Noise Plus Interference Power Estimation inAdaptive OFDM Systems

Tevfik Yucek and Hiiseyin ArslanDepartment of Electrical Engineering, University of South Florida

4202 E. Fowler Avenue, ENB-118, Tampa, FL, 33620yucek@eng.usf.edu and arslan@eng.usf.edu

Abstract- Noise variance and signal-to-noise ratio (SNR) areimportant parameters for adaptive orthogonal frequency divisionmultiplexing (OFDM) systems since they serve as a standardmeasure of signal quality. Conventional algorithms assume thatthe noise statistics remain constant over the OFDM frequencyband, and thereby average the instantaneous noise samples toget a single estimate. In reality, noise is often made up ofwhite Gaussian noise along with correlated colored noise thataffects the OFDM spectrum unevenly. This paper proposes anadaptive windowing technique to estimate the noise power thattakes into account the variation of the noise statistics acrossthe OFDM sub-carrier index as well as across OFDM symbols.The proposed method provides many local estimates, allowingtracking of the variation of the noise statistics in frequencyand time. A mean-squared-error (MSE) expression in order tochoose the optimal window dimensions for averaging in timeand frequency is derived. Evaluation of the performance withcomputer simulations show that the proposed method tracks thelocal statistics of the noise more efficiently than conventionalmethods.

I. INTRODUCTION

Orthogonal Frequency Division Multiplexing (OFDM) is amulti-carrier modulation scheme in which the wide transmis-sion spectrum is divided into narrower bands and data is trans-mitted in parallel on these narrow bands. Therefore, symbolperiod is increased by the number of sub-carriers, decreasingthe effect of inter-symbol interference (ISI). The remaining ISIeffect is eliminated by cyclically extending the signal. OFDMprovides effective solution to high data-rate transmission byits robustness against multi-path fading [1]. Parallel with thepossible data rates, the transmission bandwidth of OFDMsystems is also large. UWB-OFDM [2] and IEEE 802.16 basedwireless metropolitan area network (WMAN) [3] systems areexamples of OFDM systems with large bandwidths. Becauseof these large bandwidths, noise can not be assumed to bewhite with flat spectrum across subcarriers.The signal-to-noise ratio (SNR) is broadly defined as the

ratio of the desired signal power to the noise power andhas been accepted as a standard measure of signal quality.Adaptive system design requires the estimate of SNR in orderto modify the transmission parameters to make efficient useof system resources. Poor channel conditions, reflected bylow SNR values, require that the transmitter modify trans-mission parameters such as coding rate, modulation modeetc. to compensate channel and to satisfy certain applicationdependent constraints such as constant bit error rate (BER) and

throughput. Dynamic system parameter adaptation requires areal-time noise power estimator for continuous channel qual-ity monitoring and corresponding compensation in order tomaximize resource utilization. SNR knowledge also providesinformation about the channel quality which can be used byhandoff algorithms, power control, channel estimation throughinterpolation, and optimal soft information generation for highperformance decoding algorithms.The SNR can be estimated using regularly transmitted train-

ing sequences, pilot data or data symbols (blind estimation).In this paper, we restrict ourself to data aided estimation. Acomparison of time-domain SNR estimation techniques canbe found in [4]. There are several other SNR measurementtechniques which are given in [5] and references listed therein.In literature of OFDM SNR estimation, related work is few. Inconventional SNR estimation techniques, the noise is usuallyassumed to be white and an SNR value is calculated forall subcarriers [4], [6]. In [7], this assumption is removedby calculating SNR values for each subcarrier. However, thecorrelation of the noise variance across subcarriers is notused since noise variance is calculated for each subcarrierseparately.White noise is rarely the case in practical wireless commu-

nication systems where the noise is dominated by interferenceswhich are often colored in nature. This is more pronounced inOFDM systems where the bandwidth is large and the noisepower is not the same over all the sub-carriers. Color ofthe noise is defined as the variation of its power spectraldensity in frequency domain. This variation of spectral contentaffects certain sub-carriers more than the others. Therefore, anaveraged noise estimate is not the optimal technique to use. Inthis paper, the assumption of the noise to be white is removedand variation of the noise power across OFDM sub-carriers aswell as across OFDM symbols is allowed. The noise variancesat each subcarrier is estimated using a two dimensional slidingwindow whose size is calculated using local statistics of thenoise. These estimates are specifically useful for adaptivemodulation, and optimal soft value calculation for improvingchannel decoder performance. Moreover, it can be used todetect and avoid narrowband interference. We investigate acomputationally efficient fixed window size algorithm and anadaptive algorithm where the window dimensions are calcu-lated for each subcarrier. The adaptive algorithm is especiallysuitable for non-stationary interference scenarios. The paper

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focuses more on estimation of noise power, and assumes thatthe signal power, and hence SNR, can be estimated from thechannel estimates.

This paper is organized as follows. In next section, oursystem model is described. Section III explains the proposedfixed and adaptive windowing algorithms. Then, numericalresults are presented in Section IV and paper is concludedin Section V.

II. SYSTEM MODEL

OFDM converts serial data stream into parallel blocks ofsize N and modulates these blocks using inverse fast Fouriertransform (IFFT). Time domain samples of an OFDM symbolcan be obtained from frequency domain symbols as

x,(m) = IFFT{Sn,k}N-1

E Sn,kej2irmk/N O<m<N-1 (1)k=O

where Sn,k is the transmitted data symbol at the kth subcarrierof the nth OFDM symbol and N is the number of subcarriers.After the addition of cyclic prefix and D/A conversion, thesignal is passed through the mobile radio channel.

At the receiver, the signal is received along with noise andinterference. After synchronization and removal of the cyclicprefix', fast Fourier transform (FFT) is applied to the receivedsignal to go back to the frequency-domain. The received signalat the kth subcarrier of nth OFDM symbol can then be writtenas

Yn,k = S.,kH.,k + 'n,k + W.,k (2)Zn,k

where Hn,k is the value of the channel frequency response(CFR), In,k is the colored noise (interference), and Wn,kis the white Gaussian noise samples. We assume that theimpairments due to imperfect synchronization, transceivernon-linearities etc. are folded into Wn,k and the CFR is notchanging within the observation time.

The white Gaussian noise is modeled as Wn,k = A((O, aO)and the interference term as In,k = K((0, an2k), where Un,kis the local standard deviation. Note that although the time-domain samples of the interference signal is correlated (col-ored), the frequency-domain samples (In,k) are not correlated,but their variances are correlated [8]. Assuming that theinterference and white noise terms are uncorrelated, the overallnoise term Zn,k can be modeled as Zn,k = Ar(O, or'k2), whereUnk = +ao2 is the effective noise variance. The goal ofthis paper is to estimate an'k2 which can be used to find SNR.Note that if ao »> an,k, the overall noise can be assumed tobe white and it is colored otherwise.

IThe length of the cyclic prefix is assumed to be larger than the maximumexcess delay of the channel.

Fcc)

Time

Desired Signal

Interferer

Fig. 1. Illustration of non-stationary interference.

III. DETAILS OF THE PROPOSED ALGORITHM

The commonly used approach for noise power estimationin OFDM systems is based on finding the difference betweenthe noisy received sample in frequency domain and the besthypothesis of the noiseless received sample [6]. It can beformulated as

Z.,k = Y,k -Sn,kHn,k (3)

where Sn,k is the noiseless sample of the received symboland fin,k is the channel estimate for the kth sub-carrier ofnth OFDM symbol.

In this paper, three different scenarios for the noise processZn,k are considered: white noise, stationary colored noise andnonstationary colored noise. The first one is the commonlyassumed case, where the frequency spectrum of the noise isuniform. In the second scenario, we assume to have a stronginterferer which has larger bandwidth than the desired OFDMsignal. A strong co-channel interferer is a good example forthis case. In the third one, an interferer whose statistics are notstationary with respect to time and/or frequency is assumedto be present. Adjacent channel interference or a co-channelinterference with smaller bandwidth than the desired signal areexamples of this type of interference. A scenario where theinterference is not stationary both in time and in frequency isillustrated in Fig. 1. Here, the statistics of noise componentschange as we move along the time or the frequency axis.We propose to use a two dimensional sliding window for

obtaining the noise plus interference power. Windowing willremove the common assumption of having the noise to bewhite and it will take colored interference (both in time andin frequency) into account. In this case, the estimate of thenoise power at kth subcarrier of nth OFDM symbol can bewritten as

n+Lt/2+1 k+Lfl/2+10n,k = LELE E 4,uI

Ttf I=n-L,/2 u=k-Lf/2(4)

where Lt and Lf are the averaging window lengths in timeand frequency respectively.

Sliding window approach given in (4) requires appropriateLt and Lf values for accurate estimation of noise plusinterference power. If the window size is not chosen properly,it degrades the performance of estimation. Estimation error at

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the kth subcarrier of nth OFDM symbol can be written as

E(n, k) =n2k -n k

n+Lt/2-1 k+Lf/2-1

LtL E_ E _Z1,uI2- 2nk * (5)f 1=n-L,/2 u=k-Lf/2

Note that the instantaneous errors, (5), will be a function ofthe window size, how correlated the interference is within theaveraging window, average interference power and averagenoise power. Hence, the optimum values for window sizeswill be different for each subcarrier and OFDM symbol, i.e.Lt,,p, = Lt(n, k) and Lf,opt = Lf (n, k). In the next section,a suboptimal algorithm that uses the same window sizes foreach subcarrier is developed and it is later used to develop theoptimum algorithm which calculate the window sizes for eachlocal point.

A. Fixed Window SizeWhen the interference is stationary (with respect to time or

frequency), we propose to use a window with fixed dimensionsfor estimating the total noise power. Although the fixedwindow size algorithm is sub-optimum, it is computationallysimple than the optimum method that will be discussed in thenext section.The window dimensions can be calculated by minimizing

the mean-squared-error (MSE), i.e by minimizing the expectedvalue of the square of (5). In this case, the MSE can beformulated as

MSE = n,k {E(n, k)}

1 n+Lt/2-1 k+Lf/2-1 2

=.,£k LL On kIZl,u kLtLf I=n-Lt/2 u=k-Lf/2

(6)

where 9n,k is expectation over subcarriers and OFDM sym-

bols. By further simplification, (6) can be written in termsof the auto-correlation of the variance of the noise componentRa,2 (T, A) and the window dimensions (Lt and Lf) as shownin (7) at the bottom of the page.

Minimizing (7) achieves a trade-off between large windowsizes (for white noise dominated cases) and small windowsizes (for colored noise dominated cases). The window sizethat minimize the MSE should be chosen for averaging, i.e.

Ltjfixed = argminMSE, Lf,ficced = argminMSE . (8)Lt Lf

Note that the window size depends on the statistics of inter-ference and white noise. These statistics can be obtained byaveraging since the processes are assumed to be stationary.

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t- 0.1

O.

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0 10 20 30 40Window size

50 60 70

Fig. 2. Mean squared error as a function of window dimension in frequency.

Fig. 2 shows the MSE for different interference scenariosas a function of averaging size. In this figure, only windowingin frequency domain is considered for simplicity although thesame concept is true for windowing in time. The best averag-ing window size becomes infinity for the white noise case andit has different values depending on the auto-correlation of thepower of the total noise. As can be seen from this figure, theaveraging size that gives the minimum error decreases as thecorrelation decreases.By using a fixed sliding window, the common assumption

of having the noise to be white is removed and the coloredinterference (both in time and in frequency) is taken intoaccount. However, the noise statistics are assumed to beconstant, i.e. Ra,2 (-, A) is not changing over the estimationperiod.

B. Adaptive Window Size

In the previous section, a.fixed window size is used over thewhole subcarrier index as well as across OFDM symbols byassuming the noise statistics are constant in frequency and intime. This assumption is not valid when we the interference isnot stationary with respect to time (e.g. 802.11 interference)or with respect to frequency (e.g. narrowband interference[9]) or both. When the dominant interference statistics changeover the time or frequency, the algorithm proposed in theprevious section will degrade. In this section, we propose touse different window dimensions for each subcarrier. Thisis achieved by assuming that the interference within theneighborhood of a subcarrier is stationary, i.e. the interferenceis quasi-stationary with respect to time and frequency.

M SE=(1~t~ 1)Rg 2(0,0)~2 Lt/2-1 Lf/2-1

MSE = (1 + LtLf)R,L2(0,)- L E :E1 L,-1 Lf-1

R,,2 (1, U)+ L2L2 E E (Lt I1l)(Lf Jul)R.,2 (1, u). (7)

2 t ~f1=-Lt u=-Lf

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350 -25 -20 -15 -10 -5 0Stationary/White power ratio (dB)

Fig. 3. The averaging size obtained by semi-anlytic method and the proposedadaptive window size algorithms.

In order to be able to find the optimum window dimensionsfor each local point, we replace the correlation term in (7)with local correlation estimate Ri,2. Rf,2 is estimated usingonly the noise terms Zn,k within the window for which theMSE is calculated.

The optimum window size for each local point is found byminimizing (7). The correlation values are estimated using thenoise samples within the hypothesized window. The optimumwindow size in a subcarrier may be very large if the noise hasflat spectral content. In order to decrease the computationalcomplexity, window dimensions are found by assuming theinterference is stationary. Then, window sizes less then orequal to this value is tested. If the obtained result is equal to thethis maximum value, the maximum window size is increasedand the algorithm is repeated.

IV. NUMERICAL RESULTSAn OFDM system with 1024 subcarriers and 20MHz band-

width is considered. The stationary interference is assumed tobe caused by a co-channel user transmitting in the same bandwith the desired user, and a co-channel signal with 10MHzbandwidth is used to simulate the non-stationary interference.We use averaging over 20 OFDM symbols and considerestimation of Lf only, but the results can be generalized totwo dimensional case as well.

Fig. 3 shows the window length in frequency domain fora hypothetical non-stationary interference. Results obtainedusing the proposed adaptive algorithm and using excessivesearch are shown. As can be seen, the proposed algorithm findsthe correct window dimensions with little error. These error iscaused by the absence of enough statistics for obtaining thelocal correlations. At the edges of the interference the optimumwindow size goes to zero and it becomes larger where thenoise variance is constant.

Figs. 4 and 5 show the MSEs for the conventional, fixedsize window and adaptive window size algorithms. Fig. 4 gives

5 10 15

Fig. 4. Mean squared error for different algorithms as a function of thestationary colored noise to white noise power ratios.

the MSEs as a function of the stationary interference to whitenoise power ratio and Fig. 5 gives the MSE as a functionof the non-stationary interference to white noise power ratio.The total noise plus interference power is kept constant forboth figures. Note that when the ratio is very small (e.g. -25dB), the total noise can be considered as white noise only,and conventional algorithm performs best because its inherentwhite noise assumption is true. The estimation error increasesas the total noise becomes more colored for all three methods.As noise becomes more colored, the averaging window dimen-sions become smaller for both fixed and adaptive algorithmsincreasing the estimation error. For conventional algorithm,the increase in the MSE is expected as variation of the noisepower is more.When the interference is stationary, the performance of the

fixed window size algorithm is close to the performance ofthe adaptive window size algorithm while the performancedifference becomes more obvious in the case of non-stationaryinterference. This is because the stationarity assumption inthe derivation of fixed window size algorithm is valid for thestationary interference case (Fig. 4) whereas it is not true forthe non-stationary interference (Fig. 5).

V. CONCLUSIONIn this paper, a new noise variance estimation algorithm for

OFDM systems, which removes the common assumption ofwhite Gaussian noise and considers colored noise, is proposed.Noise variance, and hence SNR, is calculated by using a twodimensional sliding window in time and frequency. Windowswith fixed and adaptive dimensions are considered. The slidingwindow dimensions in each subcarrier position is calculatedadaptively using the local statistics of noise at that subcarrierhence considering the non-stationary interference scenarios.Although, the adaptive window size based algorithm givesthe optimum performance, it is computationally complex.Therefore, the fixed window size algorithm may be used in

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-.'-ConventionalFixed window sizeAdaptive window size

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Fig. 5. Mean squared error for different algorithms as a function of thenon-stationary colored noise to white noise power ratios.

applications where computational complexity is the limitingfactor. Simulation results show that the proposed algorithmout-performs conventional algorithm under colored noise.

ACKNOWLEDGMENT

The work in this paper was supported by Logus BroadbandWireless Solutions.

REFERENCES

[1] R. Prasad and R. Van Nee, OFDMfor Wireless Multimedia Comrnunica-tions. Boston, London: Artech House Publishers, 2000.

[2] J. Balakrishnan, A. Batra, and A. Dabak, "A multi-band OFDM systemfor UWB communication," in Proc. IEEE Conference on Ultra WidebandSystems and Technologies, Nov. 2003, pp. 354-358.

[3] IEEE Standard for Local and Metropolitan area networks Part 16: AirInterface for Fixed Broadband Wireless Access Systems, The Institute ofElectrical and Electronics Engineering, Inc. Std. IEEE 802.16, 2002.

[4] D. Pauluzzi and N. Beaulieu, "A comparison of SNR estimation tech-niques for the AWGN channel," IEEE Trans. Commun., vol. 48, no. 10,pp. 1681-1691, Oct. 2000.

[51 M. Turkboylari and G.-L. Stuiber, "An efficient algorithm for estimatingthe signal-to-interference ratio in TDMA cellular systems," IEEE Trans.Commun., vol. 46, no. 6, pp. 728-731, June 1998.

[6] S. He and M. Torkelson, "Effective SNR estimation in OFDM systemsimulation," in Proc. IEEE Global Telecommunications Conf., vol. 2,Sydney, NSW, Australia, Nov. 1998, pp. 945-950.

[7] S. Boumard, "Novel noise variance and SNR estimation algorithm forwireless MIMO OFDM systems," in Proc. IEEE Global Telecommunica-tions Conf., vol. 3, Dec. 2003, pp. 1330-1334.

[8] M. Ghosh and V. Gadam, "Bluetooth interference cancellation for 802.1 lgWLAN receivers," in Proc. IEEE Int. Conf. Commun., vol. 2, May 2003,pp. 1169-1173.

[9] R. Nilsson, F. Sjoberg, and J. P. LeBlanc, "A rank-reduced LMMSE can-celler for narrowband interfernece sppression in OFDM-based systems,"IEEE Trans. Commun., vol. 51, no. 12, pp. 2126-2140, Dec. 2003.

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