Why R–S Codes Perform Well Against Burst Noise
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8.1.2 Why R–S Codes Perform Well Against Burst Noise
Consider an (n, k) = (255, 247) R–S code, where each symbol is made up of m = 8
bits (such symbols are typically referred to asbytes ). Since n − k = 8, Equation (8.4)
indicates that this code can correct any 4 symbol errors in a block of 255. Imagine
the presence of a noise burst, lasting for 25-bit durations and disturbing one block
of data during transmission, as illustrated in Figure 8.3. In this example, notice that
a burst of noise that lasts for a duration of 25 contiguous bits, must disturb exactly 4
symbols. The R–S decoder for the (255, 247) code will correct any 4-symbol errors
without regard to the type of damage suffered by the symbol. In other words, when
a decoder corrects a byte, it replaces the incorrect byte with the correct one,
whether the error was caused by one bit being corrupted or all 8 bits being corrupted.
Thus, if a symbol is wrong, it might as well be wrong in all of its bit positions.
This gives a R–S code a tremendous burst-noise advantage over binary
codes, even allowing for the interleaving of binary codes. In this example, if the
25-bit noise disturbance had occurred in a random fashion rather than as a contiguous
burst, it should be clear that there would then be many more than 4 symbols
affected (as many as 25 symbols might be disturbed). Of course, that would be
beyond the capability of the (255, 247) code.
8.1.3 R–S Performance as a Function of Size, Redundancy, and Code Rate
For a code to successfully combat the effects of noise, the noise duration has to represent
a relatively small percentage of the codeword. To ensure that this happens
most of the time, the received noise should be averaged over a long period of time,
reducing the effect of a sudden or unusual streak of bad luck. Hence, one can expect
that error-correcting codes become more efficient (error performance improves)
as the code block size increases, making R–S codes an attractive choice
Figure 8.3 Data block disturbed by 25-bit noise burst.
Figure 8.4 Reed–Solomon, rate 7/8, decoder performance as a function of symbol size.
whenever long block lengths are desired [3]. This is seen by the family of curves in
Figure 8.4, where the rate of the code is held at a constant 7/8, while its block size
increases from n = 32 symbols (with m = 5 bits per symbol) to n = 256 symbols (with
m = 8 bits per symbol). Thus, the block size increases from 160 bits to 2048 bits.
As the redundancy of an R-S code increases (lower code rate), its implementation
grows in complexity (especially for high speed devices). Also, the bandwidth
expansion must grow for any real-time communications application. However, the
benefit of increased redundancy, just like the benefit of increased symbol size, is
the improvement in bit-error performance, as can be seen in Figure 8.5, where the
code length n is held at a constant 64, while number of data symbols decreases from
k = 60 to k = 4 (redundancy increases from 4 symbols to 60 symbols).
Figure 8.5 represents transfer functions (output bit-error probability versus
input channel symbol-error probability) of hypothetical decoders. Because there is
no system or channel in mind (only an output-versus-input of a decoder), one
might get the idea that the improved error performance versus increased redundancy
is a monotonic function that will continually provide system improvement
even as the code rate approaches zero. However, this is not the case for codes operating
in a real-time communication system. As the rate of a code varies from
minimum to maximum (0 to 1), it is interesting to observe the effects shown in
Figure 8.5 Reed–Solomon (64, k) decoder performance as a function of redundancy.
Figure 8.6. Here, the performance curves are plotted for BPSK modulation and an
R–S (31, k) code for various channel types. Figure 8.6 reflects a real-time communication
system, where the price paid for error-correction coding is bandwidth expansion
by a factor equal to the inverse of the code rate. The curves plotted show
clear optimum code rates which minimize the required Eb /N0 [4]. The optimum
code rate is about 0.6 to 0.7 for a Gaussian channel, 0.5 for a Rician-fading channel
(with the ratio of direct to reflected received signal power, K = 7 dB), and 0.3 for a
Rayleigh-fading channel. (Fading channels are treated in Chapter 15.) Why is there
an Eb /N0 degradation for very large rates (small redundancy) and very low rates
(large redundancy)? It is easy to explain the degradation at high rates compared
with the optimum rate. Any code generally provides a coding gain benefit; thus, as
the code rate approaches unity (no coding), the system will suffer worse error performance.
The degradation at low code rates is more subtle because in a real-time
communication system using both modulation and coding, there are two mechanisms
at work. One mechanism works to improve error performance, and the other
works to degrade it. The improving mechanism is the coding; the greater the redundancy,
the greater will be the error-correcting capability of the code. The degrading
mechanism is the energy reduction per channel symbol (compared with the data
symbol) which stems from the increased redundancy (and faster signaling in a
real-time communication system). The reduced symbol energy causes the demodu-
Figure 8.6 BPSK plus Reed–Solomon (31, k) decoder performance as a function of code rate.
lator to make more errors. Eventually, the second mechanism wins out, and thus, at
very low code rates the system experiences error-performance degradation.
Let us see if we can corroborate the error performance versus code rate in
Figure 8.6 with the curves in Figure 8.2. The figures are really not directly comparable
because the modulation is BPSK in Figure 8.6, while it is 32-ary MFSK in Figure
8.2. However, perhaps we can verify that R-S error performance-versus-code
rate exhibits the same general curvature with MFSK modulation as it does with
BPSK. In Figure 8.2, the error performance over an AWGN channel, improves as
the symbol error-correcting capability t increases from t = 1 to t = 4; the t = 1 and
t = 4 cases correspond to R-S (31, 29) and R-S (31, 23) with code rates of 0.94 and
0.74, respectively. However at t = 8, which corresponds to R-S (31, 15) with code
rate equal to 0.48, the error performance at PB = 10−5 degrades by about 0.5 dB of
Eb /N0, compared with the t = 4 case. From Figure 8.2, we can conclude that if we
were to plot error performance versus code rate, the curve would have the same
general shape as it does in Figure 8.6. Note that this manifestation cannot be
gleaned from Figure 8.1, since that figure represents a decoder transfer function,
which provides no information about the channel and the demodulation. There-fore, of the two
mechanisms at work in the channel, the Figure 8.1 transfer function only presents the
output-versus-input benefits of the decoder, and displays nothing about the loss of energy as
a function of lower code rate. More is said about choosing a code in concert with a modulation
type in Section 9.7.7.
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