Blind noise variance and SNR estimation for OFDM systems based on information theoretic criteria

ARTICLE IN PRESS Signal Processing 90 (2010) 2766–2772

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Blind noise variance and SNR estimation for OFDM systems based on information theoretic criteria$ Kun Wang , Xianda Zhang Department of Automation, Tsinghua University, Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China

a r t i c l e i n f o

abstract

Article history: Received 15 November 2009 Received in revised form 30 December 2009 Accepted 6 March 2010 Available online 11 March 2010

Blind noise variance and SNR estimators based on information theoretic criteria for orthogonal frequency division multiplexing (OFDM) systems are proposed in this paper. By using the redundancy induced by the cyclic-prefix, the unknown channel order is estimated via minimum length description (MDL) or Akaike information criterion (AIC), and the noise variance and SNR can be obtained subsequently. The proposed methods do not require any pilot symbols and therefore improve the spectrum efficiency. Simulation results are presented to highlight the benefit of the proposed algorithms compared with the other existing blind methods. & 2010 Elsevier B.V. All rights reserved.

Keywords: OFDM Noise variance estimation SNR estimation MDL AIC

1. Introduction Orthogonal frequency division multiplexing (OFDM) technology has received a lot of attention in the field of wireless communication due to its robustness to frequency selective fading and efficiency for transmitting high-rate data. Noise variance and signal to noise ratio (SNR) are very important parameters to evaluate the quality of the communication link. In OFDM systems, the knowledge of the noise variance and SNR is required in many algorithms such as maximum likelihood (ML) frequency offset estimation [1] and adaptive modulation and resource allocation [2], etc. And the noise variance can also be used in the process of spectrum sensing in OFDM-based

$ This work was supported by the National Natural Science Foundation of China under Grants 60975041 and by Basic Research Foundation of Tsinghua National Laboratory for Information Science and Technology (TNList).  Corresponding author. Fax: + 86 10 62786911. E-mail addresses: [email protected] (K. Wang), [email protected] (X. Zhang).

0165-1684/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.03.007

cognitive radio systems to detect the primary signal. Therefore, accurate estimation of the noise variance or SNR is of great importance to improve the overall performance of the OFDM systems. Numerous noise variance and SNR estimators have been proposed for OFDM systems, which can be classified as data-aided (DA) ones [3–7] and non-data aided (NDA) or blind ones [8,9]. Although DA methods can achieve good accuracy, they need to allocate part of the subchannels to transmit pilot symbols, resulting in low bandwidth efficiency. To overcome the drawbacks of DA algorithms, two NDA algorithms for OFDM systems have been developed in [8,9], respectively, which do not require any pilot sequence. Both the methods make use of correlation induced by the cyclic-prefix (CP) of OFDM symbols and must estimate the channel order before obtaining the noise variance and SNR estimates. However, they have some drawbacks in the process to determine the channel order, which is very important for estimating the noise variance and SNR. The method in [8] needs to set a threshold and its performance is very sensitive to this subjective value. The algorithm in [9] estimates the

ARTICLE IN PRESS K. Wang, X. Zhang / Signal Processing 90 (2010) 2766–2772

channel order through maximizing the geometric mean of individual likelihood elements and does not require any threshold. However, it lacks theoretical foundations and there exists a performance gap between this estimator and the estimator with channel order assumed to be known even in the high SNR regime. In this paper, blind noise variance and SNR estimators based on information theoretic criteria are proposed for OFDM systems. By using the redundancy induced by the CP, the order of the unknown fading channel is estimated based on MDL or AIC. And the noise variance and SNR can be estimated subsequently. Simulation results show that the proposed noise variance estimators have similar performance as the estimator with perfect knowledge of channel order for high SNR values, and the proposed SNR estimator outperforms significantly the other existing NDA algorithms in [8,9] when SNR is larger than 0 dB. 2. OFDM signal model Consider a discrete-time OFDM system consisting of N subcarriers. The complex signals of mth block, defined as Xk(m) for k = 0, 1, y, N 1, are modulated onto the subcarriers by inverse discrete Fourier transform (IDFT). We assume that Xk(0),y,Xk(N 1) are independent and identically distributed (i.i.d.) random variables with zero means and variances 1. To avoid the inter-symbol interference (ISI), a CP of length Nc is inserted at the beginning of each OFDM symbol. Then the mth OFDM symbol can be expressed as 1 1 NX xn ðmÞ ¼ pffiffiffiffi Xk ðmÞe j2pðk=NÞðnNc Þ Nk¼0

ð1Þ

for n= 0, 1, y, N+ Nc  1. In this paper, the channel is assumed to be invariant during M OFDM symbols and its impulse response is denoted as h= [h0, h1,y, hL], where L is the channel order and is assumed to be less than Nc. Assuming perfect synchronization, the received sampled signal is given by yn ðmÞ ¼

L X

hl xnl ðmÞUðnlÞ þ

l¼0

þ vn ðmÞ,

L X

hl xNc þ N þ nl ðm1ÞUðln1Þ

l¼1

n ¼ 0,1, . . . ,N þ Nc 1,

ð2Þ

where UðÞ is the step function and vn ðmÞ  CN ð0,s2v Þ is additive white Gaussian noise. And the SNR is defined as PL jh j2 SNR ¼ l ¼ 02 l : ð3Þ

sv

3. Proposed methods 3.1. Noise variance estimation The redundancy induced by the CP is used here to estimate the noise variance. Using the fact that xn(m) =xn + N(m) for n =0,y, Nc  1, it is easy to verify yk(m) = yk + N(m) for k = L,y,Nc  1 in the absence of noise. By defining y~ i ðmÞ ¼ yi þ N ðmÞyi ðmÞ,i ¼ 0,1, . . . ,Nc 1,

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one can obtain from (2) that y~ i ðmÞ ¼

L X

hl ðxN þ il ðmÞxN þ Nc þ il ðm1ÞÞUðli1Þ þ v~ i ðmÞ,

l¼1

0 ri rL1

ð4Þ

and y~ i ðmÞ ¼ v~ i ðmÞ,

Lr i rNc 1,

ð5Þ

where v~ i ðmÞ ¼ vi þ N ðmÞvi ðmÞ is a Gaussian noise with variance s2v~ ¼ 2s2v . From (5), we can see that once the channel order L is estimated, as suggested in [8], the noise variance can be obtained as

s^ 2v ¼

M N c 1 X X

1

^ 2MðNc LÞ m ¼ 1 i ¼ L^

jy~ i ðmÞj2 ,

ð6Þ

where L^ is the estimation of channel order L. It is clear that the estimator based on (6) will obtain the smallest variance when L^ equals L, thus it is very important to develop an accurate estimator of channel order L. In [8], a threshold needs to be set to estimate L, and the performance is very sensitive to this subjective choice. The algorithm in [9] estimates L through maximizing the geometric mean of individual likelihood elements and does not require any threshold. But there exists a performance gap between this noise variance estimator and the one with L assumed to be known even in the high SNR regime. To overcome the limitations of these algorithms, blind estimators based on information theoretic criteria are developed here. ~ Define new vectors yðmÞ ¼ ½y~ 0 ðmÞ,y~ 1 ðmÞ, . . . ,y~ Nc 1 ðmÞT , ~ ~ ¼ ½xNL ðmÞ vðmÞ ¼ ½v~ 0 ðmÞ,v~ 1 ðmÞ, . . . ,v~ Nc 1 ðmÞT and xðmÞ xN þ Nc L ðm1Þ, . . . ,xN1 ðmÞxN þ Nc 1 ðm1ÞT . Then Eq. (4) can be written in matrix form as ~ y~ 1:L ðmÞ ¼ HxðmÞ þ v~ 1:L ðmÞ,

ð7Þ

where a1:L denotes the vector consisting of the first L elements of a, and H is defined as 1 0 hL hL1    h1 C B hL    h2 C B0 C: ð8Þ H¼B B ^ & ^ C A @ 0 0    hL y~ 1:L ðmÞ can be modeled approximately as a complex Gaussian vector when N is large according to the central limit theorem. And its covariance matrix can be shown as n o H ð9Þ Ry~ 1:L ¼ E y~ 1:L ðmÞy~ 1:L ðmÞ ¼ HRx~ HH þ s2v~ IL , n o H ~ where Rx~ ¼ E xðmÞ x~ ðmÞ is the covariance matrix of ~ xðmÞ, which can be proven to be non-singular. Therefore, ~ the probability density function (PDF) of yðmÞ can be approximated by n o 1 H ~ ~ exp y~ 1:L ðmÞR1 f ðyðmÞÞ ¼ L y~ 1:L y 1:L ðmÞ p jRy~ 1:L j ( ) NY c 1 1 jy~ i ðmÞj2  : ð10Þ exp 2 2 i¼L

psv~

sv~

In practice, the channel order L, Ry~ 1:L and s2v~ are all unknown. Fortunately, the information theoretic criteria,

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such as MDL and AIC, provide an effective way to estimate L. MDL and AIC have been well investigated and proven to be very effective in various model selection problems. They do not need any information about the noise power or set any threshold. In these methods, different hypotheses on the unknown parameter, such as the number of signal sources or the channel order, are constructed. By choosing the variable that minimizes the objective functions under different hypotheses, the unknown parameter can be well estimated. Since L is unknown, there are Nc hypotheses. The jth hypothesis is given by Hj : L ¼ j,

j ¼ 0,1, . . . ,Nc 1:

ð11Þ

Under the jth hypothesis, the objective functions of the MDL and AIC are, respectively, defined as [10,11] ^ Þ þ 0:5klogðMÞ, MDLðjÞ ¼ logf ðYjH j

ð12Þ

^ Þ þ2k, AICðjÞ ¼ 2logf ðYjH j

ð13Þ

M Y

1

m¼1

0

j jR

p

y~ 1:j j

n o H ~ exp y~ 1:j ðmÞR1 y~ 1:j y 1:j ðmÞ

( )1 M NY c 1 Y ~ i ðmÞj2 1 j y A @ exp 2 2 m¼1 i¼j

psv~

sv~

logf ðYjRy~ 1:j ,s2v~ Þ ¼ MtrðR1 y~ 1:j R j ÞMlogjR y~ 1:j j 

m¼1 i¼j

jy~ i ðmÞj2

s2v~

ð14Þ

ð15Þ

M 1 X H y~ ðmÞy~ 1:j ðmÞ: M m ¼ 1 1:j

ð16Þ

The derivative of logf ðYjRy~ 1:j ,s2v~ Þ with respect to Ry~ 1:j is [12]     @logf YjRy~ 1:j ,s2v~ 1 1 ð17Þ ¼ M R1 y~ 1:j R j Ry~ 1:j Ry~ 1:j : @Ry~ 1:j By setting Eq. (17) equal to zero, we can obtain the ML estimate of Ry~ 1:j as R^ y~ 1:j ¼ Rj :

ð18Þ 2 v~

Similarly, the ML estimate of s can be expressed as

s^ 2v~ ¼

c 1 1 NX r, ðNc jÞ i ¼ j i

L^ ¼ argmin MDLðjÞ,

j 2 f0,1, . . . ,Nc 1g

ð22Þ

j 2 f0,1, . . . ,Nc 1g:

ð23Þ

j

or L^ ¼ argmin AICðjÞ,

Once L^ is obtained, the noise variance can be estimated according to (6). It seems complex to compute jRj j, j = 1,y,Nc  1, required by our algorithm. To reduce the computational cost, here we give an iterative algorithm to compute them. The sample covariance matrix of y~ 1:Nc 1 ðmÞ ¼ ½y~ 0 ðmÞ, . . . ,y~ Nc 2 ðmÞT is given by RNc 1 ¼

M 1 X H ðmÞy~ 1:Nc 1 ðmÞ: y~ M m ¼ 1 1:Nc 1

j ¼ 1,2, . . . ,Nc 1,

ð24Þ

ð19Þ

P 2 ~ where ri ¼ M m ¼ 1 jy i ðmÞj =M, i ¼ 0, . . . ,Nc 1, is the (i+ 1)th diagonal entry of the matrix RNc .

ð25Þ

where A1:j denotes the submatrix consisting of the first j rows and the first j columns of A. Since RNc 1 is symmetric positive definite, there exists a lower triangular matrix G 2 CðNc 1ÞðNc 1Þ with positive diagonal entries such that [13] RNc 1 ¼ GGH :

!

þ logðs2v~ Þ ,

ð21Þ

The channel order is determined by choosing the j that minimizes MDL(j) or AIC(j), i.e.,

Then Rj can be expressed as

where tr(A) denotes the trace of the matrix A and Rj is defined as Rj ¼

  2 AICðjÞ ¼ 2Mlog jRj jðs^ v~ ÞNc j þ 2j2 :

Rj ¼ ½RNc 1 1:j ,

and the log-likelihood function is given by

M N c 1 X X

and

j

^ is the maximum likelihood estimate of H , which where H j j includes all the unknown parameters under the jth hypothesis, k is the number of free adjusted parameters ~ ~ ~ yð2Þ, . . . ,yðMÞ is the matrix consisting in Hj , and Y ¼ ½yð1Þ, of all the observations. From Eq. (10), the joint PDF of Y under the jth hypothesis is formulated as f ðYjRy~ 1:j ,s2v,j ~ Þ¼

Recall that the unknown parameters under the jth hypothesis are Ry~ 1:j and s2v~ . Since Ry~ 1:j is a complex symmetric matrix, so there are j2 free parameters in it. And the total number of free parameters under jth hypothesis is k= j2 +1. By substituting (15)–(19) into (12) and (13) and omitting the items without j, we can obtain   2 ð20Þ MDLðjÞ ¼ Mlog jRj jðs^ v~ ÞNc j þ 0:5j2 logðMÞ

ð26Þ

This decomposition is known as the Cholesky decomposition. From Eqs. (25) and (26) and using the fact that G is lower triangular, one can obtain that Rj ¼ G1:j GH 1:j

ð27Þ

and jRj j ¼ jG1:j j2 ¼

j Y

gi2 ,

j ¼ 1,2, . . . ,Nc 1,

ð28Þ

i¼1

where gi is the ith diagonal entry of G. Based on Eq. (28), jRj j, j=1,y,Nc  1, can be computed iteratively as follows: jR0 j ¼ 1, jRj j ¼ jRj1 jgj2 ,

j ¼ 1, . . . ,Nc 1:

ð29Þ

The steps of the noise variance estimator based on MDL are listed below. 1. Compute the sample covariance matrix RNc 1 according to (24), then rj, j = 0,y, Nc  2, is the (j+ 1)th diagonal entry of RNc 1 , and compute rNc 1 . 2. Obtain the Cholesky decomposition of RNc 1 according to (26).

ARTICLE IN PRESS K. Wang, X. Zhang / Signal Processing 90 (2010) 2766–2772

3. Compute the MDL(j), j = 0,y,Nc  1, using the following algorithm.   N c 1 X r r0 ¼ ri , jR0 j ¼ 1, MDLð0Þ ¼ MN c log 0 Nc i¼0

In the above discussion, we did not consider the computational cost of Step 3 for the following two reasons: (a) It is much lower than the computational complexity of Steps 1 and 2. (b) The algorithms in [8,9] also have a similar step to choose the channel order L, and the complexities of all the three algorithms are comparable in this step.

for j= 1,y,Nc  1

rj ¼ rj1 rj1 , jRj j ¼ jRj1 jgj2 ,   rj MDLðjÞ ¼ MlogðjRj jÞ þ MðNc jÞlog þ 0:5j2 logðMÞ Nc j

3.2. SNR estimation It is easy to verify from (2) that Eðjyn ðmÞj2 Þ ¼ PL 2 2 2 l ¼ 0 jhl j þ sv , therefore, the signal power S ¼ l ¼ 0 jhl j can be estimated by

end 4. Obtain L^ ¼ argminj MDLðjÞ, and the noise variance is 2 ^ estimated by s^ v ¼ rL^ =2ðNc LÞ.

PL

S^ ¼

The steps of the algorithm based on AIC are similar to those of MDL and we omit them here. The major complexity of the noise variance estimator based on MDL or AIC comes from two parts: computation of the covariance matrix (Eq. (24)) and its Cholesky decomposition (Eq. (26)). For the first part, noting that RNc 1 is complex symmetric, Nc(Nc 1)M/2 multiplications (we only consider multiplications involved in the different algorithms) are needed. For the second part, (Nc  1)3/6 multiplications are sufficient [13]. The total complexity is therefore as follows: Nc ðNc 1ÞM=2 þ ðNc 1Þ3 =6:

^ ^ ¼ S : SNR s^ 2v

ð32Þ

Cui et al. α = 0.01 Cui et al. α = 0.05 Cui et al. α = 0.1 Socheleau et al. MDL AIC

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −10

0

10

ð31Þ

It is worthy to note that all the samples in the observation window are used to estimate S in our method while only a part of samples are used in [8]. Because the proposed SNR estimator uses more information to estimate S and the noise variance estimation obtained by MDL or AIC is very accurate, it outperforms the estimator in [8] in all SNR regions. The SNR estimator in [9] exploits the cyclostationary statistics induced by the CP to estimate the signal power. It can achieve good accuracy in the negative SNR regime since the cyclostationary statistics based algorithms are usually very effective when the noise power is larger than the signal power. However, when SNR is high, the noise power is negligible compared to the signal power, and the proposed estimator based on (32) is very accurate and outperforms the one in [9].

The major complexities of the estimators in [8,9] are about 3NcM and NcM, respectively. Hence, when M is larger than Nc, the complexity of the proposed noise variance estimator is less than 2Nc/9 times that of the method in [8] and 2Nc/3 times that of [9]. The computational complexity based on Eq. (30) is usually acceptable in practice, and the proposed method can be a candidate for blind noise variance estimation for OFDM systems.

Probability of Correct Detction

Nc 1 M Nþ X X 1 2 jyn ðmÞj2 s^ v MðN þ Nc Þ m ¼ 1 n ¼ 0

and the SNR is given by

ð30Þ

1

2769

20

30

40

50

SNR (dB) Fig. 1. The probability of correct detection of the channel order versus SNR, M =24.

60

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to be 0.01, 0.05 and 0.1, respectively. And all the results are averaged over 10 000 independent runs. Fig. 1 shows the probability of correct detection of the channel order using the proposed algorithms and the algorithms in [8,9] with M= 24. It can be seen that the performance of the method in [8] is very sensitive to the threshold a, which is difficult to set a prior since we have no knowledge of the channel. The proposed estimators based on MDL and AIC outperform the other two methods in the high SNR regime (20–60 dB for MDL and 10–60 dB for AIC), and the detection probability of the

4. Simulation results Simulation results are presented to evaluate the performance of the proposed estimators. An OFDM system with QPSK constellation is considered. The number of subcarriers is N = 64 and the CP length is Nc = 16. We consider a nine-tap Rayleigh fading channel with the exponentially decaying powers set as Eðjhl j2 Þ ¼ P el=3 =ð 8l ¼ 0 el=3 Þ,l ¼ 0,1, . . . ,8, so the channel order is L= 8. Here we compare our algorithms with the algorithms in [8,9]. The threshold required in [8], denoted as a, is set

−1

10

NMSE

Cui et al. α = 0.01 Cui et al. α = 0.05 Cui et al. α = 0.1 Socheleau et al. MDL AIC Perfect knowledge of L −2

10

−3

10 −10

0

10

20

30

40

50

60

SNR (dB) Fig. 2. The NMSE of the noise variance estimates versus SNR, M= 24.

−1

10

NMSE

Socheleau et al. MDL AIC Perfect knowledge of L

M=24

−2

10

M=48

M=96

−3

10 −10

0

10

20

30

40

50

SNR (dB) Fig. 3. The NMSE of the noise variance estimates versus SNR for various M.

60

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MDL-based estimator equals 1 when SNR is larger than 40 dB. Fig. 2 depicts the normalized mean square error (NMSE) of different noise variance estimators with M= 24, where 2 NMSE is defined as E½ðs^ v s2v Þ2 =s4v . The NMSE of the estimator based on Eq. (6) with L assumed to be known is also presented for comparison. We observe that the performance of the estimator in [8] degrades rapidly when SNR is high, which is because L is frequently under-estimated and a part of signals are also used to estimate the noise variance. Although under-estimation is

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greatly harmful for high SNR, it is beneficial for very low SNR ð r5 dBÞ since the noise power is predominant and more data are used to estimate the noise power [9]. As mentioned previously, there exists a performance gap between the estimator in [9] and the one with L assumed to be perfectly known even for high SNR. We can see that the proposed estimator based on MDL or AIC has almost the same performance as the one with perfect knowledge of L when SNR is larger than 25 dB, although the NMSE of the method in [9] is lower than that of MDL for 0 r SNR o 20 dB and slightly lower than AIC for 0 rSNR o10 dB.

1

10

Cui et al. α=0.01 Cui et al. α=0.05 Cui et al. α=0.1 Socheleau et al. Proposed

0

10

−1

NMSE

10

−2

10

−3

10

−4

10 −10

0

10

20

30

40

50

60

50

60

SNR (dB) Fig. 4. The NMSE of the signal power estimates versus SNR, M =24.

1

10

Cui et al. α=0.01 Cui et al. α=0.05 Cui et al. α=0.1 Socheleau et al Proposed Perfect knowledge of L and S

0

NMSE

10

−1

10

−2

10

−3

10 −10

0

10

20

30

40

SNR (dB) Fig. 5. The NMSE of the SNR estimates versus SNR, M = 24.

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Fig. 3 compares the performances of the proposed noise variance estimators and the one in [9] for various M. It can be seen that the performance is significantly improved as M increases, and the proposed estimator based on AIC outperforms the one in [9] for all SNR values when M= 96. Fig. 4 illustrates the NMSE of different signal power estimates versus SNR for M= 24. The proposed signal power estimator based on (31) exploits AIC to estimate the noise variance. We can see that the NMSEs of the other algorithms are much higher than ours when SNR Z 10 dB, and the NMSE of the algorithm in [9] is about two order of magnitude larger than that of our method for high SNR values, although it is slightly lower in negative SNR regime. This demonstrates that the estimator in [9], which exploits the cyclostationary statistics of the CP, is only effective in low SNR regime. The performances of different SNR estimators are compared in Fig. 5. The NMSE of the estimator with perfect knowledge of the channel order L and signal power S is also plotted as a lower bound for SNR estimation. We can see that the proposed method outperforms the other two methods significantly when SNR Z 5 dB, and our method has the similar performance as the estimator with perfect knowledge of L and S when SNR Z15 dB. This is mainly because the noise variance and the signal power can be well estimated using the proposed algorithms. 5. Conclusion In this paper, blind noise variance and SNR estimators based on information theoretic criteria have been proposed for OFDM systems. By using the redundancy induced by the CP, the estimation of channel order is successfully transformed into a problem of model selection, which can be effectively solved by MDL or AIC algorithm. Once the

channel order is estimated, the noise variance and SNR can be obtained subsequently. The proposed methods are blind since they do not need to transmit any pilot sequence or set any threshold. Simulation results have demonstrated the efficiency of the proposed algorithms. References [1] R. Mo, Y.H. Chew, T.T. Tjhung, New blind frequency offset estimator for OFDM systems over frequency selective fading channels, Signal Processing 87 (1) (2007) 148–161. [2] Z. Shen, J.G. Andrews, B.L. Evans, Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints, IEEE Trans. Wireless Commun. 4 (6) (November 2005) 2726–2737. [3] S. He, M. Torkelson, Effective SNR estimation in OFDM system simulation, in: IEEE GLOBECOM, 1998, pp. 945–950. [4] S. Boumard, Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems, in: IEEE GLOBECOM, 2003, pp. 1330– 1334. [5] H. Xu, G. Wei, J. Zhu, A novel SNR estimation algorithm for OFDM, in: IEEE VTC, 2005. [6] X.D. Xu, J. Yang, X.H. You, J.H. Zhao, A novel RMS delay spread estimation for wireless OFDM systems, IEICE Trans. Electron. Commun. 89 (10) (2006) 2558–2565. [7] Y. Wang, L.H. Li, P. Zhang, A new noise variance estimation algorithm for multiuser OFDM systems, in: IEEE PIMRC, 2007, pp. 1–4. [8] T. Cui, C. Tellambura, Power delay profile and noise variance estimation for OFDM, IEEE Commun. Lett. 12 (11) (January 2006) 25–27. [9] F.X. Socheleau, S. Houcke, Non data-aided SNR estimation of OFDM signals, IEEE Commun. Lett. 12 (11) (November 2008) 813–815. [10] M. Wax, T. Kailath, Detection of signals by information theoretic criteria, IEEE Trans. Acoust. Speech Signal Process. 33 (2) (April 1985) 387–392. [11] E. Fishler, M. Grosmann, H. Messer, Detection of signals by information theoretic criteria: general asymptotic performance analysis, IEEE Trans. Signal Process. 50 (5) (May 2002) 1027–1036. [12] J.R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, third ed., Wiley Series in Probability and Statistics, Chichester, 2007. [13] G.H. Golub, C.F. Van Loan, Matrix Computations, third ed., Johns Hopkins University Press, Baltimore, 1996.