Research Letter  Open Access
Brice Djeumou, Samson Lasaulce, Antoine O. Berthet, "Combining Coded Signals with Arbitrary Modulations in Orthogonal Relay Channels", Journal of Electrical and Computer Engineering, vol. 2008, Article ID 287320, 4 pages, 2008. https://doi.org/10.1155/2008/287320
Combining Coded Signals with Arbitrary Modulations in Orthogonal Relay Channels
Abstract
We consider a relay channel for which the following assumptions are made. (1) The sourcedestination and relaydestination channels are orthogonal (frequency division relay channel). (2) The relay implements the decodeandforward protocol. (3) The source and relay implement the same channel encoder, namely, a convolutional encoder. (4) They can use arbitrary and possibly different modulations. In this framework, we derive the best combiner in the sense of the maximum likelihood (ML) at the destination and the branch metrics of the trellis associated with its channel decoder for the ML combiner and also for the maximum ratio combiner (MRC), cooperativeMRC (CMRC), and the minimum meansquare error (MMSE) combiner.
1. Motivations and Technical Background
We consider orthogonal relay channels for which orthogonality is implemented in frequency [1]. Since the sourcedestination channel is assumed to be orthogonal to the relaydestination channel, the destination receives two distinct signals. In order to maintain the receiver complexity at a low level, the destination is imposed to combine the received signals before applying channel decoding. The relay is assumed to implement the decodeandforward (DF) protocol. We have at least two motivations for this choice. First, in contrast with the wellknown amplifyandforward (AF) protocol, it can be implemented in a digital relay transceiver. More importantly, whereas the AF protocol imposes the sourcerelay channel to have the same bandwidth as the relaydestination channel, the DF protocol offers some degrees of freedom in this respect. This is a critical point when the cooperative network has to be designed from the association of two existing networks. For instance, if one wants to increase the performance of a digital video broadcasting (DVB) receiver or reach some uncovered indoor areas, a possible solution is to use cell phones, say universal mobile telecommunications system (UMTS) cell phones as relaying nodes. The problem is that DVB signals use a 20โMHz bandwidth (sourcerelay channel) while UMTS signals have only a bandwidth of 5โMHz (relaydestination channel). The AF protocol cannot be used here; but the DF protocol can be used, for instance, by adapting the modulation of the cooperative signal to the available bandwidth. In this case, the destination has to combine two signals with different modulations.
In this context, one of the issues that needs to be addressed is the design of the combiner. A conventional MRC cannot be used for combining signals with different modulations (except for special cases of modulations). Even if the modulations at the source and relay are identical, the MRC can severely degrade the receiver performance because it does not compensate for the decoding noise introduced by the relay [2โ6]. This is why the authors of [2, 4] proposed a maximumlikelihood detector (MLD) for combining two BPSKmodulated signals coming from the source and relay. The authors of [6] proposed an improved MRC called CMRC which aims at maximizing receive diversity. The authors of [3] proposed a linear combiner for which the weights are tuned to minimize the raw bit error rate (BER). The main issue is that one is not always able to explicit the raw BER as a function of the combiner weights whereas the likelihood calculation is more systematic. Additionally, when some a priori knowledge is available, the ML metric can be used to calculate an a posteriori probability (APP). In the context of orthogonal relay channels, the authors of [4] derived two new combiners: the best MRC in the sense of the equivalent signaltonoise ratio and MMSE combiner. They also assessed the BER performance of the latter and MLD in the uncoded case.
Compared to the aforementioned works, this paper also aims at designing a good combiner at the destination but it differs from them on two essential points. (1) The interaction between the combiner and channel decoder is exploited in the sense that we want to express the branch metrics of the trellis associated with channel decoding for the MRC, MMSE combiner, CMRC, and especially for the ML combiner. (2) When the ML combiner is assumed, the source and relay can use arbitrary modulations (not necessarily BPSK modulations as in [2, 3, 5]) and, more importantly, these can be different.
2. Signal Model
At the source, the information bit sequence is encoded into a sequence of bits and modulated into the transmitted signal , where for all , is a finite alphabet corresponding to the modulation constellation used by the source and . At the relay, the message is decoded, reencoded with the same encoder as the source and modulated into the transmitted signal where for all , is a finite alphabet corresponding to the modulation constellation used by the relay, and . We denote by (resp., ) the number of coded bits conveyed by one source (resp., relay) symbol. By definition: and . More specifically, the information bit sequence is assumed to be encoded by a rate convolutional encoder (). As the sequence comprises symbols, we have that where is the channel encoder memory. Assuming timeselective but frequencynonselective channels, the baseband signals received by the destination from the source and relay, respectively, write and , where and are zeromean circularly symmetric complex Gaussian noises with variances and , respectively. The complex coefficients and represent the gains of the sourcedestination and sourcerelay fading channels. For insuring coherent decoding, these two gains are assumed to be known to the receiver and relay, respectively. We define , , , and , where is the gain of the sourcerelay fading channel. Note that, in order to ensure the conservation of the coded bit rate between the input and output of the relay, and have to be linked by the following compatibility relation: . In the sequel, we will use the quantity , where lcm is the least common multiple function. For simplicity, we assume that the source and relay use the same channel coder. Therefore, the relay has to use a modulation that is compatible with the source's one. We will also assume that the number of times per second the channel can be used is directly proportional to the available bandwidth. For example, if the source uses a BPSK modulation and the cooperation channel has a bandwidth equal to half the downlink bandwidth, the relay can use a QPSK modulation.
3. A New Trellis Branch Metric
3.1. When the Source and Relay Use Arbitrary and Different Modulations
In this case, the linear combiners derived by [3, 4, 6] cannot be used in general. However, provided that the above compatibility condition is met, the ML combiner can be derived as we show now. Let us denote by and the sequences of noisy symbols received by the destination from the source and relay, respectively. The discrete optimization problem the ML combiner solves is as follows: As the reception noises are assumed to be independent, . The first term easily writes as In order to express the second term, we introduce a sequence of discrete symbols denoted by which models the residual noise at the relay after the decodingreencoding process. This noise is, therefore, modeled by a multiplicative error term which is not independent of the symbols transmitted by the relay. Additionally, the statistics of this noise are assumed to be known by the destination. For this, one can establish once and for all a lookup table between the sourcerelay SNR and the bit error rate after reencoding at the relay. The cooperation signal writes then as , where and is the symbol the relay would generate if there were no decoding error at the relay. For example, when the relay uses a QPSK modulation, . Therefore, we have that . At this point, we need to make an additional assumption in order to easily derive the path metric of the ML decoder. From now on, we assume that the discrete symbols of the sequence are conditionally independent. This assumption is very realistic, for example, if the source and relay implement a bitinterleaved coded modulation (BICM) or a trelliscoded modulation (TCM). In the case of the BICM, the channel coder, which generates coded bits, and the modulator are separated by an interleaver. The presence of this interleaver precisely makes the proposed assumption reasonable. Under the aforementioned assumption, one can expand as The main consequence of this assumption is a significant reduction of the decoder complexity. If the assumption is not valid, the proposed derivation can always be used but the performance gain obtained can be marginal since the errors produced by will not be spread over the data block but rather occurs in a sporadic manner along the block.
In order to express the path metric of a given path in the trellis associated with channel decoding, we need now to link the likelihood expressed above and the likelihood associated with a given bit , where . The reason why we consider subblocks of bits is that in order to meet the rate compatibility condition, the ML combiner combines the symbols received from the source with the symbols received from the relay. Now for all , let us define the sets , a set of subblocks of consecutive bits, , and , their equivalents in the source (resp., relay) modulation space. With these notations, the bit likelihood can be expressed as follows: where we used the notation . When a BICM is used, the obtained loglikelihood sequence is then deinterleaved and given to a Viterbi decoder.
3.2. When the Source and Relay Use Arbitrary and Identical Modulations
The derivation of the codedbit likelihood in the case where the modulations used by the source and relay are the same is ready since it is special case of derivation conducted previously with . In this case, both ML and linear combiners can be used since the combination can be performed symbolbysymbol. The loglikelihood becomes , where . If we further assume that the modulations used are BPSK modulations, the likelihood on the received sequences takes a more explicit form. Indeed, it can be checked that where represents the residual bit error rate (after the decodingreencoding procedure inherent to DF protocol). Denote by the interleaver function such that and . Finally, the path metric is merely given by So the combining and channel decoding are performed jointly by modifying the branch metrics as indicated above.
When using a linear combiner, one has to compute the APP from the equivalent signal at the combiner output. This computation requires the equivalent channel gain and noise. We provide them for each linear combiner considered here. For a given combiner, denote its optimal vector of weights by and rewrite the signal at the combiner output as , where and are the equivalent channel gain and noise, respectively. The bit loglikelihood can then be easily expressed as . Table 1 summarizes the values of these quantities with the notations and .

4. Simulation Example
For Figures 1 and 2, we assume that the source and the relay implement a rate convolutional encoder (4state encoder with a free distance equal to 5). Frequency nonselective Rayleigh block fading channels are assumed and the data block length is chosen to be . First, we compare the combiners between themselves when both the relay and source use a 4QAM modulation. Figure 1 represents the BER at the decoder output as a function of . There are 6 curves: from the top to the bottom, they, respectively, represent the performance with no relay, with the relay associated with the conventional MRC, MMSE, CMRC, and ML combiners. When implementing the conventional MRC, the receiver does not significantly improve its performance with respect to the noncooperative case whereas the other combiners can provide more than an 8โdB gain and perform quite similarly. Then (see Figure 2), we evaluate the performance gain brought by the MLD when the source and relay have to use different modulations: the source implements a BPSK while the relay implements either a 4QAM or a 16QAM. The second scenario would correspond to a case where the sourcedestination channel bandwidth is 4 times larger than the relaydestination channel bandwidth (e.g., 20โMHz versus 5โMHz). We see that the MLD not only makes cooperation possible but also allows the destination to extract a significant performance gain from it. To have an additional reference, we also represented the performance of the equivalent virtual MIMO system, which is obtained for .
5. Concluding Remarks
The results provided in this letter and many other simulations performed in the coded case led us to the following conclusion: if the source and relay can use the same modulation, the CMRC generally offers the best performancecomplexity tradeoff. On the other hand, if the modulations are different, as it would be generally the case when two existing communications systems are associated to cooperate, linear combiners and thus the CMRC cannot be used in general and the ML combiner is the only implementable combiner.
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Copyright
Copyright © 2008 Brice Djeumou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.