\authorblockNSudipta Kundu and B. Sundar Rajan \authorblockADept. of ECE, IISc, Bangalore 560012, India, Email: sudiptak, bsrajan @ece.iisc.ernet.in
###### Abstract

With no CSI at the users, transmission over the two-user Gaussian Multiple Access Channel with fading and finite constellation at the input, is not efficient because error rates will be high when the channel conditions are poor. However, perfect CSI at the users is an unrealistic assumption in the wireless scenario, as it would involve massive feedback overheads. In this paper we propose a scheme which uses only quantized knowledge of CSI at the transmitters with the overhead being nominal. The users rotate their constellation without varying their transmit power to adapt to the existing channel conditions, in order to meet certain pre-determined minimum Euclidean distance requirement in the equivalent constellation at the destination. The optimal modulation scheme has been described for the case when both the users use symmetric -PSK constellations at the input, where , being a positive integer. The strategy has been illustrated by considering examples where both users use QPSK or 8-PSK signal sets at the input. It is shown that the proposed scheme has better throughput and error performance compared to the conventional non-adaptive scheme, at the cost of a feedback overhead of just bits, for the -PSK case.

## I Introduction

A multiple access channel (MAC) consists of multiple users transmitting independent information to a common destination. There is no cooperation among the users. The capacity region for a discrete memoryless MAC is well known [1] [2]. For a two-user MAC with additive white Gaussian noise (AWGN) the capacity achieving input is the continuous Gaussian alphabet. The two-user Gaussian MAC with finite input constellations like -QAM, -PSK was studied in [3] [4]. It was shown that relative rotation between input constellations [3], or a constellation power allocation scheme [4] may be employed to maximize the constellation constrained (CC) capacity regions. Trellis based coding schemes were also suggested to achieve any rate pair within the CC capacity region.

In this paper, a two-user MAC with quasi-static fading is considered, as shown in Fig. 1. The two users transmit information to a common destination. The random variables and are the channel gains for User-1 and User-2 respectively and , , where denotes the circular symmetric complex Gaussian random variable with variance . AWGN gets added to the received signal at the destination, . User- transmits a symbol from a complex finite constellation (like -QAM or -PSK) of unit average energy, i.e, . Let be the average power constraint for each user. The received signal at the destination is thus represented by

 y=√Ph1x1+√Ph2x2+z.

We assume that perfect CSI i.e. the tuple is available only at the destination.

At the destination the system can be viewed as a single user AWGN channel with the symbols drawn from a sum constellation

 Ssum =√Ph1S1+√Ph2S2 =√Ph1(S1+h2h1S2) =√Ph1(S1+γejθS2)Seff, (1)

where , and denotes the effective constellation.

Without loss of generality it can be assumed that , as destination has knowledge of both and separately. If , then at the destination the ratio can be simply reversed to compute . Which one among the two ratios is calculated is made known to the users via a single bit of feedback. For the rest of the paper, we assume that the ratio is calculated at the destination. However, the results obtained still hold when the ratio calculated is , by interchanging the roles of User-1 and User-2. For the rest of the paper a -PSK constellation refers to a symmetric PSK signal set, with , being a positive integer. The points in the -PSK signal set are of the form , where . We assume that , where is an -PSK constellation. We refer to the pair to represent and call it the fade state throughout the paper. We refer to the complex plane that represents with as the plane.

Perfect channel state information (CSI) is available at the destination only, which quantizes the plane into finite number of regions. The quantization obtained is similar to the one used for physical layer network coding in [5], which was subsequently derived analytically in [6]. This quantized knowledge of the fade state is made available to the users to adapt their modulation scheme via rotation of constellations to compensate for the possibly bad channel conditions. MAC with limited channel state information at transmitter (CSIT) has been studied from an information theoretic point of view in [7],[8]. In [9], it was shown that for a two-user discrete memoryless MAC with additional common message, finer CSIT results in increasing the capacity region. To the best of our knowledge, explicit modulation schemes with finite constellations and quantized fade state feedback has not been reported before.

The contributions and organization of this paper are as follows:

• A quantization of the plane is derived, for the case when both users use -PSK constellations at the input. We illustrate the quantization procedure by taking examples of the QPSK and 8-PSK case. (Section II-B)

• A modulation scheme is proposed for the users, which adapts according to the quantized feedback about the fade state that they receive from the destination, in order to satisfy a certain minimum distance guarantee in given in (1). The fade states which leads to violation of this minimum distance guarantee have been identified. Adaptation involves rotation of the constellation of one user relative to the other, without any change in transmit power, in order to effectively avoid these bad channel conditions. (Section III-A)

• The procedure to obtain the optimal angles for rotation is stated for the -PSK case. The optimal rotation angles are calculated in closed form for the QPSK and 8-PSK case. (Section III-B)

• An upper bound on , i.e., the maximum value of the minimum distance in the effective constellation that can be guaranteed, is derived. (Section III-C)

• Simulation results are presented to show the extent to which the proposed strategy outperforms the conventional transmission scheme without adaptation. (Section IV)

## Ii Channel Quantization for M-Psk Signal Sets

In this section we obtain a quantization of the plane into finite number of regions at the destination.

### Ii-a Distance Distribution in the effective constellation

Without loss of generality we assume that the average power constraint of each user is . It is known that the error performance for an AWGN channel is determined by the Euclidean distance distribution of the input constellation. In our case, the distance distribution of decides the error performance at the destination. For any value of , denotes the distance between the two points and , where , refer to the points and respectively with . It is given by

 d′2(s1,s2)sum↔(s′1,s′2)sum =P|h1|2|(s1−s′1)+γejθ(s2−s′2)|2 =P|h1|2d2(s1,s2)↔(s′1,s′2), (2)

where (2), denotes the distance between the points and , where refers to the points and in . Since simply scales the distances in we can focus only on

 d2(s1,s2)↔(s′1,s′2)=|(s1−s′1)+γejθ(s2−s′2)|2 (3)

as the quantity of interest.

It is clear from (3) that for certain values of the distance between points and in reduces to zero, i.e. if

 γejθ=−(s1−s′1)(s2−s′2) (4)

then . These values of are called the singular fade states [5], [6]. Singular fade states can also be defined as follows:

###### Definition 1

A fade state is said to be a singular fade state if .

Clearly, is a singular fade state, for any arbitrary signal set . For any input constellation the other non-zero singular fade states are obtained using (4). For a given input constellation , let denote the set of all singular fade states.

###### Example 1

When , where is a QPSK constellation, then the non-zero singular fade states are at

 γ =√2,θ=45∘,135∘,225∘,315∘ γ =1,θ=0∘,90∘,180∘,270∘ γ =1√2,θ=45∘,135∘,225∘,315∘

Since for an AWGN channel the error performance at the destination is dominated by the minimum distance of the input constellation, it is sufficient to study the minimum distance of . Also from Definition 1 minimum distance in reduces to zero at the singular fade states. The following lemma provides an upper bound on the minimum distance of the effective constellation .

###### Lemma 1

When both the users use any arbitrary signal set (which includes -PSK, -QAM) at the input, then for any fade state , the minimum distance between any two points in is upper bounded by the minimum distance in the input constellation .

{proof}

From the definition of , we have,

 d2min(γ,θ) =min(s1,s2)≠(s′1,s′2)∈S2|(s1−s′1)+γejθ(s2−s′2)|2 ≤min(s1,s2)≠(s′1,s′2)∈S2{|s1−s′1|2+γ2|s2−s′2|2}
 Nowd2min(γ,θ) ≤mins1≠s′1∈S|s1−s′1|2=d2min(S). (5) Alsod2min(γ,θ) ≤mins2≠s′2∈Sγ2|s2−s′2|2=γ2d2min(S). (6)
 From (5) and (6), and using the fact that γ≥1, we have d2min(γ,θ)≤min{d2min(S),γ2d2min(S)}=d2min(S).

In the following lemma, it is proved that in order to study the distance profile in it is sufficient to consider when both users use -PSK signal sets. Distance profiles for other values of can be obtained from . We use the term wedge to denote the region and on the plane. The lines and for and the arc for form the boundary of the wedge .

###### Lemma 2

To study the distance profile in when both the users use -PSK constellations, it is sufficient to consider the case . All other cases can be obtained from this.

{proof}

The proof is in two steps. First we show that the distance profile is a repetitive structure with period . Next, it is shown that within the wedge the distance profile is symmetric about the bisector of this wedge i.e., the line. We have from (1),

 Seff=S+γejθS.

For any arbitrary value of where ,

 Seff =S+γejθ′(Sej2kπM) =S+γejθ′S.

The last equality follows from the fact that rotating a -PSK constellation by an integral multiple of does not alter the distance profile of the constellation. Thus, whatever distance profiles for are obtained for the wedge , it is exactly repeated for the remaining wedges to cover the entire range of .

To show that the distance profiles are symmetric about , we need to show that and , where , have the same distance profiles. We have

 S+γej(πM−α)S =S+γej(πM−α)(Se−2πM) =S+γe−j(πM+α)S. (7)

The first equality is because i.e. rotating by gives the same constellation. Thus for , . Also due to the symmetric nature of -PSK constellation, the distance distribution of the sum constellation depends only on the relative angle of rotation between the input constellations. Thus and have same distance profiles, for any . This together with (7) proves the second part of the lemma.

From Lemma 2, it is clear that when both users use -PSK signal sets, if is a singular fade state, then there exists singular fade states at , where because distance distribution in is periodic with period . The distance distribution of is the basis for channel quantization. Also from Lemma 2, it suffices to obtain such a quantization only for the wedge . This can then be reflected along the line, to give the quantization for the wedge , which when repeated for the remaining wedges will cover the entire plane.

### Ii-B Channel Quantization for the M-PSK case

In this subsection we propose a technique to obtain the quantization of the plane, when both users use -PSK signal sets. From Lemma 1, when both users use -PSK signal sets at the input, the minimum distance in for any value of , . Now the following lemma gives the number of singular fade states in the wedge .

###### Lemma 3

When both users use -PSK signal sets at the input, the number of singular fade states in the wedge , is given by . Further, these singular fade states lie along the two lines and .

{proof}

From [6], the total number of singular fade states other than zero is . Out of these, lie on the circle . It is also known from [6], that if is a singular fade state, then is also a singular fade state. Thus, half of the total number of remaining singular fade states lie inside the circle and the other half lies outside it. This along with the fact that singular fade states are periodic, implies the number of singular fade states for the wedge , is given by

 1M⎛⎜⎝M34−M222+M⎞⎟⎠=M28−M4+1.

Also from [6], it is clear that these fade states lie along and lines. We denote this set of all singular fade states lying in the wedge by . Let .

Observe from (3), that the distance between two points in is a function of and . Let and for . Now from (3), the distance between the elements of the pair is

 d2(s1,s2)↔(s′1,s′2)=|△s1|2+γ2|△s2|2+2γ|△s1△s2|cos(θ+ϕ2−ϕ1). (8)

Consider another other pair with , and let and for . If , and or , then from (8), for all values of , even though the value of this distance changes with .

###### Definition 2

A distance class denoted by , is a subset of , which contains the pairs of the form , where and denote the complex points in , such that the distance between the two elements of a pair is same for all pairs in and this property holds for all values of and , though the value of the distance depends on .

For a given input constellation , let denote the set of the all distance classes for it.

###### Definition 3

Associated with every distance class is a function , called the class distance function, which gives the value of the distance between the two elements of a pair in for any .

###### Definition 4

For a given fade state , the function gives the value of the distance between the two elements of a pair in , for any . This is called the fade state distance function.

We use integer , to represent the point in i.e. the -PSK signal set. The integer , denotes the complex point in obtained by combining the points and of i.e. it refers to the point in . For each distance class , among all the pairs choose the one with the minimum value of to be the representative in . If more than one pair has the same value of choose the one with the lowest value of as the class representative. When the users -PSK signal set at input, there are pairwise distances in . These pairwise distances are thus partitioned into distance classes.

###### Example 2

Fig. 2, shows when both users use QPSK constellations for . There are 120 pairwise distances and 20 distance classes. These are listed in Table I, along with the corresponding class distance functions, and the class representatives.

We define the set of all class distance function, and the set of all fade state distance functions as follows,

 d¯C(γ,θ) ={dC(γ,θ)|C∈¯C} dΓ,Θ(C) ={dγ,θ(C)|(γ,θ)∈(Γ,Θ)plane}.

From Definition 1, at a singular fade state the value of at least one of the class distance functions in will reduce to zero.

###### Lemma 4

Among the set of all class distance functions that reduce to zero at the singular fade state , there is a particular one which is the minimum among that set, for all values of .

{proof}

Let be the number of class distance functions that reduce to zero at the singular fade state . Denote these by , and let be the representative element for the distance class . From (3) and (4), we have

 d2Ci(γ,θ) =d2(s1,i,s2,i)↔(s′1,i,s′2,i) =|(s1,i−s′1,i)+γejθ(s2,i−s′2,i)|2 =|s2,i−s′2,i|2|γejθ+s1,i−s′1,is2,i−s′2,i|2 =|s2,i−s′2,i|2|γejθ−γ′ejθ′|2. (9)

From (9), these class distance functions differ only in the constant coefficient . From Definition 2, all these coefficients are different. Let

 l′=argmin1≤i≤L|s2,i−s′2,i|2.

Now, from (9), is minimum among all for all values of .

###### Definition 5

The region corresponding to distance class , denotes the region in the complex plane for which the class distance function gives the minimum distance in , i.e.,

 R(C)={ (γ,θ)∈{(Γ,Θ)/H}| dC(γ,θ)≤dC′(γ,θ)for allC′≠C∈¯C}.
###### Definition 6

When both the users use -PSK constellations at the input, denotes the portion of the region lying in the wedge , i.e.,

 RW(C)=R(C)∩% wedge[0,π/M].

Note that, when both the users use -PSK constellations at the input, for some the corresponding region can be a null set, because the associated class distance function does not give the minimum distance in for any value of in . There is always a distance class for which the associated class distance function is . We denote this particular distance class as . From Lemma 1, the value of this class distance function is the upper bound for the minimum distance in . For example when both users use QPSK signal sets at the input, then from Table I, and the associated class distance function is .

The procedure to obtain the quantization of the plane, when both users use -PSK constellations at the input, is as follows:

1. Obtain the singular fade states in i.e., lying in the wedge . Each of these singular fade state is denoted by where .

2. For the singular fade state in , identify the set of class distance functions in that reduces to zero at that singular fade state . Choose the one among them, which is minimum in that set for all values of . (From Lemma 4, there is always only one such class distance function.) Let this class distance function be corresponding to distance class . Repeat this for all , to obtain a set of class distance functions , . Each reduces to zero at the singular fade state . This is the set of all possible class distance functions other than , that can possibly produce the minimum distance in .

3. To find the region , we need to obtain the values of for which where , and . The curves , , form the pairwise boundary between the regions corresponding to the two distance classes and . The curves form the pairwise boundary between the regions corresponding to distance classes and . The region is that region in the wedge excluding the complex point , which is the innermost region bounded by these pairwise boundaries, enclosing the singular fade state . For example, Fig. 6 depicts the region corresponding to the singular fade state at when both users use 8-PSK signal sets. In the figure, the curve refers to the curve , and refers to the curve . It is the innermost region (shaded in the figure) in the wedge bounded by the pair-wise boundaries and surrounding the singular fade state . Once the regions , are obtained, the region exterior to all these regions, lying within the wedge , is the region where is the minimum distance in , i.e., the region .

4. The quantization obtained in Step 3, for the wedge , can now be extended by the procedure suggested in Section II-A to cover the entire plane.

We will illustrate the procedure with two examples.

###### Example 3

Channel Quantization for QPSK signal sets.

Here we consider the scenario where both users use QPSK constellations at input, i.e. . From Lemma 3, there are two singular fade states in the wedge , i.e. and these are at and . The class distance functions in which reduce to zero at these singular fade states are identified. For the singular fade state the distance