Introduction - Signal Processing and Optimization for Transceiver Systems - page 4

q ( t )

y ( t )

x ( t )

s ( t ) s ( t )

+

F ( j ω ) H ( j ω ) G ( j ω )

prefilter or precoder LTI channel postfilter or equalizer

Figure 1.3 . The analog communication system represented in terms of frequency

responses.

1.3 Digitial communication systems

In the preceding section, the signal s ( t ) was regarded as a continuous time signal
with continuous (unquantized) amplitude. In a digital communication system,
themessagesare quantized amplitudes, transmitted in discrete time. Figure 1.4
shows the schematic of a digital communication system. Here we have a discrete-
time message or signal s ( n ) which we wish to transmit over a continuous-time
channel. The amplitudes of s ( n ) are “digitized,” that is they come from a finite
set of symbols . This collection of symbols is called a constellation .Wesha l
come to details of digitization later. 2 Since s ( n ) is a discrete-time signal and the
channel is continuous-time, the signal is first converted into a continuous-time
signal x ( t ) as indicated in the figure.
The conversion from s ( n )to x ( t ) can be described schematically in two steps.
The building block indicated as D/C is a discrete-to-continuous-time converter ,
and it converts s ( n ) to a signal s c ( t )givenby

∞

s c ( t )= s ( n ) δ c ( t − nT ) . (1 . 6)

n = −∞

Here δ c ( t )isthe impulse or Dirac delta function [Oppenheim and Willsky, 1997].
Thus, the sample s ( n ) is converted into an impulse positioned at time nT. The
sample spacing T determines the speed with which the message samples are
conveyed. Since we have 1 /T symbols per second, the symbol rate is given by

1
T

f s = Hz . (1 . 7)

The prefilter F ( jω ) at the transmitter performs a convolution to produce the
output

∞

x ( t )= s ( n ) f ( t − nT ) , (1 . 8)

n = −∞

2 Examples of constellations include PAM and QAM systems to be described in Sec. 2.2.