Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 18

estimated symbol
in the constellation

s ( n ) s ( n )

est

detector

Figure 2.5 . The detector, or decision device, at the receiver takes the reconstructed
symbol s ( n ) and maps it into a symbol s est ( n ) in the constellation, which is regarded

as an estimate of the transmitted symbol s ( n ) .

Im

Re

Figure 2.6 . Noise cloud associated with a 4-bit QAM constellation.

Figure 2.8(b) shows the threshold detector characteristics for an arbitrary num-
ber of bits. As a specific example, assume that the transmitted symbol at time
n was s ( n )=3 A, as highlighted in Fig. 2.7. If the error term e ( n )(whichisa
random variable) is such that

| e ( n ) | <A (2 . 9)

then s ( n ) is within the shaded box shown in Fig. 2.7, and the symbol estimation

is correct, that is, s est ( n )= s ( n ) . If e ( n ) has magnitude larger than A then the
symbol is estimated to be A (for e ( n ) <

− A )or5 A (for e ( n ) >A ), and there is

an error.
The probability of error in the decision can be calculated if we know the
probability density function (pdf) of the additive error term e ( n )inEq. (2.7).
Figure 2.7 shows an example of this pdf, denoted as f E ( e ) .