Encoding Information in the Frequency and Time Domains 2.4 2000/07/24 2004/08/09 11:35:04.375 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Doug Daniels rainking@alumni.rice.edu information communication frequency Fourier coefficients encoding domain time modem This module discusses the Fourier series encoding scheme. We can create an encoding scheme in the frequency domain to represent an alphabet of letters. But, as this information-encoding scheme stands, we can represent one letter for all time. However, we note that the Fourier coefficients depend only on the signal's characteristics over a single period. We could change the signal's spectrum every T as each letter is typed. In this way, we turn spectral coefficients on and off as letters 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. shows such a signal in the time-domain.

Encoding Signals The encoding of signals via the Fourier spectrum is shown over three "periods." In this example, only the third and fourth harmonics are used, as shown by the spectral magnitudes corresponding to each T -second interval plotted below the waveforms. Can you determine the phase of the harmonics from the waveform?

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