with Carrier Frequency Offset and Channel Estimation Error in Frequency Selective Fading Channels. In Proc.IEEE Int. Conf. on Communications ICC), June 27. [5] A. Mehrotra. Noise Analysis of Phase-Locked Loops. IEEE Trans. Circuits Syst. I, 499): , 22. [6] T. C. W. Schenk, R.W. van der Hofstad, E. R. Fledderus, and P. F. M. Smulders. Distribution of the ICI Term in Phase Noise Impaired OFDM Systems. IEEE Trans. Wireless Commun., 64): , 27. [7] S. K. Wilson and J. M. Cioffi. Probability density functions for analyzing multi-amplitude constellations in Rayleigh and Ricean channels. In IEEE Transactions on Communications, volume 47, March [8] S. Wu, P Liu, and Y. Bar-Ness. Phase Noise Estimation and Mitigation for OFDM Systems. IEEE Trans. Wireless Commun., 512): , 26. VI. CONCLUSIONS In this paper we present a semi-analytical tool for the performance analysis of OFDM systems subject to the simultanous effects of phase noise, channel estimation errors and frequency selective fading. Phase noise models for different practically