mohatta2015.pdf

THE IDEA 3
Figure 1.2
Sampling and digitizing speech.
a series or packet of digital symbols to a cell phone. The transmitted signal travels
through the air, often bouncing off of walls and buildings, before arriving at the cell
phone receiver. The receiver's job is to figure out what symbols were sent. This is
an example of the channel equalization problem.
To solve this problem, we would like a mathematical model of what is happening.
The model should be based on the laws of physics. Cell phone signals are transmitted
using electromagnetic (radio) waves. The signal travels through the air, along
a path to the receiver. From the laws of physics, the effect of this "channel" is
multiplication by a channel coefficient. Thus, if s is the transmitted symbol, then
cs is the received symbol, where c is a channel coefficient. To keep things simple,
we will assume c is a real number (e.g., —10), though in practice it is a complex
number with real and imaginary parts (amplitude and phase).
Sometimes the channel is dispersive, so that the signal travels along multiple
paths with different path lengths, as illustrated in Fig. 1.1. The first path goes
directly from the transmitter to the receiver and has channel coefficient c = —10.
The second path bounces off a building, so it is longer, which delays the signal like
an echo. It has channel coefficient d = 9. There is also noise present. The overall
mathematical model of the received signal values is given in (1.1). The portion of
the received signal containing the transmitted symbols is illustrated in Fig. 1.3.
Notice that the model includes terms n\, rii to model random noise. The laws
of physics tell us that electrons bounce around randomly, more so at higher temperatures.
We call this thermal noise. Such noise adds to the received signal.