A digital radio  (Page 4/8)

1. know $g$ so the sampled signal can be scaled by $1/g$ to recover the symbol values, or
2. separate $\pm g$ from $\pm 3g$ and output symbol values $\pm 1$ and $\pm 3$ .

Once the symbols have been reconstructed, then the original message can be decodedby reversing the association of letters to symbols used at the transmitter (for example, by reading [link] backwards). On the other hand, if the samples were taken at the moment oftransition from one symbol to another, then the values might become confused.

To investigate the timing question more fully, let $T$ be the sample interval and $\tau$ be the time the first pulse begins. Let $\delta$ be the time it takes for the signal to move from the transmitter to the receiver.Thus the $(k+1)$ st pulse, which begins at time $\tau +kT$ , arrives at the receiver at time $\tau +kT+\delta$ . The midpoint of the pulse, which is the best time to sample,occurs at $\tau +kT+\delta +T/2$ . As indicated in [link] , the receiver begins sampling at time $\eta$ , and then samples regularly at $\eta +kT$ for all integers $k$ . If $\eta$ were chosen so that

then all would be well. But there are two problems: the receiverdoes not know when the transmission began, nor does it know how long it takes for the signalto reach the receiver. Thus both $\tau$ and $\delta$ are unknown!

Somehow, the receiver must figure out when to sample.

Basically, some extra “synchronization” procedureis needed in order to satisfy [link] . Fortunately, in the ideal case, it is not reallynecessary to sample exactly at the midpoint; it is necessary only to avoid the edges. Even if the samples are nottaken at the center of each rectangular pulse, the transmitted symbol sequence can still be recovered.But if the pulse shape were not a simple rectangle, then the selection of $\eta$ becomes more critical.

How does the pulse shape interact with timing synchronization?

Just as no two clocks ever tell exactly the same time, no two independent oscillators Oscillators, electronic components that generate repetitive signals,are discussed at length in Chapter [link] . are ever exactly synchronized. Since the symbol period at the transmitter, call it $T_{trans}$ , is created by a separate oscillator from thatcreating the symbol period at the receiver, call it $T_{rec}$ , they will inevitably differ.Thus another aspect of timing synchronization that must ultimately be considered is how to automaticallyadjust $T_{rec}$ so that it aligns with $T_{trans}$ .

Similarly, no clock ticks out each second exactly evenly. Inevitably, there is some jitter, or wobble in thevalue of $T_{trans}$ and/or $T_{rec}$ . Again, it may be necessary to adjust $\eta$ to retain sampling near the center of the pulse shape as the clock times wiggle about.The timing adjustment mechanisms are not explicitly indicated in thesampler box in [link] . For the present idealized transmission system,the receiver sampler period and the symbol period of the transmitter are assumed to be identical(both are called $T$ in Figures [link] and [link] ) and the clocks are assumed to be free of jitter.

What about clock jitter?

Even under the idealized assumptions above, there is another kind of synchronization that is needed.Imagine joining a broadcast in progress, or one in which the first $K$ symbols have been lost during acquisition.Even if the symbol sequence is perfectly recovered after time $K$ , the receiver would not know which recoveredsymbol corresponds to the start of each frame. For example, using the letters-to-symbol code of [link] , each letter of the alphabet is translated into a sequence of four symbols.If the start of the frame is off by even a single symbol, the translation from symbols back into letterswill be scrambled. Does this sequence represent $a$ or $X$ ?