A digital radio  (Page 3/8)

Some codes are better than others. How can we tell?

To be concrete, let

• the symbol interval $T$ be the time between successive symbols, and
• the pulse shape $p\left(t\right)$ be the shape of the pulse that will be transmitted.

For instance, $p\left(t\right)$ may be the rectangular pulse

which is plotted in [link] . The transmitter of [link] is designed so that every $T$ seconds it produces a copy of $p(·)$ that is scaled by the symbol value $s[·]$ . A typical output of the transmitterin [link] is illustrated in [link] using the rectangular pulse shape. Thus the first pulse begins at some time $\tau$ and it is scaled by $s\left[0\right]$ , producing $s\left[0\right]p(t-\tau )$ . The second pulse begins at time $\tau +T$ and is scaled by $s\left[1\right]$ , resulting in $s\left[1\right]p(t-\tau -T)$ . The third pulse gives $s\left[2\right]p(t-\tau -2T)$ , and so on. The complete output of the transmitter is the sum of allthese scaled pulses:

Since each pulse ends before the next one begins, successive symbols should not interfere with each other at the receiver.The general method of sending information by scaling a pulse shape with the amplitude of the symbolsis called Pulse Amplitude Modulation (PAM).When there are four symbols as in [link] , it is called 4-PAM.

For now, assume that the path between the transmitter and receiver, which is often called the channel , is “ideal.” This implies that the signal at the receiver is the same as the transmittedsignal, though it will inevitably be delayed (slightly) due to the finite speed of the wave, and attenuated by the distance.When the ideal channel has a gain $g$ and a delay $\delta$ , the received version of the transmitted signal in [link] is shown in [link] .

There are many ways that a real signal may change as it passes from the transmitter to receiver through a real(nonideal) channel. It may be reflected from mountains or buildings. It may be diffracted as it passes through theatmosphere. The waveform may smear in time so that successive pulses overlap.Other signals may interfere additively (for instance, a radio station broadcasting at the same frequency in adifferent city). Noises may enter and change the shape of the waveform.

There are two compelling reasons to consider the telecommunication system in the simplified (idealized) case before worrying aboutall the things that might go wrong. First, at the heart of any working receiver is a structurethat is able to function in the ideal case. The classic approach to receiver design (and also the approachof Software Receiver Design ) is to build for the ideal case and later to refine so that the receiver will still work when bad things happen.Second, many of the basic ideas are clearer in the ideal case.

The job of the receiver is to take the received signal (such as that in [link] ) and to recover the original text message. This can be accomplished by an idealizedreceiver such as that shown in [link] . The first task this receiver must accomplish is to sample the signal to turn it intocomputer-friendly digital form. But when should the samples be taken? Comparing [link] and [link] , it is clear that if the received signal were sampled somewhere near the middle of eachrectangular pulse segment, then the quantizer could reproducethe sequence of source symbols. This quantizer must either