Distributed Channel Quantization for Two-User Interference Networks

# Distributed Channel Quantization for Two-User Interference Networks

Xiaoyi Leo Liu, Erdem Koyuncu, and Hamid Jafarkhani
Center for Pervasive Communications and Computing
University of California, Irvine
###### Abstract

We introduce conferencing-based distributed channel quantizers for two-user interference networks where interference signals are treated as noise. Compared with the conventional distributed quantizers where each receiver quantizes its own channel independently, the proposed quantizers allow multiple rounds of feedback communication in the form of conferencing between receivers. We take the network outage probabilities of sum rate and minimum rate as performance measures and consider quantizer design in the transmission strategies of time sharing and interference transmission. First, we propose distributed quantizers that achieve the optimal network outage probability of sum rate for both time sharing and interference transmission strategies with an average feedback rate of only two bits per channel state. Then, for the time sharing strategy, we propose a distributed quantizer that achieves the optimal network outage probability of minimum rate with finite average feedback rate; conventional quantizers require infinite rate to achieve the same performance. For the interference transmission strategy, a distributed quantizer that can approach the optimal network outage probability of minimum rate closely is also proposed. Numerical simulations confirm that our distributed quantizers based on conferencing outperform the conventional ones.

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Distributed Channel Quantization for Two-User Interference Networks

Xiaoyi Leo Liu, Erdem Koyuncu, and Hamid Jafarkhani

Center for Pervasive Communications and Computing

University of California, Irvine

## I Introduction

Channel quantization in a network with multiple receivers is fundamentally different from that in a point-to-point system. In a point-to-point system, the receiver can acquire the entire channel state information (CSI) and send the corresponding quantized feedback information to the transmitter [1, 2, 3, 4]. On the other hand, in a network with multiple receivers, each receiver only has access to its own local CSI due to different geographical locations of the different receivers. Each receiver can thus quantize only a part of the entire global CSI, which results in a distributed quantization problem.

In the existing work on distributed quantization for networks [1, 5, 6], each receiver first quantizes its local CSI independently and then sends a finite number of bits representing quantized information through feedback links to other terminals. After decoding feedback information from all receivers, each terminal reconstructs the quantized version of the global CSI. Afterwards, transmission methods such as beamforming or power control are adopted by treating the global quantized CSI as the exact unquantized CSI. For example, power control and throughput maximization for interference networks based on separate quantized feedback information from receivers are analyzed in [5, 6]. In [1], beamformers are designed for the -user MIMO interference channels with independent quantized information from each receiver. The performance of these quantizers depend on the number of feedback bits assigned for quantization to each receiver and always suffer from some loss when compared with the optimal performance.

In this paper, we propose a novel distributed quantization strategy with multiple rounds of feedback communication in the form of conferencing between receivers. Through conferencing among receivers, partial CSI from other receivers can be utilized for a better overall quantizer performance. To illustrate this, we consider the distributed quantization problem for two-user interference networks with time sharing and interference transmission strategies. The network outage probability is the performance metric. We first propose a distributed quantizer that achieves the optimal network outage probability of sum rate in both time sharing and interference transmission with only two bits of feedback information. We also propose a distributed quantizer that attains the optimal network outage probability of minimum rate in time sharing with finite average feedback rate. For the optimal network outage probability of minimum rate in interference transmission, a distributed quantizer that can approach it closely is also proposed. By numerical simulations, we show the effectiveness of the proposed quantizers by comparing them with the conventional ones.

The rest of this paper is organized as follows: In Section II, we provide a description of the system model. In Sections III and IV, we introduce and analyze the distributed quantizers for time sharing and interference transmission strategies, respectively. Numerical simulations are provided in Section V.

Notations: Bold-face letters refer to vectors or matrices. denotes the matrix transpose. , and represent the sets of complex, real and natural numbers, respectively. The set of complex -vectors is denoted by and the set of complex matrices is denoted by . represents a circulary-symmetric complex Gaussian random variable (r.v.) with mean and covariance . is the probability density function (PDF) of a r.v. . is the cardinality of the set . For sets and , . denotes the expectation and denotes the probability. For any , is the largest integer that is less than or equal to and is the smallest integer that is larger than or equal to . For any logical statement , we let when is true, and when is false. Finally, for , the real number is the base- representation of the real number .

## Ii Preliminaries

### Ii-a System strategy

Consider an interference network where transmitters and send independent signals to receivers and concurrently. Both transmitters and receivers are equipped with only a single antenna. The channel gain from to is denoted by for . We assume that and , where is the covariance of interference links. Let . Then, denotes the local CSI at receiver , and represents the entire CSI. The additive noises at the receivers are distributed as .

We assume a quasi-static block fading channel in which the channels vary independently from one block to another while remain constant within each block. Each receiver can perfectly estimate its local CSI and provide quantized instantaneous CSI to other terminals via error-free and delay-free feedback links.

### Ii-B Transmission strategies

We consider two transmission strategies in the two-user interference network, namely time sharing and interference transmission. Time sharing means either transmitter only occupies a proportion of the block to transmit while remains silent in the rest, thus no interference exists. Interference transmission refers to the scenario where both transmitters send signals within the entire block, thereby causing interference to each other. We assume that interference signals are dealt with as noises. Since we focus on the design of distributed quantizers based on conferencing, we also assume that only one strategy will be performed in the entire transmission for simplicity.

In time sharing, let be the percentage of time within the entire block in which only is active for with . The instantaneous power used by is , where and is the short-term power constraint. It is optimal for both transmitters to use full power under the condition of no interference. Therefore, for a given , the end-to-end rate at receiver is

 Rts,k(tk)≜tklog2(1+PHk,k).

In interference transmission, for and , the end-to-end rate at receiver is

 Rit,k(p1,p2)≜log2(1+pkPHk,kplPHl,k+1).

### Ii-C Network Outage Probability

Our performance measure is the network outage probability, which is the fraction of channel states at which the rate measure of the network falls below a target data rate . Such a performance metric is well-suited for applications where a given constant data rate needs to be sustained for every channel state. Two kinds of rate measurements are considered, namely sum rate and minimum rate. Our goal is to design efficient distributed quantizers that can achieve the optimal network outage probability of sum rate or minimum rate for both time sharing and interference transmission strategies.

## Iii Distributed Quantization for Network Outage Probability of Sum Rate

We first design distributed quantizers for interference transmission. The sum rate is . We define the network outage probability as111We choose the sum-rate outage threshold to be for a more fair comparison with the rate threshold that we shall specify for the minimum-rate outage threshold.

 \textmdOUTsrit≜\textmdPr{SRit(p1,p2)<2ρ}.

It is proved in [7] that the maximum sum rate is . Therefore, the optimal (minimum-achievable) network outage probability is

 \textmdOUToptsr,it=\textmdPr{max{SRit(1,0),%SRit(0,1),SRit(1,1)}<2ρ}.

In the following, we design a distributed quantizer, namely , that can achieve with only feedback bit per receiver. The quantizer consists of two local encoders and a unique decoder. The -th encoder is located at receiver and the decoder is shared by all terminals, for . The components of operate as follows:

For , maps to or according to . Accordingly, receiver will send the feedback bit “1” if , and “0” otherwise. The decoder decodes the bits fed back by receivers and recovers the values of for . The interference transmission pair is decided based on Table 1.

Denote the network outage probability achieved by as and let be the average feedback rate.222The average feedback rate in this paper is the sum of the average number of feedback bits fed back by each receiver.

###### Theorem 1.

and .

Proof: With , an outage event occurs only when for every , or equivalently when both receivers feeds back “” and the corresponding power vector from Table \@slowromancapi@ still results in outage. This shows that . Since two bits are fed back in total (one bit for either receiver), the average feedback rate is two bits per channel state.

The design of utilizes the fact that checking whether or leads to an outage event only requires the knowledge of local CSI at either receiver. Thus two bits of conferencing between receivers provides adequate information to each other for choosing the right pair to achieve the optimal performance.

We now consider the design of disributed quantizers for the time sharing strategy. In this case, we can similarly define the network outage probability of sum rate as where . Under the constraint of = 1, the maximum sum rate can easily be calculated to be . Therefore, the optimal network outage probability is

 \textmdOUToptsr,ts=\textmdPr{SRts(1,0)<2ρ,SRts(0,1)<2ρ}.

Noticing that and and using the same ideas as in the construction of , we can design a distributed quantizer for time sharing that achieves with only one bit of feedback per receiver (we omit the details). On the other hand, the equalities and also imply . Hence, we only need to consider interference transmission if our objective is to minimize the network outage probability of the sum rate.

## Iv Distributed Quantization for Network Outage Probability of Minimum Rate

We now study the design of distributed quantizers that minimize the outage probability of minimum rate. First, we determine the optimal network outage probability with time sharing or interference transmission. For time sharing, we define the network outage probability as

 \textmdOUTmr,ts≜\textmdPr{MRts(t1,t2)<ρ},

where is the minimum achievable rate of the two transmitters. In interference transmission, the network outage probability is

 \textmdOUTmr,it≜\textmdPr{MRit(p1,p2)<ρ},

where . Now, let and denote the optimal time sharing and power pairs that achieve and , respectively. We have the following two results, whose proofs can be found in Appendix A.

###### Proposition 1.

We have

 t⋆1=log2(1+PH2,2)log2(1+PH1,1)+log2(1+PH2,2),t⋆2=log2(1+PH1,1)log2(1+PH1,1)+log2(1+PH2,2). (1)
###### Proposition 2.

If , we have

 (p⋆1,p⋆2)=(√4P2H1,2H2,1H2,2+4PH2,2H1,2H1,1+1−12PH1,2,1), (2)

and otherwise, if , we have

 (p⋆1,p⋆2)=(1,√4P2H1,1H1,2H2,1+4PH1,1H2,1H2,2+1−12PH2,1). (3)

In particular, the optimal network outage probabilities of minimum rate for time sharing and interference transmission are given by and , respectively.

We now propose two distributed quantizers, namely and . For the time sharing strategy, will attain exactly with a finite average feedback rate. For interference transmission, will approach tightly with a finite average feedback rate.

### Iv-a Time Sharing

For a given , the minimum time percentage for receiver to prevent outage is given by

 tk,min=ρlog2(1+PHk,k),

which can be calculated and known by receiver , for . Denote by the time sharing pair determined by . The first task of is to determine whether or not through feedback communication between receivers. The first task is essentially a distributed decision-making problem. If holds, the second task is to find that also enables .

The quantizer is composed by two local encoders with the th encoder located at receiver and a unique decoder employed by all terminals. We add the superscript “” to indicate their operations in the -th round of conferencing for . Also, four parameters for are stored and updated at all terminals. Let represent the values of after round .

In round , maps into or via , for . Receiver will send the feedback bit “1” if , and the feedback bit “0” otherwise. Then, decodes the bits fed back by receivers and recovers the values of for . If or , an outage event is sure to happen. Then we set as the time sharing pair (in fact, any time sharing pair can be used as outage is inavoidable) and the conferencing process ends. Otherwise, and are updated as for , then continues to the next round.

In round where , maps into or according to

 \textmdENClmr,ts,k(hk)=1(tk,min≥t\textmdlb,l−1k,min+t\textmdub,l−1k,min2),

for . Receiver will send 1 bit of “1” if , and “0” otherwise. Then decodes the bits fed back by receivers and recovers the values of for .

1. If , an outage event is inavoidable. We thus set as the time sharing pair and conferencing ends.

2. If , we set as the time sharing pair, and conferencing ends.

3. If and , we let and . If and , we let and . In either case, conferencing continues to the next round.

Note that the condition is equivalent to , and determines whether holds or not. To accomplish this, either receiver quantizes its own in a finer and finer way when increases and tells the quantized feedback bits to others. The parameters serve as the lower and upper bounds on updated by conferencing between receivers. The decision of whether holds or not is made by jointly considering and . The inter-receiver conferencing process will continue until the exchanged feedback bits are adequate to make a precise decision about whether holds or not.

Let and denote the network outage probability and average feedback rate of , respectively. The following theorem shows that whenever the optimal time shairing pair in Proposition 1 can avoid outage, the time sharing pair picked by will also avoid outage with probability one, and that the average feedback rate of is finite. The proof is provided in Appendix B.

###### Theorem 2.

For any , we have

 \textmdOUT(DQmr,ts)=\textmdOUToptmr,ts, (4)

and

 \textmdFR(DQmr,ts)≤2+2e−ρlog2P(1+C0P), (5)

where is a bounded constant that is independent of .333Since we focus on showing the average feedback rate is finite for any , it is beyond the scope of our paper to derive the tightest bound, i.e., the smallest value for .

Theorem 2 shows zero-distortion in network outage probability actually can be achieved by finite average feedback rates, other than infinite number of feedback bits in the traditional view. This surprising result comes from our design for feedback communication between receivers based on conferencing.

### Iv-B Interference Transmission

For and , the maximum allowed power of transmitter that will not cause outage to receiver when transmitter uses full power can be calculated to be

 pk,max=Hl,l(2ρ−1)Hk,l−1PHk,l.

Note that can be calculated at receiver .

The proposed quantizer consists of two local encoders, two local compressors and a unique decoder. The -th encoder and -th compressor are located at receiver , while the decoder is used by all terminals. We add the superscript “” to indicate their operations in the -th round of conferencing for .

For any , let . Denote as the interference transmission pair determined by . There are at most two rounds of conferencing in .

In round , maps into a codeword in according to

 \textmdENC0mr,it,1(h1)=⎧⎪⎨⎪⎩0,p2,max≤0,argmaxx∈CM,x≤p2,maxx,p2,max>0.

Then maps the index of to a binary description in , a set of binary representations for codewords in . With fixed-length coding, bits indicating the index of are fed back by receiver 1.444The performance of can be improved by taking variable-length coding into consideration. We use fixed-length coding here for convenience. decodes them and recovers the value of , then receiver 2 will send one bit of “1” if , and “0” otherwise. If “1” is fed back by receiver 2, is the decided pair and thus, conferencing for the current channel state finishes. Otherwise, conferencing will continue to the next round.

In round , maps into a codeword in according to

 \textmdENC1mr,it,2(h2)=⎧⎪⎨⎪⎩0,p1,max≤0,argmaxx∈CM,x≤p1,maxx,p1,max>0.

Then maps the index of to a binary description in . bits indicating the index of are fed back by receiver 2. decodes them and recovers the value of , and is the final interference transmission pair.

The interference transmission pair decided by has at least one element equal to , i.e., or , which arises from the fact that the performance of any pair that does not satisfy this can be improved by multiplying the pair with a scaling factor until at least one element reaches [7]. Therefore, the proposed quantizer only needs to work on the non-one element. To do this, either receiver tries to tell others the maximum power it can tolerate for preventing outage.

Denote the network outage probability and average feedback rate of by and , respectively. The following theorem provides upper bounds on and . The proof of the theorem is provided in Appendix D.

###### Theorem 3.

For any and , we have

 \textmdOUT(\textmdDQmr,it)≤\textmdOUToptmr,it+C1M, (6)

and

 \textmdFR(\textmdDQmr,it)≤2log2(M+1)+3, (7)

where is a bounded constant that is independent of and .

From Theorem 3, it is seen that the distortion in network outage probability is inversely proportional to , while the average feedback rate is bounded by a finite constant plus the term that scales as . Letting satisfy , we can observe that the loss in outage probability due to quantization decays at least exponentially with the total feedback rate R as .

### Iv-C Time Sharing or Interference Transmission?

We recall from Section \@slowromancapiii@ that for the network outage probability of sum rate, the interference transmission is always superior to time sharing. On the other hand, for the network outage probability of minimum rate, depending on the power constraing , either one of two transmission strategies may be optimal. To illustrate this phenomenon, the network outage probabilities and are plotted versus for various in Fig. 1. The target data rate is . We can observe from Fig. 1 that for any given , there is a threshold power level (that depends on ) such that when , , and when , . In other words, we should use interference transmission when , and otherwise, if , we should utilize the time sharing strategy. The decision between time sharing and interference transmission only requires the knowledge of , which can be a prior information known by all terminals. Although it is difficult to derive a closed-form expression of , it can still be estimated through numerical simulations. For example, according to Fig. 1, we have dB when and , respectively.

## V Numerical Simulations

In this section, we present simulations to verify the theoretical results for in time sharing and in interference transmission. For each instance of and , a sufficient number of channel state realizations are generated to observe at least 5000 outage events. We have chosen .

We will compare the performance of the proposed quantizers with that of the conventional one [5, 6] denoted by in time sharing and interference transmission, respectively. For readers’ convenience, we provide a brief description of the quantizer as described in [5, 6]. For , receiver employs bits to quantize and separately based on a scalar codebook generated by Lloyd Algorithm [8] with the cardinality being . All terminals decode the feedback bits and reconstruct the quantized as . In time sharing, and are calculated according to Proposition 1 by treating as , while in interference transmission, and are computed by Proposition 2 based on . The average feedback rate of is bits per channel state. We add the subscript of “” or “” to to distinguish when it is applied in time sharing or interference transmission, respectively.

In Fig. 2 (a), the network outage probabilities of minimum rate for , (with ) and the case with no feedback (where either transmitter consumes half of the entire block to transmit, i.e., ) are plotted. It is shown that the network outage probabilities of the latter two scenarios are worse than that of (the minimum one), which substantiates that feedback is necessary as well as the proposed quantizer based on conferencing is superior. Fig. 2 (b) plots the average feedback rate of , which is finite and small in the entire interval of . Furthermore, when or , the average feedback rate approaches towards or , respectively. This corresponds to the upper bound in Theorem 2 and it can be intuitively interpreted like this: when , the probability that for , is increasing towards , then after two rounds, will be chosen as most likely. On the other hand, when , the probability that for , also goes to , thus after round , the quantization process will finish because an outage event is inevitable almost surely.

In Fig. 3, we show the distortions of network outage probability for minimum rate of , and the case with no feedback (where both transmitters will use full power, i.e., ) versus . For each , we choose a value of smaller than thus interference transmission should be applied. In order to demonstrate that outperforms even when has a higher feedback rate, we choose the number of feedback bits assigned to as is . Note that when and