4.5 Encoding information in the frequency domain

To emphasize the fact that every periodic signal has both a time and frequency domain representation, we can exploit both to encode information into a signal. Refer to the Fundamental Model of Communication . We have an information source, and want to construct a transmitter thatproduces a signal $x(t)$ . For the source, let's assume we have information to encode every $T$ seconds. For example, we want to represent typed letters produced by anextremely good typist (a key is struck every $T$ seconds). Let's consider the complex Fourier series formula in the light of trying to encodeinformation.

Assume we have $N$ letters to encode: $\{a_1, \dots , a_N\}$ . One simple encoding rule could be to make a single Fouriercoefficient be non-zero and all others zero for each letter. For example, if $a_n$ occurs, we make $c_n=1$ and $c_k=0$ , $k\neq n$ . In this way, the $n^{\mathrm{th}}$ harmonic of the frequency $\frac{1}{T}$ is used to represent a letter. Note that the bandwidth —the range of frequencies required for the encoding—equals $\frac{N}{T}$ . Another possibility is to consider the binary representation of the letter's index. For example, if theletter $a_{13}$ occurs, converting $13$ to its base 2 representation, we have $13=1101_2$ . We can use the pattern of zeros and ones to represent directlywhich Fourier coefficients we "turn on" (set equal to one) and which we "turn off."

We can create an encoding scheme in the frequency domain to represent an alphabet of letters. But, as thisinformation-encoding scheme stands, we can represent one letter for all time. However, we note that the Fourier coefficientsdepend only on the signal's characteristics over a single period. We could change the signal's spectrumevery $T$ as each letter is typed. In this way, we turn spectral coefficients on and off asletters are typed, thereby encoding the entire typed document. For the receiver (see the Fundamental Model of Communication ) to retrieve the typed letter, it would simply use the Fourier formula for the complex Fourier spectrum for each $T$ -second interval to determine what each typed letter was. [link] shows such a signal in the time-domain.

Encoding signals

In this Fourier-series encoding scheme, we have used the fact that spectral coefficients can be independently specified andthat they can be uniquely recovered from the time-domain signal over one "period." Do note that the signal representing theentire document is no longer periodic. By understanding the Fourier series' properties (in particular that coefficients aredetermined only over a $T$ -second interval, we can construct a communications system. Thisapproach represents a simplification of how modern modems represent text that they transmit over telephone lines.