Interference Mitigation in Turbo-coded OFDM Systems using Robust Statistics Sheetal Kalyani*
K. Giridhar
Motorola India Research Labs Bagmane Tech Park, CV Raman Nagar Bangalore 560093, India Email:
[email protected]
Department of Electrical Engineering Indian Institute of Technology Madras Chennai 600036, India Email:
[email protected]
Abstract— A robust cost function for log likelihood ratio (LLR) computation is proposed for turbo coded OFDM systems in the presence of narrowband interference (NBI) and co-channel interference (CCI). In systems employing standards such as the IEEE 802.16d/e, CCI behaves like NBI with the number of affected subcarriers ranging from 10% to 30%. The combined effect of NBI and thermal Gaussian noise leads to a contaminated Gaussian (CG) noise probability density function (pdf). Simulation results indicate that the proposed method performs very close to the optimal method where the optimal method computes the LLR using the CG pdf. While the optimal method requires knowledge of the NBI power, the fraction of subcarriers contaminated by NBI and the NBI pdf, the proposed method does not require knowledge of these parameters.
I. I NTRODUCTION Turbo codes have been applied as error correcting codes in numerous areas such as deep-space and satellite communications, as well as for interference limited applications such as third generation cellular and personal communication services. Turbo codes have also been proposed for use in OFDM based standards such as IEEE 802.16d/e. Our focus in this work is on mitigating the effect of narrowband interference (NBI) on the turbo decoder. While there has not been much work on turbo decoding in the presence of NBI in OFDM systems, there has been significant amount work on turbo decoding in the presence of impulsive noise [2], [3], [4] in TDMA systems. Note that in an OFDM system, the effect of NBI is similar to the effect of impulsive noise in TDMA based systems. In both OFDM and TDMA systems a contaminated Gaussian (CG) noise probability density function (pdf) is seen at the receiver due to the effect of NBI in the former and impulsive noise in the latter. Hence, it is possible to apply the ideas in [4], [2], [3] to a NBI affected OFDM system. Our earlier work in [12] proposes a weighted LLR (W-LLR) method to handle NBI in turbo coded OFDM systems. The current work improves on the W-LLR method, and at the cost of a modest increase in complexity can yield significant improvement in severe NBI affected channels. Our interest is on the effect of NBI on the turbo decoder in OFDM systems in the following applications (i) and (ii) listed below: *This work was done when the first author was with IIT Madras.
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Application (i) NBI effect due to co-channel interference (CCI) behaving like NBI in re-use one IEEE 802.16d/e (WiMAX) systems. The combined effect of frequency selectivity, power control, and diversity mapping (allocating physically non-adjacent subcarriers to a user) in systems using the IEEE 802.16d/e standards makes the CCI behave like NBI with the fraction the subcarriers () affected by NBI usually ranging from 0.1 ≤ ≤ 0.30. Estimating the time varying and NBI power is not easy since they are a function of the number of users at the cell boundary, transmitting power of these users, the channel between the user and the dominant interferer, the channel between the desired user and it’s base station, the diversity mapping scheme used and, the efficiency of the power control mechanism employed by the base station. The turbo decoder in which the LLR is computed using the CG pdf [3] would be optimal for the Gaussian NBI case; however, it would require knowledge of NBI power, fraction of tones affected by NBI, and the NBI pdf. Assuming that all these parameters are known is not practical in our application. A couple of papers [6], [7], [14] focus on rejecting or removing NBI at the receiver, without modifying the decoder block. However, these are not suitable for our applications due to the following reasons: a) In systems which employ the IEEE 802.16d/e standard, the base stations are frequency synchronized and hence the NBI due to the co-channel interferer will fall exactly on the OFDM subcarrier of the desired user. In such a case, one cannot employ the method in [14] since it does not work when the NBI lies exactly on an OFDM subcarrier. b) The method in [7] cannot be utilized since in our applications can be large. For example if = 0.25 then 25% of the modulated data symbols have to be discarded (provided one could first find the NBI locations accurately!) and even a powerful code will not be able to recover almost all the data symbols. Furthermore, NBI is present on multiple tones and is time varying both in terms of its location and power. Therefore locating the multitone NBI and then trying to null it using a notch filter as in [7] is not practical. c) The work in [6] uses excision filtering to filter out the interference before the FFT, at the receiver, but it also considers small values of and needs to estimate the carrier frequencies for each narrowband interferer. When is large
excision filtering will lead significant loss of data subcarriers since it also filters out the data symbol present at the NBI contaminated subcarrier. Application (ii) Any OFDM system operating in the unlicensed band will be affected by NBI from the other devices operating in the same band. For example, since IEEE 802.11g operates in the same unlicensed band as Bluetooth, cordless phone, and microwave oven it is affected by NBI due to these systems. Since there is interference from various sources in the unlicensed spectrum, the number of subcarriers affected by interference can range from 0% to 10%. Further, in the presence of FH systems such as Bluetooth which change their carrier frequency periodically, the NBI is time varying and has to be esimated online. Note that both [6], [7] require some finite amount of time to detect the frequency of the interferer. Further the NBI in application (ii) can be non-Gaussian in nature. For example, if a WLAN system is operating in the presence of a bluetooth network, then since there is no power control in the bluetooth network with reference to the WLAN system, the interference from bluetooth can have a alpha stable pdf. This situation is similar to the interference situation in a FH SS radio network discussed in [5]. In such a case, even the usual assumption of Gaussian NBI may not be valid. Therefore, only assuming a CG pdf with Gaussian NBI as considered in [3] can lead to poor performance of the turbo decoder. Assuming that the NBI affected channels or subcarriers are known [1] suggests the erasure of those bits to improve the performance of the turbo decoder in the presence of impulse noise in the case of ADSL. The assumption that the NBI affected subcarrier positions are known is not realistic in most cases, especially when we are looking at OFDM systems where the subcarriers positions that are affected by NBI may randomly vary from one OFDM symbol to another. Furthermore, if we do not know the location of the NBI affected subcarriers, and instead try to detect these locations, erasure based on the detected subcarriers may not be efficient. Finally, if about 10% or more subcarriers are affected by NBI, setting all the corresponding LLRs to zero could yield a poor bit error rate (BER) when compared with the proposed approach. We propose a modified LLR (M-LLR) method which treats NBI/CCI as outliers in the detection problem. The work in [4] is also based on a similar idea and proposes to use the Huber cost function based LLR (H-LLR) method for turbo decoding in alpha stable noise. However, we will show that the M-LLR method has a better simulated BER performance than the HLLR method for the simulation setup considered here. It will also be shown through simulations that the M-LLR method has a performance very similar to the optimal LLR scheme in [3] for a wide range of and NBI power values. II. S YSTEM M ODEL An OFDM system with a single transmitter and a single receiver is considered. Each OFDM symbol, has N subcarriers and a cyclic prefix of L samples. Each user is given N
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subcarriers and the FEC block for the user is over these N subcarriers. At the receiver, after the removal of the cyclic prefix (of length L) of the nth OFDM symbol, the N point FFT is applied to it to obtain the N × 1 frequency domain OFDM symbol. Out of this N × 1 vector, the N entries of interest to us can be represented as a N × 1 vector given by bn = Sn Hn + Jn + Nn
(1)
where each Nn,i , i = 1, ..., N , is a zero mean white circular Gaussian random variable with both the real and imaginary noise components have variance σ 2 /2. Sn is a N ×N diagonal matrix consisting of the turbo coded M-PSK or M-QAM modulated symbols Sn,i . Here, Hn is the vector of N values of channel frequency response (CFR) corresponding to the N subcarriers of a single user. The N × 1 NBI vector, Jn has l nonzero entries with 0 ≤ l < N/2, and Jn is given by Jn = Kn Nnb n
(2)
where each element of the N × 1 vector Nnb n is a random variable of zero mean (or zero location parameter) and specific variance (or specific geometric power). In the NBI model, Kn is a N ×N diagonal matrix with l nonzero entries, where each nonzero entry Kn (i, i) = 1. In every OFDM symbol, for each user, NBI is randomly located on l out of the N subcarriers 1 . The pdf of the effective noise component, Jn,i + Nn,i , for i = 1, 2, ..., N , is the CG pdf given by 1 − − |x|22 e σ + H (3) πσ 2 where H is the NBI pdf and = Nl . Both bn and Hn , are both given as inputs to the turbo decoder to compute the LLRs. While in this work perfect channel knowledge is assumed at the receiver, the proposed algorithms can also work with channel estimates. Channel estimation in the presence of NBI can be carried out using the methods in [10] and [11]. The bit LLR of the mth bit cn,i,m of the ith data symbol Sn,i in the nth OFDM symbol given by p(cn,i,m = 1) . (4) LLR(cn,i,m ) = ln p(cn,i,m = 0) g(x) =
where p(cn,i,m = 1) is the probability that cn,i,m = 1 given bn,i and Hn,i . In erasure decoding the LLRs corresponding to NBI affected subcarriers are set to zero which implies that p(cn,i,m = 1) = p(cn,i,m = 0). The LLR computed under the Gaussian noise pdf assumption, i.e., the Gaussian LLR can be written as B−1 |b −Cm,l Hn,i |2 2 − n,i σ 2 e l=1 LLR(cn,i,m ) = ln B−1 (5) 2 − |(bn,i −C m,l Hn,i |2 σ2 e l=1 1 In
a conventional communications system the NBI/CCI would be correlated and not distributed independently over l random subcarriers. However, in systems which employ diversity mapping and power control, the subcarriers allocated to a user need not be physically adjacent and further due to power control one may not see equally strong CCI on physically adjacent subcarriers. Hence in such a scenario NBI due to CCI will not be correlated and will be randomly spread on l out of the total N subcarriers allocated to a user.
where Cm,l is an element of the set of M-PSK or M-QAM symbols with the mth bit cm = 1. Hence, there will be 2B−1 such elements where B is number of bits per data symbol. Similarly C m,l is an element of the set of M-PSK symbols with the mth bit cm = 0. It is well known that a Gaussian pdf based estimator shows poor performance when the actual noise pdf is a CG pdf [8]. In a similar fashion, the LLR computed under the Gaussian pdf assumption is also a poor measure of the actual value the LLR can take when the pdf deviates even slightly from the Gaussian pdf.
LLR(cn,i,m ) =
ln
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e
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α ≤ |rn,i | < β (7) |rn,i | ≥ β
l=1
III. ROBUST S TATISTICS BASED LLR METHOD The W-LLR method [12] multiplies the Gaussian LLR with a weight value between one and zero to mitigate the effect of NBI. The weight value is based on extreme value theory and if it is small it indicates the presence of strong NBI at the subcarrier. However this is not equivalent to (or as good as) computing the LLR using the CG noise pdf. But this optimal LLR computation would require accurate knowledge of the CG pdf parameters and this information may not be easily available. An alternative approach is to change the cost function used in LLR computation from Gaussian to a non-Gaussian robust function. In this case, we need not know the NBI pdf, and NBI power since for us the NBI affected subcarriers are like outliers in the detection problem. Such an idea has been proposed in the context of turbo decoding in the presence of heavy tailed noise where the Gaussian LLR has been replaced by the Huber cost function in [4]. Here we propose to use a similar approach and modify the LLR cost function so that the turbo decoder is able to handle NBI. To detect whether the ith subcarrier is affected by NBI, we look at the corresponding residual rn,i = bi − Hn,i Sn,i . Since Sn,i is unknown, we hypothesize on Sn,i such that the absolute value of the residual is defined as |rn,i | = min |bn,i − Hn,i X| X∈C
(6)
where C is the set of M-PSK/M-QAM symbols. In robust statistics theory, the absolute value of the residual |rn,i | is a commonly used outlier diagnostic. The presence of NBI on the ith subcarrier is generally reflected as a large value of |rn,i |. While the Gaussian LLR is based on the l2 norm, the Huber cost function based LLR minimizes the l2 norm when the |rn,i | in (6) is smaller than a user chosen threshold β and minimizes the l1 norm when |rn,i | is greater than β. However, instead of an abrupt transition from l2 norm cost function to an l1 norm based cost function we suggest a two step transition, i.e., when |rn,i | is less than α, use the l2 norm based cost function, when α ≤ |rn,i | < β use a cost function based on l1.5 norm and when |rn,i | is greater than β use a cost function based on l1 norm for computing the LLR. The LLR in the M-LLR method is given by
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with xn,i,m,l = |bn,i − Cm,l Hn,i | and xn,i,m,l = |bn,i − C m,l Hn,i |. We compared the proposed M-LLR scheme with the H-LLR scheme (a scheme which uses the Huber cost function for the LLR computation) where the LLR is given by
LLR(cn,i,m ) =
ln
2B−1 l=1
−
e
x2 n,i,m,l σ2
x2 n,i,m,l 2B−1 − σ2 e
l=1 B−1 2βxn,i,m,l 2 − σ e l=1 ln 2B−1 2βxn,i,m,l − σ e
|rn,i | < β (8) |rn,i | ≥ β
l=1
The user-specified value of α and β represents a tradeoff between efficiency and the robustness of the method where efficiency is a measure of how well the method performs in pure Gaussian noise. When α = β → ∞, the M-LLR method reduces to a Gaussian LLR method. When α = β → 0, the MLLR cost function minimizes the l1 norm and is very robust to outliers. However, it has a much higher MSE than the Gaussian LLR method when noise is purely Gaussian. We choose β such that the methods pays less than 5% penalty if noise is purely Gaussian and at the same time are also robust to outliers (For more details on efficiency robustness tradeoff please refer to [8]). Both the M-LLR and H-LLR method work well for any symmetric NBI pdf, i.e., the NBI can be even as thick tailed as Cauchy [8]. While in the optimal LLR scheme, the LLR computation changes if the NBI pdf changes, in the case of M-LLR method the LLR computation remains the same regardless of the NBI pdf. Hence, one does not need to know the NBI pdf apriori for the M-LLR method and since NBI is treated as outliers one need not know and NBI power. IV. S IMULATION R ESULTS An OFDM system with QPSK modulation operating with bandwidth of 20 MHz is considered. Each OFDM symbol has N = 2048 subcarriers and we look at a case of 4 users with each user being given N = 480 subcarriers in each OFDM symbol. We have shown the BER performance for a single user and not all the four users. Each frame consists of 50 OFDM symbols. Pedestrian B channel[13] is used as the channel model and a Doppler frequency of 10 Hz is assumed. Similar simulation results have also been obtained with Vehicular A channel. About
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Fig. 2 compares the BER performance of various schemes when 10% of the subcarriers in each OFDM symbol is affected by Gaussian NBI of SIR = −3 dB. The relative performance of the various methods is similar to Fig. 1. The H-LLR scheme is about 2dB poorer than the M-LLR scheme at a BER of 10−5 . The proposed M-LLR scheme is about 8 dB better than the Gaussian LLR scheme at a error rate of 2 × 10−4 . 0
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2000 channel realizations were used for evaluating the BER at every value of average SNR where average SN R = E[|Sn,i |2 |Hn,i |2 ] . Turbo code parameters were chosen according σ2 to IEEE 802.16e standards [9] for rate half turbo code with block length of 480 information bits, i.e., N = 480 subcarriers contain the QPSK modulated turbo coded data symbols. The number of turbo decoder iterations is set to six. Here, l random subcarriers out of the N subcarriers allocated to a user are affected by Gaussian NBI with l taking on values l = 50 (10% of the subcarriers of each user affected by NBI) or l = 120(25% of the subcarriers affected by NBI). We have 2 2 = 0.5 (3 dB SIR), σnb = considered NBI with NBI power σnb 2 1 ( 0 dB SIR) and σnb = 2 ( −3 dB SIR). We compare the M-LLR scheme with the Gaussian LLR scheme, the H-LLR scheme, the W-LLR scheme [12] and the optimal LLR scheme [3]. We choose α = σ, β = 1.5σ and assume that for the optimal LLR scheme we know the NBI pdf, NBI power and the fraction of subcarriers contaminated by NBI. We also compare the M-LLR scheme with an erasure decoding scheme for which it is assumed that the NBI location are known exactly and the LLRs corresponding to these locations are set to zero. Fig. 1 compares the BER performance
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Fig. 1. BER when 10% of the subcarriers affected by Gaussian NBI with SIR = 0dB
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of various schemes when 10% of the subcarriers in each OFDM symbol is affected by Gaussian NBI of SIR = 0 dB. The M-LLR scheme has a performance similar to the optimal LLR scheme for SNR in the range 0 dB to 8 dB. Further, we have shown the performance of the Gaussian LLR scheme when the is no NBI. Even though M-LLR method is handling 10% NBI of SIR = 0 dB, it is about only 1 dB poorer than the performance of the Gaussian LLR scheme which has no NBI, for SNRs in the range 0 to 8 dB. For SNRs beyond 8 dB, the optimal LLR scheme shows error free transmission for 2000 independent frames while the M-LLR schemes does not have any bit errors for the same 2000 frames at SNRs greater than 12 dB. Note that the proposed M-LLR scheme also outperforms the W-LLR scheme [12] and the Erasure decoding scheme. It also outperforms the H-LLR scheme at
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Fig. 3. BER when 25% of the subcarriers affected by Gaussian NBI with SIR = 0dB
Fig. 3 compares the BER performance of various schemes when 25% of the subcarriers in each OFDM symbol is affected by Gaussian NBI of SIR = 0 dB. Inspite of the fact that 25% of the subcarriers are affected by NBI, the method proposed by us is only 3 to 4 dB poorer than the no NBI case. Note that applying erasure of the data symbols as suggested by [1] for the 25% NBI case gives a very poor result when compared with the M-LLR scheme. In fact the erasure decoding scheme is nearly as bad as the Gaussian LLR scheme in its performance. The W-LLR scheme is better than the erasure decoding scheme. However, it is also quite poor
when compared with the M-LLR scheme. At a BER of 10−4 , the M-LLR scheme is atleast 3 dB better than the W-LLR scheme while the erasure decoding scheme does not achieve a BER of 10−4 even at 16dB. The performance of the M-LLR scheme is quite close to that of the optimal LLR scheme. 0
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band. The method suggested by us is not specific to the turbo decoder and can be used for LLR computation in any error correcting code. The M-LLR method has a performance close to the optimal scheme in both Gaussian and Cauchy NBI even though it does not have knowledge of the NBI percentage, power and pdf. Even when the M-LLR scheme is used for = 0 (no NBI) case the performance degradation is negligible when compared with the Gaussian LLR scheme (which is the optimal scheme for the no NBI case).
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Fig. 4. BER when 25% of the subcarriers affected by Cauchy NBI of GSIR = 3 dB
Figure 4 compares the BER performance of various schemes when 25% of the subcarriers are affected by Cauchy NBI of GSIR = 3dB2 . The Gaussian LLR method shows significant degradation in the presence of Cauchy NBI. The M-LLR and H-LLR schemes show similar performance and are quite close to the optimal scheme. Similar results have been obtained when 10% of the subcarriers have been affected by Cauchy NBI. Simulations have also been performed for 0.25 < < 0.5 and both the H-LLR and M-LLR schemes still work well even though the fraction of subcarriers affected by NBI is large. Further the proposed M-LLR scheme also works well even if actual NBI power is as high as 10 dB (i.e., SIR=-10dB). We also have simulation results where the M-LLR scheme is used in the absence of NBI and in such a case its performance is very close to the “NO NBI” curves in the figures. Hence, the loss in performance of the M-LLR scheme in a scenario where the Gaussian LLR is optimal (i.e., in the absence of NBI) is negligible while the reverse (using Gaussian LLR in the presence of NBI) is not true. However, these simulation results have not been shown in the paper due to space constraint. V. S UMMARY We suggest the use of the M-LLR method for computing the LLR in the turbo decoder for OFDM systems which employ standards such as the IEEE 802.16d/e and also for other multicarrier or OFDM systems operating in the unlicensed 2 Since the variance is not defined for the Cauchy pdf, in order to have a fair comparison of the performance of the symbol detectors for the different CG distributions we keep the same geometric signal to interference ratio (GSIR) for both the Cauchy NBI based CG pdf and the Gaussian NBI based CG pdf [11]. For detail on computing the GSIR for Cauchy NBI refer to [11]. In the case of Gaussian NBI the SIR is same as GSIR.
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[1] L. Zhang and A. Yongacoglu, “Turbo Decoding with Erasures for Highspeed Transmission in the Presence of Impulse Noise,” in International seminar on Broadband Commmunications Access, Feb 2002. [2] J. Mitra and L. Lampe, “Robust Decoding for Channel with Impulse Noise,” in Proc. of IEEE Global Telecommunications Conference, Nov. 2006, pp. 1-6. [3] T. Faber, T. Scholand and P. Jung, “Turbo decoding in impulsive noise environments,” Electronics Letters, vol. 39, pp. 1069 - 1071, July 2003. [4] T. C. Chuah, “Robust iterative decoding of turbo codes in heavy-tailed noise,” IEE Proceedings Communications, vol. 152, pp. 29 - 38, Feb. 2005. [5] J. Ilow, D. Hatzinakos and A. N. Venetsanopoulos, “Performance of FH SS radio networks with interference modeled as a mixture of Gaussian and alpha-stable noise,” IEEE Trans. on Communications, vol. 46, pp. 509 - 520, Apr. 1998. [6] A. J. Coulson, “Bit error rate performance of OFDM in narrowband interference with excision filtering, IEEE Transactions on Wireless Communications, vol. 5, pp. 2484-2492, Sept. 2006. [7] K. Shi, Y. Zhou, B. Kelleci, T. W. Fischer, E. Serpedin and A. I. Karsilayan, “ Impacts of Narrowband Interference on OFDM-UWB Receivers: Analysis and Mitigation,” IEEE Trans. Sig. Proc., vol. 55, pp. 1118 - 1128, Mar. 2007. [8] P.J. Huber, Robust Statistics. John Wiley and sons, New York, 1981 [9] IEEE P802.16-2004, Standard for local and metropolitan area networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems, 2004. [10] S. Kalyani and K. Giridhar, “Extreme Value Theory Based OFDM Channel Estimation In The Presence Of Narrowband Interference, ” in Proc. of IEEE Global Telecommunications Conference, Nov. 2006, pp. 1-5 [11] S. Kalyani and K. Giridhar, “OFDM channel estimation in the presence of NBI and the effect of misspecified NBI model,” in Proc. of IEEE 8th Workshop on Signal Processing Advances for Wireless Communications, SPAWC 2007, 17-20 June 2007, pp. 1 - 5. [12] S. Kalyani, V. Raj and K. Giridhar, “Narrowband Interference Mitigation in Turbo-coded OFDM Systems,” in Proc. of IEEE International Conference on Communications, ICC 2007, Jun. 24-28, 2007, pp. 1059 - 1064. [13] Recommendation ITU-R M.1225, “Guidelines for Evaluation of Radio Transmission Technologies for IMT-2000, ” 1997. [14] D. Darsena, G. Gelli, L. Paura and F. Verde, “NBI-resistant zero-forcing equalizers for OFDM systems, ” IEEE Communications Letters, vol. 9, pp.744-746, Aug 2005.
