0.7 Bits to symbols to signals

How much will two bits be worth in the digital marketplace?

—Hal Varian, Scientific American , Sept. 1995

Any message, whether analog or digital, can be translated into a string of binary digits.In order to transmit or store these digits, they are often clustered or encoded into a more convenientrepresentation whose elements are the symbols of an alphabet . In order to utilize bandwidth efficiently, thesesymbols are then translated (again!) into short analog waveforms called pulse shapes that are combined to form the actual transmitted signal.

The receiver must undo each of these translations. First, it examines the received analog waveform anddecodes the symbols. Then it translates the symbols back into binary digits, from which the original messagecan (hopefully) be reconstructed.

This chapter briefly examines each of these translations, and the tools needed to make the receiver work.One of the key ideas is correlation which can be used as a kind of pattern matching toolfor discovering key locations within the signal stream. "Correlation" shows how correlation can be viewed as a kind of linear filter, and hence its properties can be readily understoodin both the time and frequency domains.

Bits to symbols

The information that is to be transmitted by a communication system comes in many forms: a pressure wavein the air, a flow of electrons in a wire, a digitized image or sound file, the text in a book.If the information is in analog form, then it can be sampled (as in Chapter [link] ). For instance, an analog-to-digital converter can transform the outputof a microphone into a stream of numbers representing the pressure wave in the air, or it can turn measurements of the current in thewire into a sequence of numbers that are proportional to the electron flow. The sound file, which is already digital,contains a long list of numbers that correspond to the instantaneous amplitude of the sound.Similarly, the picture file contains a list of numbers that describe the intensity and color of the pixels in the image.The text can be transformed into a numerical list using the ASCII code. In all these cases,the raw data represent the information that must be transmitted by the communication system. The receiver, in turn, mustultimately translate the received signal back into the data.

Once the information is encoded into a sequence of numbers, it can be reexpressed as a string of binary digits 0 and 1.This is discussed at length in Chapter [link] . But the binary 0–1 representation is not usually very convenientfrom the point of view of efficient and reliable data transmission. For example, directly modulating a binary string with a cosinewave would result in a small piece of the cosine wave for each 1 and nothing (the zero waveform) for each 0.It would be very hard to tell the difference between a message that contained a string of zeroes, andno message at all!

The simplest solution is to recode the binary $0,1$ into binary $\pm 1$ . This can be accomplished using either the linear operation $2x-1$ (which maps 0 into $-1$ , and 1 into 1), or by $-2x+1$ (which maps 0 into 1, and 1 into $-1$ ). This “binary” $\pm 1$ is an example of a two-element symbol set. There are many other common symbol sets.In multilevel signaling , the binary terms are gathered into groups.Regrouping in pairs, for instance, recodes the information into a four-level signal.For example, the binary sequence might be paired thus: