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28 Introduction Chap. 1
where B is the signal bandwidth. L’Hospital’s rule is used to evaluate this limit:
1
C lim b
(1–13)
1 + (E b N 0 T b )x ¢ E b
E b
≤ log 2 er =
N 0 T b N 0 T b 1n 2
=
x:0
If we signal at a rate approaching the channel capacity, then P e : 0, and we have the
maximum information rate allowed for P e : 0 (i.e., the optimum system). Thus, 1/T b = C,
or, using Eq. (1–13),
or
E b N 0 = ln 2 =-1.59 dB
(1–14)
This minimum value for E b /N 0 is -1.59 dB and is called Shannon’s limit. That is, if
optimum coding/decoding is used at the transmitter and receiver, error-free data will be
recovered at the receiver output, provided that the E b /N 0 at the receiver input is larger than
-1.59 dB. This “brick wall” limit is shown by the dashed line in Fig. 1–8, where Pe jumps
1
from 0 (10 -q ) to
2 (0.5 * 100 ) as E b /N 0 becomes smaller than - 1.59 dB, assuming that the
ideal (unknown) code is used. Any practical system will perform worse than this ideal system
described by Shannon’s limit. Thus, the goal of digital system designers is to find practical
codes that approach the performance of Shannon’s ideal (unknown) code.
When the performance of the optimum encoded signal is compared with that of BPSK
without coding ( 10 -5 BER), it is seen that the optimum (unknown) coded signal has a coding
gain of 9.61- (-1.59) = 11.2 dB. Using Fig. 1–8, compare this value with the coding gain of
8.8 dB that is achieved when a turbo code is used. Table 1–4 shows the gains that can be
obtained for some other codes.
Since their introduction in 1993, turbo codes have become very popular because they
can perform near Shannon’s limit, yet they also can have reasonable decoding complexity
[Sklar, 1997]. Turbo codes are generated by using the parallel concatenation of two simple
convolutional codes, with one coder preceded by an interleaver [Benedetto and Montorsi,
1996]. The interleaver ensures that error-prone words received for one of the codes
corresponds to error-resistant words received for the other code.
All of the codes described earlier achieve their coding gains at the expense of
bandwidth expansion. That is, when redundant bits are added to provide coding gain, the
overall data rate and, consequently, the bandwidth of the signal are increased by a multiplicative
factor that is the reciprocal of the code rate; the bandwidth expansion of the coded system
relative to the uncoded system is 1/R = n/k. Thus, if the uncoded signal takes up all of the
available bandwidth, coding cannot be added to reduce receiver errors, because the coded
signal would take up too much bandwidth. However, this problem can be ameliorated by
using trellis-coded modulation (TCM).
Trellis-Coded Modulation
1
T b
=
E b
N 0 T b ln 2
Gottfried Ungerboeck has invented a technique called trellis-coded modulation (TCM) that
combines multilevel modulation with coding to achieve coding gain without bandwidth