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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.

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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

11 = 6

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Satellite location Company Continental DBSC DirecTV Dominion Echostar Echostar/Directsat MCI TCI/Tempo USSB Unassigned

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C.8.1 C.8.2 C.8.3 C.8.4 C.8.5 C.8.6 C.8.7 C.8.8 C.8.9 C.8.10 C.8.11 C.8.12 C.8.13 C.8.14 C.9.1 C.9.2 C.9.3 C.9.4 C.9.5 Trigonometric Functions Cartesian to Polar Functions Polar to Cartesian Functions Logarithmic and Exponential Functions Sign Conversion Functions Float to Integer Conversion Functions Number and String Conversion Functions String Manipulation Functions Time and Date Functions Probability Functions Statistical Functions Array Functions Other Functions Formatting Codes and Logic General Information Simulator Commands Simulator Functions Simulator Commands Listed Alphabetically Simulator Functions Listed Alphabetically

Strategy: In calculating the volume of a solid of revolution we usually apply either the disk formula (or the washer formula) or the cylindrical shells formula (or the difference of cylindrical shells formula). To decide which formula to use: (1) Decide along which axis you are going to integrate. This depends on the shape and position of the region R that is revolved. (2) (i) Use the disk formula (or the washer formula) if the region R is revolved perpendicular to the axis of intergration. (ii) Use the cylindrical shells formula (or the difference of cylindrical shells formula) if the region R is revolved parallel to the axis of integration.

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