Decision-Aided Compensation of Severe Phase ... - IEEE Xplore
1614
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
Decision-Aided Compensation of Severe Phase-Impairment-Induced Inter-Carrier Interference in Frequency-Selective OFDM Konstantinos Nikitopoulos, Member, IEEE, Stelios Stefanatos, and Aggelos K. Katsaggelos, Fellow, IEEE
Abstract—A new, reduced complexity algorithm is proposed for compensating the Inter-Carrier Interference (ICI) caused by severe PHase Noise (PHN) and Residual Frequency Offset (RFO) in OFDM systems. The algorithm estimates and compensates the most significant terms of the frequency domain ICI process, which are optimally selected via a Minimum Mean Squared Error (MMSE) criterion. The algorithm requires minimal knowledge of the phase process statistics, the estimation of which is also considered. The scheme outperforms previously proposed compensation methods of similar complexity, when severe phase impairments are present. Index Terms—Frequency offset, inter-carrier interference, OFDM, phase noise.
I. I NTRODUCTION
I
T is well known that OFDM is very sensitive to phase impairments, namely Phase Noise (PHN) and Residual Frequency Offset (RFO) [1], which induce a phase rotation, common to all subcarriers of the same OFDM symbol, plus Inter-Carrier Interference (ICI) [1]–[5]. In the presence of severe PHN and high order constellations, the provided Symbol Error Rate (SER) is lower bounded by the presence of ICI, rendering the employment of an ICI compensation scheme necessary. Since the effect of the ICI on the system’s performance is not only a function of the actual PHN and RFO values but it also depends on the transmitted constellation, we herein define the phase process as “severe” whenever it produces a symbol error rate (SER) performance floor larger than 10−3 for the target transmitted constellation without ICI compensation. Several approaches have been proposed in the literature for ICI compensation. A pilot based estimator is proposed in [6] for estimating the most significant terms of the PHN/RFO process. The latter is expanded on specially selected basis functions, via a least squares method. However, the performance of the scheme in frequency selective channels, as well as the optimal number of PHN/RFO terms to be estimated are not addressed. Decision-aided MMSE-based methods have been also proposed in [7], [8], where perfect knowledge of the correlation matrix of the phase noise process is assumed and
Manuscript received September 18, 2007; revised March 3, 2008 and September 5, 2008; accepted November 11, 2008. The associate editor coordinating the review of this letter and approving it for publication was D. Dardari. The authors are with the National & Kapodistrian University of Athens, Department of Physics, Athens, Greece (e-mail: [email protected]). This work has been performed in the framework of the project “PYTHAGORAS II - Support to Research Groups in Universities,” co-funded by the European Social Fund and Greek National Resources (EPEAEK II). Digital Object Identifier 10.1109/TWC.2009.071029
computationally intensive matrix inversions are required. In [7] all ICI components are estimated and compensated, making the approach very complex compared to the one proposed here. In [8] the complexity is reduced by estimating only a subset of, heuristically selected, significant ICI coefficients and the estimation is based on decisions which are made on a reduced set of sub-carriers. This approach results in increased vulnerability to decision errors, especially when the corresponding sub-carriers experience a deep fade. In the herein proposed algorithm the optimal number of the compensated PHN/RFO terms is calculated based on a Minimum Mean Squared Error (MMSE) criterion, and the computationally intensive matrix inversions are avoided by involving Discrete Fourier Transform (DFT) computations, which are intrinsic operations in an OFDM system. II. S YSTEM M ODEL The incoming information, during the m-th OFDM symbol is encoded into complex QAM or PSK symbols X(k) with k = 0, 1, . . . , N − 1 (herein assumed as independent, identically distributed), of average energy σx2 , which modulate N sub-carriers by an Inverse Discrete Fourier Transform (IDFT). A cyclic prefix longer than the duration of the Channel Impulse Response (CIR) is employed in order to avoid intersymbol interference. After discarding the cyclic prefix, the received OFDM symbol y(n) can be expressed as y(n) = u(n)d(n) + w(n),
(1)
where d(n) = IDFT{X(k)H(k)} = −1 j2πkn/N X(k)H(k)e with H(k) the channel (1/N ) N k=0 frequency response at the k-th sub-carrier of the timeinvariant (during a frame) channel, w(n) is the complex 2 , and Additive White Gaussian Noise (AWGN) of variance σw PHN RFO u(n) = exp{j(φ (n) + φ (n)} is the complex rotation due to the PHN and RFO processes. The PHN process is herein modeled as a discrete time Wiener process, equivalent to the sampled version of the continues-time process, with zero-mean Gaussian increments of variance σφ2 [5]. Note, however, that the proposed scheme can be employed under any typical PHN model with unknown phase statistics, as shown in section III. C. Demodulation of the received OFDM symbol is performed via the DFT, whose output equals Y (k) = X(k)H(k)U (0) + I(k) + W (k), (2) N −1 where I(k) = l=0,l=k X(l)H(l)U (l−k) represents the ICInoise term, U (0) is the Common Error (CE) term, W (k) = DFT{w(n)} and U (k) = IDFT{u(n)} represents the ICI
c 2009 IEEE 1536-1276/09$25.00
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
1615
coefficients. Note that U (k) (upper case) is defined as the inverse DFT of u(n) (lower case) with a slight abuse of notation. III. D ESCRIPTION OF THE P ROPOSED A LGORITHM First, the CE is estimated and compensated, using any one of the various methods already proposed in the literature (see, for instance, [5]). Equalization and hard decision ˆ (t) (k) detection follows, thus providing tentative decisions X of the transmitted symbols that will be used for calculatˆ (t) (k)H(k)} ˆ which is needed for ing dˆ(t) (n) = IDFT{X the tentative phase process estimation in the time domain u ˆ(t) (n). Estimation refinement, which is discussed in detail ˆ (t) (k) = IDFT{ˆ later, is applied to U u(t) (n)} resulting in ˆ U (k). The latter is employed for providing the final phase ˆ process estimate uˆ(n) = DFT{U(k)}. Compensation of u(n) is then achieved by de-rotating the time-domain received symbols. After PHN/RFO compensation, a second decision round follows, which provides the final estimates. The various building blocks of the proposed system are described in detail in subsections III.A and III.B, assuming known phase process statistics and perfect channel estimation. The case of unknown statistics is discussed in subsection III.C. A. Tentative Phase Process Estimation After a tentative d(n) decision is reached, denoted by dˆ(t) (n), Zero Forcing (ZF) estimation follows (i.e., by dividing y(n) with dˆ(t) (n)). In order to avoid unreliable ZF estimates for values of dˆ(t) (n) very close to zero, ZF estimation is performed only for those values of n belonging to the set 2 N = n : dˆ(t) (n) ≥ ρ2 where ρ is a pre-specified threshold. These estimates are subsequently used for the estimation / N , by interpolation. Thus, an initial estimate of u ˆ(t) (n), n ∈ of the phase process u(n) results, which is equal to uˆ(t) (n) = u(n) + e(n),
(3)
where e(n) is the estimation error given in Eq. 4, shown at the top of the next page, where einterp (n) is an error term which in general depends on the employed interpolation scheme and the accuracy of the ZF estimates. Under the assumption of perfect tentative decisions and no interpolation (i.e., ρ is chosen sufficiently small), it is shown in the Appendix that the sequence e(n) is white, zero-mean process with variance (5) σe2 = exp ρ¯2 Ei ρ¯2 /SNR, 2 , where Ei(·) is the exponential integral, SNR = σd2 /σw 2 2 σd = σx /N , and ρ¯ = ρ/σd . Based on our experimental evidence, even if the assumptions do not hold exactly, (5) provides a good approximation for high SNR values (so that the estimate dˆ(t) (n) is sufficiently close to d(n)) and such values of ρ that interpolation is performed for a very small subset of sub-carriers. A more detailed discussion on the choice of the design parameter ρ is provided in the simulations section.
Fig. 1. Representative power spectral densities E |U (k)|2 and 2 E |E(k)| .
B. Phase Process Estimation Refinement Phase process estimation and compensation is perˆ (t) (k) = formed in the frequency domain by calculating U (t) IDFT{ˆ u (n)} = U (k) + E(k), with E(k) = IDFT{e(n)} and E{|E(k)|2 } = σe2 /N for all k. Representative plots of E{|U (k)|2 } and E{|E(k)|2 } are shown in Fig. 1. It is depicted that most of the power of the ICI coefficients is concentrated in a few frequency bins. Additionally, there are many low power ICI coefficients, which are practically “buried” in noise, making their estimation unreliable. We thus re-estimate u(n) by discarding (nulling) the unreliable estimates of the lowestpower ICI coefficients. By denoting by K ⊆ {0, 1, . . . , N − 1} the set of those values of k corresponding to the K (cardinality of K) highest-power ICI coefficients, we define the ˆ (k) = U ˆ (t) (k) for k ∈ K and ˆ (t) (k) as U refined estimate of U ˆ U (k) = 0 for k ∈ / K, which is used for calculating the final ˆ estimate u ˆ(n) = DFT{U(k)}. The optimal size of K (K) has now to be specified. After estimation refinement,
U (k)e−j2πnk/N + E(k)e−j2πnk/N , u ˆ(n) = u(n) − k∈K /
k∈K
(6) for n = 0, 1, . . . , N − 1. The second term on the right hand side of (6) represents the error due to the nulling operation, which imposes an information loss due to the filtering of the frequency components of the phase process and the last term represents the corresponding noise effect. It is clear that K controls the tradeoff between information loss (the larger K the smaller the information loss) and noise suppression (the smaller K the larger the noise suppression). Selection of K is performed by minimizing an appropriate cost function. The MSE of the estimate in (6) is herein selected for this purpose, defined as C(K) as shown in Eq. 7 at the top of the next page. Since the first term on the right-hand side of (7) is a non-increasing function of K and the second term
1616
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
e(n) =
ezf (n) w(n) + u(n) d(n) − dˆ(t) (n) /dˆ(t) (n), n ∈ N n∈ /N einterp (n), Δ
C(K) =
N −1 σ2 1 E |u(n) − u ˆ(n)|2 = E |U (k)|2 + e K N n=1 N
(4)
(7)
k∈K /
C. Unknown Phase Process Statistics selection of K requires knowledge of The optimal 2 . The latter can be estimated E |U (k)|2 and σw by an SNR estimator, while the ensemble average E |U (k)|2 can be esˆ (t) (k)|2 over a window of timated from the time average of |U M OFDM symbols. Specifically, for the m-thOFDM symbol, ˆ (t) (k)|2 , is this time average estimate, denoted by Tm |U computed as m
ˆ (t) (k)|2 = 1 ˆμ(t) (k)|2 , Tm |U |U (11) M μ=m−M+1
ˆμ(t) (k) U
Fig. 2.
for μ = m−M +1, . . . , m (μ is the where the value of OFDM symbol index), has already been calculated in the “estimationrefinement” step of the proposed algorithm. For large (t) 2 ˆ ˆμ(t) (k)|2 = E |U (k)|2 + → E |U M , Tm |Uμ (k)| σe2 /N and therefore, E |U (k)|2 can be estimated as ˆ (t) (k)|2 − σe2 /N. E |U (k)|2 ≈ Tm |U (12)
Composition of C (K) as a function of K.
is a (linearly) non-decreasing function of K, C(K) has a unique minimum for a value K = Kopt which is the cardinality of the set Kopt = k : E |U (k)|2 ≥ σe2 /N, 0 ≤ k ≤ N − 1 . (8) The validity of (8) as the minimizer of C(K) can be shown as follows: Consider any set K = Kopt . The cost function for the set K is equal to
σe2 2 K , (9) C(K) = C(Kopt ) ± E |U (k)| − N k∈K /
where the plus sign holds for the case K ⊂ Kopt , with Kopt = K ∪ K , K ∩ K = , and the minus sign holds for the case where K ⊃ Kopt , with K = Kopt ∪ K , Kopt ∩ K = . It readily follows from (8) that the second term in (9) is positive for K ⊂ Kopt and negative for K ⊃ Kopt . Therefore C(Kopt ) < C(K) for all K = Kopt .
(10)
Figure 2 shows the composition of C(K) as per (7) for the same parameters as in Fig. 2. Note that C(K) for K = 1 (i.e., K = {0}) corresponds to CE estimation. As can be seen, a small offset from the optimal value of K does not result in a large variation in the value of C(K), i.e., C(K) is relatively flat around its minimum value. This same observation was made for C(K) plots resulting from extensive experimentation with the parameter setting.
Using (12), (8) is now modified to ˆ (t) (k)|2 ≥ 2σ 2 /N, 0 ≤ k ≤ N − 1 . K opt = k : Tm |U e (13) In order to avoid selecting values of k corresponding to low power frequency bins due to imperfect E |U (k)|2 estimation, only the values of K corresponding to low frequencies are kept. D. Complexity Requirements A set of (tentative) decisions is needed at first for any scheme targeting the compensation of more ICI components than the number of pilots. Kopt can be calculated once, or even off-line. The additional, run-time, complexity of the algorithm lies in the a) Phase Process Estimation, which involves two FFT calculations (complexity 2N log2 N ), 2N multiplications 2 ˆ (t) (k)H(k) and dˆ(t) (n) , for the calculation of A(k) = X
and at maximum N divisions of y(n) by dˆ(t) (n) (i.e., nulling and zero-order interpolation, which are used for simulations, are of negligible complexity), and b) Phase Process Compensation, which involves two additonal FFTs, and de-rotation. Thus, the total complexity is, approximately, 4N (log2 N + 1) independently of the number of ICI calculated coefficients. Schemes like [8] need O(q 3 ) calculations for phase estimation, at least (2q − 1) calculations for the A(k) terms (i.e., when the estimation takes place in adjacent subcarriers) and N q calculations for compensation, where q denotes the number of ICI coefficients to be estimated. This dependency on q makes this scheme computationally inefficient when a large number
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
1617
of ICI coefficients is targeted (see Sec. IV for an example). Finally, the prior knowledge on the phase noise second order statistics needed by other algorithms [7], [8] can be replaced by the computationally efficient method of Section III.C. IV. S IMULATIONS In this section, the performance of the proposed algorithm is demonstrated by extensive simulation results. The OFDM system has N = 256 sub-carriers out of which 4 are pilot subcarriers that are employed in CE estimation. Data sub-carriers are modulated by 16-QAM symbols. In all simulations a fixed realization of a time invariant, frequency selective channel has been assumed, which describes a Non-Line-Of-Sight channel for a fixed wireless access system in a small urban scenario at 5.8 GHz. In the implementation of the proposed algorithm, perfect SNR knowledge was considered, a value of ρ¯ = 0.3 was chosen in all cases (unless stated otherwise), and the simplest possible, zero order interpolation was performed in all simulations. It is noted that uncoded data transmission was considered. In practice, appropriate error control coding would be applied in order to shift the performance curves of the uncoded system to the desired operational point. In Fig. 3 the performance of the proposed algorithm is shown (in terms of SER) in contrast to other previously proposed approaches. Two different cases of PHN/RFO statistics have been assumed: a) fRFO = 0.01, σφ = 0.005 and b) fRFO = 0.01, σφ = 0.012. The performance of pilot-based CE estimation/compensation along with the ideal case (in the absence of PHN or RFO and compensation scheme) are depicted for reference. The pilot-aided algorithm proposed in [6] is shown to be of rather poor performance due to the decreased number of sub-carriers employed for estimation (which increases vulnerability to the frequency selectivity of the channel) and due to the limited estimated ICI coefficients (equal to the number of pilots). The performance of [6] is shown only for the case where σφ = 0.012. The results for the σφ = 0.005 case are similar. A realization of [8] is also depicted which employs the q = 17 strongest sub-carriers (and their neighboring sub-carriers) and estimates q = 17 ICI coefficients. For the adopted N size, this is the maximum number of ICI coefficients that can be estimated/compensated by [8] when restricting its complexity to the one of the herein proposed approach (see Sect. III.D). It is shown that the proposed approach outperforms [8], despite its minimal statistical knowledge requirements. This performance gain increases with SNR since the number of coefficients which can be reliably estimated also increases and, finally, becomes much larger than q. Additionally, it is noted that despite their similar complexity, the proposed scheme results in more efficient implementations (with increased area efficiency) since it reemploys the same modules ((I)FFTs). The MMSE and MLE algorithms of [4] are not depicted since the first is four orders of magnitude more complex (due to matrix inversion), whereas the second one is equivalent to the proposed one without the estimation refinement step (i.e., all ICI coefficients are estimated and compensated), with obvious performance loss. Selection of K according to (8) results from the minimization of the MSE. However, the figure of merit for a communication system is the SER. In Fig. 4 the performance
Fig. 3. Performance comparison of the proposed algorithm, perfect CE compensation and algorithms in [6] and [8].
Fig. 4.
SER dependence on the choice of K.
of the system using the proposed algorithm is depicted in terms of the SER for the two cases of PHN/RFO described above and for various values of K, with SNR = 34 dB. The value that satisfies (8) is marked with an arrow and as can be seen, it results in performance close to the best achievable one. It can be seen that nearly best performance can be achieved for a large range of K values which indicates that the algorithm can provide nearly-optimal results even in cases where the statistics of the PHN/RFO process are not perfectly known. The same conclusions were drawn for other PHN/RFO statistics by extensive simulation not shown here. The performance of the proposed scheme in the case of unknown PHN/RFO statistics is examined in this section. Figure 5 depicts the performance of the practical version of the algorithm, which estimates the unknown statistics, as proposed in Section III.C. The parameter M is set equal to 1, in order to demonstrate the worst-case scenario performance of the proposed algorithm. It is shown, that despite the small value, the algorithm is robust, and presents performance very close
1618
Fig. 5.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
System performance for unknown phase process statistics.
to a system employing perfect PHN/RFO knowledge. This observation implies that larger values of M do not significantly increase the performance of the algorithm. The accuracy of the calculated cost function and thus the applicability of the proposed scheme relies on the nointerpolation assumption. However, interpolation does take place, resulting in deviation from the presented modeling. Figure 6 depicts, the theoretical (Eq. 7) and actual (i.e., simulated) C(Kopt ) values as a function of ρ¯, for fRFO = 0.01 and σφ = 0.005. It becomes apparent that significant deviations appear for high ρ¯ and high SNR values, simultaneously, where the interpolation errors become the dominant noise factor and impose a performance floor, which prevents further improvement of the system. In this latter case the proposed scheme cannot be directly applied. The cost function has to be extended in order to include the effect of the interpolation errors in it, which has to be minimized by jointly deciding on the optimal values of ρ¯ and K. However, from Fig. 6, it becomes apparent that the MSE of the estimated vector is highly improved for ρ¯ values up to 0.3–0.4, while larger values of ρ¯ lead to a slight MSE performance enhancement. Simulations have shown that this MSE improvement has practically no effect on the SER performance (as long as the interpolation errors are small). This practically means that by assigning small values to the design parameter ρ¯, the model can be trustworthy without significant performance loss and without adopting more complicated model descriptions. V. C ONCLUSIONS A new algorithm is proposed for OFDM systems operating in significant composite phase impairments. The proposed scheme is shown to be more computational efficient than the ones proposed in the literature, when the compensation of a large number of ICI terms is targeted. A PPENDIX The second-order properties of the error sequence ezf (n), n ∈ N , are calculated under the assumption of perfect channel estimation and tentative decisions during CE
Fig. 6.
Theoretical and actual C Kopt as a function of ρ¯.
compensation. In this case, ezf (n) = w(n)/d(n), n ∈ N , is clearly a white random sequence. In order to calculate its mean and variance the first-order probability density function (p.d.f.) of the sequence |d(n)|, n ∈ N , is determined. For large values of N , the sequence d(n), n = 0, 1, . . . , N − 1, is a complex, zero mean, Gaussian process with variance σd2 = σx2 /N and independent real and imaginary parts. Therefore, the first-order p.d.f. of the process |d(n)|, n = 0, 1, . . . , N − 1, is Rayleigh [9], given by f (|d(n)|) = 2|d(n)|/σd2 exp −|d(n)|2 /σd2 , 0 ≤ |d(n)| ≤ ∞. (14)
The first order p.d.f of the sequence |d(n)|, n ∈ N , equals the p.d.f of (14) conditioned on the event {|d(n)| > ρ} shown in (15) at the top of the next page, where ρ¯ ρ/σd and Pr{A} is the probability of event A. Using (15) the mean and variance of ezf (n), n ∈ N , can be obtained. The mean equals E {ezf (n)} = E {w(n)} E {1/d(n)} = 0, n ∈ N ,
(16)
since w(n) and d(n) are independent and |d(n)| > 0 for n ∈ N . The variance can be calculated as in (17) shown 2 and at the top of the next page, where SNR = σd2 /σw Ei(·) is the exponential integral. The above results are valid for describing the second-order properties of the sequence e(n), n = 0, 1, . . . , N − 1, exactly when no interpolation has taken place and approximate when interpolation takes place. R EFERENCES [1] T. Pollet, M. van Bladel and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise," IEEE Trans. Commun., vol. 43, pp. 191-193, Feb./Mar./Apr. 1995. [2] A. G. Armada, “Understanding the effects of phase noise in orthogonal frequency division multiplexing," IEEE Trans. Broadcast., vol. 47, pp. 153-159, June 2001. [3] P. Robertson and S. Kaiser, “Analysis of the effects of phase noise in OFDM systems," in Proc. IEEE Int. Conf. Commun., (ICC95), Seattle, WA, June 1995, pp. 1652-1657. [4] S. Wu and Y. Bar-Ness, “A phase noise suppression algorithm for OFDM-based WLANs," IEEE Commun. Lett., vol. 6, pp. 535-537, Dec. 2002. [5] K. Nikitopoulos and A. Polydoros, “Phase-impairment effects and compensation algorithms for OFDM systems," IEEE Trans. Commum. vol. 53, no 3, Mar. 2005.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
f (|d(n)| | {|d(n)| > ρ}) = f (|d(n)|) /Pr {|d(n)| > ρ} = exp ρ¯2 f (|d(n)|) , for n ∈ N E |ezf (n)|2
= =
E |w(n)|2 E 1/|d(n)|2 ∞ 1 2 σw f (|d(n)| | {|d(n)| > ρ}) d (|d(n)|} = exp ρ¯2 Ei ρ¯2 /SNR, for n ∈ N 2 |d(n)| ρ
[6] R. A. Casas, S. L. Biracree, and A. E. Youltz, “Time domain phase noise correction for OFDM signals," IEEE Trans. Broadcast., vol. 48, pp. 230-236. Sept. 2002. [7] S. Wu and Y. Bar-Ness, “Phase noise estimation and mitigation for OFDM systems," IEEE Trans. Wireless Commun., vol. 5, pp. 3616-3625, Dec. 2006.
1619
(15)
(17)
[8] D. Petrovic, W. Rave, and G. Fettweis, “Effects of phase noise on OFDM systems with and without PLL: characterization and compensation," IEEE Trans. Commun., vol. 55, pp. 1607-1616. Aug. 2007. [9] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. McGraw-Hill, 2002.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
Decision-Aided Compensation of Severe Phase-Impairment-Induced Inter-Carrier Interference in Frequency-Selective OFDM Konstantinos Nikitopoulos, Member, IEEE, Stelios Stefanatos, and Aggelos K. Katsaggelos, Fellow, IEEE
Abstract—A new, reduced complexity algorithm is proposed for compensating the Inter-Carrier Interference (ICI) caused by severe PHase Noise (PHN) and Residual Frequency Offset (RFO) in OFDM systems. The algorithm estimates and compensates the most significant terms of the frequency domain ICI process, which are optimally selected via a Minimum Mean Squared Error (MMSE) criterion. The algorithm requires minimal knowledge of the phase process statistics, the estimation of which is also considered. The scheme outperforms previously proposed compensation methods of similar complexity, when severe phase impairments are present. Index Terms—Frequency offset, inter-carrier interference, OFDM, phase noise.
I. I NTRODUCTION
I
T is well known that OFDM is very sensitive to phase impairments, namely Phase Noise (PHN) and Residual Frequency Offset (RFO) [1], which induce a phase rotation, common to all subcarriers of the same OFDM symbol, plus Inter-Carrier Interference (ICI) [1]–[5]. In the presence of severe PHN and high order constellations, the provided Symbol Error Rate (SER) is lower bounded by the presence of ICI, rendering the employment of an ICI compensation scheme necessary. Since the effect of the ICI on the system’s performance is not only a function of the actual PHN and RFO values but it also depends on the transmitted constellation, we herein define the phase process as “severe” whenever it produces a symbol error rate (SER) performance floor larger than 10−3 for the target transmitted constellation without ICI compensation. Several approaches have been proposed in the literature for ICI compensation. A pilot based estimator is proposed in [6] for estimating the most significant terms of the PHN/RFO process. The latter is expanded on specially selected basis functions, via a least squares method. However, the performance of the scheme in frequency selective channels, as well as the optimal number of PHN/RFO terms to be estimated are not addressed. Decision-aided MMSE-based methods have been also proposed in [7], [8], where perfect knowledge of the correlation matrix of the phase noise process is assumed and
Manuscript received September 18, 2007; revised March 3, 2008 and September 5, 2008; accepted November 11, 2008. The associate editor coordinating the review of this letter and approving it for publication was D. Dardari. The authors are with the National & Kapodistrian University of Athens, Department of Physics, Athens, Greece (e-mail: [email protected]). This work has been performed in the framework of the project “PYTHAGORAS II - Support to Research Groups in Universities,” co-funded by the European Social Fund and Greek National Resources (EPEAEK II). Digital Object Identifier 10.1109/TWC.2009.071029
computationally intensive matrix inversions are required. In [7] all ICI components are estimated and compensated, making the approach very complex compared to the one proposed here. In [8] the complexity is reduced by estimating only a subset of, heuristically selected, significant ICI coefficients and the estimation is based on decisions which are made on a reduced set of sub-carriers. This approach results in increased vulnerability to decision errors, especially when the corresponding sub-carriers experience a deep fade. In the herein proposed algorithm the optimal number of the compensated PHN/RFO terms is calculated based on a Minimum Mean Squared Error (MMSE) criterion, and the computationally intensive matrix inversions are avoided by involving Discrete Fourier Transform (DFT) computations, which are intrinsic operations in an OFDM system. II. S YSTEM M ODEL The incoming information, during the m-th OFDM symbol is encoded into complex QAM or PSK symbols X(k) with k = 0, 1, . . . , N − 1 (herein assumed as independent, identically distributed), of average energy σx2 , which modulate N sub-carriers by an Inverse Discrete Fourier Transform (IDFT). A cyclic prefix longer than the duration of the Channel Impulse Response (CIR) is employed in order to avoid intersymbol interference. After discarding the cyclic prefix, the received OFDM symbol y(n) can be expressed as y(n) = u(n)d(n) + w(n),
(1)
where d(n) = IDFT{X(k)H(k)} = −1 j2πkn/N X(k)H(k)e with H(k) the channel (1/N ) N k=0 frequency response at the k-th sub-carrier of the timeinvariant (during a frame) channel, w(n) is the complex 2 , and Additive White Gaussian Noise (AWGN) of variance σw PHN RFO u(n) = exp{j(φ (n) + φ (n)} is the complex rotation due to the PHN and RFO processes. The PHN process is herein modeled as a discrete time Wiener process, equivalent to the sampled version of the continues-time process, with zero-mean Gaussian increments of variance σφ2 [5]. Note, however, that the proposed scheme can be employed under any typical PHN model with unknown phase statistics, as shown in section III. C. Demodulation of the received OFDM symbol is performed via the DFT, whose output equals Y (k) = X(k)H(k)U (0) + I(k) + W (k), (2) N −1 where I(k) = l=0,l=k X(l)H(l)U (l−k) represents the ICInoise term, U (0) is the Common Error (CE) term, W (k) = DFT{w(n)} and U (k) = IDFT{u(n)} represents the ICI
c 2009 IEEE 1536-1276/09$25.00
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
1615
coefficients. Note that U (k) (upper case) is defined as the inverse DFT of u(n) (lower case) with a slight abuse of notation. III. D ESCRIPTION OF THE P ROPOSED A LGORITHM First, the CE is estimated and compensated, using any one of the various methods already proposed in the literature (see, for instance, [5]). Equalization and hard decision ˆ (t) (k) detection follows, thus providing tentative decisions X of the transmitted symbols that will be used for calculatˆ (t) (k)H(k)} ˆ which is needed for ing dˆ(t) (n) = IDFT{X the tentative phase process estimation in the time domain u ˆ(t) (n). Estimation refinement, which is discussed in detail ˆ (t) (k) = IDFT{ˆ later, is applied to U u(t) (n)} resulting in ˆ U (k). The latter is employed for providing the final phase ˆ process estimate uˆ(n) = DFT{U(k)}. Compensation of u(n) is then achieved by de-rotating the time-domain received symbols. After PHN/RFO compensation, a second decision round follows, which provides the final estimates. The various building blocks of the proposed system are described in detail in subsections III.A and III.B, assuming known phase process statistics and perfect channel estimation. The case of unknown statistics is discussed in subsection III.C. A. Tentative Phase Process Estimation After a tentative d(n) decision is reached, denoted by dˆ(t) (n), Zero Forcing (ZF) estimation follows (i.e., by dividing y(n) with dˆ(t) (n)). In order to avoid unreliable ZF estimates for values of dˆ(t) (n) very close to zero, ZF estimation is performed only for those values of n belonging to the set 2 N = n : dˆ(t) (n) ≥ ρ2 where ρ is a pre-specified threshold. These estimates are subsequently used for the estimation / N , by interpolation. Thus, an initial estimate of u ˆ(t) (n), n ∈ of the phase process u(n) results, which is equal to uˆ(t) (n) = u(n) + e(n),
(3)
where e(n) is the estimation error given in Eq. 4, shown at the top of the next page, where einterp (n) is an error term which in general depends on the employed interpolation scheme and the accuracy of the ZF estimates. Under the assumption of perfect tentative decisions and no interpolation (i.e., ρ is chosen sufficiently small), it is shown in the Appendix that the sequence e(n) is white, zero-mean process with variance (5) σe2 = exp ρ¯2 Ei ρ¯2 /SNR, 2 , where Ei(·) is the exponential integral, SNR = σd2 /σw 2 2 σd = σx /N , and ρ¯ = ρ/σd . Based on our experimental evidence, even if the assumptions do not hold exactly, (5) provides a good approximation for high SNR values (so that the estimate dˆ(t) (n) is sufficiently close to d(n)) and such values of ρ that interpolation is performed for a very small subset of sub-carriers. A more detailed discussion on the choice of the design parameter ρ is provided in the simulations section.
Fig. 1. Representative power spectral densities E |U (k)|2 and 2 E |E(k)| .
B. Phase Process Estimation Refinement Phase process estimation and compensation is perˆ (t) (k) = formed in the frequency domain by calculating U (t) IDFT{ˆ u (n)} = U (k) + E(k), with E(k) = IDFT{e(n)} and E{|E(k)|2 } = σe2 /N for all k. Representative plots of E{|U (k)|2 } and E{|E(k)|2 } are shown in Fig. 1. It is depicted that most of the power of the ICI coefficients is concentrated in a few frequency bins. Additionally, there are many low power ICI coefficients, which are practically “buried” in noise, making their estimation unreliable. We thus re-estimate u(n) by discarding (nulling) the unreliable estimates of the lowestpower ICI coefficients. By denoting by K ⊆ {0, 1, . . . , N − 1} the set of those values of k corresponding to the K (cardinality of K) highest-power ICI coefficients, we define the ˆ (k) = U ˆ (t) (k) for k ∈ K and ˆ (t) (k) as U refined estimate of U ˆ U (k) = 0 for k ∈ / K, which is used for calculating the final ˆ estimate u ˆ(n) = DFT{U(k)}. The optimal size of K (K) has now to be specified. After estimation refinement,
U (k)e−j2πnk/N + E(k)e−j2πnk/N , u ˆ(n) = u(n) − k∈K /
k∈K
(6) for n = 0, 1, . . . , N − 1. The second term on the right hand side of (6) represents the error due to the nulling operation, which imposes an information loss due to the filtering of the frequency components of the phase process and the last term represents the corresponding noise effect. It is clear that K controls the tradeoff between information loss (the larger K the smaller the information loss) and noise suppression (the smaller K the larger the noise suppression). Selection of K is performed by minimizing an appropriate cost function. The MSE of the estimate in (6) is herein selected for this purpose, defined as C(K) as shown in Eq. 7 at the top of the next page. Since the first term on the right-hand side of (7) is a non-increasing function of K and the second term
1616
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
e(n) =
ezf (n) w(n) + u(n) d(n) − dˆ(t) (n) /dˆ(t) (n), n ∈ N n∈ /N einterp (n), Δ
C(K) =
N −1 σ2 1 E |u(n) − u ˆ(n)|2 = E |U (k)|2 + e K N n=1 N
(4)
(7)
k∈K /
C. Unknown Phase Process Statistics selection of K requires knowledge of The optimal 2 . The latter can be estimated E |U (k)|2 and σw by an SNR estimator, while the ensemble average E |U (k)|2 can be esˆ (t) (k)|2 over a window of timated from the time average of |U M OFDM symbols. Specifically, for the m-thOFDM symbol, ˆ (t) (k)|2 , is this time average estimate, denoted by Tm |U computed as m
ˆ (t) (k)|2 = 1 ˆμ(t) (k)|2 , Tm |U |U (11) M μ=m−M+1
ˆμ(t) (k) U
Fig. 2.
for μ = m−M +1, . . . , m (μ is the where the value of OFDM symbol index), has already been calculated in the “estimationrefinement” step of the proposed algorithm. For large (t) 2 ˆ ˆμ(t) (k)|2 = E |U (k)|2 + → E |U M , Tm |Uμ (k)| σe2 /N and therefore, E |U (k)|2 can be estimated as ˆ (t) (k)|2 − σe2 /N. E |U (k)|2 ≈ Tm |U (12)
Composition of C (K) as a function of K.
is a (linearly) non-decreasing function of K, C(K) has a unique minimum for a value K = Kopt which is the cardinality of the set Kopt = k : E |U (k)|2 ≥ σe2 /N, 0 ≤ k ≤ N − 1 . (8) The validity of (8) as the minimizer of C(K) can be shown as follows: Consider any set K = Kopt . The cost function for the set K is equal to
σe2 2 K , (9) C(K) = C(Kopt ) ± E |U (k)| − N k∈K /
where the plus sign holds for the case K ⊂ Kopt , with Kopt = K ∪ K , K ∩ K = , and the minus sign holds for the case where K ⊃ Kopt , with K = Kopt ∪ K , Kopt ∩ K = . It readily follows from (8) that the second term in (9) is positive for K ⊂ Kopt and negative for K ⊃ Kopt . Therefore C(Kopt ) < C(K) for all K = Kopt .
(10)
Figure 2 shows the composition of C(K) as per (7) for the same parameters as in Fig. 2. Note that C(K) for K = 1 (i.e., K = {0}) corresponds to CE estimation. As can be seen, a small offset from the optimal value of K does not result in a large variation in the value of C(K), i.e., C(K) is relatively flat around its minimum value. This same observation was made for C(K) plots resulting from extensive experimentation with the parameter setting.
Using (12), (8) is now modified to ˆ (t) (k)|2 ≥ 2σ 2 /N, 0 ≤ k ≤ N − 1 . K opt = k : Tm |U e (13) In order to avoid selecting values of k corresponding to low power frequency bins due to imperfect E |U (k)|2 estimation, only the values of K corresponding to low frequencies are kept. D. Complexity Requirements A set of (tentative) decisions is needed at first for any scheme targeting the compensation of more ICI components than the number of pilots. Kopt can be calculated once, or even off-line. The additional, run-time, complexity of the algorithm lies in the a) Phase Process Estimation, which involves two FFT calculations (complexity 2N log2 N ), 2N multiplications 2 ˆ (t) (k)H(k) and dˆ(t) (n) , for the calculation of A(k) = X
and at maximum N divisions of y(n) by dˆ(t) (n) (i.e., nulling and zero-order interpolation, which are used for simulations, are of negligible complexity), and b) Phase Process Compensation, which involves two additonal FFTs, and de-rotation. Thus, the total complexity is, approximately, 4N (log2 N + 1) independently of the number of ICI calculated coefficients. Schemes like [8] need O(q 3 ) calculations for phase estimation, at least (2q − 1) calculations for the A(k) terms (i.e., when the estimation takes place in adjacent subcarriers) and N q calculations for compensation, where q denotes the number of ICI coefficients to be estimated. This dependency on q makes this scheme computationally inefficient when a large number
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
1617
of ICI coefficients is targeted (see Sec. IV for an example). Finally, the prior knowledge on the phase noise second order statistics needed by other algorithms [7], [8] can be replaced by the computationally efficient method of Section III.C. IV. S IMULATIONS In this section, the performance of the proposed algorithm is demonstrated by extensive simulation results. The OFDM system has N = 256 sub-carriers out of which 4 are pilot subcarriers that are employed in CE estimation. Data sub-carriers are modulated by 16-QAM symbols. In all simulations a fixed realization of a time invariant, frequency selective channel has been assumed, which describes a Non-Line-Of-Sight channel for a fixed wireless access system in a small urban scenario at 5.8 GHz. In the implementation of the proposed algorithm, perfect SNR knowledge was considered, a value of ρ¯ = 0.3 was chosen in all cases (unless stated otherwise), and the simplest possible, zero order interpolation was performed in all simulations. It is noted that uncoded data transmission was considered. In practice, appropriate error control coding would be applied in order to shift the performance curves of the uncoded system to the desired operational point. In Fig. 3 the performance of the proposed algorithm is shown (in terms of SER) in contrast to other previously proposed approaches. Two different cases of PHN/RFO statistics have been assumed: a) fRFO = 0.01, σφ = 0.005 and b) fRFO = 0.01, σφ = 0.012. The performance of pilot-based CE estimation/compensation along with the ideal case (in the absence of PHN or RFO and compensation scheme) are depicted for reference. The pilot-aided algorithm proposed in [6] is shown to be of rather poor performance due to the decreased number of sub-carriers employed for estimation (which increases vulnerability to the frequency selectivity of the channel) and due to the limited estimated ICI coefficients (equal to the number of pilots). The performance of [6] is shown only for the case where σφ = 0.012. The results for the σφ = 0.005 case are similar. A realization of [8] is also depicted which employs the q = 17 strongest sub-carriers (and their neighboring sub-carriers) and estimates q = 17 ICI coefficients. For the adopted N size, this is the maximum number of ICI coefficients that can be estimated/compensated by [8] when restricting its complexity to the one of the herein proposed approach (see Sect. III.D). It is shown that the proposed approach outperforms [8], despite its minimal statistical knowledge requirements. This performance gain increases with SNR since the number of coefficients which can be reliably estimated also increases and, finally, becomes much larger than q. Additionally, it is noted that despite their similar complexity, the proposed scheme results in more efficient implementations (with increased area efficiency) since it reemploys the same modules ((I)FFTs). The MMSE and MLE algorithms of [4] are not depicted since the first is four orders of magnitude more complex (due to matrix inversion), whereas the second one is equivalent to the proposed one without the estimation refinement step (i.e., all ICI coefficients are estimated and compensated), with obvious performance loss. Selection of K according to (8) results from the minimization of the MSE. However, the figure of merit for a communication system is the SER. In Fig. 4 the performance
Fig. 3. Performance comparison of the proposed algorithm, perfect CE compensation and algorithms in [6] and [8].
Fig. 4.
SER dependence on the choice of K.
of the system using the proposed algorithm is depicted in terms of the SER for the two cases of PHN/RFO described above and for various values of K, with SNR = 34 dB. The value that satisfies (8) is marked with an arrow and as can be seen, it results in performance close to the best achievable one. It can be seen that nearly best performance can be achieved for a large range of K values which indicates that the algorithm can provide nearly-optimal results even in cases where the statistics of the PHN/RFO process are not perfectly known. The same conclusions were drawn for other PHN/RFO statistics by extensive simulation not shown here. The performance of the proposed scheme in the case of unknown PHN/RFO statistics is examined in this section. Figure 5 depicts the performance of the practical version of the algorithm, which estimates the unknown statistics, as proposed in Section III.C. The parameter M is set equal to 1, in order to demonstrate the worst-case scenario performance of the proposed algorithm. It is shown, that despite the small value, the algorithm is robust, and presents performance very close
1618
Fig. 5.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
System performance for unknown phase process statistics.
to a system employing perfect PHN/RFO knowledge. This observation implies that larger values of M do not significantly increase the performance of the algorithm. The accuracy of the calculated cost function and thus the applicability of the proposed scheme relies on the nointerpolation assumption. However, interpolation does take place, resulting in deviation from the presented modeling. Figure 6 depicts, the theoretical (Eq. 7) and actual (i.e., simulated) C(Kopt ) values as a function of ρ¯, for fRFO = 0.01 and σφ = 0.005. It becomes apparent that significant deviations appear for high ρ¯ and high SNR values, simultaneously, where the interpolation errors become the dominant noise factor and impose a performance floor, which prevents further improvement of the system. In this latter case the proposed scheme cannot be directly applied. The cost function has to be extended in order to include the effect of the interpolation errors in it, which has to be minimized by jointly deciding on the optimal values of ρ¯ and K. However, from Fig. 6, it becomes apparent that the MSE of the estimated vector is highly improved for ρ¯ values up to 0.3–0.4, while larger values of ρ¯ lead to a slight MSE performance enhancement. Simulations have shown that this MSE improvement has practically no effect on the SER performance (as long as the interpolation errors are small). This practically means that by assigning small values to the design parameter ρ¯, the model can be trustworthy without significant performance loss and without adopting more complicated model descriptions. V. C ONCLUSIONS A new algorithm is proposed for OFDM systems operating in significant composite phase impairments. The proposed scheme is shown to be more computational efficient than the ones proposed in the literature, when the compensation of a large number of ICI terms is targeted. A PPENDIX The second-order properties of the error sequence ezf (n), n ∈ N , are calculated under the assumption of perfect channel estimation and tentative decisions during CE
Fig. 6.
Theoretical and actual C Kopt as a function of ρ¯.
compensation. In this case, ezf (n) = w(n)/d(n), n ∈ N , is clearly a white random sequence. In order to calculate its mean and variance the first-order probability density function (p.d.f.) of the sequence |d(n)|, n ∈ N , is determined. For large values of N , the sequence d(n), n = 0, 1, . . . , N − 1, is a complex, zero mean, Gaussian process with variance σd2 = σx2 /N and independent real and imaginary parts. Therefore, the first-order p.d.f. of the process |d(n)|, n = 0, 1, . . . , N − 1, is Rayleigh [9], given by f (|d(n)|) = 2|d(n)|/σd2 exp −|d(n)|2 /σd2 , 0 ≤ |d(n)| ≤ ∞. (14)
The first order p.d.f of the sequence |d(n)|, n ∈ N , equals the p.d.f of (14) conditioned on the event {|d(n)| > ρ} shown in (15) at the top of the next page, where ρ¯ ρ/σd and Pr{A} is the probability of event A. Using (15) the mean and variance of ezf (n), n ∈ N , can be obtained. The mean equals E {ezf (n)} = E {w(n)} E {1/d(n)} = 0, n ∈ N ,
(16)
since w(n) and d(n) are independent and |d(n)| > 0 for n ∈ N . The variance can be calculated as in (17) shown 2 and at the top of the next page, where SNR = σd2 /σw Ei(·) is the exponential integral. The above results are valid for describing the second-order properties of the sequence e(n), n = 0, 1, . . . , N − 1, exactly when no interpolation has taken place and approximate when interpolation takes place. R EFERENCES [1] T. Pollet, M. van Bladel and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise," IEEE Trans. Commun., vol. 43, pp. 191-193, Feb./Mar./Apr. 1995. [2] A. G. Armada, “Understanding the effects of phase noise in orthogonal frequency division multiplexing," IEEE Trans. Broadcast., vol. 47, pp. 153-159, June 2001. [3] P. Robertson and S. Kaiser, “Analysis of the effects of phase noise in OFDM systems," in Proc. IEEE Int. Conf. Commun., (ICC95), Seattle, WA, June 1995, pp. 1652-1657. [4] S. Wu and Y. Bar-Ness, “A phase noise suppression algorithm for OFDM-based WLANs," IEEE Commun. Lett., vol. 6, pp. 535-537, Dec. 2002. [5] K. Nikitopoulos and A. Polydoros, “Phase-impairment effects and compensation algorithms for OFDM systems," IEEE Trans. Commum. vol. 53, no 3, Mar. 2005.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
f (|d(n)| | {|d(n)| > ρ}) = f (|d(n)|) /Pr {|d(n)| > ρ} = exp ρ¯2 f (|d(n)|) , for n ∈ N E |ezf (n)|2
= =
E |w(n)|2 E 1/|d(n)|2 ∞ 1 2 σw f (|d(n)| | {|d(n)| > ρ}) d (|d(n)|} = exp ρ¯2 Ei ρ¯2 /SNR, for n ∈ N 2 |d(n)| ρ
[6] R. A. Casas, S. L. Biracree, and A. E. Youltz, “Time domain phase noise correction for OFDM signals," IEEE Trans. Broadcast., vol. 48, pp. 230-236. Sept. 2002. [7] S. Wu and Y. Bar-Ness, “Phase noise estimation and mitigation for OFDM systems," IEEE Trans. Wireless Commun., vol. 5, pp. 3616-3625, Dec. 2006.
1619
(15)
(17)
[8] D. Petrovic, W. Rave, and G. Fettweis, “Effects of phase noise on OFDM systems with and without PLL: characterization and compensation," IEEE Trans. Commun., vol. 55, pp. 1607-1616. Aug. 2007. [9] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. McGraw-Hill, 2002.
