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Sec. 7–11 Study-Aid Examples 551
7–10 SUMMARY
The performance of digital systems has already been summarized in Sec. 7–6, and the performance
of analog systems has been summarized in Sec. 7–9. The reader is invited to review
these sections for a summary of this chapter. However, to condense these sections, we will
simply state that there is no “best system” that provides a universal solution. The solution
depends on the noise performance required, the transmission bandwidth available, and the
state of the art in electronics that might favor the use of one type of communication system
over another. In addition, the performance has been evaluated for the case of an additive white
Gaussian noise channel. For other types of noise distributions or for multiplicative noise, the
results would be different.
7–11 STUDY-AID EXAMPLES
SA7–1 BER for a Brickwall LPF Receiver Referring to Section 7–2 and parts (a) and (b) of
Fig. 7–4, let a unipolar signal plus white Gaussian noise be the input to a receiver that uses a
low-pass filter (LPF). Using the assumptions leading up to Eq. (7–24a), it is shown that the BER
for the data out of a receiver that uses an LPF (not the optimum matched filter) is approximately
A 2
P e = Q¢ where A is the peak value of the input unipolar signal, N 0 /2 is the PSD of the
C 4N 0 B ≤,
noise, and B is the equivalent bandwidth of the LPF. In obtaining this result, it is argued that the
sample value for a filtered binary-1 signal, s 01 (t 0 ), is approximately A, provided that the equivalent
bandwidth of the LPF is B 7 2/T, where T is the width of the transmitted rectangular pulse
when a binary-1 is sent. R = 1/T is the data rate. If the receiver uses an ideal LPF (i.e., a brickwall
LPF) with
H(f) =ßa f
2B b
(7–147)
show that the approximation s 01 (t 0 ) ≈ A is valid for B Ú 2/T.
Solution The MATLAB solution is shown by the plots in Fig. 7–28. Fig. 7–28a shows the
unfiltered rectangular pulse of amplitude A = 1 and width T = 1. Figs. 7–28b, 7–28c, and 7–28d
show the filtered pulse when the bandwidth of the brickwall LPF is B = 1/T, B = 2/T, or B = 3/T,
respectively. For B Ú 2/T, it is seen that the sample value is approximately A, where A = 1, when
the sample is taken near the middle of the bit interval. Also, note that there is negligible ISI
provided that B Ú 2/T. Furthermore, as shown in Fig. 7–28b for B = 1/T, it is seen that s 01 (t 0 ) ≈
1.2·1 = 1.2A. Consequently, it is tempting to use s 01 (t 0 ) ≈ 1.2A (which will give a lower BER) in the
formula for the BER and specify the filter equivalent bandwidth to be exactly B = 1/T. However, if
this is done, the BER formula will not be correct for the cases when B 7 1/T. In addition (as shown
in SA7–2), if a RC LPF is used (instead of a brickwall LPF, which is impractical to build), we do
not get s 01 (t 0 ) = 1.2A for B = 1/T. That is, if s 01 (t 0 ) = 1.2A is used to obtain the BER formula, the
formula would not be valid for the RC LPF. Consequently, Eq. (7–24a), which assumes s 01 (t 0 ) ≈ A
for B Ú 2/T, is approximately correct for all types of practical LPFs that might be used.
Furthermore, if s 01 (t 0 ) is not equal to A for a particular filter, then just replace A in (7–24a) by the
correct sample value for that filter, and the exact (i.e., not an approximate) result will be obtained.