# Combining Coded Signals with Arbitrary Modulations in Orthogonal Relay Channels

Authors: Brice Djeumou; Samson Lasaulce; Antoine O. Berthet
Journal: Research Letters in Communications
 (2008)

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## Abstract

We consider a relay channel for which the following assumptions
are made. (1) The source-destination and relay-destination
channels are orthogonal (frequency division relay channel). (2) The
relay implements the decode-and-forward 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),
cooperative-MRC (C-MRC), and the minimum mean-square error
(MMSE) combiner.


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## Body

## 1. Motivations and Technical Background

We consider  orthogonal relay channels for which
orthogonality is implemented in frequency [1]. Since the source-destination channel is assumed to be
orthogonal to the relay-destination 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 decode-and-forward (DF)
protocol. We have at least two motivations for this choice. First, in contrast
with the well-known amplify-and-forward (AF) protocol, it can be implemented in
a digital relay transceiver. More importantly, whereas the AF protocol imposes
the source-relay channel to have the same bandwidth as the relay-destination
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 20 MHz bandwidth (source-relay
channel) while UMTS signals have only a bandwidth of 5 MHz (relay-destination
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 [2–6]. This is why the authors of
[2, 4] proposed a
maximum-likelihood detector (MLD) for combining two BPSK-modulated signals
coming from the source and relay. The authors of [6] proposed an improved MRC
called C-MRC 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 N-relay channels, the authors of [4] derived two new combiners:
the best MRC in the sense of the equivalent signal-to-noise ratio and MMSE
combiner. They also assessed the BER performance of the latter and MLD in the
uncoded case.Compared to the afore-mentioned 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, C-MRC,
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, theL-information bit sequence m¯ is encoded into a sequence of bits b¯ and modulated into the transmitted signal x¯=(x(1),…,x(T)),
where for all t∈{1,…,T},x(t)∈𝒳, 𝒳 is a finite alphabet corresponding to the
modulation constellation used by the source and 𝔼|x(t)|2≤P0.
At the relay, the message is decoded, reencoded with the same encoder as the
source and modulated into the transmitted signal x¯1=(x1(1),…,x1(T1)) where for all t1∈{1,…,T1},x1(t1)∈𝒳1, 𝒳1 is a finite alphabet corresponding to the
modulation constellation used by the relay, and 𝔼|x1(t1)|2≤P1.
We denote by s (resp., r) the number of coded bits conveyed by one
source (resp., relay) symbol. By definition: s=log2|𝒳| and r=log2|𝒳1|.
More specifically, the information bit sequence is assumed to be encoded by a 1/q-rate convolutional encoder (q∈ℕ*). As the sequence x¯ comprises T symbols, we have that q(k+ν)=sT where ν is the channel encoder memory. Assuming
time-selective but frequency-nonselective channels, the baseband signals
received by the destination from the source and relay, respectively, write y0(t)=h0x(t)+z0(t) and y1(t1)=h1x1(t1)+z1(t1),
where z0 and z1 are zero-mean circularly symmetric complex
Gaussian noises with variances σ02 and σ12,
respectively. The complex coefficients h0 and h1 represent the gains of the source-destination
and source-relay fading channels. For insuring coherent decoding, these two
gains are assumed to be known to the receiver and relay, respectively. We
define γ0=𝔼|h0|2(P/σ02), γ1=𝔼|h1|2(P/σ12), γ1′=𝔼|h1′|2(P/σ02),
and ρ1=𝔼|X1X*|/P,
where h1′ is the gain of the source-relay fading
channel. Note that, in order to ensure the conservation of the coded bit rate
between the input and output of the relay, s and r have to be linked by the following
compatibility relation: sT=rT1.
In the sequel, we will use the quantity k=lcm(s,r),
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 y¯0 and y¯1 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:
(1)m¯^=argmaxm¯∈𝔽2LpML=argmaxm¯∈𝔽2Lp(y¯0,y¯1|x¯).As the reception noises are
assumed to be independent, pML=p(y¯0|x¯)p(y¯1|x¯).
The first term easily writes as
(2)p(y¯0|x¯)=∏t=1T1πσ02exp(−|y0(t)−h0x(t)|2σ02).In order to express the second
term, we introduce a sequence of T1 discrete symbols denoted by e¯1 which models the residual noise at the relay
after the decoding-reencoding 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 source-relay SNR and the bit error rate after reencoding at
the relay. The cooperation signal writes then as y1(t1)=h1x1(t1)+z1(t1),
where x1(t1)=e¯1(t1)x˜1(t1) and x˜1(t1) 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, e1∈{1,ej(π/2),ejπ,ej(3π/2)}.
Therefore, we have that p(y¯1|x¯)=p(y¯1|x¯˜1)=∑e¯1p(y¯1,e¯1|x¯˜1)=∑e¯1p(e¯1|x¯˜1)p(y¯1|x¯˜1,e¯1).
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 e¯1 are conditionally independent. This assumption
is very realistic, for example, if the source and relay implement a
bit-interleaved coded modulation (BICM) or a trellis-coded 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 afore-mentioned
assumption, one can expand p(y¯1|x¯) as
(3)p(y¯1|x¯)=∑e¯1∏t1=1T1p(e1(t1)|x˜1(t1))p(y1(t1)|x˜1(t1),e1(t1))=∑e¯1∏t1=1T1p(e1(t1)|x˜1(t1))×1πσ12exp(−|y1(t1)−h1e1(t1)x˜1(t1)|2σ12). 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 bitbj,
where j∈{1,…,k}.
The reason why we consider subblocks of k bits is that in order to meet the rate
compatibility condition, the ML combiner combines the ks=k/s symbols received from the source with the kr=k/r symbols received from the relay. Now for all (i,j)∈{0,1}×{1,…,k},
let us define the sets ℬi(k)(j)={b¯k=(b1,…,bk)∈{0,1}k,bj=i},
a set of subblocks of rs consecutive bits, 𝒳i(ks)(j)={x¯ks∈𝒳ks,bj=i},
and 𝒳1,i(kr)(j)={x˜¯1kr∈𝒳1kr,bj=i},
their equivalents in the source (resp., relay) modulation space. With these
notations, the bit likelihood can be expressed as follows:
(4)λ(bj=i)=ln[∑b¯1k∈ℬi(k)(j)p(y¯0,1ks|x¯1ks)p(y1,1kr|x˜¯1,1kr)]=ln[∑b¯1k∈ℬi(k)(j)∏t=1ksp(y0(t)|x(t))∏t1=1krp(y1(t1)|x˜1(t1))], where we used the notation v¯1n=(v(1),…,v(n)).
When a BICM is used, the obtained log-likelihood sequence is then
de-interleaved and given to a Viterbi decoder.


### 3.2. When the Source and Relay Use Arbitrary and Identical Modulations

The derivation of the coded-bit 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 withk=s=r.
In this case, both ML and linear combiners can be used since the combination
can be performed symbol-by-symbol. The log-likelihood becomes λ(bj=i)=ln[∑b¯1s∈ℬi(s)(j)p(y0(t)|x(t))p(y1(t)|x(t))],
where 1≤j≤s.
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
(5)ln[p(y¯0,y¯1|m¯)]=−kqln(2π)−Tln(σ02σ12)−∑t=1T[|y0(t)−h0x(t)|2σ02−ln[∑e=−1,+1Pr[ϵ(t)=e]×exp(−|y1(t)−h1ex(t)|2σ12)]],
where Pr[ϵ=−1] represents the residual bit error rate (after
the decoding-reencoding procedure inherent to DF protocol). Denote by π the interleaver function such that t=π(t0) and t0=π−1(t).
Finally, the path metric is merely given by(6)μx=∑t0=1T[|y0(t0)−h0x(t0)|2σ02−ln[∑e=−1,+1Pr(ϵ(t0)=e)×exp(|y1(t0)−h1ex(t0)|2σ12)]].
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 byw¯=(w0,w1) and rewrite the signal at the combiner output
as y=∑i=01wiyi=heqx+zeq,
where heq and zeq∼𝒩(0,σeq2) are the equivalent channel gain and noise,
respectively. The bit log-likelihood can then be easily expressed as λ(bj=i)=ln[∑x¯∈𝒳i(s)(j)p(y|heq,x¯)].
Table 1 summarizes the values of these quantities with the notations a0=|h0|2 and a1=|h1|2.Table 1
Equivalent channel parameters for the linear combiners.
heqσeq2MRCa0σ02+a1σ12a0σ02+a1σ12MMSEa0σ02+a1ρ12σ12+a1P0(α12−ρ12)a0σ02+a1ρ12σ12+a1P0(α12−ρ12)C-MRCa0+min{γ1′,γ1}γ1a1a0σ02+min{γ1′,γ1}γ1a1σ12


## 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 y¯0 and y¯1 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:
(1)m¯^=argmaxm¯∈𝔽2LpML=argmaxm¯∈𝔽2Lp(y¯0,y¯1|x¯).As the reception noises are
assumed to be independent, pML=p(y¯0|x¯)p(y¯1|x¯).
The first term easily writes as
(2)p(y¯0|x¯)=∏t=1T1πσ02exp(−|y0(t)−h0x(t)|2σ02).In order to express the second
term, we introduce a sequence of T1 discrete symbols denoted by e¯1 which models the residual noise at the relay
after the decoding-reencoding 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 source-relay SNR and the bit error rate after reencoding at
the relay. The cooperation signal writes then as y1(t1)=h1x1(t1)+z1(t1),
where x1(t1)=e¯1(t1)x˜1(t1) and x˜1(t1) 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, e1∈{1,ej(π/2),ejπ,ej(3π/2)}.
Therefore, we have that p(y¯1|x¯)=p(y¯1|x¯˜1)=∑e¯1p(y¯1,e¯1|x¯˜1)=∑e¯1p(e¯1|x¯˜1)p(y¯1|x¯˜1,e¯1).
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 e¯1 are conditionally independent. This assumption
is very realistic, for example, if the source and relay implement a
bit-interleaved coded modulation (BICM) or a trellis-coded 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 afore-mentioned
assumption, one can expand p(y¯1|x¯) as
(3)p(y¯1|x¯)=∑e¯1∏t1=1T1p(e1(t1)|x˜1(t1))p(y1(t1)|x˜1(t1),e1(t1))=∑e¯1∏t1=1T1p(e1(t1)|x˜1(t1))×1πσ12exp(−|y1(t1)−h1e1(t1)x˜1(t1)|2σ12). 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 bitbj,
where j∈{1,…,k}.
The reason why we consider subblocks of k bits is that in order to meet the rate
compatibility condition, the ML combiner combines the ks=k/s symbols received from the source with the kr=k/r symbols received from the relay. Now for all (i,j)∈{0,1}×{1,…,k},
let us define the sets ℬi(k)(j)={b¯k=(b1,…,bk)∈{0,1}k,bj=i},
a set of subblocks of rs consecutive bits, 𝒳i(ks)(j)={x¯ks∈𝒳ks,bj=i},
and 𝒳1,i(kr)(j)={x˜¯1kr∈𝒳1kr,bj=i},
their equivalents in the source (resp., relay) modulation space. With these
notations, the bit likelihood can be expressed as follows:
(4)λ(bj=i)=ln[∑b¯1k∈ℬi(k)(j)p(y¯0,1ks|x¯1ks)p(y1,1kr|x˜¯1,1kr)]=ln[∑b¯1k∈ℬi(k)(j)∏t=1ksp(y0(t)|x(t))∏t1=1krp(y1(t1)|x˜1(t1))], where we used the notation v¯1n=(v(1),…,v(n)).
When a BICM is used, the obtained log-likelihood sequence is then
de-interleaved and given to a Viterbi decoder.


## 3.2. When the Source and Relay Use Arbitrary and Identical Modulations

The derivation of the coded-bit 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 withk=s=r.
In this case, both ML and linear combiners can be used since the combination
can be performed symbol-by-symbol. The log-likelihood becomes λ(bj=i)=ln[∑b¯1s∈ℬi(s)(j)p(y0(t)|x(t))p(y1(t)|x(t))],
where 1≤j≤s.
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
(5)ln[p(y¯0,y¯1|m¯)]=−kqln(2π)−Tln(σ02σ12)−∑t=1T[|y0(t)−h0x(t)|2σ02−ln[∑e=−1,+1Pr[ϵ(t)=e]×exp(−|y1(t)−h1ex(t)|2σ12)]],
where Pr[ϵ=−1] represents the residual bit error rate (after
the decoding-reencoding procedure inherent to DF protocol). Denote by π the interleaver function such that t=π(t0) and t0=π−1(t).
Finally, the path metric is merely given by(6)μx=∑t0=1T[|y0(t0)−h0x(t0)|2σ02−ln[∑e=−1,+1Pr(ϵ(t0)=e)×exp(|y1(t0)−h1ex(t0)|2σ12)]].
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 byw¯=(w0,w1) and rewrite the signal at the combiner output
as y=∑i=01wiyi=heqx+zeq,
where heq and zeq∼𝒩(0,σeq2) are the equivalent channel gain and noise,
respectively. The bit log-likelihood can then be easily expressed as λ(bj=i)=ln[∑x¯∈𝒳i(s)(j)p(y|heq,x¯)].
Table 1 summarizes the values of these quantities with the notations a0=|h0|2 and a1=|h1|2.Table 1
Equivalent channel parameters for the linear combiners.
heqσeq2MRCa0σ02+a1σ12a0σ02+a1σ12MMSEa0σ02+a1ρ12σ12+a1P0(α12−ρ12)a0σ02+a1ρ12σ12+a1P0(α12−ρ12)C-MRCa0+min{γ1′,γ1}γ1a1a0σ02+min{γ1′,γ1}γ1a1σ12


## 4. Simulation Example

For Figures1
and 2, we assume that the source and the relay implement a 1/2-rate convolutional encoder (4-state 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 1024.
First, we compare the combiners between themselves when both the relay and
source use a 4-QAM modulation. Figure 1 represents the BER at
the decoder output as a function of γ0=γ1.
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, C-MRC, 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 8 dB 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 4-QAM or a 16-QAM. The second scenario would correspond to a
case where the source-destination channel bandwidth is 4 times larger than the
relay-destination channel bandwidth (e.g., 20 MHz versus 5 MHz). 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 1×2 MIMO system, which is obtained for γ1=+∞.Figure 1
BER at the destination withγ1′=γ0, γ1=γ0.Figure 2
BER at the destination withγ1′=γ0−10,  γ1 = γ0+10.


## 5. Concluding Remarks

The results
provided in this letter and many other simulations performed in thecoded case led us to the following conclusion: if the source and relay can use the
same modulation, the C-MRC generally offers the best performance-complexity
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 C-MRC cannot be used in
general and the ML combiner is the only implementable combiner.



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*Source: 287320-2008-08-05.xml*