Distributed Beamforming in Wireless Multiuser Relay-Interference Networks with Quantized Feedback
We study quantized beamforming in wireless amplify-and-forward relay-interference networks with any number of transmitters, relays, and receivers. We design the quantizer of the channel state information to minimize the probability that at least one receiver incorrectly decodes its desired symbol(s). Correspondingly, we introduce a generalized diversity measure that encapsulates the conventional one as the first-order diversity. Additionally, it incorporates the second-order diversity, which is concerned with the transmitter power dependent logarithmic terms that appear in the error rate expression. First, we show that, regardless of the quantizer and the amount of feedback that is used, the relay-interference network suffers a second-order diversity loss compared to interference-free networks. Then, two different quantization schemes are studied: First, using a global quantizer, we show that a simple relay selection scheme can achieve maximal diversity. Then, using the localization method, we construct both fixed-length and variable-length local (distributed) quantizers (fLQs and vLQs). Our fLQs achieve maximal first-order diversity, whereas our vLQs achieve maximal diversity. Moreover, we show that all the promised diversity and array gains can be obtained with arbitrarily low feedback rates when the transmitter powers are sufficiently large. Finally, we confirm our analytical findings through simulations.
Wireless relay network, beamforming, interference, distributed vector quantization, symbol error probability, diversity gain, array gain.
While it has been demonstrated in several studies that cooperation can greatly improve the performance and reliability of wireless network communications[1, 2, 3, 4, 5], interference still remains to be a fundamental issue in cooperative network design. Most of the previous work on cooperative networks relies on orthogonal channel allocation so that different transmitters do not interfere with each other. However, allocating orthogonal channels for each user may not be desirable due to time and bandwidth limitations[6, 7]. In such cases, one should explore effective ways to deal with interference while preserving cooperative diversity gains.
Multiple antenna interference cancelation techniques are very effective when dealing with interference in cooperative networks. They offer reasonable performance with low decoding complexity. In this work, we consider a different approach. To be able to study the ultimate performance limits, we do not put any restrictions on our decoders. We would like to design a cooperation scheme that achieves maximal diversity benefits, and thus provides high reliability, even in the presence of multiuser interference.
For networks with a single transmitter-receiver pair and no interference, network beamforming using amplify-and-forward (AF) relays has shown to achieve the maximal spatial diversity[9, 10]. However, the optimal beamforming policy requires one or two real numbers to be broadcasted from the receiver to the relays. Using distributed beamforming with quantized instantaneous channel state information (CSI), it is possible to obtain both maximal diversity, as well as high array gain with only a few feedback bits from the receiver[11, 12, 13]. A special case of quantized feedback for cooperative networks is the relay selection scheme[14, 15, 16]. It has been formally shown in  that, for a network with parallel relays, the relay selection scheme provides the maximum diversity .
Quantized feedback schemes have also been studied for non-cooperative multiuser interference networks. In , the author considers zero-forcing beamforming with finite rate feedback in multiple-input multiple-output (MIMO) broadcast channels. Interference alignment for multiuser interference networks with limited feedback has been studied in . Unlike what we shall study in this work, where we seek to optimize the reliability of the system in terms of the diversity gain, the goal of the above two papers was to optimize the data transmission rate in terms of the multiplexing gain. A common conclusion that we can infer from both studies is that, in order to achieve the same multiplexing gain as a system with perfect CSI, the feedback rate should be increased at least logarithmically with the transmitter power; any constant feedback rate results in a complete loss of multiplexing gain. This is unlike point-to-point systems where feedback is not even necessary to achieve the maximal multiplexing gain, and a few feedback bits is usually sufficient to transmit with rates that are close to the one with perfect CSI. The feedback requirements of interference networks appears to be considerably higher than that of interference-free networks.
What are the feedback requirements if instead we would like to ensure maximal reliability in the presence of interference? One goal of this paper is to answer this question for cooperative networks with transmitters, receivers, and parallel AF relays. We assume that each transmitter and each relay has its own short term power constraint. The transmitters do not have any CSI. Each receiver knows its own receiving channels and the channels from the transmitters to the relays. Each relay only knows the magnitudes of its own receiving channels. Each relay and each receiver also has partial CSI provided by feedback. The feedback information represents a quantized beamforming vector. In that sense, this paper is also a generalization of single-user quantized network beamforming  to multiuser interference networks. On the other hand, such a generalization is quite challenging because of the distributed nature of the network. Let us now describe some of these challenges and our approaches to address them.
In interference networks, the relays amplify both noise and interference, which results in completely different problem formulations and solutions. Second, there are multiple receivers that have different optimal beam directions. As a result, it is difficult to design a scheme that can provide a reasonable performance to all the users.
Another difficulty is related to acquiring feedback information from several separated receivers. The optimal beamforming policy requires the full CSI of the interference network. In practice however, none of the receivers can obtain such information via training methods. We thus consider two different quantization schemes: In the first scheme, the feedback information is provided by a global quantizer (GQ) that knows the entire CSI. We use this hypothetical quantizer to analyze the performance limits of network beamforming in the presence of interference. In the more practical second scheme, we use distributed local quantizer (LQ) encoders at each receiver. Each receiver can access only a part of the CSI, and provides its own feedback information for the relays and the other receivers.
In , we introduced a general systematic LQ design method, called localization, in which one synthesizes an LQ out of an existing GQ using high-rate scalar quantization combined with entropy coding. In the same work, we described an application of the method to MIMO broadcast channels. In this work, we apply it to design LQs for our network model. Therefore, our GQ has another important purpose other than the one we have previously mentioned: It will also serve as the basis of our LQs.
We would also like to note that the LQ design in this paper distinguishes itself from the one in  in several ways, even though the underlying localization method will be the same. First, we need to consider a totally different and much more complicated distortion function. Second, the high-rate scalar quantizers, that form the crucial part of the method, should be designed accordingly. Third, the performance analysis of the resulting LQs is thus different and more complicated. As a result, in this work, we will only analyze the performance of localization for a particular class of GQs that are based on relay selection.
Our performance measure is what we call the network error rate (NER). Given a fixed channel state, it is the probability that at least one user incorrectly decodes its desired symbol(s). In that sense, any receiver can be interested in the symbols transmitted by any subset of transmitters.
We use a generalized diversity measure to characterize the asymptotic behavior of the NER as the transmitter powers grow to infinity. In what follows, we describe this measure together with its motivations: Suppose that a wireless communication system achieves an error rate of , where is the transmitter power constraint and is a constant that is independent of . Then, we call and , the first-order and the second-order diversity gains, respectively, and say that the scheme achieves diversity . Such a definition of diversity is more precise than the traditional one as we demonstrate by an example: For two hypothetical communication systems with diversity gains , and , where and , the former always outperforms the latter for all sufficiently large. On the other hand, the traditional definition, according to which the diversity gain is for both systems, fails to distinguish between the asymptotic performance of the two.
The main contributions of this paper can be summarized as follows: First, we show that, regardless of the quantizer and the amount of feedback that is used, the maximal achievable diversity of our network model is when , whereas it is when .111The case corresponds to a relay-broadcast network that does not suffer any multiuser interference. Even though our main goal in this paper is to analyze interference networks, we present the extension of our results to broadcast networks, so as to demonstrate the detrimental effects of interference in a comparative manner. In other words, the relay-interference network suffers from a second-order diversity loss compared to an interference-free network that can achieve diversity with . Then, we construct a relay-selection based fixed-length GQ (fGQ) that can achieve maximal diversity for any . Next, using our fGQ and the localization method, we design both fixed-length and variable-length LQs (fLQs and vLQs). Our fLQs can achieve diversity when , and diversity when , using feedback bits per receiver. They show that it is possible to achieve very high reliability using a fixed number of feedback bits. On the other hand, our vLQs can achieve maximal diversity gain for any . Moreover, the feedback rate they require decays to zero as the transmitter powers grow to infinity. Therefore, they provide a very fortunate answer to the question that we have posed earlier: In a relay-interference network, it is possible to achieve maximal reliability using arbitrarily low feedback rates per receiver, when the transmitter powers are sufficiently large. Another desirable property of our vLQs is the fact that the array gain they provide can be made arbitrarily close to the one provided by the fGQ.
The rest of the paper is organized as follows: In Section II, we introduce our network model, performance and diversity measures, and problem definition. In Section III, we show that the maximal diversity of our network model is . In Sections IV and V, we introduce our GQ and LQ designs, respectively. Numerical results are provided in Section VI. In Section VII, we draw our major conclusions. An upper bound on the probability density function (PDF) and the cumulative distribution function (CDF) of a frequently used random variable (RV) is provided in Appendix A. Some other technical proofs are provided in Appendices B through E.
Notation: For a logical statement , “ is true for sufficiently large” means that there exists such that for all , is true. indicates the 2-norm, is the infinite norm, is the inner product. , and represent the sets of complex numbers, real numbers, and positive integers, respectively. is the determinant of a square matrix . , denote the transpose and the Hermitian transpose of , respectively. represents the probability. is the PDF, and is the CDF of an RV . is the expected value of . means that is a Gamma RV with for and for , . For any sets and , is the set of elements in , but not in . is the cardinality of . , , is the cartesian power. is the Euler-Mascheroni constant, , and is the empty set. For a real-valued function with , let . Then, is the unique vector with the property that , and “” represents some partial ordering (e.g. lexicographical ordering) of complex vectors. We define in a similar manner. Finally, is the natural logarithm, is the logarithm to base , is the hyperbolic cosine, is the Gaussian tail function, is the gamma function, is the exponential integral, and is the modified Bessel function of the second kind of order .
Ii Network Model and Problem Statement
Ii-a System Model
The block diagram of the system is shown in Fig. 1. We have a relay network with transmitters, receivers, and parallel relays. The cases and correspond to a relay-broadcast network and a relay-interference network, respectively. We assume that there is no direct link between the transmitters and the receivers.
Denote the channel from the th transmitter to the th relay by and the channel from the th relay to the th receiver by . Let denote the channel state of the entire network. We assume that the entries of are independent and distributed as , with finite variances . For brevity, let , which denotes all the channels from the relays to the th receiver.
Only the short-term power constraint is considered, which means that for every symbol transmission, the average power levels used at the th transmitter and the th relay are no larger than and , respectively.
We assume a quasi-static channel model; the channel realizations vary independently from one channel state to another, while within each channel state the channels remain constant. We assume that the th receiver knows and each relay knows the magnitudes of its own receiving channels, i.e. the th relay knows . Some possible procedures to reveal the channel states to the receivers can be found in [13, 11]. For completeness, we give an outline of one possible way: The th destination can acquire the knowledge of by training from the th relay. The th relay can acquire the knowledge of using training sequences from the th source. It can also amplify and forward its received training signal from the source to the destination, so that the destination can estimate the product of and . As is known by the destination, can be estimated.
Each relay and each receiver also has partial CSI provided by feedback. In this paper, we consider two different feedback schemes, namely the global and local quantization schemes.
Ii-B Global Quantization
Our global quantizer is defined by a global encoder and a global decoder, as described in Fig. 2. The global encoder consists of two parts. For each channel state, first, a GQ encoder maps the channel realization to an index in , the index set of the codebook elements. Then, a lossless global compressor maps this index to a binary description.
Let denote the length of a binary description . We call a fixed-length GQ (fGQ) if . Otherwise, we call a variable-length GQ (vGQ).
In either case, the global encoder feeds back , using bits. The feedback bits are received by the global decoders without any errors or delays.
There is a unique global decoder at each relay and each receiver, which comprises of the complementary parts to the global encoder: A lossless decompressor and a quantizer decoder. First the decompressor reconstructs the quantization index from the received binary description. It is followed by the quantizer decoder which maps the quantization index to a codebook element. The codebook has elements, . Without loss of generality, for , we set . For the rest of this paper, we will use the well-known notation . Therefore, , and , for some .
In the most general case, the th relay may make use of the side information in the process of decoding the feedback information. However, in order to keep the relay operation as simple as possible, we do not consider such a scenario in this paper.
Ii-C Local Quantization
We define our local quantizer by local encoders, with the th encoder at the th receiver, and a unique local decoder at each receiver and relay, as described in Fig. 3. The th local encoder comprises of two parts: An LQ encoder and a lossless local compressor . Note that the domain of each LQ encoder is different from the domain of the GQ encoder. For the th encoder, the domain corresponds to the channel states from the transmitters to the relays and from the relays to the th receiver, represented by the concatenation vector .
The th receiver feeds back , using bits. We call an fLQ if, . Otherwise, we call it a vLQ. For the latter case, the feedback rate of the th receiver can be expressed as .
After all the feedback messages are exchanged between the receivers and the relays, each of them decodes the feedback bits using the local decoder. The local decoder is the composition of a decompressor and a quantizer decoder . Overall, . Thus, , and , for some .
Ii-D Transmission Scheme
We use a two-step AF protocol[10, 11]. In the first step, the th transmitter selects a symbol from a constellation , where , , and sends . We normalize as . Thus, the average power used at the th transmitter is . During the first step, there is no reception at the receivers, but the th relay receives
Suppose that a quantizer , global or local, is employed in the network, and , for some . Then, the relays use the beamforming vector to adjust their transmit power and transmit phase. During the second step, the transmitters remain silent, but the th relay transmits
where the relay normalization factor is given by
The average power used at the th relay can be calculated to be . We require as a result of the short term power constraint. The channel state dependent normalization factors ensure that the instantaneous transmit power of each relay remains within its power constraint with high probability.222Because of the noise at its received signal, a relay can exceed its transmit power constraint at some instants. The phrase “short-term” comes from the observation that, regardless of the channel states, the relay always obeys its power constraint when its transmit power is averaged over the transmitted symbols and the noise.
Also, note that within the restriction of , is the maximal normalization factor that we can use. In other words, if a factor satisfies for some , then it violates the short term power constraint. Still, one can employ another factor with (e.g. ). We shall discuss later in Section III whether or not such a different choice of the normalization factor can improve the network performance.
After the two steps of transmission that has been described above, the received signal at the th receiver can be expressed as:
where is the noise at the th receiver. We assume that the noises , and are independent.
Ii-E Performance Measure
The th receiver attempts to decode the symbols of the transmitters with indices given by an arbitrary but fixed set . As an example, for a network with and , let and . Then, the first receiver is interested only in the symbols of the first and the second transmitters, while the second receiver is interested only in the symbols of the second and the third transmitters. In general, we assume that . This guarantees that at least one receiver is interested in the symbols of the th transmitter. In particular, for , we have .
Let us call the vector of transmitted symbols as the super-symbol relevant to the th receiver, and be its decoded version. We say that an error event occurs at a receiver if it incorrectly decodes its desired super-symbol. In this case, the optimal decoder at the th receiver is an individual maximum likelihood (ML) decoder333In the literature, the phrase “individual” usually refers to the cases in which the a posteriori probability is maximized over a single transmitter alphabet. Note that, in our case, the maximization is over the product alphabet that represents the set of all super-symbols that the th receiver is interested in. given by , where is the relevant super-symbol alphabet. For a fixed channel state , and beamforming vector , let denote the conditional super-symbol error rate (SER) of the th receiver with the individual ML decoder.
Let us now define a single quantity that represents the SER performance of all the receivers. We define the conditional network error rate (conditional NER, or CNER), denoted by , as the probability that at least one receiver incorrectly decodes its desired super-symbol.
Our performance measure, the NER, is the expected value of the CNER. Given a quantizer global or local, the NER can thus be expressed as
Ii-F Diversity Measure
Let us also define a unique diversity measure for our network. Let , , where . In other words, we allow the power constraint of each transmitting terminal to grow linearly with . Then, the first-order diversity achieved by a quantizer is given by
One problem with this conventional definition of diversity is that it fails to characterize the asymptotic effect of possible sub-linear -dependent terms (e.g. logarithmic terms) in the error rate expression. In order to properly handle such cases, we define the second-order diversity as
Note that the first-order diversity is always positive, while the second-order diversity can be negative.
Now, the diversity (gain) achieved by a quantizer is given by .
With these definitions, the asymptotic performance with a quantizer , as grows to infinity, can be expressed as
where the factor is the array gain. It is sublogarithmic in the sense that . Also, we use it only when we compare the performance of two quantizers that provide the same diversity gain.
Finally, for two diversity gains , and , we say that is higher than (or ) if either or .
Ii-G Problem Statement
Our goal is to design the quantizer , given a limited feedback rate, such that the NER is minimized. We consider this problem for both GQs and LQs.
To achieve our goal, we first determine the maximal possible diversity with our network model. Then, we design structured fGQs that can achieve this diversity. Finally, we use our observations on fGQs to systematically design fLQs that achieve maximal first order diversity, and then, vLQs that achieve maximal diversity.
We would like to note that, as demonstrated in , the numerical optimization of our quantizers is always possible by using algorithms such as the Generalized Lloyd Algorithm [21, 22]. These algorithms can be used to improve the array gain performance, or in some particular cases, the second-order diversity performance of our structured codebook designs. We will not consider such optimizations in this paper since they are straightforward.
Iii Lower Bounds on Quantizer Performance
Before we attempt to design a high-performance low-rate quantizer, it is natural to determine the best possible performance we can expect with any quantizer. In this section, we find lower bounds on the NER for both relay-interference and relay-broadcast networks that hold for any quantizer , global or local.
Let represent the set of all beamforming vectors. Then, we have
Let with . Then, there are constants that are independent of both and , such that for all , and for all sufficiently large,
Moreover, the bounds in (9) hold for any relay normalization factor that satisfies .
Please see Appendix B. ∎
In other words, for relay-broadcast networks, the maximal diversity gain is . Indeed, for a network with , it was shown in  that diversity is achievable.
On the other hand, for relay-interference networks, the maximal diversity gain is . Since , interference results in a second order diversity loss in our network model.
Theorem 1 also shows that a different relay normalization factor cannot improve the diversity upper bounds, provided that it satisfies the short-term power constraint, and a codebook is employed. Thus, for the rest of this paper, we will only consider as our relay normalization factor.
An immediate question that stems from Theorem 1 is whether there exists finite rate quantizers that can achieve maximal diversity. In the next section, we construct an fGQ that provides an affirmative answer.
Iv Maximal Diversity with an fGQ
In order to determine an fGQ that can achieve maximal diversity, let us first determine, for any , the optimal GQ given a fixed codebook with finite cardinality.
Given a fixed codebook with , the optimal GQ is given by .
Let . We have
Thus, performs at least as good as any quantizer with codebook . ∎
Therefore, given that we employ an optimal GQ encoder given by Proposition 1, the GQ codebook uniquely determines the system performance. But, there is one complication: If we ever want to implement the optimal GQ encoder, we should be able to evaluate , for any given and . Unfortunately, a closed form characterization of the CNER is very difficult, if not impossible. For that reason, we design a suboptimal quantizer that, instead of the actual CNER, uses an upper bound on the CNER. Fortunately, this suboptimal quantizer will be powerful enough to achieve maximal diversity for any .
Iv-a An Upper Bound on the CNER
For the th receiver, instead of the individual ML decoder described in Section II-E, suppose that we employ a joint ML decoder , where . Recall that, for the individual ML decoder at the th receiver, the a posteriori probability was maximized over . For the joint ML decoder, the maximization is over at all the receivers.
Let denote the error rate of the joint ML decoder. Then, we have . Also, from (4),444Note that, in order to be able to perform ML decoding, the receivers should know which beamforming vector is used by the relays. In other words, for each , the receivers should know . This explains why we need to have a quantizer decoder at each receiver as well as each relay.
In (11) and (12), the decoded symbol vector for each receiver is obviously different, i.e. , though we have omitted the dependence on for brevity. Furthermore, from now on, we shall omit the condition in the summations as it is clear from the context.
Now, using a union bound over all the receivers, it follows for the CNER that
This upper bound can easily be evaluated for any constellation and thus, it is good enough for our purposes. However, for clarity of exposition in the rest of the paper, we seek a much simpler bound. First, let us define
Then, (15) can be further bounded as
Iv-B Diversity Analysis of the Relay Selection Scheme
For , we have shown in  that a feedback scheme based on relay selection can achieve diversity . Here, we generalize this result to any .
For , due to both multiuser interference and its manifestation in Theorem 1, it is not clear whether diversity would be achievable. The main goal of this section is to show that it is indeed achievable with a GQ that maximizes the NSNR, and surprisingly, again using a simple relay selection codebook.
The relay selection codebook can be defined as , where for , and for . Then, for any and , We define our fGQ as
where, for any relay selection vector , we have from (12) that
Note that chooses the relay selection vector that maximizes the NSNR.
In the following theorem, we show that, for both relay-broadcast and relay-interference networks, achieves maximal diversity by finding an upper bound on the NER:
There are constants that are independent of such that for all sufficiently large,
Please see Appendix C. ∎
In other words, the relay selection scheme with an fGQ achieves maximal diversity for any . It is remarkable that full diversity is achieved regardless of the number of transmitters and receivers.
Note that our selection scheme requires feedback bits. With feedback bits, where , diversity orders and are achievable for , and , respectively, simply by considering the selection scheme for any fixed of the relays and disregarding the others.
In practical networks, we may not have a GQ that knows the entire CSI of the network. In such situations, we would like to characterize the achievable performance using LQ encoders that know only a part of the CSI.
V Diversity with LQs
In the previous section, we showed that a GQ using relay selection can achieve full diversity. Motivated by this result, we expect that a relay selection based LQ will achieve high diversity orders. In this section, we design two such LQs: An fLQ that achieves maximal first-order diversity, and a vLQ that achieves maximal diversity. Both quantizers will have similar structures. We construct them using the localization method, in which we synthesize an LQ out of an existing GQ. The synthesized LQ and the GQ share the same codebook. For our particular quantization scheme, we use the GQ in (23) as the basis of our LQs. Since is based on relay selection, all of our LQs will be based on relay selection as well555In principle, the localization method itself is applicable to any GQ with any codebook; it is not limited to relay selection based GQs. However, for a general GQ, it is very difficult to analytically determine the performance of the synthesized LQ. Therefore, we focus only on the localization of relay selection based GQs..
Let denote a generic localization of . For the synthesized quantizer , the superscript indicates whether it is fixed-length () or variable-length(); and are design parameters that we shall specify later on. For a particular channel state , the components of the synthesized quantizer operate as follows:
V-A1 LQ Encoders
For notational convenience, . The th LQ encoder calculates . In other words, it calculates its own contribution to the NSNR for all possible relay selection vectors. Then, it quantizes each of the possible contributions using a scalar quantizer
Its output message is the concatenation of sub-messages .
V-A2 An Illustration of the LQ Encoders
Let us now illustrate the operation of the LQ encoders with a simple example with , and , as shown in Fig. 4. For some fixed channel variances, power constraints, and channel state , suppose that , , , , , and . In the figure, each of these local NSNR values are represented by a disk () on the real axis. Since we are using an LQ, can be calculated only by the first receiver, and similarly, can be calculated only by the second receiver. Note that the GQ has access to all the local SNRS and in this example, selects the relay with index .
After the LQ encoder calculates its local NSNR values, it quantizes them using a scalar quantizer that is uniquely determined by the parameters and . In our example, we use bins and set . Each bin is represented by a half open interval ( ) on the real axis. The output message of the LQ encoder is the concatenation of its quantized local NSNR values (submessages), shown as frames with a dashed outline, on the right hand side of the figure.
In general, there are sub-messages, each with possible values. Therefore, for a fixed-length synthesis , at each channel state, each receiver feeds back bits without any compression.
For a variable-length synthesis , we use a lossless compressor that produces an empty codeword (of length ) whenever , and otherwise a codeword of length bits that can uniquely represent each . In other words, for a given channel state, the number of feedback bits produced by any receiver is either bits or bits666If the empty codeword is not allowed, one can use a “” (a codeword of length bit) instead of the empty codeword, and append a “” to each remaining codeword of length bits. The resulting codewords are uniquely decodable as well. Then, all of the results in this paper will hold for the case where the empty codeword is forbidden, given that the required feedback rates are increased by bit. Also, note that one can achieve a better compression by using entropy encoders instead of the suboptimal compressors that we employ. Even though the localization method was introduced originally with entropy encoders, the compressors that we use in this paper will be good enough for our purposes..
After all the feedback messages of the receivers are exchanged between the receivers and the relays, each of them decodes the feedback bits using the local decoder. The decoder operation is the same for each receiver and relay.
First, a decompressor perfectly recovers all the submessages from all the receivers, . All of these submessages are passed to the LQ decoder.
V-A5 An Illustration of the LQ Decoder
For clarity of exposition, let us first present the LQ decoder for the example scenario in Section V-A2 and the same channel state . A more formal description of the general LQ decoder operation will be presented afterwards.
In general, the main goal of the LQ decoder is to imitate the GQ as good as possible. For our particular example, the GQ selects the relay with index , where . Then, the first goal of the LQ decoder should be to determine . However, the LQ decoder only knows the quantized local NSNR values, , as shown in Fig. 4. Therefore, it cannot determine the exact value of . However, as we shall describe in what follows, it can perfectly determine a subset of where resides.
For any , , and . We can use these facts to determine the possible locations of the local NSNR values, as represented in Fig. 5 by half-open intervals ( ) of .
Since , and we know for sure that and , we should have . Using the same arguments for all , we can obtain , and . We have thus determined the possible locations of , as shown in Fig. 6, by having access only to the quantized versions of .
The LQ decoder’s main goal was to find . Using the possible locations of that we have found, it is now clear that the third relay should provide the best NSNR. The LQ decoder’s output will be . Note that this is the same output as the GQ output. Therefore, for this particular channel state, the LQ operates in the same manner as the GQ.
However, the LQ decoder will not be this lucky in general. As an example, another channel state might result in and . In this case, the LQ decoder will know for sure that both the second relay and the third relay provides a larger NSNR than the first relay. On the other hand, it cannot determine which one of the second and the third relays provides the best NSNR. Therefore, it chooses one of them, and its decision may not be the optimal one that would instead be provided by the GQ. We shall quantify the effect of such suboptimal decisions later on.
V-A6 LQ Decoder
We now give the general and formal description of the LQ decoder.
Let denote the set of indices from which our GQ in (23) produces its output.777 is not necessarily a singleton, but our definition of the guarantees that the GQ output is unique. In other words, is the set of indices of relays that provide the maximal NSNR. Also, let . Note that . Moreover, due to the structure of , not only
Therefore, , and can be easily calculated by the LQ decoder.
Since , the LQ decoder can determine which relay selection vector(s) can possibly provide the maximal NSNR. In general, it can choose any one of the relay selection vectors that are indicated by . But, to be more precise, we define
V-A7 Localization Distortion
Let us now study two possible cases of interest regarding the LQ output: If , then the LQ output provides the same NSNR as the GQ output. Otherwise, the LQ might make a suboptimal decision. This results in what we call the localization distortion (LD), given by
A useful upper bound on the LD can be calculated as:
where is the upper bound on the localization distortion, given by
V-B Maximal First-Order Diversity with an fLQ
Our main result concerning the fLQs is given by the following theorem:
Let , and . Then, for sufficiently large, the NER with , which uses a fixed feedback bits per receiver per channel state, is upper bounded by
where are constants that are independent of .
Please see Appendix D. ∎
In other words, using a fixed feedback bits per receiver per channel state, we can achieve diversity for , and diversity for . Since for the broadcast network, and for the interference network, our fLQ has a second-order diversity loss compared to the optimal performance for both types of networks. Also, it is straightforward to show that, using bits, where , we can achieve diversity gains and in relay-broadcast networks and relay-interference networks, respectively.
The scalar quantizer resolution for our fLQ is bit per local NSNR. In what follows, we show that, by appropriately increasing the resolution with , one can achieve maximal diversity, while the compressors make sure that the feedback rate remains bounded.
V-C Maximal Diversity with a vLQ
For vLQs equipped with entropy coding, we have the following result:
Let be a fixed constant that is independent of . For any that satisfies , let , and
Then, for sufficiently large, we have
and, in addition, the feedback rate of the th receiver satisfies