A digital radio  (Page 5/8)

Thus proper decoding requires locating where the frame starts, a step called frame synchronization.Frame synchronization is implicit in [link] in the choice of $\eta$ , which sets the time $t$ ( $=\eta$ with $k=0$ ) of the first symbol of the first (character) frame of the message of interest.

How to find the start of a frame?

In the ideal situation, there must be no other signals occupying the same frequency range as the transmission.What bandwidth (what range of frequencies) does the transmitter [link] require? Consider transmitting a single $T$ -second wide rectangular pulse. Fourier transform theory shows that any such time-limitedpulse cannot be truly band limited, that is, cannot have its frequency content restricted to a finite range.Indeed, the Fourier transform of a rectangular pulse in time is a sinc function in frequency(see [link] in [link] ).The magnitude of this sinc is overbounded by a function that decays as the inverse of frequency(peek ahead to [link] ). Thus, to accommodate this single pulse transmission,all other transmitters must have negligible energy below some factor of $B=1/T$ . For the sake of argument, suppose that a factor of 5 is safe, that is,all other transmitters must have no significant energy within $5B$ Hz. But this is only for a single pulse. What happens whena sequence of $T$ -spaced, $T$ -wide rectangular pulses of various amplitudes is transmitted?Fortunately, as will be established in [link] , the bandwidth requirements remain about the same, at leastfor most messages.

What is the relation between the pulse shape and the bandwidth?

One fundamental limitation to data transmission is the trade-off between the data rate and thebandwidth. One obvious way to increase the rate at which data are sentis to use shorter pulses, which pack more symbols into a shorter time. This essentially reduces $T$ . The cost is that this would require excludingother transmitters from an even wider range of frequencies since reducing $T$ increases $B$ .

What is the relation between the data rate and the bandwidth?

If the safety factor of $5B$ is excessive, other pulse shapes that would decay faster as afunction of frequency could be used. For example, rounding the sharp corners of a rectangularpulse reduces its high frequency content. Similarly, if other transmitters operated at high frequenciesoutside $5B$ Hz, it would be sensible to add a low pass filter at the front end of the receiver.Rejecting frequencies outside the protected $5B$ baseband turf also removes a bit of the higher frequency content ofthe rectangular pulse. The effect of this in the time domain is that thereceived version of the rectangle would be wiggly near the edges. In both cases, the timing of the samplesbecomes more critical as the received pulse deviates further from rectangular.

One shortcoming of the telecommunication system embodied in the transmitter of [link] and the receiver of [link] is that only one such transmitter at a time can operate in any particular geographicalregion, since it hogs all the frequencies in the baseband, that is, all frequencies below $5B$ Hz. Fortunately, there is a way to have multiple transmittersoperating in the same region simultaneously. The trick is to translate the frequency content so that insteadof all transmitters trying to operate in the 0 and $5B$ Hz band, one might use the $5B$ to $10B$ band, another the $10B$ to $15B$ band, etc. Conceivably, this could be accomplished by selecting a different pulseshape (other than the rectangle) that has no low frequency content, but the most common approach is to“modulate” (change frequency) by multiplying the pulse shaped signal by a high frequency sinusoid.Such a “radio frequency” (RF) transmitter is shown in [link] , though it should be understood that the actual frequencies used may place it in the television band orin the range of frequencies reserved for cell phones, depending on the application.