Iterative correction of clipped OFDM signals with unknown clipping levels Wilson D. Wellisch
Ian A. Ulian and André Noll Barreto
Agência Nacional de Telecomunicações - ANATEL Brasília - Brazil (
[email protected])
Universidade de Brasília Brazil (
[email protected],
[email protected])
Abstract—Iterative correction can be employed to mitigate the distortion caused by nonlinear power amplifiers in OFDM systems. However, proposals found in the literature suppose that the receiver has perfect knowledge of the nonlinear conditions encountered in transmission. In this paper, we propose the use of the Extendend Kalman Filter and the Unscented Kalman Filter to estimate the nonlinear characteristics of the amplifier before performing the iterative correction, and see that by using these filters, the amplifier clipping level can be correctly estimated even without a pilot signal. Keywords— iterative correction with hard detection, nonlinear distortion, orthogonal frequency division multiplexing (OFDM).
I.
INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) signals are composed by the sum of many independently modulated subcarriers, and have therefore a large amplitude dynamic range. The amplitude variation, is usually quantified by the peak-to-average power ratio (PAPR) [1][2]. The high PAPR of OFDM signals is arguably one of its main disadvantages, as, because of this, OFDM systems suffer from non-linear effects common to power amplifiers [3]–[5]. Several techniques are proposed in the literature to combat this problem. Most of them process the signal before transmission, aiming at decreasing its PAPR value, as for instance [6],[7]. They all have a considerable complexity, often involving the calculation of several IFFTs or optimization of transmission parameters. Furthermore, PAPR is a poor performance metric, and a more thorough metric considering the relationship between backoff and bit-errorrate, such as the total degradation, shows only moderate gains for most methods, particularly when coding is considered [8]. The high PAPR is more critical in the uplink, where the transmitter must be cheap and power thrifty. Hence, it is of interest to add complexity to the receivers instead, whereas in the transmitter a simple clip-and-filter approach can be taken before amplification. Information-theoretic investigations show that even severe clipping results in only a moderate reduction of system capacity [9], as long as an adequate nonlinear receiver is employed. A maximum-likelihood detector can be computationally expensive [10] [11], but an alternative way is to use an iterative decision-feedback receiver to estimate the nonlinear distortion in the signal [12][13]. This approach shows good performance both in uncoded and coded
system. However, proposals in the literature assume knowledge by the receiver of the nonlinear response at the transmitter and of the backoff level in signal transmission. To overcome this issue, we propose the use of two techniques to estimate at the receiver, the nonlinear conditions encountered in transmission: the Extended Kalman Filter and the Uscented Kalman Filter. We see that performance results of the iterative correction receiver using the estimated parameters are extremely close to the idealized cases. Section II introduces the system model, including the iterative correction with hard detection (ICHD) receiver. The Extended Kalman Filter is introduced in section III, and the Uscented Kalman Filter in section IV. Estimation results using the filters and performance results of the receiver using the parameters estimated by the filters are shown in section VI. SYSTEM MODEL
II.
The complex baseband OFDM signal can be represented as ∑
∑
/
,
,
/
(1)
where , is the transmitted modulation symbol at the n-th OFDM block and at the l-th subcarrier with frequency / . is the useful OFDM symbol interval, and is the total symbol interval including the guard interval , which is filled up by a cyclic prefix. is the shaping pulse, which is a rectangular pulse with width . This signal can be implemented by the inverse fast Fourier transform (IFFT) and, dropping the symbol index n, the time samples of an OFDM symbol can be denoted by ∑
/
/
/
,0
1,
(2)
To model the presence of a non-linear power amplifier, the signal can be submitted to a Rapp amplifier, that models a solid state amplifier [16], whose AM/AM conversion is (3)
/
where is the signal amplitude, and is the smoothness factor.
978-1-4673-6337-2/13/$31.00 ©2013 IEEE
0
is the saturation amplitude
The signal is usually deliberately clipped application, which can be represented by a limiter: , ,
0 |
|
| |
.
before
received signal to obtain a signal version for the next iteration: ,
(4)
where
(10)
is the residual nonlinear distortion.
The clipped signal can be modeled as the sum of the attenuated signal and a nonlinear noise component [2]: , where with 1
(5)
is the non linear distortion, which is uncorrelated , and α is the attenuation factor [2] erfc √
√
.
(7)
can be seen as a sum of random variables that are not necessarily statistically independent. However, based on the central limit theorem, we assume to be Gaussian, because this theorem still holds for a large numbers of dependent variables [21].This model has been widely used to investigate nonlinear distortions, and has shown results with good accuracy, especially for a large number of subcarriers N. Immediately after clipping, the signal is amplified and sent to the channel. The received signal can be written as ,(8) where are the noise samples and is the impulse channel response. Also, we assume perfect channel estimation and perfect channel synchronization. Thus, the received signal has, besides the receiver thermal noise , also the clipping or nonlinear noise . The ICHD receiver estimates the nonlinear noise and attempts to cancel it, by detecting the received signal, regenerating the transmitted signal and clipping it. This way, we can estimate the clipping noise, and cancel it out. The signal without the estimated clipping noise can then be detected again, and this can be repeated in several iterations. The iterative method for a coded signal is described in Fig. are reencoded, mapped to a 1 The decoded symbols modulation symbol, and modulated using OFDM, generating an estimate of the transmit signal X. The estimated transmitted sequence is then processed through two branches. One of them regenerates the attenuated version of the unclipped signal α . The other one tries to regenerate the clipped signal by passing the estimated signal through the same non linear conditions encountered in transmission. The . By clipped estimation can be denoted by calculating the difference between the signals coming from the two branches, we obtain an estimate of the clipping noise: ,
(9)
The Fourier transform of the estimated clipping noise is multiplied by the channel response H and, subtracted from the
Fig. 1. Receiver with iterative nonlinear noise cancellation
We get a more accurate estimate of the nonlinear distortion with every additional iteration, but we observe in simulations that usually no more than three iterations are requireed. We can emphasize that the clipping of the regenerated signal is done assuming the clipping level at the transmitter is known. Figure 2 shows simulation results using input backoff IBO = 0dB in transmission over an AWGN channel, with 3 receiver iterations, but with different IBO estimates at the receiver. We can observe that the algorithm performance can be severely degraded if the wrong estimate is used.
Fig. 2. BER performance with error in IBO estimation.
III.
THE EXTENDED KALMAN FILTER
The Kalman Filter [14] is a recursive estimator for linear stochastic models. However, when the system is nonlinear, the Extended Kalman Filter (EKF) can be employed. It uses the Kalman Filter principles, but linearizes, through Taylor series expansions, the estimation of a state around the current estimate using the partial derivatives of the functions involved in the algorithm. It is necessary to model the system by two equations. Firstly, we need the model equation: 1,
1
,
(11)
where models the system process, that relates the states with is a vector of Gaussian random variables the system inputs, with zero mean and covariance matrix . , with
the identity matrix with order NP (number of parameters to be models the estimated), and k is the discrete time index. uncertainties associated to the model imperfections. Secondly, we have the measurement equation ,
,
(12)
represents the observation function, related to the where is a vector of zero-mean gaussian system states, and random variables and covariance matrix . , which represents the measurement noise. We consider that the receiver knows the amplifier model, as in (3), but not its parameters. The filter tries to estimate two parameters of the function that models the amplifier: 0 and p. The function that models the process is given by ,
.
(14)
Thus, the model equation becomes
A. Unscented Transform (UT) The UT is responsible for the sigma points calculation (that will be used to estimate mean and a posteriori covariance) [18]. Some important parameters in the sigma points determination are defined as follow: 1) Spread Factor (α): Responsible for the sigma points spread around the mean. Generally defined with a small value (typically 1 10 ). 2) Secondary Spread Factor (κ): Adjustable factor that helps in the regulation of the spread. (typically set to zero). 3) Distribution factor (β): related to a previously knowledge of the distribution model for an accurately estimation (β=2 is an optimal value for Gaussian Distribution). 4) Parameter vector dimension (L). Considering , where θ is the parameter vector with Gaussian Distribution to be determined and . a non-linear its covariance. Then, we function of θ, is its mean and can determinate the following vector Θ with 2L+1 dimensions:
(15)
Θ Θ
,
The measured signal y is the absolute value of the time signal that arrives at the receiver after signal equalization:
Θ
,
.
1
,
(16)
The signal measurement is affected by the nonlinear amplifier in the transmission and white noise. The function , that models the system measurement is given by | |
,
/
| |
,
2
1
1/2
,
1, … , 1, … ,2 .
where
(21) represents the ith
squared root matrix of . The elements of the vector are the sigma points. The sigma points are weighted through the following weight factors: (22) 1
(17)
(23) (24)
and, thus, the measurement equation is | |
(19) (20)
(18)
B. UKF application The process follows the model below, in which the parameter vector is represented by instead of to avoid confusion:
0
, IV.
,
THE UNSCENTED KALMAN FILTER
Like the EKF, the Unscented Kalman Filter (UKF) is a recursive filter used to estimate parameters through non-linear functions. Unlike the EKF, that uses a Gaussian Random Variable propagated through a first order linearization of the non-linear function, the UKF uses a deterministic sample technique that chooses specific points around the mean, known as the Unscented Transform (UT). The chosen points of the UT, called sigma points, are propagated through the non-linear function and weighted to obtain a third-order Taylor series expansion, for Gaussian inputs, for every nonlinearity [17]. The UKF also has the advantage that the calculation of the Jacobian matrix is not needed. This is more relevant when the system function has a derivate of high complexity or when complex numbers are involved.
(25) (26)
,
where and are Gaussian random variables that take in account noise sources inherent to the system. In the simulated case, the vector parameter was considered as a constant, so 0. was considered as the Gaussian Noise in the receptor. . refers to the OFDM signal, in time domain, after the nonlinear power amplifier. Then, the process follows the prediction and measurement update steps. Prediction Steps: 1) Calculation of sigma points: Θ (27) / / Θ
/
/
/
(28)
Θ
/
/
/
(29)
V. 2) Propagation of sigma points through the model process : Θ
Θ
/
, with
/
0, … , 2.
(30)
3) Weighing and value combination of Θ /
∑
/
∑
Θ
/
and , Figures 3 and 4 show, respectively, the values of 1 and 3. estimated by EKF, for an amplifier with Each step corresponds to one sample in the time domain.
: (31)
/
Θ
RESULTS
Θ
/
/
/
/
(32)
Measurements Update Steps: 1) New sigma points calculation by the values obtained in the prediction step: Θ
(33)
/
/
Θ
/
/
/
(34)
Θ
/
/
/
(35)
Fig. 3.
0
estimation using EKF
2) Propagation of the new sigma points through the model function of the measurement (amplification function): Θ
φ
, with
/
0, … , 2.
(36)
3) weighing and value combination of φ : ∑
φ
(37)
∑
φ
∑
Θ
φ /
(38) /
φ
(39)
4) Calculation of the Kalman Gain: (40)
Fig. 4. p estimation using EKF
than for , but, the The convergence is much faster for estimate quickly converges close to the real values. There was estimation and about 10% for an error of 3% for estimation. For both parameters, it is interesting to observe that the values set initially were very distant from the real ones, with an error of more than 200%. Figures 5 and 6 show the UKF estimation for A0 and p. With UKF, the estimation converges even faster to a very close value of the real saturation level, with an error of only 3% in the p parameter.
5) Update of estimated parameters and related variance: (41) (42)
/ /
To avoid that an outlier estimate misrepresents the final estimation of the parameter, causing the syntax to diverge, the following pseudo-syntax was used as control information in the saturation parameter: 2
;
Fig. 5.
0
estimation using UKF
1
And the following to the p parameter: 1.2 ; Because the p parameter estimation was more sensitive to outliers than the saturation parameter, it was necessary to establish a more restrictive control information.
Fig. 6. p estimation using UKF
In order to verify the use of the parameter estimation algorithm with the ICHD method we use the limiter as the amplifier model and try to estimate only its clipping level. This estimated value is used by the ICHD method to recreate the nonlinear conditions had in transmission. We consider an OFDM system with 256 subcarriers, of which 210 transmit data. The DC and the outermost subcarriers are null. The subcarriers are separated by 15 kHz and the cyclic prefix corresponds to 1/16 of the useful symbol interval. A convolutional encoder with rate 1/2 was employed. Rate 3/4 is also considered with puncturing. The modulation employed in all simulations was 16-QAM. We investigate the performance of ICHD in two situations: when the receiver knows the transmitter nonlinearities, and when the non-linearity parameters are estimated using EKF at the receiver (curves using the UKF were omitted, because both EKF and UKF result in almost the same values). Figure 7 shows a situation in which 2dB at the transmitter and 1/2 over an AWGN channel. ICHD method improves the system performance even with only one iteration. Moreover, the BER performance with estimated parameters is extremely close to the idealized situation.
VI.
We have seen that the ICHD algorithm is sensitive to errors in the estimated nonlinear parameters of the power amplifier. We suggest the use of nonlinear filters, like the Extended Kalman Filter and the Unscented Kalman Filter, to estimate these parameters at the receiver. Both methods proved to be effective and the BER curves using the estimated parameters were shown to be very close to those with the real parameters. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Fig. 7. BER performance with IBO = 2dB, and R=1/2, AWGN channel.
In Figure 8, we used again 2 at the transmitter, but now 3/4, in a multipath Rayleigh fading UMTS-A channel model [20]. Again, ICHD improves the system performance, but not so pronounced as in an AWGN channel. This happens because the effecst of the non-linearities on the error performance of the OFDM system subject to frequency selective fading are reduced [21]. Furthermore, we are able to see again that the performance of ICHD in conjunction with EKF is extremely close to the idealized situation.
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Fig. 8. BER performance with IBO = 2dB and R= ¾, multipath.
CONCLUSION
H. Ochiai and H. Imai, ”Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., vol. 50, pp. 89-101, Jan. 2002 P. Banelli and S. Cacopardi, ”Theoretical analysis and performance of OFDM signals in nonlinear AWGN channels,” IEEE Transactions on Communications, vol. 48, no. 3, pp. 430-441, Mar. 2000. C. van den Bos, M.H.L. Kouwenhoven, and W.A. Serdijin, ”Effect of smooth nonlinear distortion on OFDM symbol error rate,” IEEE Transactions on Communications, vol. 49, pp. 1510-1514, Sept. 2001 B. S. Krongold and D. L. Jones, ”PAR reduction in OFDM via active constellation extension,” in IEEE Trans. Broadcasring, Apr. 2002. S. H. Mueller e J.B. Huber, ”OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” Electronics Letters, vol. 33, no. 5, pp. 368369, Feb. 1997. L. J. Cimini, Jr. e N. R. Sollenberger, ”Peak-to-average power ratio reduction of an OFDM signal using partial transmit sequences with embedded side information,” in Proc. Globecom 2000. T. Jiang et al., ”PAPR Reduction of OFDM Signals Using Partial Transmit Sequences With Low Computational Complexity,” in IEEE Trans. Broadcasting, 53(3), set. 2007. I. Ulian and A. N. Barreto ”On the Impact of Non-Linear High Power Amplifiers on Coded OFDM,” Intl. Telecommun. Symposium, Set. 2010. P. Zillmann and G. Fettweis, ”On the Capacity of Multicarrier Transmission over Nonlinear Channels,” in Proc. of the 61st IEEE VTC 2005, May 2005. W. Rave, P. Zillmann, and G. Fettweis, ”Iterative Correction and Decoding of OFDM Signals affected by Clipping,” in Proc. MC-SS 2005, pp. 443452, Oct. 2005. H. D. Han and P. Hoeher, ”Simultaneous Predistortion and Nonlinear Detection for Nonlinearly Distorted OFDM Signals,” in Proc. of IST Summit 200, June 2005. J. Tellado, L. Hoo, and J. M. Cioffi, ”ML detection of nonlinearly distorted multicarrier symbols by iterative decoding,” in IEEE Transactions on Communication, Vol. 51, No. 2, pp. 218228, 2003. H. Chen, and A. M. Haimovich, ”Iterative estimation and cancellation of clipping noise for OFDM,” IEEE Commun. Letters, 7(7), 2003. S. Haykin, ”Adaptive Filter Theory,” Prentice Hall, 1996. RECOMMENDATION ITU-R M.1225, ”Guidelines for Evaluatoin of Radio Transmission Technologies for IMT-2000”, 1997. Rapp, C. “Effects of hpa-nonlinearity on a 4-DPSK/OFDM signal for a digital sound broadcasting system. Tech. Conf. ECSC, October 1991.” E. A. Wan, R. van derMerwe, and A. T. Nelson, Dual Estimation and the Unscented Transformation, In: Neural Information Processing Systems 12.2000, pp. 666-672, MIT Press. E. Wan, R. van der Merwe, “The Unscented Kalman Filter for Nonlinear Estimation,” in Proc. Of IEEE Symposium 2000 (AS-SPCC), Lake Louise, Alberta, Canada, Oct. 2000. W. Feller, ”An Introduction to Probability Theory and its Applications”, 2nd ed. New York: Wiley, 1957, vol. 1. ITU-R M.1225, ”Guidelines for Evaluation of Radio Transmission Technologies for IMT-2000”, 1997. K. R. Panta and J. Armstrong, ”‘Effects of Clipping on the Error Performance of OFDM in Frequency Selective Fading Channels” in IEEE Trans. Wireless Commun., vol. 3, No. 2, Mar. 2004.
