On the Estimation of Sampling Clock Frequency Offset in Single and Multiple Carrier Systems K. Shi and E. Serpedin Dept. of Electrical Engineering Texas A&M University, College Station, TX 77843, USA. Email:
[email protected]
Abstract— The aim of this correspondence is to report in a closed-form expression a lower bound on the performance of sampling clock frequency offset (SFO) estimators for both single carrier and multi-carrier linearly modulated signals. This benchmark is expressed in terms of the so-called modified CramerRao bound (MCRB), which is shown to coincide asymptotically asymptotically (large sample) with the Cramer-Rao bound (CRB) for scalar parameter estimation. Comparison of the MCRB with the performance of existing SFO estimators gives rise to several challenging research problems: design of efficient SFO estimators and development of a framework to assess correctly the performance of the second-order feedback SFO estimators with respect to the MCRB.
I. I NTRODUCTION Due to the uncertainty of local oscillators and dispersive wireless channels, a synchronizer is a necessary and important component in any digital communication receiver. References [6] and [7] present an excellent overview on the carrier and symbol delay estimators that have been reported in the literature. Since the Cramer-Rao lower bound (CRB) [5] of carrier and timing delay estimators can not be evaluated in closed-form expression, a looser bound, the so-called modified Cramer-Rao bound (MCRB) was introduced in [1]. This lower bound is much easier to calculate than CRB, and approximates well the true CRB in some situations. The MCRB was applied to carrier frequency, carrier phase and symbol timing offset estimation for both linearly modulated systems [1] and nonlinearly modulated systems [6]. The MCRB was further extended to vector parameter estimation in [4]. The relationship between the asymptotic CRB (ACRB) and MCRB was clarified in [8]. In general, the received signal may be subject to an additional source of error, namely the sampling clock frequency offset (SFO) introduced by the analog-to-digital (A/D) converter. In single carrier systems, the SFO can be estimated either explicitly using the feedforward estimator [2] or implicitly in terms of a scheme which assumes a timing error detector and a second-order tracking loop [6]. Although, in single carrier transmissions, that assume reduced-length packets, the effects of SFOs can be neglected, in multi-carrier systems, accurate SFO estimators are very important in order to avoid possible inter-carrier and inter-symbol interference. Some explicit estimators were proposed in [10], [12]. However, to the best of our knowledge, there is no lower bound available to quantify the performance of these SFO estimators.
The goal of this paper is to derive simple closed-form expressions for the MCRB of SFO estimators for both single carrier and multi-carrier systems. The closed-form expressions of the asymptotic (large sample) Cramer-Rao bounds (ACRB) of the SFO for both scalar and vector parameter estimation in single carrier systems are also derived. This result verifies the tightness of MCRB for scalar parameter estimation. Also, in this paper, several open research problems pertaining to the design of efficient SFO estimators are stated. To design efficient SFO estimators, it appears that optimal second-order feedback schemes for tracking both the symbol time phase offset and clock SFO are needed for single-carrier transmissions that assume long packets. As the tracking performance of closedloop schemes assumes a linearized variance model, which does not depend explicitly on the number of data samples used, additional research results are needed in order to correctly compare the performance of second-order feedback schemes with the MCRB. It is interesting to remark that several practical applications such as passive listening, automatic recognition of modulation schemes, synchronization of OFDM-like systems that assume a large number of subcarriers, and blind synchronization of high-speed broadcast networks ask for efficient SFOestimators. The rest of the paper is organized as follows. Both the MCRB and ACRB for SFO estimators in single carrier systems are introduced in Section II. In Section III, the MCRB of SFO estimators in multi-carrier (OFDM) systems is presented. Section IV shows some comparisons of the MCRB with the performance of some practical estimators for multicarrier systems. Final conclusions are drawn in Section V. II. MCRB AND ACRB OF SFO E STIMATORS IN S INGLE C ARRIER S YSTEMS The received linearly modulated signal r(t) assumes the standard baseband model: r(t) = ej(2πΩt+θ) ai g(t − iT − τ ) + w(t) (1) i
where Ω is a carrier frequency offset, θ denotes a carrier phase offset, τ ∈ [0, T ) is the timing phase offset, a = {a i } denotes zero mean independently and identically distributed (i.i.d) random data with unit variance E{|a i |2 } = 1, g(t) represents the pulse shaping filter with the symbol period T ,
and w(t) stands for complex additive white Gaussian noise (AWGN) with variance N o = E{|w(t)|2 }. To detect correctly the data symbols, accurate estimates of (τ, θ, Ω) are needed. Numerous estimators, dealing with the joint or scalar estimation of (τ, θ, Ω), can be found in the textbook [6]. The well known Cramer-Rao lower bound [5] provides the performance limit of these estimators. Since finding closed-form expressions for CRB appears intractable, a looser bound, referred to as the MCRB for scalar estimation was introduced in [1]. The MCRB is much easier to evaluate than CRB and approximates well the CRB in some cases. The MCRB was applied to the problem of scalar estimation of (τ, θ, Ω) in [1], [6]. Besides the parameters (τ, θ, Ω), there might be an additional synchronization parameter, namely the SFO. In the receiver, r(t) is oversampled by Ts = T /P (P being the oversampling ratio). Due to the small uncertainty of the sampling clock oscillator used by the A/D converter, T is not identical but very close to the symbol period T . The sampling clock frequency offset is defined by v = (T − T )/T and its value can be up to ±50ppm (±5e-5) in practice. The oversampled signal is expressed as: r(nTs )=ej(2πΩnTs +θ)
ai g(nTs −iP Ts (1−v)−τ )+w(nTs )
i
(2) where it is assumed that the approximation 1/(1 + v) ≈ 1 − v holds true for small v. From (2), we infer that in addition to the constant timing phase offset τ , the SFO v introduces an accumulated timing offset nvT from the correct data strobe. In burst transmission systems with short packets, the accumulated timing offset, being insignificant compared to τ , is generally neglected. However, in long packet transmissions, the accumulated timing offset can be large enough to introduce significant inter-symbol interference (ISI) if only the timing phase offset τ is compensated. To deal with this problem, some explicit feedforward SFO estimators such as [2] could be used. Alternatively, the SFO can be implicitly estimated and compensated by a closed-loop tracking scheme that assumes a timing error detector and a second-order filter [6]. Normally, the performance of the latter approach is much better than that of the feedforward approach. To derive a lower bound for the clock SFO estimator, we resort to the evaluation of MCRB(v) for scalar estimation [1] at first. Later, we show that MCRB(v) equals the ACRB(v), evaluated assuming a joint estimation of the SFO and symbol delay phase offset. Denoting by T 0 the observation interval, we replace the probability density function (pdf) of received data p(r|u, v), conditioned with respect to all the unwanted and unknown parameters u and SFO v, with the likelihood function 1 Λ(v, u) = Cexp − 2Re{r(t)s∗ (t)} − |s(t)|2 dt N0 T0 (3) where C is a constant independent of {v, u}, and s(t) is given
by
s(t) = ej(2πΩt+θ)
ai g(t − iT + ivT − τ )
(4)
i
From [1], the expression of MCRB for v is given by
MCRB(v) = Eu
N0 /2
∂s(t) 2 dt T0 ∂v
(5)
This equation proves useful in keeping the length of this correspondence to a minimum and in deriving the MCRB of SFO for multi-carrier systems later. Thanks to the absolute value operator in (5),MCRB(v) is
independent of (θ, Ω). ∂s(t) 2 Using (4), we get E a T0 ∂v dt ⎧ ⎫ ⎨ ∂g(t − iT − τ ) 2 ⎬ iT dt Ea ai ⎭ ⎩ ∂t T0 i ∂g(t − iT − τ ) 2 2 2 i T dt ∂t T 0 i
= =
(6)
If LT , the length of T 0 , is much larger than the length of pulse filter g(t), it is reasonable to omit the edge effects on the integration. Similarly, we can omit the small τ for sufficiently large L. Thus, we can make the following approximation ∂g(t − iT − τ ) 2 dt ∂t T0 2 +∞ ∂g(t−iT ) dt, 0 ≤ i < L ∂t −∞ ≈ 0, otherwise +∞ = (u[i] − u[i − L + 1]) 4π 2 f 2 |G(f )|2 df (7) −∞
where G(f ) is the Fourier transform of g(t) and u[n] denotes the unit step function. Therefore, (6) becomes ∂s(t) 2 Ea ∂v dt T0 L(L − 1)(2L − 1)T 2 +∞ 2 2 4π f |G(f )|2 df (8) ≈ 6 −∞ As pointed out in [6], MCRB(v) is affected by the beginning of observation vector T 0 and in this paper we assume that T 0 always starts from the time origin. Defining the signal energy per symbol by E s = +∞ 2 −∞ |G(f )| df and +∞ 2 f |G(f )|2 df 2 −∞ (9) ξ = T +∞ |G(f )|2 df −∞ and substituting (8)-(9) into (5), we obtain: MCRB(v) ≈
4π 2 ξL(L
3(1 + v)2 − 1)(2L − 1)Es /N0
(10)
For small SFO v and large L, (9) can be approximated by 3 (11) MCRB(v) ≈ 2 3 8π ξL Es /N0
Normally the MCRB is looser than CRB [1], [8], and someone might argue if the MCRB(v) (11) could be reached by any practical estimator. Furthermore, the result (11) based on the assumption of scalar estimation might not be fit for the joint estimation of (τ, v), which normally requires evaluation of a joint CRB. To check the tightness of the scalar MCRB (11), we need to derive the asymptotic CRB, assuming a joint estimation of the timing phase offset and SFO (τ, v), and compare it with the MCRB(v). If we assume that ak ’s are known, the Fisher information matrix I(λ, a) corresponding to joint estimation of λ = (λ1 λ2 λ3 λ4 )T = (τ v θ Ω)T , is defined by [5] ∂s(t) ∂s∗ (t) dt . (12) Ii,j (λ, a) = ∂λi ∂λj T0 For sufficiently large L, from [1]-[8], we can approximate I(λ, a) as an averaged (over random data) E a [I(λ, a)]. After simple manipulations, one can easily find that the symbol timing parameters (τ, v) and carrier parameters (θ, Ω) are decoupled [8] Ii,j (λ, a) = 0,
i ∈ [1, 2] and j ∈ [3, 4]
(13)
Therefore, in the Fisher information matrix, only the part related to symbol timing parameters (τ, v) needs to be computed (ACRB(v)). Following the steps (6)-(9), we find that A0 A1 , (14) I(τ, v, a) = A1 A2 where
∂g(t − lT ) 2 (lT )i dt ∂t T 0 l +∞ L−1 i ≈ (lT ) 4π 2 f 2 |G(f )|2 df .
=
Ai
l=0
I
−1
l
(15)
−∞
Inverting (14) yields 1 (τ, v, a) = 2 A1 − A0 A2
−A2 A1
A1 −A0
.
(16)
Then, the asymptotic (large L) CRB (ACRB) is found to be ACRB(v) = (2/No )I −1 (τ, v, a)22 ≈
3 2π 2 ξL3 Es /N0
, (17)
which is as four times as (11). This result indicates the modified CRB (11) is not tight. However, we also derived the ACRB for scalar estimation of v, which is found to be the same result as (11). III. MCRB ON SFO E STIMATION FOR M ULTI -C ARRIER OFDM S YSTEMS In a multi-carrier OFDM system, the transmitted complex baseband signal is given by x(t) = al,k ej2π(k/Tu )(t−Tg −lT ) g(t − lT ) , (18) l
k∈C
where C = [−K/2, 0) ∪ (0, K/2] is the collection of data subcarriers, al,k denotes the unit-variance complex data symbol modulated on the f k = k/Tu subcarrier of the l th OFDM symbol. To avoid ISI in frequency-selective channels, each symbol is preceded by a guard interval (cyclic prefix) of length Tg , which is a copy of the last portion of the symbol. To simplify the transmitter, a discrete time implementation (sampling by T s = Tu /N of x(t)) is assumed since it can be easily generated by inverse fast Fourier transforming (IFFT) blocks of size N . Therefore, the symbol period is T = T g +Tu , which corresponds to M = N + N g samples. The pulse shaping filter g(t) is a raised cosine window defined by ⎧ 0.5 + 0.5cos (π + tπ/(ρT )) , 0 ≤ t < ρT ⎪ ⎪ ⎨ 1, ρT ≤ t < T g(t) = 0.5 + 0.5cos ((t − T )π/(ρT )) , T ≤ t ≤ (1 + ρ)T ⎪ ⎪ ⎩ 0, otherwise. (19) Compared to the rectangular window, this raised cosine widow can decrease the side-lobes of x(t) dramatically [13]. Normally, the value of ρ is between 0.025 and 0.05and we can +∞ approximate the energy per data subcarrier E s = −∞ g 2 (t)dt by T. Assuming that coarse frame acquisition is achieved and taking into account a small SFO v = (T u − Tu )/Tu = (T − T )/T and using 1/(1 + v) ≈ 1 − v, we express the equivalent received signal as r(t) = s(t) + w(t), where s(t) is given by N −lM j2πk (t−τT)(1+v) − gN u g(t−τ −lT (1−v)) s(t)= al,k e k∈C
(20) where τ is a constant timing offset. As argued in the last section, both MCRB(v) and ACRB(v) are independent of carrier synchronization parameters and we omit the possible effects of carrier phase/frequency offsets (θ, Ω) in (20) for simplicity. Assuming that the observation vector contains L OFDM symbols and K >> 1, we obtain ∂s(t) 2 Ea ∂v dt T0 LT 4π 2 k 2 t2 = g 2 (t − lT ) T u l k∈C 0 2 2 ∂g(t − lT ) +l2 T dt ∂t ≈
π 2 KL(L + 1)(2L + 1)T π 2 K 3 L3 M 2 T . (21) + 9N 2 24ρ
As indicated by (20) and [11], two effects are caused by the SFO: the subcarrier phase rotation and the OFDM symbol window shift. In (21), we can associate the first term to the subcarrier phase rotation and the second term to the symbol window shift. For normal values of K and ρ (K ≥ 50 and ρ ∈ [0.025, 0.05]), one can easily find that the second term is much smaller than the first term and can be omitted.
Substituting (21) into (5), we obtain the expression of MCRB(v) for multi-carrier systems: MCRB(v) ≈
9 . 2π 2 K 3 L3 (M/N )2 Es /N0
IV. P ERFORMANCE OF E XISTING E STIMATORS After obtaining the MCRB of SFO estimators for both single and multi-carrier systems, a natural question arises: how far is the performance of existing estimators from the MCRB. For single carrier systems, the SFO is normally obtained by means of a feedback scheme that assumes a second-order loop filter and a timing error detector such as [3], [9]. Due to the property of loop filter, the feedback scheme involves an infinite-length non-uniformly weighted observation vector (all the past samples up to the current time). For this reason, comparing the performance of feedback synchronizers with MCRB represents a difficult problem. To overcome this challenge, usually the proposed solution relies on finding an equivalent feedforward synchronizer that operates on a finite length observation vector and whose mean-square error (MSE) performance matches the performance of the feedback scheme (see e.g., [6, pp.115-140], [7, pp. 340-347]). However, to the best of our knowledge such techniques were reported only for first-order feedback structures. We have found that building equivalent feedforward models for second-order feedback schemes is a very difficult problem. Therefore, further research work is needed to design efficient feedback schemes as well as adequate metrics to assess the performance of feedback synchronizers for single carrier systems. As for multi-carrier systems, explicit estimation of SFO is possible. A decision-directed (DD) estimator was proposed in [10] and its performance shown to be much better than that of the data-aided (DA) estimator [12], where only the information provided by the pilot subcarriers was utilized. After coarse frame and carrier pre-FFT synchronization, the one-shot DD post-FFT estimation takes the form: ϕl,2 − ϕl,1 1 2πM/N K/2 + 1
(23)
where ϕl,(1|2) Al,(1|2)
= =
arg Al,(1|2) , ∗ zl,k a ˆ∗l,k zl−1,k a ˆl−1,k ,
vl . vˆl = vˆl−1 + γ˜
(22)
Most SFO estimators [10], [12], are based on post-FFT processing and assume that the OFDM symbol synchronization for τ is done before SFO estimation. Since no joint estimation is needed, one can easily find that the MCRB equals the ACRB for scalar estimation.
v˜l =
estimation results, the one-shot estimate is further filtered by a first-order closed-loop lowpass filter with step-size γ (25)
Reference [10] presents the linearized mean-square error performance of the closed-loop estimator during tracking: MSE(ˆ v) =
4γ 2 /(2 − γ) . π 2 (M/N )2 K(K + 2)2 Es /No
(26)
To compare the performance of this estimator with MCRB, we follow the standard approach and build an equivalent feedforward synchronizer that assumes a finite length observation vector. Similar to [6, pp.115-140], [7, pp. 340-347], in the equivalent feedforward model, the one-shot estimator outputs are averaged over L symbols: vˆ =
L
v˜l ,
(27)
l=1
and the MSE of this feedforward estimator is found to be 4 . (28) MSE(ˆ v) = 2 π (M/N )2 K(K + 2)2 L2 Es /No Comparing (28) with (26), we find that the closed-loop estimator and the feedforward model achieve the same MSE if √ 2−γ . (29) L= γ This equivalent observation length L is used to compare the performance of the closed-loop scheme with respect to the MCRB. Similar analysis results for the DA estimator [12] can be found in [10]. In the simulations for DD and DA schemes, we have assumed an OFDM system with N = 128 subcarriers and guard interval of length 16. In the case of the DA scheme, there are 10 pilot subcarriers inserted into the 120 QPSK data modulated subcarriers, while no pilot is inserted in the DD scheme. To obtain robust data decisions, we use a rate 1/2 convolutional encoder with generator polynomial (133,171). For each realization, one long packet containing 150 OFDM symbols is assumed. The true value of SFO is 40ppm, and 500 Monte-Carlo runs are conducted for each SNR value. As shown in Fig. 1, the performance of DD estimator is 10dB better than that of DA estimator because the DD estimator utilizes all the data subcarriers. However, the performance of DD estimator is still far away from MCRB(v). This is not a surprising result since (28) indicates that MSE(ˆ v ) of DD estimator is proportional to 1/L 2 and is much larger than MCRB(v) (which is proportional to 1/L 3 , see (22)). V. C ONCLUSIONS
(24)
k∈C(1|2)
zl,k denotes the kth output of the FFT of the l th symbol, a ˆl,k stands for the data decision after a Viterbi decoder, and C1 = [−K/2, −1], C2 = [1, K/2] denote the first and the second half of data subcarriers, respectively. To obtain better
This paper derived the modified Cramer-Rao lower bounds on the estimation of sampling clock frequency offset for both single carrier and multi-carrier systems. These lower bounds are found to be equal to the asymptotic CRB for scalar estimation. For multi-carrier systems, the performance analysis results indicate that the mean-square error of proposed estimators is proportional to 1/L 2 , while the MCRB is
−8
10
DA estimation DD estimation MCRB −9
Mean−Square Error
10
γ=0.07 QPSK −10
10
−11
10
−12
10
−13
10
5
10
15
20
Es/No (dB)
Fig. 1. Performance comparison of SFO estimators for multicarrier OFDM systems with the MCRB(v)
proportional to 1/L 3 . This result suggests looking for more efficient estimators and for techniques to correctly assess the performance of second-order feedback schemes with respect to the MRCB. R EFERENCES [1] A. N. D’Andrea, U. Mengali and R. Reggiannini, “The modified CramerRao bound and its application to synchronization problems,” IEEE Trans. Commun., vol. 42, no. 2-4, Feb./Mar./April, 1994. [2] P. Ciblat, P. Loubaton, E. Serpedin and G. B. Giannakis, “Asymptotic analysis of blind cyclic correlation based symbol rate estimators,” IEEE Trans. Inform. Theory, vol. 48, no. 7, pp. 1922-1934, July 2002. [3] F. M. Gardner, “A BPSK/QPSK timing error detector for sampled receivers,” IEEE Trans. on Comm., vol. 34, no. 5, pp. 423-429, May 1986. [4] F. Gini, R. Reggiannini, and U. Mengali, “The modified Cramer-Rao bound in vector parameter estimation”, IEEE Trans. Commun., vol. 46, no. 1, pp. 52-60, Jan. 1998. [5] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, 1993. [6] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers, Plenum Press, New York, 1997. [7] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, John Wiley & Sons Inc., 1998. [8] M. Moeneclaey, “On the true and modified Cramer-Rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters,” IEEE Trans. Commun., vol. 46, no. 11, Nov. 1998. [9] K. H. Mueller and M. Muller, “Timing recovery in digital synchronous data receivers,” IEEE Trans. Commun., vol. 24. no. 5, pp. 516-531, May 1976. [10] K. Shi, E. Serpedin, and P. Ciblat, “Decision-directed fine synchronization for coded OFDM systems,” IEEE Trans. Commun., Sept. 2003 (submitted). Available online: http://ee.tamu.edu/∼serpedin/papernew.pdf [11] M. Speth, S. A. Fechtel, G. Fock, and H. Meyr, “Optimum Receiver Design for Wireless Broad-Band Systems Using OFDM–Part I,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1668-1677, Nov. 1999. [12] M. Speth, S. A. Fechtel, G. Fock, and H. Meyr, “Optimum Receiver Design for Wireless Broad-Band Systems Using OFDM–Part II,” IEEE Trans. Commun., vol. 49, no. 4, pp. 571-578, April 2001. [13] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications, Artech House, 2000.
