Attenuation and Channel Characterization

1. For one of the problems on the second homework assignment, I've asked you to examine an interesting property of sine waves.

2. Dispersion has the effect of distorting the shape of most waves transmitted through a wire or fiber.

   • What starts as a nice square wave tends to turn into a rounded little bump.

3. This sort of distortion does not affect sine waves.

   • If a pure sine wave is transmitted through a noiseless channel, the signal received at the other end will be a sine wave of the same frequency.

   • The amplitude of the wave will be reduced (a process known as attenuation, but the shape (sine) and frequency will not change!

4. As you will learn through the homework problem, this behavior is a consequence of the linearity of transmission systems.

5. If the transmitted signal is a sine wave, two pieces of information are needed to predict the received signal.

   • The phase of the received signal will depend on the total propagation time. This, as we just discussed, will depend on the signal frequency.

   • The amplitude of the signal will depend on the degree to which it is attenuated as it passes through the transmission medium. This is also dependent on the waves frequency.

6. One can characterize the behavior of a transmission medium by determining two functions:

   1. A function that maps a sine wave's frequency into its propagation speed (or alternately the phase shift that will be observed when it is received).

   2. A function that maps a sine wave's frequency into the factor by which it will be attenuated.
   Given an arbitrary signal to be transmitted through the medium one can then apply these functions to the transmitted signal's Fourier series to determine its form on arrival.