Symbol Timing Recovery

10.3. Synchronization 201 10.3.4 Symbol Timing Recovery Once carrier synchronization is acheived, we need to find correct sampling instances for the sampling of symbols according to Chapter 5. This is what this section is about. There is a multitude of different methods around for this purpose, and we will only consider a few in some detail. According to the introduction in Mueller & Müller [12], early on in history, symbol synchronization for one-dimensional signalling was done using various analog methods. That would be up to the first half of the 1970s. Franks [3] describes several of those. Some were based on threshold crossings, i.e. when the output of the matched filter at hand passes a threshold somewhere between two sampling instances. If the signal passes several thresholds, then the mean of allthose time instances wouldbe used. This wouldthen markthe time instance mid-way between two sampling instances. Some methods used the derivative of the signal, which is correlated to the estimated data. Yet some were based on non-linear preprocessing of the signal, resulting in a signal containing a sinusoidal that is related to the actual symbol clock, which is then extracted using a narrow BP-filter. The first discrete time method for symbol synchronization was published in 1976 by Mueller & Müller [12]. It uses the intended symbol samples and the corresponding detected symbol values, i.e. it uses one sample per symbol interval. From those, it generates a discrete time signal that can be used to adjust the timing of future samples. This started a series of published discrete time synchronization methods using one or a few samples per symbol interval, some of which are simple modifications of the Mueller & Müller method, and some that are unique ideas of their own but seemingly inspired by Mueller & Müller. In the following, we will describe the original method of Mueller & Müller, together with some modifications of the method, which primarily make the method useful also for two and more dimensions. Additionally, we will describe another basic method that many have built upon, published in 1986 by Gardner [4], which uses two samples per symbol interval, but does not use detected symbol values. Mueller & Müller Original Idea Consider communicating using antipodal signalling, as in Section 10.2, with the two signals s 0 (t) = A, 0 t < T, and s 1 (t) = A, 0 t < T. Then we have an eye pattern as in Figure 10.3 on Page 188, and the ideal sampling instance for the k-th signal interval is at t = kt. Now, if our actual sampling instance for the k-th signal interval is something else than that then

202 Chapter 10. Practical Aspects the effective distance to the threshold 0 is reduced in half of the cases, resulting not only in increased symbol error probability, but also in Inter-Symbol Interference, which is covered in more detail in Chapter 7. Let x k be the sample at the k-th sampling instant, and let a k be the corresponding quantized value. In our antipodal example with threshold 0, we have a k = sgn(x k ). The method proposed by Mueller & Müller [12] suggests that we should use an appropriately scaled version of k = x k a k 1 x k 1 a k (10.1) to update the next sampling instance. Actually, they also gives a more general expression involving more sampling instances, but this is the only one that they state explicitly, and for which they give examples of implementation. This is therefore usually what is meant by Mueller & Müller timing recovery. Let us analyze this for the binary situation given above, assuming no added noise. First, see Figure 10.16for the case where the sampling takes place too early. The transitions between one sample and the next depends on three consecutive symbols. When we sample too early, these are the symbols in symbol intervals number k 2, k 1 and k. Each of these symbols can take any of two values, whichmeansthatwehave2 3 = 8possibletransitions. Forfourofthem, wehave k = 0. Those transitions are indicated in Figure 10.16a. For the other four transitions, we have k > 0. Those transitions are indicated in Figure 10.16b. This means that the expected value of k is positive if we sample too early. Second, see Figure 10.17 for the case where the sampling takes place too late. Again, there are 2 3 = 8 possible transitions. When we sample too late, these are the symbols in symbol intervals number k 1, k and k +1. Again, there are four transitions, for which we have k = 0. Those transitions are indicated in Figure 10.17a. And for the other four transitions, we have k < 0. Those transitions are indicated in Figure 10.16b. This means that the expected value of k is negative if we sample too late. For the signals that we have considered here, all transitions from one ideal sampling instance in the eye pattern to the next are straight lines. This means that the expected value of k is proportional to the sampling instance error. Correcting the sampling interval with a value that is proportionalto k is then reasonable. Let us consider the reasonable situation where there is added WGN with zero mean to the received signal, which after the initial matched filter is no longer white, but still Gaussian with zero mean. The result is then that the timing correction term k is also subject to zero-mean noise. This means that

10.3. Synchronization 203 (a) k 1 k (b) k 1 k Figure 10.16: An eye pattern for a binary modulation scheme, with sampling instances k 1 and k, where the samples are taken too early. The eight possible transitions depend on symbol intervals k 2, k 1 and k. (a) The four transitions that result in k = 0. (b) The remaining four transitions, resulting in k > 0. (a) k 1 k (b) k 1 k Figure 10.17: An eye pattern for a binary modulation scheme, with sampling instances k 1 and k, where the samples are taken too late. The eight possible transitions depend on symbol intervals k 1, k and k +1. (a) The four transitions that result in k = 0. (b) The remaining four transitions, resulting in k < 0.

204 Chapter 10. Practical Aspects the expected value ofthe resulting errordue to this noiseis zero. The effect this noise has on the timing recovery is that the actual timing will fluctuate even when the timing recovery has converged. Strictly speaking, this added noise is no longer Gaussian due to the involved nonlinear mathematical operations. Mueller & Müller [12] performs a thorough analysis of this timing recovery, which is fairly intricate and outside the scope of this presentation. Mueller & Müller Modifications Mueller& Müller s original method is intended for one-dimensional modulation, i.e. various versions of ASK (see Section 6.3.3). As for PSK, it is only intended for BPSK. Cowley & Sabel [1] realized that by adjusting Equation 10.1 slightly, we get a version of Mueller & Müller s method that works for any M-PSK constellation. The new equation that is used to update the timing is then k = Re{x k a k 1 x k 1 a k}. (10.2) As we have seen in Figures 10.16 and 10.17, there are symbol sequences that do not update the timing at all if there is no noise. The only thing that would happen in a real situation is that the channel/receiver noise would cause an update that is completely independent of the communicated signal. Jennings & Clarke [8], as well as Jin et. al. [9] suggest that it is unnecessary to actually update the timing if the detected symbols constitute such a sequence. The resulting variance of the timing error is then reduced compared to if we always update the timing. Hang & Renfors [7] adjusts Mueller & Müller s method for QAM, while Louveaux et. al. [11] adjusts the method for multi-dimensional signalling. Gardner s Method Gardner s method [4] is intended for BPSK and QPSK. The description here is for QPSK, with BPSK as a special case at the end. QPSK is a two-dimensional modulation method that easily lends itself to a baseband representation. Then we have an in-phase signal and a quadrature signal, which are outputs from matched filters. The method is based on sampling each of them twice per symbol interval, one of them is the intended sample for one symbol and the other is a sample half-way through the sampling interval. Let x I,k,1 and x Q,k,1 denote the samples half-way through the k-th symbol interval, and let x I,k,2 and x Q,k,2 denote the samples at the end of the k-th symbol interval. Gardner s method uses the equation k = x I,k,1 (x I,k,2 x I,k 1,2 )+x Q,k,1 (x Q,k,2 x Q,k 1,2 ). (10.3)

10.3. Synchronization 205 It should be noted that k above is independent of the phase-shift of the I/Qdemodulator. This means that this method can be used to synchronize the symbol timing even before the phase synchronization has converged. For BPSK, assuming modulation in the I-component, there is no information in the Q-component once the phase synchronization has converged. Thus, once the phase synchronization has converged, Equation 10.3 can be replaced by the simpler k = x I,k,1 (x I,k,2 x I,k 1,2 ), (10.4) since the Q-component only contains noise in that case.

206 Chapter 10. Practical Aspects

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208 Bibliography [10] Rolf Johannesson and Kamil Sh. Zigangirov. Fundamentals of Convolutional Coding. IEEE Press, New York, 1999. [11] J. Louveaux, L. Cuvelier, L. Vandendorpe, and T. Pollet. Baud Rate Timing Recovery Scheme for Filter Bank-Based Multicarrier Transmission. IEEE Transactions on Communications, 51(4):652 63, April 2003. [12] Kurt H. Mueller and Markus Müller. Timing Recovery in Digital Synchronous Data Receivers. IEEE Transactions on Communications, 24(5):516 31, May 1976. [13] Mikael Olofsson. Signal Theory. Studentlitteratur, Lund, Sweden, 2011. [14] Claude E. Shannon. A Mathematical Theory of Communication. Bell System Technical Journal, 27:379 423 and 623 656, July and October 1948. [15] Andrew J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory, 13(2):260 269, April 1967. [16] Wikipedia. Decibel. http://en.wikipedia.org/wiki/decibel. Visited 2013-05-27.