Optimal Quadratic Non-Assisted Parameter Estimation for Digital Synchronisation Javier Villares and Gregori V´azquez Department of Signal Theory and Communications, Polytechnic University of Catalonia E-mail: fjavi,
[email protected] Abstract— This paper deals with the optimal design of quadratic NonData-Aided (NDA) open- and closed-loop estimators. The new approach supplies the Best Quadratic Unbiased Estimator (BQUE), i.e., that one with the minimum variance, without the need of assuming a given statistics for the nuisance parameters, that is, avoiding the common adoption of the gaussian assumption, which does not apply in digital communications. Alternatively, if the unbiased constraint is relaxed, a bayesian open-loop estimator is presented and its performance compared with the open-loop BQUE solution. On the other hand, the closed-loop BQUE is developed, showing that it outperforms the well-known “ad hoc” Gaussian Stochastic Maximum Likelihood (GSML) scheme for short observation windows, that is, for low-complexity implementations, only converging to the Gaussian Unconditional Cramer-Rao Bound UGCRB asymptotically. Afterwards, the BQUE deduction is applied to obtain robust NDA estimators of the signal parameters in the presence of a frequency-selective channel. Finally, the quadratic analysis is naturally extended to higherorder techniques which exhibit a better performance for high SNRs. Keywords— Synchronization, parameter estimation, continuous phase modulation, non-data-aided, carrier and timing recovery.
N
I. I NTRODUCTION
ON -Data-Aided (NDA) synchronisation has received lately a lot of attention because it simplifies the system protocols and makes unnecessary the transmission of training sequences (preambles) that reduce significantly the spectral efficiency. So far, most algorithms have been devised by heuristic reasoning. In [1] [2] the authors presented a general framework allowing the formulation of any NDA synchroniser based on second order moments under a Maximum Likelihood (ML) perspective. There are two basic reasons for limiting the analysis to quadratic synchronisers. It is shown [1] that the stochastic ML solution becomes quadratic for low SNRs, whereas it is still unknown for moderate to high SNRs because the difficult treatment of the unknown transmitted symbols. The second reason is that quadratic algorithms allow efficient digital implementations. In this paper, we propose a totally different approach to the design of NDA discriminators that does not have to cope with the symbols extraction problem as in [1]. Making use of basic concepts from the estimation theory [3], we have deduced the Best Quadratic Unbiased Estimator (BQUE), that is, an estimator of the desired parameters that is quadratic, unbiased and exhibits minimum variance. Its name has been chosen by analogy with the classical Best Linear Unbiased Estimator (BLUE) [3]. The BQUE approach allows to unify the design of openand closed-loop synchronizers by constraining the value and/or
0-7803-7257-3/02/$10.00 2002 IEEE.
slope of the discriminator S-curve at certain points and by optimizing its performance within the expected operating range. Furthermore, the study is extended to address the problem of synchronization in mobile radio communications systems wherein a frequency-selective time-variant random channel distorts seriously the transmitted wave. The optimal NDA BQUE obtained in the case of an ideal propagation medium is modified in order to take into account the statistical knowledge about the channel at hand. The results for the usual WSSUS (Wide-Sense Stationary Uncorrelated Scattering) channel in [4] show that: 1) the multipath degrades the estimator performance in proportion to the channel delay spread, as expected; 2) this degradation is bounded and less significant as the delay spread becomes larger than one symbol period. The structure of the paper is the following. The signal model and the problem statement are presented in Section II. Section III introduces the algebraic notation used through the paper. Section IV deduces the exact solution for the open-loop BQUE and an analytical expression is deduced with the help of a discrete approximation. In Section V the unbiased constraint is relaxed and an alternative open-loop bayesian discriminator having minimum mean squared error is presented. Section VI deduces the closed-loop BQUE for the parameter tracking. Section VII compares the BQUE and Gaussian Stochastic Maximum-Likelihood (GSML) feedback detectors with the UGCRB. Section VIII includes into the study the presence of a slow-variant frequency-selective channel. Section IX extends the results to higher-order estimators. Simulations results and their comments can be found in Section X and, finally, conclusions are drawn in Section XI. II. D ISCRETE - TIME S IGNAL M ODEL The noisy samples of the received complex envelope for any linearly modulated signal can be expressed as follows:
r(nTs ) =
X
i
xi ej!nTs g(nTs iT
()
) + w(nTs )
(1)
where g t is the shaping pulse, x i are the complex symbols T=Nss of a given alphabet transmitted every T seconds, T s is the sampling period and w t the additive noise term having E fw t w t g. an arbitrary correlation function R w Equation (1) includes the effect of the synchronism errors under consideration, i.e., the carrier frequency offset ! and the timing error . The timing error is supposed to be bounded to
46-1
()
=
( )=
() ( + )
T=2 seconds because NDA timing estimates have an ambiguity equal to the symbol period T . Likewise, the maximum frequency offset is limited by the receiver (two-sided) bandwidth (W ), provided the Nyquist criterion is respected. All the synchronization techniques proposed in the paper shall process blocks of N received samples. It is easy to show that all the blocks starting at multiples of the symbol interval T can be represented by the vector r:
r = A x + w
()
=
The following notation is introduced herein to facilitate the deduction of the proposed BQU estimators: o
(3) b is the sample covariance matrix and the operator where R vec M stands for the column stacking of any matrix one underneath the other. Using the notation defined in the previous section, the generic equation of a quadratic discriminator is:
( )
b ) = MH R b ^ = rH Mr = T r(MR
^
M = vec(
=
)
(5)
The minimum variance unbiased (MVU) estimator M if the actual parameter is has this expression:
M = argminM E subject to MH R is the following:
j^j2
o
(6)
= and, thus, the cost function to minimize
C () = MH Q M
Rb Rb H
o
and the Lagrange multiplier e
=
=0
=
=
=
= Rij Rlk + Rik Rlj +
+ (ai p) alT (ak p) aj H (ai al p p) (aj ak)H
(8) where Rij is the element i; j of R , an the n-th row of A , 4 E fjxn j4 g (8n), pn E fx2n g E f xn 2 g the n-th element of the row vector p and stands for the Hadamard product of matrices. Any statistical a priori knowledge of the parameter of interest 2 fjj g can be introduced in the optimization process by means of a bayesian approach, that is, by averaging the cost function in (7) with respect the adopted prior:
( ) =
=
=
Col
=
( )
= E fC ()g = R C ()W d R
R
MH Q W d M MH R d
(9)
where W e and all the irrelevant terms have been wiped out from the last equation. The weighting function W f is the prior of the parameter of interest. If no a priori knowledge is available, then, a uniform prior shall be 1 adopted within the operative range , that is, W 2 . The minimization of (9) yields the following expression for the open-loop BQU estimator :
= ()
MH R e
(7)
=
M = Q 1R (10) R R with Q = Q W d and R = R d. The value of the multipliers (8 2 ) that force the unbiased solution
within the whole interval , requires the solution to the following system of integral equations:
MH R = RH Q 1 R =
Z
RH d Q 1 R =
(11)
At that point, we have opted for a discrete approximation of the integral in (11) considering only a finite set of constraints s 1 ; ; L T . Thus, we obtain that R R s and 1 s (after some manipulations) (11) becomes R H s Q Rs incorporating the definitions below:
=[
IV. O PEN -L OOP (OL) B EST Q UADRATIC U NBIASED E STIMATOR (OL-BQUE)
n
n
Q (Ni + j; Nk + l) = E ri rj rk rl + 4 222 (ai al ) (aj ak )H
(4)
where T r ) is the trace operator, M is the matrix containing the discriminator complex coefficients, and where M is defined as:
MH
E
H
b = A A H + Rw ; R = E R = E xx @ R =D A H +A D H ; D = @ A S = @ @ Rb = vec(Rb ) ; R = vec(R ) ; S = vec(S )
(
=
=
III. N OTATION
n
Q
imposes the unbiased constraint M H R . The fourthorder moments contained in Q can be computed as follows for any rotational symmetric constellation (E fx i jxi j2n g E fxi jxi j2n g 8i; n) with uncorrelated and identically 2 I) of mean power 2 E fjxn j2 g: distributed symbols (
(2)
where the columns of A are those T -seconds delayed versions of the shaping pulse g t entering inside the observation window. Besides, matrix A contains the timing and frequency errors, being the parameter of interest (the other parameter is assumed to be perfectly known). Finally, covariance the noise matrix is known and equal to R w E wwH . Finally, let us recall that binary Continuous Phase Modulations (CPM) can be seen as the superposition of a set of PAM (pulse amplitude modulation) signals by resorting to the Laurent expansion [5][6]. Therefore, the Laurent decomposition allows to extend the model in (2) to any binary CPM modulation. For more details, the reader is referred to [1].
b = rrH R
where
]
=
Rs = [R1 ; ; RL ] = [ 1 ; ; L ]T
(12)
The discrete approximation of the solution is therefore: M = Q 1 (Rs ) = Q 1 Rs RHs Q 1 Rs # s
(13)
1 It turns out that the matrix R H s Q Rs in (13) can be singular if the oversampling (N ss ) and the length of r are not large enough. In that case, the set of constraints cannot be exactly fulfilled and the pseudo-inverse operator # will supply the least-squares fitting.
46-2
()
V. U NCONSTRAINED O PEN -L OOP BAYESIAN E STIMATOR (OL-BAYES) In this section we present an alternative criterion to design open-loop estimators from a bayesian approach when relaxing the unbiased constraint. The discriminator coefficients M will be those that minimize the following cost function: n
Cuol = E E kMH Rb
k2
o
MH QM MH Re
=
=
e and Q admit analytical solutions for uniform priors, and both R 1 i.e., W 2 .
=
VI. C LOSED -L OOP (CL) B EST Q UADRATIC U NBIASED E STIMATOR (CL-BQUE) In this section the estimator is required to track the parameter fluctuations around =0 with minimum variance for a given loop bandwidth, i.e., for a given value of the S-curve slope at =0. The pretended BQU discriminator of the parameter errors will be that minimizing the following cost function:
MH S0 1 0
(16)
where the Lagrange multipliers 0 and 0 impose the unbiased and MH S0 at =0. constraints MH R0 It can be shown that the constraint 0 is always fitted due to the discriminator symmetry and the CL-BQUE solution reduces to:
=0
=1
1
M = 0 Q0 1 S0 = S HQQ0 S10S
0 0 0
VIII. S YNCHRONISATION IN THE PRESENCE OF
(14)
where the last equivalenceR only preserves the terms dependent e on MH and where R R W d and Q was introduced in (10). The expression of the discriminator M minimizing (14) is therefore: M Q 1Re (15)
Ccl = MH Q0 M + MH R0 0 +
below that of the GSML and the UGCRB is not attained. This fact confirms the UGCRB as a suitable bound for quadratic (unbiased) NDA discriminators but it also proves that it can not be attained if the observation window is not large enough. In that case, the performance of the CL-BQUE (18) becomes a tighter bound valid for any quadratic unbiased NDA discriminator.
(17)
MULTIPATH
In this section we extend the above theory in order to address the problem of parameter estimation in the presence of a frequency-selective channel h t . The channel is assumed to have a slow temporal variation so that it can be considered almost constant within the block duration. Moreover, the channel response h t is practically limited to the receiver bandwidth W so that:
()
()
1
S0H Q0 1 S0
2
(
(18)
Other suitable approach to the problem is to treat the received data (r) as if they were gaussian. The resulting discriminator, called Gaussian Stochastic Maximum Likelihood (GSML), has the following structure in the uniparametric case [2] [7]: b ) T r(R0 1 S0 R0 1R ^ = 1 2 T r (R0 S0 )
(19)
This discriminator is known to attain the Gaussian Unconditional Cramer-Rao Bound (UGCRB) if N ! 1 [1][7]. In Section X simulations have shown that the CL BQUE (Section VI) becomes asymptotically equivalent to the GSML and, hence, both attain the UGCRB (asymptotically). However, when the length of r is short, the variance of the CL-BQUE is
)
(
r(nTs )
VII. CL-BQUE VS . M AXIMUM L IKELIHOOD APPROACH
L X1
of the channel [8]. Hereafter, we will consider that the receiver (bandpass) bandwidth W is set to =T (100% excess of bandwidth) in order to admit the majority of modulations. The channel taps h k=W will be modeled as zero-mean complex gaussian variables with their envelope and phase following a Rayleigh and uniform distribution, respectively. The Rayleigh distribution is adopted hereafter because it corresponds to a worst-case situation. Anyway, it could be possible to assume for the first coefficients of h k=W a Ricean distribution in order to model with more accuracy the line-of-sight (LoS) components [4]. Having in mind the above considerations, the received discrete complex envelope in equation (1), adding the multipath distortion, becomes [8]:
and its tracking error variance is therefore:
2^ =
1
h(k=W )sinc(W t k) (20) W k=0 with sinc(t) = sin(t)=(t) and L=W the effective duration h(t) =
L
1
= P xi P h(k=W )ej!nTs g(nTs i k=0 +w(nTs )
)
k=W
iT
)+
(21) It can be readily shown, with the same premises stated in (2), that all the blocks r of received samples share the following expression if the product W T is an integer number:
r = A Hx + w = A y + w
1
(22)
where the columns of H are =W -seconds delayed versions of the channel impulse response h k=W . It is really important to realize that the inclusion of a random channel h t yields directly to the same model in (2), except that the proposed estimators will have to cope with the extended, correlated vector of symbols y Hx. Therefore, the channel has two negative effects: a) Firstly, the unknown vector of symbols y is about W T times longer than x in (2). The increment of the nuisance parameters will put a limit on the variance of blind synchronizers when
46-3
=
(
)
()
dealing with a time dispersive channel. b) Secondly, the channel modifies the transmitted symbols covariance matrix , that is:
= E HxxH HH = 2E HHH
(23)
where we have assumed uncorrelated symbols x i . In a uncorrelated scattering (US) scenario [4], the correlation of the channel taps is also null, thus yielding to a diagonal covariance matrix: I
1
= 2E fHHH g = 2 P E hihHi = i=0 = diag f (0); : : : ; (K 1)g
1 X i=
1
P DP (k=W
=
iT )
(25)
n
(26)
It is worth noting that the channel makes the sequence y k to be
k (for any ). cyclostationary so that k W T The diagonal structure of the covariance matrix allows to obtain a simple expression for the matrix Q in (8):
( +
)= ( )
Q (Ni + j; Nk + l) = E ri rj rk rl = Rij Rlk + Rik Rlj + + ai a a al (m4 2m2 m2) k
(27)
where the k-th element of the vectors m 2 and m4 is:
= E jyk j2 = (k) m4 (k) = E jyk j4 = 2 (k) + 1 X + 2 4 22 P DP 2 (k=W m2 (k)
i=
1
(28)
iT )
=
=0
t
(30)
2 k=W T
0
is integer
else
(31)
and hence the equation (22) reduces to that we had in the case of an ideal channel (2). In that case, the channel would only change the distribution of those symbols which are different from zero in (31). 2) Highly frequency-selective channel ( ! 1):
= W2T I
(32)
and, therefore, the channel is implicitly increasing the vector of unknown symbols x in (2) by a factor of W T , as well as changing their distribution. Notice that this expansion may require to oversample the received signal in order to guarantee the matrix A has more rows (received samples) than columns (unknown symbols). Otherwise, any NDA estimator of the parameter will suffer from self-noise, resulting in a variance floor at high SNRs. This fact forces the designer to always hold the inequality Nss > W T . IX. E XTENSION TO HIGHER - ORDER DISCRIMINATORS
In the calculation of (27), the last two in (8) depending terms E yk 2 is always on p have been removed because E yk2 zero due to the normal distribution of the channel taps h k=W . On the other hand, m 4 k in (28) has been computed using the equation (8) in the absence of noise, substituting the matrix A by H and setting all the indices i,j ,k ,l to the same value, say k. The lack of correlation of the channel taps has allowed to cancel again the last two terms in (8). Equations (23), (3) and (27) make possible to obtain the blind open- and closed-loop estimators proposed in sections IV, V and VI when the existing channel can be modeled as above and the knowledge of its PDP is quite accurate. An important difference must be remarked in the case of the CL-BQUE: the unimposed by the Lagrange multiplier biased constraint at
()
0
j
)
where C is a constant determined by the received mean power and is the well-known delay spread magnitude. Depending on the channel delay spread two asymptotic situations have been detected: 1) Flat fading channel ( ! ):
(k ) =
o
+
P DP (t; ) = Cexp
where the Power Delay Profile (PDP) is defined as [4]:
P DP (t) = E jh(t)j2
(
with 0 and 0 those scalars holding the respective constraints in (16). In some cases a limited set of parameters is sufficient to describe totally the PDP function. Moreover, in a lot of scenarios the mobile radio channel is pretty well modeled by adopting a (decreasing) exponential PDP [4]:
(24)
being hi the i-th column of H (I is the number of symbols in x). If the product W T is an integer number, the k-th element in the diagonal of is found to be:
(k) = 2
0 cannot be omitted and, therefore, the solution to (16) is given by: M Q0 1 0 R0 0 S0 (29)
(
)
In this section the procedure for designing optimal openand closed-loop estimators is extended to n-order discriminators (with n>2). For the n-order case, equation (4) can be rewritten as:
^ = MH Rb (n) where
Rb (n) = Rb Rb (n 2) n > 2
(33)
(even)
(34)
and stands for the Kronecker product of matrices. Odd values of n are not considered because all modulation schemes in practice are symmetric with respect to the origin and so the odd moments are always null.
46-4
−4
−1
10
10
Tracking Error Variance
−3
Normalized Tracking Error Variance
GSML UCRBG BQUE 4th−Order Data−Aided (MCRB)
−2
10
10
−4
10
−5
10
GSML CL−BQUE −5
10
−6
10
−7
10
−6
0
5
10
15
20 EbNo
25
30
35
2 =T 2 ) for the GSML and CL-BQUE Fig. 1. Normalized timing variance ( discriminators N =4. The fourth-order discriminator proposed in reference 3. [8] is also plotted and compared with the MCRB [9][6]. Bn T
= 5 10
All the expressions included in previous sections are then usable if the following substitutions are done previously:
Rb ! Rb (n) n o R ! R(n) = E Rb (n) S ! S(n) = @@ R(n) H Q ! Q(n) = E R(n) R(n)
(35)
Generally, the complexity and the minor improvement reflected in the system BER with respect to quadratic algorithms, do not justify the utilization of higher-order synchronisers in communications systems. However, when the purpose is not strictly synchronisation, but the exact estimation and/or tracking of the time of arrival and/or the Doppler offset of the incoming signal, which is the case of advanced navigation and positioning systems (DGPS, GNSS, etc.), they could be taken into account despite their complexity. In any case, the higherorder study herein is valuable because it supplies new bounds that give information on the potential improvement these techniques can yield (Figure 1). X. S IMULATION RESULTS The simulations have been carried out for the MSK (Minimum Shift Keying) modulation as a particular case of the binary Continuous Phase Modulations CPM. This transmission scheme is adopted because it allows a simple extension to any linear digital modulation and to any multiple access modulation, as well [11]. Recall that the Laurent expansion [5] [6] allows the formulation of binary CPM signals in terms of the model presented in Section II. The simulations have been done for additive white gaussian noise (AWGN) and two samples per ). symbol (Nss - Figure 1: the GSML and CL-BQUE are compared with the UGCRB when the vector of samples r is short (N =4). It is clear
=2
10
40
0
10
20
30 N
40
Fig. 2. Normalized timing variance as a function of CL-BQUE discriminators (EbNo=40dB).
50
60
N for the GSML and
that in this case the variance of both discriminators is above the UGCRB. The discrepancies are more important for high EbNo because the gaussian assumption becomes more exact as the EbNo is reduced. Figure 1 also shows how fourth-order detectors [10] are capable to be below the UGCRB and nearer the DA performance. It is also remarkable that for low SNRs the UGCRB becomes a valid bound for the variance of any NDA detector irrespective of its order. - Figure 2: the asymptotic convergence of the CL-BQUE and the GSML is shown. The variance depicted in the figure is for an open-loop configuration when ' (also Fig. 4). The corresponding closed-loop tracking variance is approximately L0 : =BnT times lower if the normalized loop bandwidth (Bn T ) is very small and L 0 N . - Figure 3 and 4: the two open-loop schemes proposed in the paper (Sections IV and V) are compared. Figure 4 shows how the OL-BAYES can reduce the mean squared error within the designed interval (even for a extremely high EbNo=40 dB) because it is not forced to be unbiased (see figure 3). The behaviour of the studied closed-loop discriminators (GSML and CL-BQUE, Sec. VI) for 6 is also depicted. (tracking). Figures 3 and 4 show their specialization for It is also notorious that the CL-BQUE has a better behaviour ). than the GSML outside the steady-state situation ( 6 - Figure 5: the CL-BQUE is evaluated for the extreme scenarios stated in equations (31) and (32). The channel variability is modeled making use of the U-shaped doppler spectrum [4] with maximum doppler spread equal to 833 Hz. This value corresponds to an European GSM 1800 handset moving at an uncommon speed of 500 Km/h in an urban area. The figure reflects the robustness of the proposed NDA estimator against the multipath, concluding that the maximum losses induced by the channel are bounded. Another significant conclusion is that the degradation caused by the channel is first manifested at high SNRs even if the delay spread is quite small (T= ), whereas
46-5
0
=05
=0
=0
=0
10
1.5 −1
10
GSML CL−BQUE OL−BQUE OL−BAYES
1
ideal channel σ=0.01T σ=0.08T σ=0.1T σ=T σ=5T
−2
Normalized Timing Error Variance
10
S−curve
0.5
0
−0.5
−3
10
−4
10
−5
10
−1 −6
10
−1.5 −1
−0.8
−0.6
−0.4
−0.2 0 0.2 Normalized Frequency Error (λ)
0.4
0.6
0.8
0
1
Fig. 3. Frequency S-curves for the CL-BQUE, GSML, OL-BQUE and OL: and N =4. BAYES for
=05
5
10
15
20 EbNo
25
30
35
40
Fig. 5. Normalized timing variance as a function of the received EbNo for the CL-BQUE discriminator (N =8, Nss =4) and different values of the channel 3. delay spread. Bn T
= 5 10
2
10
vergence proved empirically. However, the BQUE is observed to outperform the GSML for short data vectors and high SNRs. The extension of the results to communication systems presenting frequency-selective fading has revealed that, whenever the multipath is uncorrelated, the degradation is bounded (for a given signal bandwidth) and is a function of the temporal dispersion of the channel (delay spread), which is known (or estimated) at the receiver. Finally, the formulation of the paper is extended to higherorder synchronisers and their utilization discussed.
1
10
0
Normalized Mean Squared Error
10
−1
10
−2
10
−3
10
GSML CL−BQUE OL−BQUE OL−BAYES
−4
10
R EFERENCES
−5
10
[1]
−6
10
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 Normalized Frequency Error (λ)
0.4
0.6
0.8
1
Fig. 4. Normalized mean square error as a function of for the CL-BQUE, GSML, OL-BQUE and OL-BAYES with =0.5. N =16 and Nss =4
for low SNRs the performance loss increases rapidly as the delay spread gets close to T . XI. C ONCLUSIONS This paper presented a new, versatile approach for designing both open- and closed-loop optimal synchronisers with constraints on the S-curve shape (unbiased restrictions). If a little amount of bias is tolerated, a very simple, elegant bayesian estimator was formulated in Sec. V which is found to reduce the mean squared error within the designed range of the parameter. For the closed-loop case, an optimal (unbiased) parameter error detector is obtained whose tracking variance is a lower bound for any NDA quadratic unbiased discriminator with independence of the number of samples it processes. A comparison with the classical GSML is carried out and their asymptotic con-
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