Page 324 - Satellite Communications, Fourth Edition

304 Chapter Ten

achieved by making the transmit and receive filters identical, each
having a frequency response which is the square root of the raised-cosine
response. Having identical filters is an advantage from the point of view
of manufacturing.
The most commonly encountered type of noise has a flat frequency
spectrum, meaning that the noise power spectrum density, measured in
joules (or W/Hz), is constant. The noise spectrum density will be denoted
by N . When the filtering is designed to maximize the received signal-
0
to-noise ratio, the maximum signal-to-noise voltage ratio is found to be
equal to 22E >N 0 , where E is the average bit energy. The average bit
b
b
energy can be calculated knowing the average received power P and
R
the bit period T b.
E P T b (10.17)
b
R
The probability of the detector making an error as a result of noise is
given by

1 E b
P e erfca b (10.18)
2 Å N 0

where erfc stands for complementary error function, a function whose
value is available in tabular or graphic form in books of mathematical
tables and as built-in functions in many computational packages. A
related function, called the error function, denoted by erf(⋅) is some-
times used, where

erfc(x) 1 erf(x) (10.19)

Equation (10.18) applies for polar NRZ baseband signals and for
BPSK and QPSK modulation systems. The probability of bit error is also
referred to as the bit error rate (BER). A P of 10 6 signifies a BER of
e
1 bit in a million, on average. The graph of P versus E /N in decibels
b
e
0
is shown in Fig. 10.17. Note carefully that the energy ratio, not the
decibel value, of E /N must be used in Eq. (10.18). This is illustrated
b
0
in the following example.
Example 10.1 The average power received in a binary polar transmission is 10
mW, and the bit period is 100 s. If the noise power spectral density is 0.1 J, and
optimum filtering is used, determine the bit error rate.
Solution From Eq. (10.17):

E b 5 10 3 10 23 3 100 3 10 26
26
5 10 J