0.1 A telecommunication system  (Page 10/15)

Digital methods are not new. Morse code telegraphy (which consists of a sequence of dashes and dots coded into long and short tone bursts)became widespread in the 1850s. The early telephone systems of the 1900s were analog,with digitization beginning in the 1960s.

Digital communications (relative to fully analog) have the following advantages:

• digital circuits are relatively inexpensive,
• data encryption can be used to enhance privacy,
• digital realization tends to support greater dynamic range,
• signals from voice, video, and data sources can be merged for transmission over a common system,
• digital signals can be easily compressed
• noise does not accumulate from repeater to repeater over long distances,
• low error rates are possible, even with substantial noise,
• errors can be corrected via coding.

In addition, digital receivers can easily be reconfigured or upgraded, because they are essentiallysoftware driven. For instance, a receiver built for one broadcast standard(say for the American market) could be transformed into a receiver for the European market with little additional hardware.

But there are also some disadvantages of digital communications (relative to fully analog), including the following:

• more bandwidth (is generally) required than with analog,
• synchronization is required.

Pulse shaping

In order to transmit a digital data stream, it must be turned into an analog signal.The first step in this conversion is to clump the bits into symbols that lend themselvesto translation into analog form. For instance, a mapping from the letters of the Englishalphabet into bits and then into the 4-PAM symbols $\pm 1$ , $\pm 3$ was given explicitly in [link] . This was converted into an analog waveform usingthe rectangular pulse shape [link] , which results in signals that look like [link] . In general, such signals can be written

where the $s\left[k\right]$ are the values of the symbols, and the function $p\left(t\right)$ is the pulse shape. Thus, each member of the 4-PAM data sequenceis multiplied by a pulse that is nonzero over the appropriate time window. Adding all the scaled pulses results inan analog waveform that can be upconverted and transmitted. If the channel is perfect (distortionless and noise free),then the transmitted signal will arrive unchanged at the receiver. Is the rectangular pulse shape a good idea?

Unfortunately, though rectangular pulse shapes are easy to understand, they can be a poor choice for a pulse shapebecause they spread substantial energy into adjacent frequencies. This spreading complicates the packing of users in frequencydivision multiplexing, and makes it more difficult to avoid having different messages interfere with each other.

To see this, define the rectangular pulse

as shown in [link] .

The shifted pulses [link] are sometimes easier to work with than [link] , and their magnitude spectra are the same by the time shifting property [link] . The Fourier transform can be calculated directly fromthe definition [link]