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The eccentricity and the semimajor axis are two of the orbital parameters specified for satellites (spacecraft) orbiting the earth. For an elliptical orbit, 0 e 1. When e 0, the orbit becomes circular. The geometrical significance of eccentricity, along with some of the other geometrical properties of the ellipse, is developed in App. B.

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2.3 Kepler s Second Law Kepler s second law states that, for equal time intervals, a satellite will sweep out equal areas in its orbital plane, focused at the barycenter. Referring to Fig. 2.2, assuming the satellite travels distances S1 and S2 meters in 1 s, then the areas A1 and A2 will be equal. The average velocity in each case is S1 and S2 m/s, and because of the equal area law, it follows that the velocity at S2 is less than that at S1. An important

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codes, (known as Turbo Product codes or TPCs, see Comtech, 2002) However the more common arrangement is to use parallel concatenation using convolution encoders Because of the continuous nature of convolution coding, data and code sequences rather than words are involved The convolution encoders shown in Fig 1111 (Burr, 2001) are recursive convolution encoders They differ from the convolution encoder of Fig 112 in that feedback is employed (This recursive feedback is part of the encoding process and is quite separate from the iterative feedback to be described shortly, which gives turbo codes their name) It can be shown that the recursive feedback (see Burr, 2001) assists in maintaining a large minimum Hamming distance (see Sec 112) for code sequences Another difference between the encoders of Fig 1111 and that of Fig 11.

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2 is that the data sequence is fed directly to the multiplexer of Fig 1111, making the output systematic (see Sec 112) Parity-1 bits are generated directly from the data bits and parity-2 bits from the interleaved data bits, so that two independent streams of parity bits are generated Interleaving as described in Sec 115, was used there as a means of combating bursty errors With turbo encoding the purpose of interleaving is different, it is used to provide independent parity bits for the same input A number of different methods of interleaving are available, and the design of the interleaver is a crucial aspect of turbo code design The output coded sequence is data, parity-1, parity-2 Since each encoder generates a parity bit for every data bit the code rate is 1/3 This is relatively low but can be increased by puncturing, as described in Sec.

10

114 and shown in Fig 1111 For example, one parity bit might be discarded in turn from each of the encoders, resulting in a 1/2 code rate Other rates are possible with puncturing If puncturing is used with the encoder, dummy bits are inserted at the decoder to replace.

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A turbo encoder with puncturing. The symbol indicates an interleaver, and the { symbol indicates modulo-2 addition. (Courtesy of A Burr and the IEEE.)

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4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 Variables The Keyboard The Mouse Output to the Screen Loops Functions

those discarded. The dummy bit level is set midway between the binary 1 and 0 levels, so they do not affect the decoding process. The block schematic for the decoder is shown in Fig. 11.12 (Burr, 2001). The demultiplexer provides outputs for the data sequence and parity-1 and parity-2 sequences. These outputs are soft, meaning that some measure of the bit level is used rather than a hard decision output of binary 1 or 0. For example, assuming a threshold decision level of 0.5V, the demultiplexer output might be 0.9V, 0.7V 0.1V, 0.2V, 0.9V, 0.65V, 0.3V, suggesting a hard decision binary sequence of 1 1 0 0 1 1 0. However the hard decision output does not make use of the likelihood of the hard decision being correct. The 0.9V level is obviously more likely to be a binary1 than the 0.65V level. A statistical measure termed the log-likelihood ratio (LLR) is most commonly used. For a given received value r, let p(1/r) represent the probability that a 1 was transmitted, and p(0/r) the probability that a 0 was transmitted. The log likelihood ratio is defined as LLR loge a p(1>r) p(0>r) b (11.23)