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Comparing it with the stored version of 000 results in [0.5 ( 1)]
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The distance determined in this manner is often referred to as the Euclidean distance in acknowledgment of its geometric origins, and the distance squared is known as the Euclidean distance metric. On this basis, the received codeword is closest to the 000 codeword, and the decoder would produce a binary 0 output. Soft decision decoding results in about a 2-dB reduction in the [Eb/N0] required for a given BER (Taub and Schilling, 1986). This reference also gives a table of comparative values for soft and hard decision coding for various block and convolutional codes. Clearly, soft decision decoding is more complex to implement than hard decision decoding and is only used where the improvement it provides must be had. 11.10 Shannon Capacity In a paper on the mathematical theory of communication (Shannon, 1948) Shannon showed that the probability of bit error could be made arbitrarily small by limiting the bit rate Rb to less than (and at most equal to) the channel capacity, denoted by C. Thus Rb C (11.17)
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For random noise where the spectrum density is flat (this is the N0 spectral density previously introduced) the channel capacity is given by C W log2 a1 S b N (11.18)
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Here, W is the baseband bandwidth, and S/N is the baseband signal to noise power ratio (not decibels). Shannon s theorem can be written as Rb W log2 a1 S b N (11.19)
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Letting PR represent the average signal power, and Tb the bit period then as shown by Eq. (10.17) the bit energy is Eb PRTb. The noise power is N WN0 and the signal to noise ratio is S N PR N Eb TbWN0 (11.20)
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The bit rate is Rb 1/Tb. Substituting this in Eq. (11.20) then in the inequality (11.19) gives Rb W log2 a1 R b Eb W N0 b (11.21)
As noted in Sec. 10.6.3, the ratio of bit rate to bandwidth (Rb/W in the inequality (11.21) is an important characteristic of any digital system. The greater this ratio, the more efficient the system. The limiting case is when the inequality sign is replaced by the equal sign. Rb W log2 a1 R b Eb W N0 b (11.22)
Keep in mind that this relationship is for the condition of arbitrarily small probability of bit error. A plot of Rb/W as a function of Eb/N0 is shown in
6 5.75
Bit Rate/Bandwidth
QPSK
BPSK
3 Eb N0
5 (dB)
Figure 11.10 Graph showing the Shannon limit, Eq. (11.22). The units for the y-axis are (bits/s)/Hz (a dimensionless ratio in fact). The points for BPSK and QPSK are evaluated for a Pe 10 5.
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Fig. 11.10. Note that although the graph shows Eb/N0 in decibels ([Eb/N0] in our previous notation) the power ratio must be used in evaluating Eq. (11.22). In any practical system there will be a finite probability of bit error, and to see how this fits in with the Shannon limit, consider the BER graph of Fig. 10.17, which applies for BPSK and QPSK. From Fig. 10.17 [or from calculation using Eq. (10.18)], for a probability of bit error (or 5 BER in this case) of 10 the [Eb/N0] is about 9.6 dB. (See also the uncoded curve of Fig. 11.8). As shown in Sec. 10.6.3 the bit rate to bandwidth ratio is 1/(1 ) for BPSK and 2/(1 ) for QPSK. For purposes of comparison ideal filtering will be assumed, for which 0. Thus on Fig. 11.10, the points (1, [9.6]) for BPSK and (2, [9.6]) can be shown. At an [Eb/N0] of 9.6 dB the Shannon limit indicates that a bit rate to bandwidth ratio of about 5.75 : 1 should be achievable, and it is seen that BPSK and QPSK are well below this. Alternatively, for a bit-rate/bandwidth ratio of 1, the Shannon limit is 0 dB (or an Eb/N0 ratio of unity), compared to 9.6 dB for BPSK.
11.11 Turbo Codes and LDPC Codes Till 1993 all codes used in practice fell well below the Shannon limit. In 1993, a paper (Berrou et al., 1993) presented at the IEEE International Conference on Communications made the claim for a digital coding method that closely approached the Shannon limit (a pdf file for the paper will be found at www-elec.enst-bretagne.fr/equipe/ berrou/ Near%20Shannon%20Limit%20Error.pdf). Subsequent testing confirmed the claim to be true. This revitalized research into coding, resulting in a number of turbo-like codes, and a renewed interest in codes known as low density parity check (LDPC) codes (see Summers, 2004). Turbo codes and LDPC codes use the principle of iterative decoding in which soft decisions (i.e., a probabilistic measure of the binary 1 or 0 level) obtained from different encoding streams for the same data, are compared and reassessed, the process being repeated a number of times (iterative processing). This is sometimes referred to as soft input soft output to describe the fact that during the iterative process no hard decisions (binary 1 or 0) are made regarding a bit. Each reassessment generally provides a better estimate of the actual bit level, and after a certain number of iterations (fixed either by convergence to a final value, or by a time limit placed on the process) a hard decision output is generated. Turbo codes are so named because the iterative or feedback process was likened to the feedback process in a turbo-charged engine (see Berrou et al., 1993). The turbo principle can be applied with concatenated block
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