On the Performance of Generalized Likelihood Ratio Test for Data-Aided Timing Synchronization of MIMO Systems Yi Zhou1 , Erchin Serpedin1 (Corresponding author)∗ , Khalid Qaraqe2 and Octavia Dobre3 1 Texas A&M University, Dept. of Electrical Engineering, College Station, Texas, USA 2 Texas A&M University at Qatar, Electrical Engineering Program, Doha, Qatar 3 Memorial University, Faculty of Engineering and Applied Science, Canada ∗ Corresponding author (E-mail:
[email protected])
hypothesis over an uncountable set of hypotheses simplifies to a detection or decision problem over a finite set of hypotheses. Due to the presence of nuisance parameters in the a priori conditional probability density functions, the simplified dataaided timing synchronization as a detection problem is a composite hypotheses test [2], in which an optimal test may not exist. It is believed that GLRT is asymptotically optimal in the situation where no uniformly most powerful (UMP) test exists [3]. Adopting the GLRT approach, the performance of the timing detector can be analyzed in an asymptotic sense and performance benchmarks could be established. This paper studies the asymptotic performance of GLRTbased timing synchronizers for MIMO systems in the presence of flat-fading channels. An upper bound for the detection probability is provided and shown to behave well as a performance benchmark for sufficiently large number of observations. Computer simulations are conducted to illustrate the influence of various design parameters on the performance of GLRTbased timing synchronizers. The rest of the paper is organized as follows. Section II introduces the time detection statistic for MIMO systems in frequency-flat channels. The asymptotic performance of this test statistic is evaluated and compared for various MIMO configurations. An upper bound for detection probability in the presence of a large number of observations is also derived. In Section III, the asymptotic performance of the test is analyzed analytically and via computer simulations.
Abstract—This paper studies the performance of the generalized likelihood ratio test (GLRT) proposed in [1] in the context of data-aided timing synchronization for multiple-input multiple-output (MIMO) communications systems and flat-fading channels. Herein paper, the asymptotic performance of the GLRT test is derived, and an upper bound for the detection probability is provided and shown to behave well as a benchmark for sufficiently large number of observations. Computer simulations are presented to corroborate the developed analytical performance benchmarks. In addition, choice of some system design parameters to improve the synchronization performance is discussed via computer simulations. Key words - GLRT; Hypothesis Testing; MIMO; Timing Synchronization.
I. I NTRODUCTION This paper studies the performance of the generalized likelihood ratio test (GLRT) based timing synchronization schemes in multiple-input multiple-output (MIMO) systems. The usage of GLRT for timing synchronization of MIMO systems appears to be reported for the first time in [1]. Because this paper exploits the preliminary results reported in [1], we will adopt throughout the paper the same assumptions, modeling framework and notations as in [1]. Therefore, herein paper it is assumed that the carrier frequency offset is small enough or is corrected separately. The interest in GLRTbased synchronization algorithms stems from their superior performance relative to other schemes in the presence of noise and interference as was mentioned in [1]. In digital receivers, as long as certain conditions are met [5], with proper methods the precision of timing synchronization can be improved linearly by increasing the sampling rate. However, due to the implementation complexity constraint, in practical receivers, the accuracy of timing synchronization is required only within a fraction of a sample period. Therefore, it is sufficient to consider the potential timing offset in a discrete set and naturally treat the timing synchronization problem as a multiple statistical hypotheses test [1], [2]. At each potential timing offset, a test statistic is evaluated given the observed data. Synchronization is declared if the test statistic threshold exceeds a pre-chosen threshold [1]. Hence, the original timing estimation problem as choosing a
,(((
II. GLRT P ERFORMANCE FOR MIMO C HANNELS Herein we adopt a frequency-flat MIMO signaling model as in [1], i.e., the coherence bandwidth of the channel is much larger than the signal bandwidth. Therefore, all frequency components of the signal experience the same magnitude of fading [4] and correspondingly in the time domain the multipath propagation cannot be resolved at the receiver. Assuming 𝑛𝑡 transmit antennas, 𝑛𝑟 receive antennas and 𝑛𝑠 complex baseband samples, the channel is modeled as [1]: Z = HS + N ,
(1)
where Z ∈ ℂ𝑛𝑟 ×𝑛𝑠 is the sampled received signal matrix, with each row containing the 𝑛𝑠 samples received from one
ˆ 1 are the unrestricted maximum-likelihood esˆ 1, Σ where H ˆ 0 is timates of H and Σ, respectively, under ℋ1 , and Σ the restricted maximum-likelihood estimate under ℋ0 when H = 0. 𝛾 is a threshold depending on the false alarm rate [2]. In [1], the GLRT statistic was expressed as:
of the 𝑛𝑟 receive antennas. Matrix H ∈ ℂ𝑛𝑟 ×𝑛𝑡 stands for the flat fading channel transfer matrix, S ∈ ℂ𝑛𝑡 ×𝑛𝑠 denotes the transmitted signal matrix, and N ∈ ℂ𝑛𝑟 ×𝑛𝑠 represents the additive noise samples. The noise at each receive antenna is modeled as a zero mean Gaussian random variable with variance 𝜎𝑛2 . The channel is temporally quasi-static, i.e., it can be regarded as constant during 𝑛𝑠 sampling periods. The transmitted and received complex baseband signal samples at some delay 𝜏 are defined as in [1]:
𝐿𝐺 (𝜏 ) = ∣I𝑛𝑠 − PS PZ𝜏 ∣−𝑛𝑠 ,
where PS = S† (SS† )−1 S and PZ = Z† (ZZ† )−1 Z. To investigate the performance of the maximum-likelihood (ML) estimator of the channel matrix H, we will vectorize the channel matrix H, by stacking its columns into a vector. The probability density function (pdf) of the complex matrix N can be represented by } { exp −tr{Ω−1 (N − M)† Σ−1 (N − M)} , 𝑝(N∣Ω, Σ) = 𝜋 𝑛𝑟 𝑛𝑠 ∣Ω∣𝑛𝑟 ∣Σ∣𝑛𝑠 (4) where Ω = E{N† N}/𝑛𝑟 ∈ ℂ𝑛𝑠 ×𝑛𝑠 , Σ = E{NN† }/𝑛𝑠 ∈ ℂ𝑛𝑟 ×𝑛𝑟 and M ∈ ℂ𝑛𝑟 ×𝑛𝑠 denotes the mean matrix. An equivalent definition involving the Kronecker product ⊗ and the vectorization operator vec(⋅) assumes the form: N ∼ 𝒞𝒩 𝑛𝑟 ,𝑛𝑠 (M, Ω, Σ) if vec(N) ∼ 𝒞𝒩 𝑛𝑟 𝑛𝑠 (vec(M), Ω ⊗ Σ). Assuming the noise samples are statistically independent at different sampling time instants, i.e., the column vectors in matrix N are independent, it turns out that Ω = I𝑛𝑠 . Therefore, the pdf simplifies to } { exp −tr{N† Σ−1 N} , (5) 𝑝(N∣Σ) = 𝜋 𝑛𝑟 𝑛𝑠 ∣Σ∣𝑛𝑠
S = [s(𝑇𝑠 ) s(2𝑇𝑠 ) ⋅ ⋅ ⋅ s(𝑛𝑠 𝑇𝑠 )], Z𝜏 = [z(𝑇𝑠 − 𝜏 ) z(2𝑇𝑠 − 𝜏 ) ⋅ ⋅ ⋅ z(𝑛𝑠 𝑇𝑠 − 𝜏 )], where 𝑇𝑠 is the sampling period, and s(𝑡) and z(𝑡) are the continuous transmitted and received vectors as a function of time 𝑡. In the case of a flat fading channel, the sampling period is much greater than the multipath delay spread, and a single channel filter tap is sufficient to represent the channel. Therefore, the MIMO channel matrix at a relative delay 𝜏 , H𝜏 , will be assumed of the form [1] { H, 𝜏 = 𝜏0 H𝜏 = 0, otherwise where the correct delay in terms of receiver’s clock is 𝜏0 , and 0 denotes the null matrix. This represents the key assumption under which the results of this paper will be built upon. Extensions of this channel modeling framework to other channel modeling set-ups such as frequency selective (MIMO-OFDM) or assuming the presence of other modeling factors are beyond the scope of this paper, and will considered somewhere else. GLRT is a likelihood ratio test for composite hypotheses in which the parameters of the probability density function are unknown a priori. The principle is straightforward [3]: it consists of finding the maximum-likelihood estimate of the unknown parameters under each hypothesis, and then plugging the estimate in the probability distribution of the corresponding hypothesis and treating the detection problem as if the estimated values were correct. This common sense test yields good results in general. In our timing synchronization problem, the null hypothesis is that the synchronization (or pilot) signal is absent or misaligned, and the alternative is that the synchronization (or pilot) signal is present and aligned correctly in time. Hence, the parameter test in a formal statistics convention is
and equivalently vec(N) ∼ 𝒞𝒩 𝑛𝑟 𝑛𝑠 (0, I𝑛𝑠 ⊗ Σ). Let z = vec(Z), h = vec(H), and n = vec(N). Taking into account: vec(IAB) = (B𝑇 ⊗ I)vec(A)
z = (S𝑇 ⊗ I𝑛𝑟 )h + n.
(7)
Using the equivalent vector model of the matrix Gaussian distribution, the pdf when the synchronization signal is present and properly aligned can be represented by
{
𝑇
†
−1
exp −[z − (S ⊗ I𝑛𝑟 )h] C 𝜋 𝑛𝑟 𝑛𝑠 ∣C∣
𝑝(z∣S; h, C) = } [z − (S𝑇 ⊗ I𝑛𝑟 )h]
(8)
where C = I𝑛𝑠 ⊗ Σ. Taking the complex conjugate gradient of the log-pdf with respect to h, and setting it to zero, it turns out that
The parameter matrix Σ is the received signal spatial covariance matrix and is a set of nuisance parameters, which are unknown but the same under either hypothesis. As one can find, the above hypothesis test is two-sided. It has been proved that there is no uniformly most powerful (UMP) test in a twosided test [6]. However, it can be shown that the GLRT is UMP among all tests that are invariant [7]. The GLRT for this problem is to decide ℋ1 if ˆ 1, Σ ˆ 1) 𝑝(Z𝜏 ∣S; H > 𝛾, ˆ 0) 𝑝(Z𝜏 ∣H0 = 0, Σ
(6)
and recalling Eq. (1), it follows that:
ℋ0 : H𝑛𝑟 ×𝑛𝑡 = 0, Σ ℋ1 : H𝑛𝑟 ×𝑛𝑡 ∕= 0, Σ.
𝐿𝐺 (𝜏 ) =
(3)
= =
∂ ln 𝑝(z∣S; h, C) ∂h∗ ∂ − ∗ [z − (S𝑇 ⊗ I𝑛𝑟 )h]† C−1 [z − (S𝑇 ⊗ I𝑛𝑟 )h] ∂h (S𝑇 ⊗ I𝑛𝑟 )† C−1 [z − (S𝑇 ⊗ I𝑛𝑟 )h] = 0.
Hence, the maximum-likelihood estimator of h is (2)
ˆ = [(S𝑇 ⊗ I𝑛 )† C−1 (S𝑇 ⊗ I𝑛 )]−1 (S𝑇 ⊗ I𝑛 )† C−1 z. h 𝑟 𝑟 𝑟
Recalling C = I𝑛𝑠 ⊗ Σ, and Kronecker product properties: (A ⊗ B)† = A† ⊗ B† , (A ⊗ B)−1 = A−1 ⊗ B−1 , and (A ⊗ B)(X ⊗ Y) = (AX) ⊗ (BY), it follows that
is available, one can approximate the GLRT statistic with chi-squared random variables under either hypothesis ℋ0 or ℋ1 . Since the asymptotic pdf under ℋ0 does not depend on ˆ = [(S𝑇 ⊗ I𝑛 )† (S𝑇 ⊗ I𝑛 )]−1 (S𝑇 ⊗ I𝑛 )† (I𝑛 ⊗ Σ)−1 z any unknown parameters, the threshold required to maintain a h 𝑟 𝑟 𝑠 constant false alarm rate (CFAR) can be found, i.e, the CFAR )𝑟 ( = [S† (SS† )−1 ]𝑇 ⊗ I𝑛𝑟 z. detector exists [2]. However, since the nuisance parameter is present in the model, we can provide only an upper bound Recalling (6), one can convert the estimator from the vector for the detection probability or equivalently a lower bound † † −1 ˆ space to the matrix space: H = ZS (SS ) , which is the for the missing rate. And since the GLRT is considered same as the initial state-space model. Thus, the equivalence asymptotically optimal in the situation where no uniformly between the vector space model and matrix space model most powerful (UMP) test exists [3], this asymptotic bound has been set up. The unbiasedness of the ML estimator can also serve as a benchmark when comparing various tests follows immediately. The Fisher information matrix takes the developed through different approaches. The bound can be expression: obtained as follows: } { † I. For any given false alarm rate 𝑃𝐹 𝐴 , determine the corre∂ ln 𝑝(z∣S; h, C) ∂ ln 𝑝(z∣S; h, C) ∫∞ I(h) = E sponding threshold 𝑇 such that 𝑇 𝑝1 (𝑥)𝑑𝑥 = 𝑃𝐹 𝐴 , where ∂h∗ ∂h∗ =
(S𝑇 ⊗ I𝑛𝑟 )† C−1 (S𝑇 ⊗ I𝑛𝑟 )
=
(S𝑇 ⊗ I𝑛𝑟 )† (I𝑛𝑠 ⊗ Σ)−1 (S𝑇 ⊗ I𝑛𝑟 ) (SS† )𝑇 ⊗ Σ−1 .
=
𝑟 𝑥 𝑟 𝑟 𝑝1 (𝑥) = 𝑥 2 −1 exp(− )2 2 Γ( )𝑢(𝑥), 2 2 is the central chi-squared pdf with 𝑟 = 2𝑛𝑟 𝑛𝑡 degrees of freedom, 𝑢(𝑥) denotes the∫ unit-step signal, and Γ(𝑢) is the ∞ Gamma function: Γ(𝑢) = 0 𝑡𝑢−1 exp(−𝑡)𝑑𝑡. II. For the given SNR, obtain the noncentrality parameter 𝜆 using (12) or (11). An upper bound for detection probability in the asymptotic case, 𝑃𝐷,𝑎 , which is the detection probability ∫∞ can be computed as 𝑃𝐷,𝑎 = 𝑇 𝑝2 (𝑥)𝑑𝑥, where ( 𝜆𝑥 )𝑘 ∞ 𝑟 𝑥+𝜆 𝑟 ∑ −1 4 2 2 )2 𝑢(𝑥), exp(− 𝑝2 (𝑥) = 𝑥 2 𝑘!Γ( 2𝑟 + 𝑘)
(9)
As 𝑛𝑠 → ∞, the modified GLRT statistic 2 ln 𝐿𝐺 for complex parameters admits the pdf [2] { 𝜒22𝑛𝑟 𝑛𝑡 under ℋ0 𝑎 2 ln 𝐿𝐺 (𝜏 ) ∼ ′2 𝜒 2𝑛𝑟 𝑛𝑡 (𝜆) under ℋ1 where “𝑎" denotes an asymptotic pdf, 𝜒2𝑟 denotes a central 2 chi-squared pdf with 𝑟 degrees of freedom, and 𝜒′ 𝑟 denotes a noncentral chi-squared pdf with 𝑟 degrees of freedom and noncentrality parameter 𝜆. The noncentrality parameter is given by 𝜆 = 2(h1 − h0 )† I(h0 , Σ)(h1 − h0 ),
𝑘=0
stands for the noncentral chi-squared pdf with 𝑟 = 2𝑛𝑟 𝑛𝑡 degrees of freedom and noncentrality parameter 𝜆. Equivalently, a lower bound of the missing rate is 𝑃𝑚𝑖𝑠𝑠,𝑙𝑏 = 1 − 𝑃𝐷,𝑎 .
(10)
where h1 and Σ are the parameters’ true values under ℋ1 . Note that Eq. (10) holds for the case without nuisance parameters. When nuisance parameters are present, the noncentrality parameter 𝜆 is decreased and the chi-squared pdf is more concentrated to the left (positive skew) for the same degrees of freedom as one can find by plotting the noncentral chi-squared pdf. Hence with the same threshold, the detection probability is decreased. Intuitively, this is the price paid for having to estimate extra parameters for use in the detector. Considering the EVDs: SS† = UΛU† and Σ = VΓV† , with U and V unitary matrices, it follows that: [ ] 𝜆 = 2h† (SS† )𝑇 ⊗ Σ−1 h { } = 2tr Σ−1 HSS† H† . (11)
III. S IMULATION R ESULTS In the following, the performances of the GLRT statistic developed for flat fading channels are shown through computer simulations. The performances are illustrated in ROC curves. The probabilities on axes are displayed for potential correct or incorrect temporal alignment tests. The probability of false alarm measures the fraction of false alarms given the synchronization sequence is absent or misaligned. The probability of a missing is the rate of omission an event when the synchronization sequence is correctly aligned in time. For a MIMO wireless communication link with four transmit antennas and four receive antennas, four different synchronization sequences, each of 𝑛𝑠 symbols, are transmitted in parallel. These sequences are constructed randomly from a quadrature phase-shift-keying (QPSK) constellation. For each synchronization test, the receiver collects 𝑛𝑠 received vector samples from the four antennas. The SNR is defined in Eq. (13). The channel assumes Rayleigh frequency-flat fading. The elements in the MIMO channel matrix H are sampled from circular complex Gaussian distribution with zero mean and unit variance. Fig. 1 shows the ROC curves for a 4-by-4 MIMO link with various SNRs. There are four synchronization sequences of length 16. The SNRs investigated are 0, -1, -2, and -3
Assuming the noise is spatially uncorrelated, i.e., Σ = 𝜎𝑛2 I𝑛𝑟 , one obtains 𝜆=
2∥HS∥2𝐹 = 2𝑛𝑟 𝑛𝑠 SNR, 𝜎𝑛2
(12)
where the signal-to-noise ratio SNR is defined as SNR = ∥HS∥2𝐹 /(𝜎𝑛2 𝑛𝑟 𝑛𝑠 ).
(13)
Assuming knowledge of the true values for the MIMO channel H (or equivalently h) and the covariance matrix Σ
n =1,n =1,n =32
dB. The proposed GLRT works well, e.g., with SNR=0dB, for 𝑃𝐹 𝐴 = 1%, the missing rate is 6 × 10−4 . As the SNR decreases, the missing rate increases (the detection probability decreases), which follows the intuition.
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Figure 3. Performance of the detector for a SISO link in Rayleigh flat fading environment with 𝑛𝑠 = 32. −4
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fact that the asymptotic bound serves a good benchmark for the case of more than 64 observations. Extensions of this work to MIMO-OFDM systems [8], [9] or other communications systems [10], [11], [12] represent interesting and challenging research problems.
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FA
Figure 1. Comparison of ROCs for 4 × 4 MIMO link in Rayleigh flat fading environment with different SNRs.
Fig. 2 show the asymptotic behavior for a 4-by-4 MIMO link with the synchronization sequence length equal to 64. As the synchronization length increases, the lower bound for missing rate gets tighter. Although the asymptotic bound theoretically requires an infinite number of observations, it still serves as a good lower bound with sufficiently large observation window, such as 𝑛𝑠 = 64. Fig. 3 corroborates this behavior in the SISO case.
ACKNOWLEDGEMENTS This work was made possible by the support offered by NPRP grants No. 09-341-2128 and No. 08-101-2-025 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. R EFERENCES
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[1] D.W. Bliss and P.A. Parker, Temporal synchronization of mimo wireless communication in the presence of interference, IEEE Transactions on Signal Processing, vol. 58, no. 3, Mar. 2010. [2] S. Kay, Fundamentals of Statistical Signal Processing, Volume II: Detection Theory, Prentice Hall, New Jersey, 1998. [3] Bernard C. Levy, Principles of Signal Detection and Parameter Estimation, Springer, New York, 2008. [4] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Prentice Hall, Jan. 2002. [5] H. Meyr, M. Oerder, and A. Polydoros, On Sampling Rate, Analog Prefiltering, and Sufficient Statistics for Digital Receivers, IEEE Transactions on Communications, vol. 42, no. 12, 1994. [6] Sir M. Kendall and A. Stuart, The Advanced Theory of Statistics, vol. 2, Macmillan, New York, 1979. [7] E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, Springer, New York, 3rd edition, 2005. [8] H.-G. Jeon and E. Serpedin, A Novel Simplified Channel Tracking Method for MIMO-OFDM Systems with Null Sub-carriers, Signal Processing, Elsevier, Volume 88, Issue 4, April 2008. [9] H.-G. Jeon and E. Serpedin, Walsh Coded Training Signal Aided Time Domain Channel Estimation for MIMO-OFDM Systems, IEEE Trans. on Communications, vol. 56, no. 9, Sept. 2008. [10] X. Li, Y.-C. Wu and E. Serpedin, Timing Synchronization in DecodeandForward Cooperative Communication Systems, IEEE Trans. on Signal Processing, Volume 57, no. 4, April 2009. [11] E. Serpedin, G. B. Giannakis, A. Chevreuil, and P. Loubaton, Blind Joint Estimation of Carrier Frequency Offset and Channel Using NonRedundant Periodic Modulation Precoders, Proceedings of the Ninth IEEE Statistical Signal and Array Processing Workshop 1998, Portland, OR, Sept. 1998. [12] T. Fischer, B. Kelleci, K. Shi, E. Serpedin, and A. Karsilayan, An Analog Approach to Suppressing In-Band Narrowband Interference in UWB Receivers, IEEE Trans. On Circuits and Systems-Part I, Volume 54, Issue 5, May 2007.
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Figure 2. Performance of the detector for a 4 × 4 MIMO system in Rayleigh flat fading environment with 𝑛𝑠 = 64.
IV. C ONCLUSIONS In this paper, timing synchronization was treated as a multiple hypotheses testing problem. The performance of generalized likelihood ratio test statistic was investigated for MIMO systems in frequency-flat environment due to its superiority in the presence of nuisance parameters. The performance of the GLRT test was assessed both analytically as well as via computer simulations . Computer simulations illustrate the
