Equalizer-Based Symbol-Rate Timing Recovery for Digital Subscriber Line Systems

Transcription

1 Equalizer-Based Symbol-ate Timing ecovery for Digital Subscriber Line Systems Sven Haar, Dirk Daecke, oland Zukunft, and Thomas Magesacher Institute for Integrated Circuits - BIDGELAB Munich University of Technology Arcisstraße 1, D-89 Munich, Germany Abstract - This paper derives a class of equalizer-based symbolrate timing recovery algorithms, which are to be employed in digital subscriber line (DSL) systems. These algorithms are intended to break the coupling of the adaptive decision-directed timing recovery and the baud-spaced equalizer in tracking mode, once acquisition has been reached. As the derivation is based on a generalized complex-valued system model, the results can be applied for all common DSL systems employing baseband and passband modulation schemes, like PAM, QAM, and CAP. Exemplarily, an algorithm based on the ratio of the largest precursor tap weight and the cursor tap weight is derived for very high data-rate DSL (VDSL) based on QAM transmission. It is shown by means of simulations, that destructive coupling is prevented. A simple modification of the complex-valued equalizer structure with cursor-normalized tap weights allows an easy implementation of the derived timing recovery algorithm. I. INTODUCTION Commonly, an adaptive equalizer is employed in digital subscriber line (DSL) receivers, as e.g. in SHDSL and VDSL systems, to mitigate the influence of intersymbol interference (ISI), which is caused by channel distortion, and additive noise. Equalization at symbol rate significantly reduces the overall implementation costs of a DSL receiver, compared to fractionally-spaced equalization, as the functional blocks feeding the equalizer have to generate output signals only at symbol rate. For baseband DSL systems, as e.g. SHDSL, the complexity of the echo canceller, which dominates implementation costs, can be reduced by more than one half. The drawback of symbolspaced equalization is possibly a slightly lower system performance and being of higher concern throughout this paper the need for a high accuracy of sampling phase synchronization [1]. Most timing recovery (T) algorithms make, in some form, use of the derivative of the received signal. But a suitable approximation of a signal s derivative requires at least two samples per symbol period T for an excess bandwidth of < r < 1, caused by commonly used (root) raised cosine filters with roll-off r []. An important class of symbol-rate timing error detectors (TED) derives the timing information from the relative magnitude of precursor and postcursor samples [3]-[6]. They perform well in acquisition mode, when the TED is fed with input samples of the non-converged adaptive equalizer. In tracking mode, however, a problem arises, when for ñ k e ϕ a k x k h( τϕ, ) LEQ c y k z k Fig. 1. System model in complex baseband representation. reasons of self-noise reduction the output signal of the adaptive equalizer is used as input for the TED. If the equalizer is flexible enough to compensate for small timing delays, unpredictable interaction between adaptive T and equalizer is generally inevitable [4], [7]. Commonly, this problem is solved by freezing the equalizer adaptation process, once acquisition has been achieved. But then the equalizer cannot track changes of channel and noise characteristics anymore. In this paper, we derive a TED directly from the adaptive equalizer with the intention to avoid the interaction problem in tracking mode. As timing information is contained in the sampled signals at receiver side, the basic idea is that timing information can also be obtained from the equalizer s tap weights. Since our derivation is based on a generalized complex-valued system model, the results can be applied to all common DSL baseband and passband transmission schemes like PAM, QAM, and CAP. II. SYSTEM AND ECEIVE MODEL The applied system model is shown in Fig. 1. The complexvalued transmit symbols { a k } are filtered by the equivalent discrete-time channel h( τϕ, ) which represents transmit and receive filters, twisted pair loop, and the receiver s digital matched filter (MF), which runs on high sampling rate 1 T S to satisfy the sufficient statistics condition []. The matched filter is followed by decimation at symbol rate 1 T at time instants t= kt + τ. Thus the equivalent channel h depends on the sampling phase τ. MF-filtered noise ñ k is added at the input of the LEQ. For QAM transmission, demodulation from carrier frequency f C to baseband will cause an unknown carrier phase offset ϕ. As we are interested in deriving a timing algorithm for post-acquisition period only, we can expect that acquisition of ϕ has been achieved and that the imum carrier phase offset ϕ is known. Thus the carrier phase offset can be completely compensated by multiplication with e ϕ. - FBF b â k

2 III. SYSTEMATIC DEIVATION OF TIMING ECOVEY ALGO- ITHMS WITH MAXIMUM LIELIHOOD ESTIMATION TECH- NIQUES For clarity, the presented derivation will focus on the linear feedforward equalizer (LEQ). In section IV it will be shown by means of simulations that the derived timing algorithms can also be applied for a nonlinear decision feedback equalizer employing the feedback filter (FBF), shown in Fig. 1. To drop the phase rotation term we set ϕ =. Then, we obtain z k = y k. The LEQ is represented by the coefficient set { c }, with [ M, N] and M, N. The cursor coefficient, i.e. the LEQ s tap weight of highest magnitude, will be denoted as c. Then the LEQ causes a delay of M symbol periods. The cascade of the equivalent discrete-time channel h( τ) and the LEQ can be represented by a single equivalent filter with impulse response q n ( τ) = c ( τ) h n ( τ). (1) The timing phase dependent slicer input z k is z k ( τ) = q i ( τ) a k i + n k, () i where n k is MF- and LEQ-filtered gaussian noise. For the timing algorithm derivation, we assume correct decisions and unbiased equalization ( â k = a k ). Since we focus on the tracking mode, the estimate τˆ for timing correction is close to the imum sampling phase τ. Following approaches of [] and [8], we start with the likelihood function for a given sequence of components, which is determined by p( z τ) 1 1 = exp z. (3) k ( τ) â k πσ n σ n k = 1 Taking the natural logarithm of the likelihood function, we obtain the log-likelihood function L( τ) 1 1 L( τ) = ln (4) z k ( τ) â k πσ n σ n k = 1 Since we are only interested in terms dependent on τ, we simplify L( τ) and obtain the obective function L 1 ( τ) L 1 ( τ) = z k ( τ) e{ â k z k ( τ) }. (5) k = 1 k = 1 To maximize the log-likelihood function, we differentiate the obective function L 1 ( τ). As the first term of (5) depends only weakly on τ compared to the second term, its derivative with respect to τ is negligible. Then, we obtain L1 ( τ) = e â τ k q ia k i + n' k k = 1 i, (6) with n' k = ċ i ( τ)ñ and and denoting the partial i k i ċ q i i derivatives of q i and c i with respect to τ, respectively. Since the summation will be performed in the loop filter, we drop the summation over to obtain timing error signal e τ at time kt [] e τ ( kt) e â k q ia k i + n' = k. (7) i This timing error signal exhibits the general structure of schemes which are based on the cross correlation of a noisy sequence and a noiseless reference sequence (in our case the detected symbols â k ) [7]. To obtain the stable tracking point of the error signal e τ ( kt) we take the expected value over data and noise. For an independent and identically distributed data sequence, we get the timing function f ( τ) f ( τ) = E[ e τ ( kt) ] = E[ a k ] e{ }. (8) q The derivative of q can be written as =. (9) q ċ h + c ḣ In tracking mode, sampling phase τ is close to the imum sampling phase τ. The magnitudes of precursor and postcursor samples h i vary almost linearly relative to a change of τ [4]. Likewise, the respective imum equalizer tap weights change almost linearly with τ. These linear characteristics are a common property of almost any digital subscriber loop. Due to this linearity, the derivatives ċ and ḣ are approximately constant for τ close to τ. So f ( τ) can be expressed as f ( τ) = const e α h ( τ) + β c ( τ). (1) The first summation term of timing function f ( τ) represents the general form of baud-rate timing functions which derive the timing phase from the relative magnitude of the precursor h 1 [5], [6], postcursor h 1 [9], or a combination of precursor and postcursor weights [3], [4]. If the timing error signal is obtained by differentiation of a likelihood function, the derivative of the input sequence or the derivative of the impulse response of the received signal (before or after equalization) is needed. If the TED operates at symbol rate, the derivative of its input sequence cannot be determined without additional information. Characteristics of the gradient or derivative ḣ or ċ have to be known in order to obtain a timing function without differentiating the input sequence. IV. DEMONSTATION FO A QAM-BASED VDSL SYSTEM A. DSL System and Scenario A QAM-based VDSL system has been selected to illustrate and verify the derivation results of section III. Exemplarily, a 5m VDSL1-TP1 loop [1], [11], i.e. a one-segment.4mm underground cable, has been chosen. Transmit and noise power spectral densities are 6 dbm/hz and

3 1 dbm/hz, respectively. The transmission scheme is 64QAM on the lower upstream frequency band ( MHz) of ETSI s standard plan [1], [11]. Transmit and receive root raised cosine filters with roll-off r =% are used. Within the receiver a decision feedback equalizer (DFE), adapted by the least mean square (LMS) algorithm, with 1 symbol-spaced feedforward taps and 1 feedback taps is employed. An important DFE design parameter is the position of the cursor tap weight c. Evaluations for the described and similar scenarios by analytical means of [1] have shown the best results for N =, meaning that the feedforward tap closest to the slicer is the cursor tap. This lowers the accuracy requirements for the T, because the maximum achievable signal-tonoise-ratio SN ( τ) at the slicer input remains almost constant for small sampling phase offsets τ = τ τ, as shown in Fig.. This figure also illustrates that for the presented scenario symbol-spaced equalization causes no principle performance degradation compared to fractionally-spaced equalization. The matched filter bound SN is 34.9 db. B. Equalizer-Based Timing ecovery Algorithm As we are interested in deriving an equalizer-based T algorithm, we focus on the right-hand sum of (1) and neglect the other sum. Then, the generalized timing function is defined by f 1 ( τ) = e β c ( τ). (11) As already mentioned in section III, a working T algorithm can be designed by a suitable combination or subset of different products { β k c k ( τ) }, k { } based on additional information about the channel characteristics. Since some symbol rate T algorithms [5], [6] for DSL systems exploit the change of the relative magnitude of channel precursor h 1 ( τ) with respect to channel cursor h ( τ) within the left-hand sum of (1), we will derive a similar criterion based on the right-hand sum of (1), with the difference that the TED input does not have to be fed with samples from the equalizer. Some algorithms of this kind have been introduced before, e.g. in [13] and [14], but based on rather ad-hoc methods for specific scenarios, lacking a generalized approach. The restriction to the equalizer s precursor tap weight c 1 ( τ) and cursor tap weight c ( τ) simplifies the timing function to f ( τ) = e { c 1 ( τ) + β c ( τ) }, with β = β β 1. (1) Division of (1) by c ( τ) does not change the timing function s stable tracking point, but makes it look more convenient f 3 ( τ) e c 1( τ) β + c ( τ) e c 1( τ) = = c ( τ) + β,, (13) with β, = e{ β }. Taking the real part of the complex division term, (13) can be expressed differently in cartesian SN in db symbol-spaced DFE DFE with T/-spaced forward filter τ Sampling phase τ Fig.. Maximum achievable SN as a function of sampling phase τ and the sampling factor of the 1 tap LEQ filter (for imum DFE tap weights { c () τ, b () τ } and no decision error propagation) T 3 1 τ 1 T c ( τ) max τ { c ( τ) } c c c Sampling phase τ Fig Normalized magnitude of the LEQ s cursor tap weight and normalized complex-valued precursor tap weights as a function of the sampling phase (solid=e{.}, dashed=im{.}). representation: c 1, ( τ)c, ( τ) + c 1, I ( τ)c, I ( τ) f 3 ( τ) = β, (14) c ( τ) +, with c i, ( τ) = e{ c i ( τ) } and c i, I ( τ) = Im{ c i ( τ) }. It can be observed from (14), that the timing function consists of an offset-term β, and a linear combination of the real and imaginary parts of the complex-valued sampling phase dependent precursor and cursor equalizer tap weights scaled by the sampling phase dependent magnitude of cursor weight c. C. Modified Equalizer Structure for Simplified Implementation Since the denominator in (14) depends on τ, we can not formally omit it. Because a division operation is generally highly undesirable for digital implementation, we will propose a different and more convenient way to use the equalizer s coefficients for T in QAM transmission schemes. The LEQ performs linear multiply and accumulate opera- τ

4 tions. Their order can be changed. Let the ( M + N + 1) -element row-vector c represent the LEQ s coefficient set. Then c can be written as c = { c M, c M + 1,, c,, c N }, (15) = { c M, c M + 1,, 1,, c N } c e arg{ c } = c c e arg{ c }, (15) with c i = c i c. For QAM transmission, the phasor multiplication e arg{ c } can be combined with the carrier phase offset multiplication to e ( arg { c } ϕ ). As depicted in Fig. 3, c is nearly constant for the applied scenario for a wide range of τ around the imum sampling phase τ. Thus the scalar adaptive tap gain multiplication might be omitted in tracking mode. Although our initial motivation for a modified equalizer structure, which was to avoid the division by c ( τ), has thus become partly obsolete, we will further apply the modified structure. Then, the timing function can be obtained very easily from the normalized equalizer s precursor tap weight c 1 ( τ) : f 3 ( τ) e c 1( τ) = (16) c ( τ) + β, = c 1, ( τ) + β, D. Determining the Unknown Offset Parameter For unbiased timing error estimation, the offset parameter β, must be chosen so that f 3 ( τ ) = c 1, ( τ ) + β, =. (17) Then the imum offset parameter is determined by the imum precursor tap weight at imum sampling phase: β, = c 1, ( τ ). (18) For the applied scenario, the in minimum mean squared error sense imum coefficient sets { c ( τ), b( τ) } have been calculated by analytical means of [1]. The normalized precursors tap weights of highest impact are shown in Fig. 3. For the presented scenario it can be seen that β, = -.9. E. TED Characteristics In general, an offset parameter that depends on unknown channel and scenario characteristics is very restrictive for implementation, even more if it is related to the imum tap weight of an adaptive equalizer. However, as c 1 ( τ) represents a c ( τ) -normalized coefficient, we make use of a linear combination of two complex-valued equalizer tap weights and determine their relative magnitude as a function of the sampling phase. Any β, c 1, ( τ ) will result in a timing bi- as. Our evaluations for various VDSL scenarios have shown that β, depends mostly on the twisted-pair cable s signal attenuation level and is approximately in the range of [-.; ] for loop lengths ranging from m to 1m. The attenuation level can easily be obtained from the receiver s automatic gain control (AGC) circuit. Then β, can be chosen suitably from a constrained set of values. Furthermore, if an FBF is employed, the maximum achievable SN ( τ) is nearly constant around τ, so that an approximate β, β, will mean a negligible overall system performance loss. The forward tap weights of the DFE are adusted by means of the LMS algorithm once every symbol period according to c i ( k + 1) = c i ( k) + µx i ( k)e D ( k), (19) with e D ( k) = â( k) - zk ( ) and step size µ. Due to the recursive nature of (19), the estimates are effectively averaged during the course of adaptation [15]. Thus the equalizer coefficients, which are used to derive timing information, already represent a mean value. The coefficients fluctuations around their mean values, which will cause a timing itter for the proposed algorithm, are influenced by the step size of the LMS algorithm. Details on the fluctuation behavior can be found in [15] and [16]. For real DSL systems, most likely a simplified LMS algorithm, like e.g. the sign-sign version, with adaptive step size will be employed, so that the coefficient fluctuations depend on the state of convergence. F. Simulation esults Time-discrete tracking-mode simulations have been performed for the described scenario based on the modified equalizer structure which has been presented in subsection C. DFE, T, and carrier phase recovery (suspending the simplified assumptions on ϕ, taken in section III) have been employed in their adaptive modes. The conventional receiver uses equalized and detected samples for timing recovery by means of the M&M TED ( f MM ( τ) = e{ z k 1 ( τ)â k z k ( τ)â k 1 }) [], [3] in combination with a second order loop filter (normalized loop bandwidth = 1 4, damping factor = 1.) [17]. As shown in figures 4-6, T and adaptive DFE interact destructively. The TED s output signal shows a constant zero mean for frozen (in figures 4-6 labeled as T MM ) and adaptive equalizer (labeled as T MM +EQ), as depicted in Fig. 4. But for the latter scenario, the sampling phase drifts due to the equalizer adaptation, as shown in Fig. 5. The resulting SN degradation in db, which is equivalent to the negative MSE, is shown in Fig. 6. The system can be stabilized by limiting the flexibility of either the T or the adaptive equalizer. If the equalizer-based T algorithm (labeled as T EQ +EQ) of (16) is employed (with β, ), no destructive interaction occurs, as shown in Fig. 5 and Fig. 6. Note, that a itter comparison based on Fig. 5, showing the sampling phase over time, is not too meaningful, because the tracking characteristics of the M&M TED and the equalizerbased TED might be different. To treat the dynamic behavior of the equalizer-based T algorithm, more sophisticated analysis is needed.

5 TED-Output MSE in db τ T Fig M&M-TED-Output (= loop filter input) as a function of time. τ ---- T T MM T MM +EQ T MM T EQ +EQ symbols Fig Sampling phase normalized to symbol period T over time Fig Mean square error (MSE) (= -SN) in db as a function of time. (Length of averaging gliding window: 1 symbols) V. SUMMAY AND CONCLUSIONS T MM +EQ symbols 1 3 symbols T MM +EQ T MM T EQ +EQ We have derived an equalizer-based class of symbol-rate timing recovery algorithms for DSL systems. As a generalized complex-valued system model is used, the derivation can be applied to baseband and passband modulation schemes. The derived timing algorithms consist of a suitable linear combination of the equalizer s tap weights. Additional knowledge on the channel characteristics is needed to design a working timing function. Exemplarily, it is shown for a VDSL system with QAM transmission, how to determine these unknown parameters. The implementation of the proposed timing recovery algorithm can be simplified by employing a slightly modified equalizer structure based on cursor-normalized tap weights. Exemplary simulation results show that the use of an equalizer-based timing recovery algorithm prevents the destructive coupling of decision-directed timing recovery and adaptive symbol-spaced equalizer in tracking mode. EFEENCES [1] S.U.H. Qureshi, Adaptive equalization, in Proceedings of the IEEE, vol. 73, no. 9, Sept. 1985, pp [] H. Meyr, M. Moeneclaey, and S.A. Fechtel, Digital communication receivers, John Wiley & Sons, [3].H. Mueller and M. Müller, Timing recovery in digital synchronous data receivers, IEEE Trans. on Comm., vol. 4, pp , May [4] P. Gysel and D. Gilg, Timing recovery in high bit-rate transmission systems over copper pairs, IEEE Trans. on Comm., vol. 46, no. 1, pp , Dec [5] A.M. Gottlieb, P.M. Crespo, J.L. Dixon, and T.. Hsing, The DSP implementation of a new timing recovery technique for high-speed digital data transmission, in Proc. Int. Conf. on Acoust., Speech, and Sig. Proc., Albuquerque, NM, 199, pp [6] C.P.J. Tzeng, D.A. Hodges, and D.G. Messerschmitt, Timing recovery in digital subscriber loops using baudrate sampling, IEEE J. Select. Areas Comm., vol. 4, pp , Nov [7] J.W.M. Bergmanns and H.W. Wong-Lam, A class of data-aided timing recovery schemes, IEEE Trans. on Comm., vol. 43, no. -4, pp , Apr [8] U. Mengali and A.N. D Andrea, Synchronization techniques for digital receivers, Plenum Press, [9].W. Lucky, J. Salz, and E.J. Weldon Jr., Principles of data communication, McGraw-Hill, New York, [1] T1E1.4, VDSL metallic interface - part1: functional requirements and common specification, Draft Trial-Use Standard, Feb. 1. [11] ETSI TM, VDSL - part1: transceiver specification, TS v1.1.3, Sept.. [1] N. Al-Dhahir and J.M. Cioffi, MMSE decision-feedback equalizers: finite length results, IEEE Trans. Inform. Theory, no. 7, pp , July [13] C.A. Ehrenbard and M.F. Tompsett, A baud-rate line-interface for twowire high-speed digital subscriber loops, in Proc. Global Telecomm. Conf., Miami, FL, 198, pp [14] S. Haar,. Zukunft, and F. Vogelbruch, A timing recovery criterion derived from the tap weights of a decision feedback equalizer for QAM digital subscriber line systems, in Proc. 14th Int. Conf. on Dig. Sig. Proc., Santorini, Greece, July, pp [15] S. Haykin, Adaptive Filter Theory, Prentice Hall, [16] O. Macchi, Adaptive Processing, John Wiley & Sons, [17] H. Meyr and G. Ascheid, Synchronization in digital communications, John Wiley & Sons, 199.