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

1Texas A&M University, Dept. of Electrical Engineering, College Station, Texas, USA2Texas A&M University at Qatar, Electrical Engineering Program, Doha, Qatar

3Memorial University, Faculty of Engineering and Applied Science, Canada∗Corresponding author (E-mail: serpedin@ece.tamu.edu)

Abstract—This paper studies the performance of the gen-eralized likelihood ratio test (GLRT) proposed in [1] in thecontext of data-aided timing synchronization for multiple-inputmultiple-output (MIMO) communications systems and flat-fadingchannels. Herein paper, the asymptotic performance of theGLRT test is derived, and an upper bound for the detectionprobability is provided and shown to behave well as a bench-mark for sufficiently large number of observations. Computersimulations are presented to corroborate the developed analyticalperformance benchmarks. In addition, choice of some systemdesign parameters to improve the synchronization performanceis discussed via computer simulations.

Key words - GLRT; Hypothesis Testing; MIMO; TimingSynchronization.

I. INTRODUCTION

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 GLRT-

based 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

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 data-

aided 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 GLRT-

based timing synchronizers for MIMO systems in the presence

of flat-fading channels. An upper bound for the detection prob-

ability 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 GLRT-

based 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.

II. GLRT PERFORMANCE FOR MIMO CHANNELS

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 transmit antennas, ! receive antennas and "

complex baseband samples, the channel is modeled as [1]:

Z = HS+N , (1)

where Z ∈ ℂ#r×# is the sampled received signal matrix,

with each row containing the " samples received from one

of the ! receive antennas. Matrix H ∈ ℂ#r×#t stands for

the flat fading channel transfer matrix, S ∈ ℂ#t×# denotes

the transmitted signal matrix, and N ∈ ℂ#r×# represents

the additive noise samples. The noise at each receive antenna

is modeled as a zero mean Gaussian random variable with

variance "2#. The channel is temporally quasi-static, i.e., it

can be regarded as constant during " sampling periods. The

transmitted and received complex baseband signal samples at

some delay � are defined as in [1]:

S = [s(#") s(2#") ⋅ ⋅ ⋅ s( "#")],

Z� = [z(#" − �) z(2#" − �) ⋅ ⋅ ⋅ z( "#" − �)],

where #" is the sampling period, and s($) and z($) are the

continuous transmitted and received vectors as a function of

time $. 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 � ,

H� , will be assumed of the form [1]

H� =

{

H, � = �0

0, otherwise

where the correct delay in terms of receiver’s clock is �0,

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

ℋ0 : H#r×#t = 0,Σℋ1 : H#r×#t ∕= 0,Σ.

The parameter matrix Σ is the received signal spatial covari-

ance 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 two-

sided 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

LG(�) =&(Z� ∣S; H1, Σ1)

&(Z� ∣H0 = 0, Σ0)> ', (2)

where H1, Σ1 are the unrestricted maximum-likelihood es-

timates of H and Σ, respectively, under ℋ1, and Σ0 is

the restricted maximum-likelihood estimate under ℋ0 when

H = 0. ' is a threshold depending on the false alarm rate

[2]. In [1], the GLRT statistic was expressed as:

LG(�) = ∣I# −PSPZ�∣−# , (3)

where PS = S†(SS†)−1S and PZ = Z†(ZZ†)−1Z.

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

&(N∣Ω,Σ) =exp

{

−tr{Ω−1(N−M)†Σ−1(N−M)}}

�#r# ∣Ω∣#r ∣Σ∣# ,

(4)

where Ω = E{N†N}/ ! ∈ ℂ# ×# , Σ = E{NN†}/ " ∈

ℂ#r×#r and M ∈ ℂ

#r×# denotes the mean matrix. An

equivalent definition involving the Kronecker product ⊗ and

the vectorization operator vec(⋅) assumes the form: N ∼()#r,# (M,Ω,Σ) if vec(N) ∼ ()#r# (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 = I# .

Therefore, the pdf simplifies to

&(N∣Σ) =exp

{

−tr{N†Σ−1N}}

�#r# ∣Σ∣# , (5)

and equivalently vec(N) ∼ ()#r# (0, I# ⊗ Σ). Let z =vec(Z), h = vec(H), and n = vec(N). Taking into account:

vec(IAB) = (B% ⊗ I)vec(A) (6)

and recalling Eq. (1), it follows that:

z = (S% ⊗ I#r )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

&(z∣S;h,C) =

exp{

−[z− (S% ⊗ I#r )h]†C−1[z− (S% ⊗ I#r )h]

}

�#r# ∣C∣(8)

where C = I# ⊗Σ. Taking the complex conjugate gradient

of the log-pdf with respect to h, and setting it to zero, it turns

out that

∂ ln &(z∣S;h,C)

∂h∗

= −∂

∂h∗[z− (S% ⊗ I#r )h]

†C−1[z− (S% ⊗ I#r )h]

= (S% ⊗ I#r )†C−1[z− (S% ⊗ I#r )h] = 0.

Hence, the maximum-likelihood estimator of h is

h = [(S% ⊗ I#r )†C−1(S% ⊗ I#r )]

−1(S% ⊗ I#r )†C−1z.

Recalling C = I# ⊗ 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

h = [(S% ⊗ I#r )†(S% ⊗ I#r )]

−1(S% ⊗ I#r )†(I# ⊗Σ)

−1z

=(

[S†(SS†)−1]% ⊗ I#r)

z.

Recalling (6), one can convert the estimator from the vector

space to the matrix space: H = ZS†(SS†)−1, which is the

same as the initial state-space model. Thus, the equivalence

between the vector space model and matrix space model

has been set up. The unbiasedness of the ML estimator

follows immediately. The Fisher information matrix takes the

expression:

I(h) = E

{

∂ ln &(z∣S;h,C)

∂h∗∂ ln &(z∣S;h,C)

∂h∗

†}

= (S% ⊗ I#r )†C−1(S% ⊗ I#r )

= (S% ⊗ I#r )†(I# ⊗Σ)

−1(S% ⊗ I#r )

= (SS†)% ⊗Σ−1. (9)

As " →∞, the modified GLRT statistic 2 lnLG for complex

parameters admits the pdf [2]

2 lnLG(�)&∼

{

�22#r#t

under ℋ0

�′2

2#r#t(*) under ℋ1

where “+" denotes an asymptotic pdf, �2! denotes a central

chi-squared pdf with , degrees of freedom, and �′2

! denotes

a noncentral chi-squared pdf with , degrees of freedom and

noncentrality parameter *. The noncentrality parameter is

given by

* = 2(h1 − h0)†I(h0,Σ)(h1 − h0), (10)

where h1 and Σ are the parameters’ true values under ℋ1.

Note that Eq. (10) holds for the case without nuisance param-

eters. When nuisance parameters are present, the noncentrality

parameter * 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:

* = 2h†[

(SS†)% ⊗Σ−1]

h

= 2tr{

Σ−1HSS

†H†}

. (11)

Assuming the noise is spatially uncorrelated, i.e., Σ = "2#I#r ,

one obtains

* =2∥HS∥2F

"2#= 2 ! "SNR, (12)

where the signal-to-noise ratio SNR is defined as

SNR = ∥HS∥2F /("2

# ! "). (13)

Assuming knowledge of the true values for the MIMO

channel H (or equivalently h) and the covariance matrix Σ

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

any unknown parameters, the threshold required to maintain a

constant false alarm rate (CFAR) can be found, i.e, the CFAR

detector exists [2]. However, since the nuisance parameter is

present in the model, we can provide only an upper bound

for the detection probability or equivalently a lower bound

for the missing rate. And since the GLRT is considered

asymptotically optimal in the situation where no uniformly

most powerful (UMP) test exists [3], this asymptotic bound

can also serve as a benchmark when comparing various tests

developed through different approaches. The bound can be

obtained as follows:

I. For any given false alarm rate -FA, determine the corre-

sponding threshold # such that∫∞

%&1(.)/. = -FA, where

&1(.) = .r

2−1 exp(−

.

2)2

r

2Γ(,

2)0(.),

is the central chi-squared pdf with , = 2 ! degrees of

freedom, 0(.) denotes the unit-step signal, and Γ(0) is the

Gamma function: Γ(0) =∫∞

0$u−1 exp(−$)/$.

II. For the given SNR, obtain the noncentrality parameter *using (12) or (11). An upper bound for detection probability

-',&, which is the detection probability in the asymptotic case,

can be computed as -',& =∫∞

%&2(.)/., where

&2(.) = .r

2−1 exp(−

.+ *

2)2

r

2

∞∑

(=0

(

�)4

)(

1!Γ( !2+ 1)

0(.),

stands for the noncentral chi-squared pdf with , = 2 ! de-

grees of freedom and noncentrality parameter *. Equivalently,

a lower bound of the missing rate is -*+"",l, = 1− -',&.

III. SIMULATION RESULTS

In the following, the performances of the GLRT statistic

developed for flat fading channels are shown through com-

puter 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 trans-

mit antennas and four receive antennas, four different syn-

chronization sequences, each of " 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 " 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

dB. The proposed GLRT works well, e.g., with SNR=0dB,

for -FA = 1%, the missing rate is 6 × 10−4. As the SNR

decreases, the missing rate increases (the detection probability

decreases), which follows the intuition.

10−3

10−2

10−1

100

10−4

10−3

10−2

10−1

100

PFA

Pm

iss

nt=4, n

r=4, n

s=16

SNR= 0dB

SNR=−1dB

SNR=−2dB

SNR=−3dB

Figure 1. Comparison of ROCs for 4×4 MIMO link in Rayleigh flat fadingenvironment 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 " = 64. Fig. 3 corroborates

this behavior in the SISO case.

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

10−1

100

PFA

Pm

iss

nt=4,n

r=4,n

s=64

SNR=−10dB

SNR=−10dB Asymptotic

SNR=−13dB

SNR=−13dB Asymptotic

Figure 2. Performance of the detector for a 4×4 MIMO system in Rayleighflat fading environment with ns = 64.

IV. CONCLUSIONS

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 superi-

ority 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

10−3

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

PFA

Pm

iss

nt=1,n

r=1,n

s=32

SNR= −3dB

SNR= −3dB Asymptotic

SNR= −6dB

SNR= −6dB Asymptotic

Figure 3. Performance of the detector for a SISO link in Rayleigh flat fadingenvironment with ns = 32.

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.

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 Founda-

tion). The statements made herein are solely the responsibility

of the authors.

REFERENCES

[1] D.W. Bliss and P.A. Parker, Temporal synchronization of mimo wirelesscommunication in the presence of interference, IEEE Transactions onSignal Processing, vol. 58, no. 3, Mar. 2010.

[2] S. Kay, Fundamentals of Statistical Signal Processing, Volume II: Detec-tion Theory, Prentice Hall, New Jersey, 1998.

[3] Bernard C. Levy, Principles of Signal Detection and Parameter Estima-tion, Springer, New York, 2008.

[4] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nded. Prentice Hall, Jan. 2002.

[5] H. Meyr, M. Oerder, and A. Polydoros, On Sampling Rate, Analog Pre-filtering, and Sufficient Statistics for Digital Receivers, IEEE Transactionson 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 TrackingMethod for MIMO-OFDM Systems with Null Sub-carriers, Signal Pro-cessing, Elsevier, Volume 88, Issue 4, April 2008.

[9] H.-G. Jeon and E. Serpedin, Walsh Coded Training Signal Aided TimeDomain Channel Estimation for MIMO-OFDM Systems, IEEE Trans. onCommunications, vol. 56, no. 9, Sept. 2008.

[10] X. Li, Y.-C. Wu and E. Serpedin, Timing Synchronization in Decodeand-Forward Cooperative Communication Systems, IEEE Trans. on SignalProcessing, Volume 57, no. 4, April 2009.

[11] E. Serpedin, G. B. Giannakis, A. Chevreuil, and P. Loubaton, BlindJoint Estimation of Carrier Frequency Offset and Channel Using Non-Redundant Periodic Modulation Precoders, Proceedings of the NinthIEEE Statistical Signal and Array Processing Workshop 1998, Portland,OR, Sept. 1998.

[12] T. Fischer, B. Kelleci, K. Shi, E. Serpedin, and A. Karsilayan, AnAnalog Approach to Suppressing In-Band Narrowband Interference inUWB Receivers, IEEE Trans. On Circuits and Systems-Part I, Volume54, Issue 5, May 2007.