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11.6 Concatenated Codes Codes designed to correct for burst errors can be combined with codes designed to correct for random errors, a process known as concatenation. Figure 11.6 shows the general form for concatenated codes. The input data are fed into the encoder designed for burst error correction. This is the outer encoder. The output from the outer encoder is fed into the encoder designed for random error correction. This is the inner encoder. The signal is then modulated and passed on for transmission. At the receiver, the signal is demodulated. The inner decoder matches the inner encoder and follows the demodulator. The output from the inner decoder is fed into the outer decoder, which matches the outer encoder. The term outer refers to the outermost encoding/decoding units in the equipment chain, and the term inner refers to the innermost encoding/decoding unit. In digital satellite television, the outer code is a R-S code, and the inner code is a convolutional code. The inner decoder utilizes Viterbi decoding.
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Rb Concatenated coding (see Sec. 11.6).
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11.7 Link Parameters Affected by Coding Where no error control coding is employed, the message will be referred to as an uncoded message, and its parameters will be denoted by the subscript U. Figure 11.7a shows the arrangement for an uncoded message. Where error control coding is employed, the message will be referred to as a coded message, and its parameters will be denoted by the subscript C. Figure 11.7b shows the arrangement for a coded message. For comparison purposes, the [C/N0] value is assumed to be the same for both situations. The input bit rate to the modulator for the uncoded message is Rb, and for the coded message is Rc. Since n code bits must be transmitted for every k data bits, the bit rates are related as Rb Rc rc
[C/N0]
(11.10)
Modulator
Demodulator
Pe Rb
[C/N0]
Modulator
Demodulator
Rc BER Encoder Rb
(b) Figure 11.7
Decoder Rb
Comparing links with and without FEC.
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Since rc is always less than unity, then Rc Rb always. For constant carrier power, the bit energy is inversely proportional to bit rate (see Eq. 10.22), and therefore, Ec Eb rc (11.11)
where Eb is the average bit energy in the uncoded bit stream (as introduced in Chap. 10), and Ec is the average bit energy in the coded bit stream. Equation (10.18) gives the probability of bit error for binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK) modulation. With no coding applied, Eb is just the Eb of Eq. (10.18), and the probability of bit error in the uncoded bit stream is Eb 1 erfc N0 2
(11.12)
For the coded bit stream, the bit energy is Ec Eq. (10.18) becomes rcEb 1 erfc N0 2
rcEb, and therefore,
(11.13)
This means that PeC PeU, or the probability of bit error with coding is worse than that without coding. It is important to note, however, that the probability of bit error applies at the input to the decoder. For the error control coding to be effective, the output BER should be better than that obtained without coding. More will be discussed about this later. The limitation imposed by bandwidth also must be considered. If the time for transmission is to be the same for the coded message as for the uncoded message, the bandwidth has to be increased to accommodate the higher bit rate. The required bandwidth is directly proportional to bit rate (see Eq. 10.16), and hence it has to be increased by a factor 1/rc. If, however, the bandwidth is fixed (the system is band limited), then the only recourse is to increase the transmission time by the factor 1/rc. For a fixed number of bits in the original message, the bit rate Rb entering into the encoder is reduced by a factor rc compared with what it could have been without coding. As an example, it is shown in Sec. 10.4 that the TI message rate is 1.544 Mb/s. When 7/8 FEC is applied, the transmission rate becomes
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