A codeword of a parallel concatenated code consists modulation transmit the uncoded information only once. Each puncturing pattern of the second trellis code is non-uniform and depends decoder performs weighted soft decoding of the input sequence.
Bit error on the particular choice of interleaver. This method was proposed in [8]. Parallel concate- The method uses symbol interleaving, and the reliability of punctured nated convolutional codes yield very large coding gains dB at the symbols may not be reproducible at the decoder.
Then do tistage decoding [7]. Recently [8], punctured versions of Ungerboeck the same to the second constituent code, but select only those systematic codes were used to construct turbo codes for 8PSK modulation. In this bits which were punctured in the first encoder. For block by block encoding a trellis termination method Natural mapping 00 01 10 11 as discussed in [4] is also shown in the same Figure. As we these methods the reliability of the punctured symbols is reproducible at mentioned in the previous section, for two parallel concatenated the decoder.
TCM we should use at least two interleavers. The ulation should be designed based on the Euclidean distance.
To achieve very low bit error rates, one should maximize The input vector u can be decomposed into k subsequences as the effective free distance of turbo code [9] [4]. Here k is a multiple of number of encoders criterion to nonbinary modulation.
Effective free Euclidean distance — Choose the constituent TCM the permutation operation and additions are modulo In Tables 1 and 2 signal 2 d2,2 wk ,. Selection of codes with memory m — Referring to Fig. The reordered mapping was used in [10] for select the feedback polynomial h0 to be primitive. To better understand the reordered mapping, con- For feedforward connections we use the following setups: For sider an 8PSK constellation which has eight cosets c0 , c1 ,..
Swap the last from a state and when remerging to a state. Then recompose the eight cosets into the reordered cosets c0 , c1 , c2 , c3 , c6 , c7 , c4 , c5. Note that the reordered mapping for 4-level signals is the same as natural III. Bit by Bit Iterative Decoding for Parallel mapping. Concatenated Trellis Codes 3. A systematic set-partitioning method for QAM constellations is given.
Peak constraints can be used to limit the constellation expansion ratio and peak-to-average power ratio PAPR. The proposed set-partitioning method can be used for systems with more than four transmit antennas directly.
Furthermore, its decoding complexity is low, thanks to the new designed inner block codes. Several design examples are presented. Author : Tolga M.
In this chapter, we introduce space - time trellis These space - time trellis codes had a high decoding complexity and required a vector Viterbi algorithm at the Author : Jerry R. Those are, for examples, the multiple-input multiple-output MIMO , space time coding , and orthogonal-frequency Author : Jerry D.
From this list of states, which represent the actual data points on the trellis diagram, the input data bits are recovered by using a table that relates how each input affects transitions between states. It will be understood from this simple example that decoding of TCM data symbols in a practical configuration is much more complex. Lower-case k and n denote the number of input and output bits processed by a convolutional encoder, indicated at Encoding begins by decomposing the K-bit input symbol into two sets of bits.
The remaining K-k bits, as indicated at 32 , are combined with the n-bit output of the encoder, as indicated at Then the n-bit portion and the K-k -bit portion are each partitioned, as indicated at 36 , into equal numbers of sets. Numerically, the number of sets is equal to half of the TCM dimension. A receiver that receives TCM symbols that are each defined by a single pair of I and Q values is said to be two-dimensional 2D.
If two pairs of I,Q are received per TCM symbol, the TCM is said to be four-dimensional 4D , three pairs means six-dimensional 6D Partitioning is illustrated or the 6D case, there being three sets for the coded and uncoded bits.
Finally, as indicated at 36 , these sets are mapped to m-ary constellation symbols so as to produce maximum free Euclidean distance , which are then fed to a modulator not shown in this figure , as indicated by the arrows in the figure.
The number of encoded bits assigned to each signal constellation symbol, depends in part on whether two-dimensional 2D , four-dimensional 4D or six-dimensional 6D TCM is employed.
Three bits of each input symbol are encoded to provide four output bits e 0 through e 3 , which are combined with the uncoded bits d 3 through d 6 of the input symbol. In set partitioning the ary signal constellation is divided into four sets of constellation symbols defined by the two coded bits.
The symbol indicated by X[0] includes coded bits e 0 and e 1 and uncoded bits d 3 and d 4. The symbol indicated by X[1] includes coded bits e 2 and e 3 and uncoded bits d 5 and d 6. In this map, the two least significant bits LSBs of a ary symbol denote the particular coset and the most significant bits MSBs identify a member within a coset.
For example, the codes and are members of the same coset ending in 01 and are also indicated by the same graphical symbol a diamond. Similarly, all the codes indicated by a triangle are in the same coset, as are all those indicated by a square and all those indicated by a circle. It will be observed that the minimum distance between members of the same coset is maximized.
Ideally, for optimal TCM code performance constellations must be mapped in this manner. That is, adjacent members of the same coset should differ by as few bits as possible. For completeness, a convolutional decoder of this configuration is shown in FIG. The final pairs of modulo-two adders provide A and B outputs that represent logical combinations of the various stages of the encoder.
The first polynomial 1 2 indicates that the A output is a logical combination of the input bit, the first through the third stages and the last stage of the encoder shift register. The other polynomial 1 2 indicates that the B output is a logical combination of the input bit, the second and third stages, and the fifth and sixth stages of the encoder shift register. As indicated in FIG. If, however, selected A and B bits are discarded, as indicated by the X symbols in the figure, the encoder effectively operates at another selected code rate.
The arrangement of used bits in the final n-bit output is somewhat arbitrary so long as both encoder and decoder follow the same assignments. This encoder is used in the preferred embodiment of the present invention and is shown in FIG. In the description that follows, it is assumed that there are at most two coded bits, C 1 and C 2 , associated with each constellation symbol.
The decoding methods described below can easily be generalized to other situations. The pragmatic trellis coded modulation PTCM decoding algorithm makes use of soft decision values, which are labeled g 1 and g 2. The soft decision value g 1 is associated with coded bit C 1 and the soft decision value g 2 is associated with coded bit C 2. There are methods other than the LLR method of calculating these soft decisions.
The LLR method is just one example. The steps of the typical PTCM decoding algorithm are described below. Please note that this description assumes that there are no more than two coded bits per constellation symbol, the algorithm is applicable to other situations as well:.
In de-puncturing, the previously discarded symbols in the encoder are replaced with median metrics erasure values that will not influence the decoding one way or another. By performing a hard decision minimum distance decode to the closest member within the determined coset, the uncoded bits represented by an I,Q value are determined, as indicated in block The uncoded bits are delayed, as indicated by block 64 , to synchronize them with the coded bits.
The inventive decoder processes the signal in much the same way as a PTCM decoder, in the sense that the coded and the uncoded bits are processed separately. Thus it maintains much of the simplicity of a PTCM decoder. The nature of this improvement is described below after an example of a method for calculating the soft decision values is given.
The inventive decoding method described below can easily be generalized to other situations. The inventive decoding algorithm also makes use of soft decision values, but the soft decision values are defined differently from in the typical PTCM method described above. There are four soft decision variables, which are labeled g 1 , g 2 , g 3 , and g 4. Each soft decision value is associated with both coded bit C 1 and coded bit C 2.
One method of calculating this relialility value is the minimum distance method, which like the previously described log-likelihood ratio LLR method, results in a maximum likelihood decoder. In the minimum distance method, the values assigned to the soft decisions for a particular I,Q pair is the minimum squared distance from the I,Q point to the closest constellation point in the coset associated with the soft decision being determined.
For the example situation in which there are two coded bits per constellation symbol, there are four cosets. There are methods other than the minimun squared distance method of calculating these soft decisions, the minimum squared distance method is just one example.
Performance Advantage:. As mentioned, the inventive decoding method does not require de-puncturing. As a result, in generating soft decision values all of the coded bits of a TCM symbol can be considered simultaneously, as described above. This leads to performance advantages over the typical PTCM method, in which code rates that involve puncturing require that each coded bit of the TCM symbol be considered separately in the generation of soft decision values.
Consider the situation portrayed in FIG. The PTCM decoder would have to use de-puncturing in this situation, and thus it would have to consider each coded bit separately when generating soft decision values.
The PTCM method would consider, then, that it was very likely that the coded bit C 1 was transmitted as a zero and that it was also very likely that coded bit C 2 was also transmitted as a zero.
Thus, it would determine that a point from coset 0 was the most likely to have been received. It would determine that the received I,Q value was very unlikely to have come from coset 3 , which does not have any coded bits with a zero value. But since points in coset 1 and coset 2 each have a coded bit that has value 0, the PTCM soft decision generation would determine that the received I,Q value was moderately likely to have come from a point from either coset 1 or coset 2 , even though the I,Q point is relatively far from a point in these cosets.
The inventive decoding method does not experience this weakness. Because it can consider both coded bits at the same time it would determine that the received I,Q value was very likely to have come from coset 0 , but very unlikely to have come from either coset 1 , coset 2 or coset 3.
As can be seen from this example, the inventive decoding method generates a truer assessment of the likelihoods, which in turn leads to a better BER performance. As mentioned above, the inventive decoding method uses a full-rate decoder. This means that the trellis it uses in its Viterbi algorithm is a full-rate trellis.
Note that on the figure, for the sake of simplicity, only transitions that eventually result in a transition into the zero state have been shown. The figure also illustrates that the inventive decoder can be used with standard PTCM encoders.
The inventive decoder would simply use the full-rate trellis that is equivalent to the PTCM encoder's punctured trellis. As a result, if an existing communications link that uses PTCM methods requires a performance improvement, the PTCM decoder could be replaced with a decoder based on the inventive decoding method without having to replace the encoder.
This could have a great advantage in situations where the cost of replacing the encoder is prohibitive, such as in satellite down links. This section describes an inventive technique for calculating the soft decision and branch metric values for the inventive decoding method.
A common practice is to use a large lookup table to store the soft decision or branch metric values associated with a given received I,Q value. The use of a lookup table allows the decoder to work with various constellation types since the values stored in the table can be changed to the values required by a new constellation. The innovative method described in this section maintains the flexibility provided by a large lookup table, but greatly reduces the amount of memory storage required.
The letters in the following explanation refer to the letters on this diagram. The two received symbol values enter the calculator at point A. The tables at stage C each store the co-ordinates for all the constellation points in a given coset. The phase input to a coset table point F plays the role of index k, and steps the table through all of the constellation points in the coset.
Stage D squares these values and adds them together to produce the squared distance between the received I,Q value and the k th point of the coset.
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