0.11 Timing recovery  (Page 12/11)

All we have to decide is what to do with the time given us.

—Gandalf, in J. R. R. Tolkien's Fellowship of the Ring

When the signal arrives at the receiver, it is a complicated analog waveform that must be sampled in order to eventuallyrecover the transmitted message. The timing offset experiments of [link] showed that one kind of “stuff” that can “happen” to the received signal is that thesamples might inadvertently be taken at inopportune moments. When this happens, the “eye” becomes “closed” and thesymbols are incorrectly decoded. Thus there needs to be a way to determine when to take the samples at the receiver. In accordance with the basic system architectureof Chapter [link] , this chapter focuses on baseband methods oftiming recovery (also called clock recovery). The problem is approached in a familiar way:find performance functions which have their maximum (or minimum) at the optimal point (i.e., at the correct sampling instantswhen the eye is open widest). These performance functions are then used to define adaptive elementsthat iteratively estimate the sampling times. As usual, all other aspects of the system are presumedto operate flawlessly: the up and down conversions are ideal, there are no interferers, and the channelis benign.

The discussion of timing recovery begins in "The Problem of Timing Recovery" by showing how a sampled version of the received signal $x\left[k\right]$ can be written as a function of the timing parameter $\tau$ , which dictates when to take samples. "An Example" gives several examples that motivate several different possible performance functions,(functions of $x\left[k\right]$ ) which lead to “different” methods of timing recovery.The error between the received data values and the transmitted data (called the source recovery error ) is an obvious candidate, but it can be measured only when the transmitted data are knownor when there is an a priori known or agreed-upon header (or training sequence).An alternative is to use the cluster variance , which takes the square of the difference between the received data valuesand the nearest element of the source alphabet. This is analogous to the decision directed approachto carrier recovery (from [link] ), and an adaptive element based on the cluster varianceis derived and studied in "Decision-Directed Timing Recovery" . A popular alternative is to measure the power of the $T$ -spaced output of the matched filter. Maximizing this power (by choice of $\tau$ ), also leads to a good answer, and an adaptive element based on output powermaximization is detailed in "Timing Recovery via Output Power Maximization" .

In order to understand the various performance functions, the error surfaces are drawn. Interestingly, in many cases,the error surface for the cluster variance has minima wherever the error surfacefor the output power has maxima. In these cases, either method can be used as the basis for timingrecovery methods. On the other hand, there are also situations when the errorsurfaces have extremal points at different locations. In these cases, the error surface provides a simple wayof examining which method is most fitting.