An Information Theoretic Analysis of Single Transceiver Passive RFID Networks

Yücel Altuğ, S. Serdar Kozat, M. Kıvanç Mıhçak Y. Altuğ is with the School of Electrical and Computer Engineering of Cornell University, Ithaca, NY, 14853, USA. (e-mail: ya68@cornell.edu), M. K. Mıhçak are with the Electrical and Electronic Engineering Department of Boğaziçi University, Istanbul, 34342, Turkey (e-mail: kivanc.mihcak@boun.edu.tr), S. S. Kozat is with the Electrical and Electronic Engineering Department of Koç University, Rumeli Feneri Yolu, Sarıyer, Istanbul, 34450, Turkey (e-mail: skozat@ku.edu.tr)Y. Altuğ is partially supported by TÜBİTAK Career Award no. 106E117; M. K. Mıhçak is partially supported by TÜBİTAK Career Award no. 106E117 and TÜBA-GEBIP Award.

Abstract

In this paper, we study single transceiver passive RFID networks by modeling the underlying physical system as a special cascade of a certain broadcast channel (BCC) and a multiple access channel (MAC), using a “nested codebook” structure in between. The particular application differentiates this communication setup from an ordinary cascade of a BCC and a MAC, and requires certain structures such as “nested codebooks”, impurity channels or additional power constraints. We investigate this problem both for discrete alphabets, where we characterize the achievable rate region, as well as for continuous alphabets with additive Gaussian noise, where we provide the capacity region. Hence, we establish the maximal achievable error free communication rates for this particular problem which constitutes the fundamental limit that is achievable by any TDMA based RFID protocol and the achievable rate region for any RFID protocol for the case of continuous alphabets under additive Gaussian noise.

I Introduction

In this paper, we deal with a multiuser communication setup which consists of “cascade” of a broadcast channel (BCC) and a multiple access channel (MAC). The encoder of BCC part and the decoder of the MAC part is the same transceiver, and the decoders of the BCC part and the encoders of the MAC part are the mobile units of the system. The ultimate goal of the communication system considered in the paper is the following: transceiver¹¹1In practical RFID systems, the problem of reader collusion is also considered, which amounts to having multiple transceivers in our setup. In our case, we concentrate on the “single reader (transceiver)” setup as first step. wants to “find out” some specific information possessed by the mobile units and for this purpose it first broadcasts the “type” of the information it seeks to receive from each mobile unit. Then every mobile unit “sends” the corresponding information of the received type to the transceiver. The specific type of information phenomenon differentiates the system at hand from the ordinary cascade of BCC and MAC, because in order to model this situation we employ a nested codebook structure at the MAC encoders, i.e. at the mobile units, which will be explained in detail in Section II-B.

Beyond its promising structure to model wireless communication networks, the problem at hand gives the fundamental limits of RFID protocols in two different ways, supposing the transceiver is RFID reader, mobile units are RFID tags and the RFID reader knows the set of the IDs of the RFID tags in the environment:

(i)

The above mentioned communication problem gives the fundamental limits achievable in TDMA based RFID protocols, since the transceiver sends the TDMA time slots, which are designated to allow communication in a collusion free manner, using the BCC part and then mobile units uses their corresponding time slot information in order to transmit their data to the RFID reader. Supposing equal information rate, say $R^{ID}$ , at each BCC branch, the maximum number of RFID tags that can be handled is $2^{R^{ID}}$ and the maximum data rate from tags to reader is the maximum rate that can be achieved using TDMA at the MAC part of the communication system.
(ii)

The above mentioned communication problem gives the fundamental limits of any RFID protocol, since the RFID reader transmits “on-off” message²²2This on-off message also meaningful in practice as far as passive RFID tags are concerned, since they need to facilitate an external energy in order to operate from the BCC to tags, and then tags communicate back their data through the MAC simultaneously to the reader. The achievable rate region of the MAC part is the fundamental limit of any RFID protocol under the assumption that receiver knows the set of the IDs of the RFID tags in the environment.

The nested codebook structure used in the MAC part of this paper is similar to the “pseudo users” concept introduced in [4], where the authors investigate a special notion of capacity for time slotted ALOHA systems by combining multiple access rate splitting and broadcast codes. However, in [4], the authors explicitly investigate the ALOHA protocol over a degraded additive Gaussian noise channel, where users communicate over a common channel using data packets with predefined collusion probability. Unlike [4], our codes achieve the capacity in the usual sense, where the codewords are sent with arbitrarily small error probability. We also investigate a cascade structure including a BCC in the front and a different MAC in the end. We study this setup both for discrete alphabets using imperfection channels to model the impurities of the actual physical system as well as for continuous alphabets over additive Gaussian noise channel by including appropriate power constraints.

We note that the nested codebook structure used in this paper differs from the nested codes defined in [5, 6]. In [5] nested codebooks, especially nested lattices codes, are explicitly defined with a multi-resolution point of view, where the nesting of codes provide progressively coarser description to finer description of the intended information. Here, our nested codebooks are independent from each other and convey different information.

Organization of the paper is as follows: In Section II we state the notation followed throughout the paper and formulate the communication problem considered in the paper. Section III devoted to derive an achievable rate region of the problem for the case of discrete alphabets, by also including “imperfection channels” in order to model the practical phenomenon better. In Section IV, we state the capacity region of the problem for the case of Gaussian BCC and Gaussian MAC by also incorporating suitable power constraints. Paper ends with the conclusions given in Section V.

II Notation and Problem Statement

II-A Notation

Boldface letters denote vectors; regular letters with subscripts denote individual elements of vectors. Furthermore, capital letters represent random variables and lowercase letters denote individual realizations of the corresponding random variable. The sequence of $\left\{a_{1},a_{2},\ldots,a_{N}\right\}$ is compactly represented by $\mathbf{a}^{N}$ . The abbreviations “i.i.d.”, “p.m.f.” and “w.l.o.g.” are shorthands for the terms “independent identically distributed”, “probability mass function” and “without loss of generality”, respectively.

II-B Problem Statement

In this paper, our major concern is finding maximum achievable error-free rates for the following multiuser communication problem (For the sake of simplicity, we define the problem for the case of two mobile units, however all of the results can easily be generalized to $M$ users using the same arguments employed in the paper): A transceiver first acts as a transmitter and broadcasts a pair of messages, $(W_{1},W_{2})\in{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ , to mobile units through the first memoryless communication channel. Mobile units decode the messages intended to them, i.e. first (resp. second) mobile unit decides ${\hat{W}}_{1}$ (resp. ${\hat{W}}_{2}$ ), and then choose their messages accordingly, i.e. first (resp. second) mobile unit chooses $M_{1}\in{\mathcal{M}}_{1}^{{\hat{W}}_{1}}$ (resp. $M_{2}\in{\mathcal{M}}_{2}^{{\hat{W}}_{2}}$ ), and simultaneously sends to transceiver, which this time acts as a receiver, through the second memoryless communication channel.

Next, we give the quantitative definition of the communication system considered:

Definition II.1

The above-mentioned communication system consists of the following components:

(i)

Eight discrete finite sets ${\mathcal{X}}$ , ${\mathcal{Y}}_{1}$ , ${\mathcal{Y}}_{2}$ , ${\mathcal{Q}}_{1}$ , ${\mathcal{Q}}_{2}$ , $\hat{{\mathcal{Q}}_{1}}$ , $\hat{{\mathcal{Q}}_{2}}$ , ${\mathcal{S}}$ .
(ii)

A one-input two-output, discrete memoryless communication channel, termed as “broadcast channel part” or shortly BCC part from now on, modeled by a conditional p.m.f. $p(y_{1},y_{2}|x)\in{\mathcal{Y}}_{1}\times{\mathcal{Y}}_{2}\times{\mathcal{X}}$ . Using the memoryless property, we have the following expression for the n-th extension of the BCC part:

$p({\mathbf{y}}_{1}^{n},{\mathbf{y}}_{2}^{n}|{\mathbf{x}}^{n})=\prod_{k=1}^{n}p(y_{1k},y_{2k}|x_{k}).$ (1)
(iii)

The memoryless “imperfections channel”, which models the impurities and the instantaneous erroneous behavior at the mobile units (especially useful in the modeling of the RFID tags), given by a conditional p.m.f. $p({\hat{q}}_{i}|q_{i})\in\hat{{\mathcal{Q}}}\times{\mathcal{Q}}_{i}$ . Using the memoryless property, we have the following expression for the n-th extension of the i-th imperfection channel

$p({\hat{\mathbf{q}}}^{n}_{i}|{\mathbf{q}}^{n}_{i})=\prod_{k=1}^{n}p({\hat{q}}_{i,k}|q_{i,k}),$ (2)

for $i\in\{1,2\}$ .
(iv)

A two-input one-output, discrete memoryless communication channel, termed as “multiple access channel part” or shortly MAC part from now on, given by a conditional p.m.f. $p(s|{\hat{q}}_{1},{\hat{q}}_{2})\in{\mathcal{S}}\times\hat{{\mathcal{Q}}_{1}}\times\hat{{\mathcal{Q}}_{2}}$ . Using the memoryless property, we have the following expression for the n-th extension of the MAC part:

$p({\mathbf{s}}^{n}|{\hat{\mathbf{q}}}_{1}^{n},{\hat{\mathbf{q}}}_{2}^{n})=\prod_{k=1}^{n}p(s_{k}|{\hat{q}}_{1,k},{\hat{q}}_{2,k}).$ (3)

Next, we state the code definition

Definition II.2

An $\left(2^{nR_{1}^{ID}},2^{nR_{2}^{ID}},2^{nR_{1}^{Data}},2^{nR_{2}^{Data}},n\right)$ code for the communication system given above consists of the following parts:

(i)

Pair of transmitter messages, termed as “broadcast channel messages” or shortly BCC messages from now on, to mobile units given as $(W_{1},W_{2})\in{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ , where ${\mathcal{W}}_{i}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{1,\ldots,2^{nR_{i}^{ID}}\right\}$ for $i\in\{1,2\}$ .

(ii)

The transceiver’s encoding function, termed as “broadcast channel encoder” or shortly BCC encoder from now on, given as

X^{BCC}\;:\;{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}\rightarrow{\mathcal{X}}^{n},\textrm{ such that }X^{BCC}\left(W_{1},W_{2}\right)={\mathbf{x}}^{n}(W_{1},W_{2}).

(4)

(iii)

The mobile units’ decoding functions, termed as “broadcast channel decoders” or shortly BCC decoders from now on, given by $g_{i}^{BCC}\;:\;{\mathcal{Y}}_{i}^{n}\rightarrow{\mathcal{W}}_{i}\cup\{0\}$ , such that $g_{i}^{BCC}({\mathbf{Y}}_{1}^{n})={\hat{W}}_{i}$ , for $i\in\{1,2\}$ , where $\{0\}$ corresponds to “miss-type” error event.
(iv)

The mobile units’ messages corresponding to decoded BCC messages ${\hat{W}}_{i}$ , termed as “multiple access channel messages” or shortly MAC messages from now on, $M_{i}\in{\mathcal{M}}_{i}^{{\hat{W}}_{i}}$ , where ${\mathcal{M}}_{i}^{{\hat{W}}_{i}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{1,\ldots,2^{nR_{i}^{Data}}\right\}$ , for $i\in\{1,2\}$ . Note that this is the message part of a “nested codebook structure” corresponding to the decoded message ${\hat{W}}_{i}$ at each mobile unit.
(v)

The mobile units’ encoding function, termed as “multiple access channel encoders” or shortly MAC encoders from now on, given by $Q_{i}^{MAC}\;:\;{\mathcal{M}}_{i}^{{\hat{W}}_{i}}\rightarrow{\mathcal{Q}}_{i}^{n}$ , for $i\in\{1,2\}$ , such that $Q_{i}^{MAC}(M_{i})={\mathbf{q}}_{{\hat{W}}_{i}}^{n}\left(M_{i}\right)$ . Note that ${\mathbf{q}}_{{\hat{W}}_{i}}^{n}\left(M_{i}\right)$ ’s are the codewords of the “nested codebook structure” corresponding to the decoded message ${\hat{W}}_{i}$ at each mobile unit.
(vi)

The transceiver’s decoding function, termed as “multiple access channel decoder” or shortly MAC decoder from now on, given by $g^{MAC}\;:\;{\mathcal{S}}^{n}\rightarrow{\mathcal{M}}_{1}^{{\mathcal{W}}_{1}}\times{\mathcal{M}}_{2}^{{\mathcal{W}}_{2}}$ .
(vii)

Decoded messages at the transceiver: $\left({\hat{M}}_{1},{\hat{M}}_{2}\right)\in{\mathcal{M}}_{1}^{W_{1}}\times{\mathcal{M}}_{2}^{W_{2}}$ . Note that since transceiver knows $(W_{1},W_{2})$ pair and tries to “learn” the corresponding $(M_{1},M_{2})$ pairs simultaneously, hence it chooses $(M_{1},M_{2})$ -th messages from the set ${\mathcal{M}}_{1}^{W_{1}}\times{\mathcal{M}}_{2}^{W_{2}}$ .

Obviously, the communication system may be intuitively considered as a cascade of a two user “broadcast channel”[1] and a two user “multiple access channel”[1] with the following modifications: first the employment of the nested codebook structure at the MAC encoders and the imperfections channels included. The aforementioned modified cascade, including the encoders, codewords and decoders at both BCC and MAC part is shown in Figure 1 below:

Figure 1: Block Diagram Representation of the multiuser communication system considered in the paper.

Now, we state following “probability of error” related definitions, which will be used throughout the paper.

Definition II.3

(i)

The conditional probability of error, $\lambda_{i}$ , for the communication system is defined by:

\lambda_{w_{1},w_{2},m_{1},m_{2}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}1-\Pr\left(\left[({\hat{W}}_{1},{\hat{W}}_{2})=(w_{1},w_{2})|(W_{1},W_{2})=(w_{1},w_{2})\right]\wedge\left[({\hat{M}}_{1},{\hat{M}}_{2})=(m_{1},m_{2})|(M_{1},M_{2})=(m_{1},m_{2})\right]\right),

(5)

and the maximal probability of error, $\lambda^{(n)}$ , for the communication system is defined by:

\lambda^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\max_{w_{1},w_{2},m_{1},m_{2}}\lambda_{w_{1},w_{2},m_{1},m_{2}}.

(6)

(ii)

The conditional probability of error for the BCC part, $\lambda_{i}^{BCC}$ , is defined by:

\lambda_{BCC}^{w_{1},w_{2}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\Pr\left(({\hat{W}}_{1},{\hat{W}}_{2})\neq(w_{1},w_{2})|(W_{1},W_{2})=(w_{1},w_{2})\right),

(7)

and the average probability of error for the BCC part, $P_{e,BCC}^{(n)}$ , is defined by:

P_{e,BCC}^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\Pr\left(\left({\hat{W}}_{1},{\hat{W}}_{2}\right)\neq\left(W_{1},W_{2}\right)\right),

(8)

(iii)

The conditional probability of error for the MAC part, $\lambda_{i}^{MAC}$ , is defined by:

\lambda_{MAC}^{m_{1},m_{2}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\Pr\left(({\hat{M}}_{1},{\hat{M}}_{2})\neq(m_{1},m_{2})|(M_{1},M_{2})=(m_{1},m_{2}),({\hat{W}}_{1},{\hat{W}}_{2})=(w_{1},w_{2})\right),

(9)

and the average probability of error for the MAC part, $P_{e,MAC}^{(n)}$ is defined by:

P_{e,MAC}^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\Pr\left(\left({\hat{M}}_{1},{\hat{M}}_{2}\right)\neq\left(M_{1},M_{2}\right)|\left({\hat{W}}_{1},{\hat{W}}_{2}\right)=\left(w_{1},w_{2}\right)\right).

(10)

Note that, using (5),(7) and (9) we conclude that

\lambda_{w_{1},w_{2},m_{1},m_{2}}=1-(1-\lambda_{BCC}^{w_{1},w_{2}})(1-\lambda_{MAC}^{m_{1},m_{2}}).

(11)

Next, achievability is defined as

Definition II.4

Any rate quadruple $\left(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data}\right)$ is said to be achievable if there exists a sequence of codes $\left(2^{nR_{1}^{ID}},2^{nR_{2}^{ID}},2^{nR_{1}^{Data}},2^{nR_{2}^{Data}},n\right)$ such that $\lambda^{(n)}\rightarrow 0$ as $n\rightarrow\infty$ .

III Discrete Case

In this section, we deal with the problem stated in Section II-B under the discrete random variables assumption.

III-A Achievable Region for The General Case

The main result of this section is the following theorem:

Theorem III.1

(Achievability-Discrete Case) Any quadruple $\left(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data}\right)\in{\mathcal{R}}_{0}$ is achievable, where

	$\displaystyle{\mathcal{R}}_{0}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data})\;:\;R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data}\geq 0,\;R_{1}^{ID}<I\left(U;Y_{1}\right),\;R_{2}^{ID}<I\left(V;Y_{2}\right),\right.$
	$\displaystyle R_{1}^{ID}+R_{2}^{ID}<I\left(U;Y_{1}\right)+I\left(V;Y_{2}\right)-I\left(U;V\right),\;R_{1}^{Data}<I({\hat{Q}}_{1};S\|{\hat{Q}}_{2}),\;R_{2}^{Data}<I({\hat{Q}}_{2};S\|{\hat{Q}}_{1}),$
	$\displaystyle R_{1}^{Data}+R_{2}^{Data}<I({\hat{Q}}_{1},{\hat{Q}}_{2};S),\textrm{ for some }p(u,v,x)\textrm{ on }{\mathcal{U}}\times{\mathcal{V}}\times{\mathcal{X}}\textrm{ and }p(q_{1},q_{2},s)\textrm{ on }{\mathcal{Q}}_{1}\times{\mathcal{Q}}_{2}\times{\mathcal{S}},$
	$\displaystyle\left.\textrm{ where }p(q_{1},q_{2},s)\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\sum_{{\hat{q}}_{1},{\hat{q}}_{2}}p(s\|{\hat{q}}_{1},{\hat{q}}_{2})p({\hat{q}}_{1}\|q_{1})p({\hat{q}}_{2}\|q_{2})p(q_{1})p(q_{2}),\textrm{ for some }p(q_{1}),p(q_{2})\textrm{ on }{\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\textrm{ respectively}\right\}.$		(12)

Proof:

Proof follows combining arguments from [2] and [1] for BCC and MAC parts, respectively; by also taking imperfection channels and nested codebook structure into account.

W.l.o.g. we suppose $\epsilon\in(0,1)$ . ³³3Since we want to show that $\lambda^{(n)}\rightarrow 0$ as $n\rightarrow\infty$ , this will suffice. To see this, observe that in the proof of the theorem, we show that for any sufficiently large $n$ and for any $\epsilon\in(0,1)$ , $\lambda^{(n)}\leq\epsilon$ , which directly implies $\lambda^{(n)}\leq\epsilon^{\prime}$ for any $\epsilon^{\prime}\geq 1$ .

First, define $A_{{\epsilon}}^{(n)}(U)$ (resp. $A_{{\epsilon}}^{(n)}(V)$ ) as the set of ${\epsilon}$ -typical sequences [1] $\mathbf{u}^{n}\in{\mathcal{U}}^{n}$ (resp. $\mathbf{v}^{n}\in{\mathcal{V}}^{n}$ ) for any given $p(u)$ (resp. $p(v)$ ) on ${\mathcal{U}}$ (resp. ${\mathcal{V}}$ ).

Next, for $w_{1}\in\{1,\ldots,2^{nR_{1}^{ID}}\}$ , we define following cells:

B_{w_{1}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left[(w_{1}-1)2^{n(I(U;Y_{1})-R_{1}^{ID}-{\epsilon})}+1,\;w_{1}2^{n(I(U;Y_{1})-R_{1}^{ID}-{\epsilon})}\right].

Similarly, for resp. $w_{2}\in\{1,\ldots,2^{nR_{2}^{ID}}\}$ , we define:

C_{w_{2}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left[(w_{2}-1)2^{n(I(V;Y_{2})-R_{2}^{ID}-{\epsilon})}+1,\;w_{2}2^{n(I(V;Y_{2})-R_{2}^{ID}-{\epsilon})}\right],

w.l.o.g. supposing that $2^{n(I(U;Y_{1})-R_{1}^{ID}-{\epsilon})},2^{n(I(V;Y_{2})-R_{2}^{ID}-{\epsilon})}\in{\mathbb{Z}}^{+}$ .

Encoding at BCC part:

i)

Generation of the codebook: Generate the codebook $\mathcal{C}_{BCC}\in{\mathcal{X}}^{2^{nR^{ID}_{1}}}\times{\mathcal{X}}^{2^{nR^{ID}_{2}}}\times{\mathcal{X}}^{n}$ such that $(i,j,m)$ -th element is $x_{m}(i,j)$ and $x_{m}(i,j)$ s are i.i.d. realizations of $X$ of which distribution is $p(x)=\sum_{u,v}p(u,v,x)$ for all $i,j,m$ and reveal the codebook to both mobile units and transceiver.
ii)

Choose an $(W_{1},W_{2})\in{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ uniformly over ${\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ , i.e. $\Pr(W_{1}=w_{1},W_{2}=w_{2})=1/\left(2^{nR_{1}^{ID}}2^{nR_{2}^{ID}}\right)$ , for all $(w_{1},w_{2})\in{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ .

iii)

Next, generate $2^{n(I(U;Y_{1})-{\epsilon})}$ , i.i.d. ${\mathbf{u}}^{n}$ , such that

p({\mathbf{u}}^{n})=\left\{\begin{array}[]{cl}\frac{1}{|A_{\epsilon}^{(n)}(U)|}&\hbox{, if }{\mathbf{u}}^{n}\in A_{\epsilon}^{(n)}(U)\\ 0&\hbox{, otherwise}\end{array}\right.

Similarly, generate $2^{n(I(V;Y_{2})-{\epsilon})}$ , i.i.d. ${\mathbf{v}}^{n}$ , such that

p({\mathbf{v}}^{n})=\left\{\begin{array}[]{cl}\frac{1}{|A_{\epsilon}^{(n)}(V)|}&\hbox{, if }{\mathbf{v}}^{n}\in A_{\epsilon}^{(n)}(V)\\ 0&\hbox{, otherwise}\end{array}\right.

Label these ${\mathbf{u}}^{n}(k)$ (resp. ${\mathbf{v}}^{n}(l)$ ), $k\in\left[1,2^{n(I(U;Y_{1})-{\epsilon})}\right]$ (resp. $l\in\left[1,2^{n(I(V;Y_{2})-{\epsilon})}\right]$ ).

iv)

If a message pair $(w_{1},w_{2})$ is to be transmitted, pick one pair $({\mathbf{u}}^{n}(k),{\mathbf{v}}^{n}(l))\in A_{\epsilon}^{(n)}(U,V)\cap B_{w_{1}}\times C_{w_{2}}$ . Then, find an ${\mathbf{x}}(w_{1},w_{2})$ which is jointly ${\epsilon}$ -typical with $(w_{1},w_{2})$ pair and designate it as the corresponding codeword of $(w_{1},w_{2})$ . Send over the BCC part, $p(y_{1},y_{2}|x)$ .

Decoding at BCC part:

i)

Find the indexes $\hat{k}$ (resp. $\hat{l}$ ) such that $({\mathbf{u}}^{n}(\hat{k}),{\mathbf{y}}_{1})\in A_{\epsilon}^{(n)}(U,Y_{1})$ (resp. $({\mathbf{v}}^{n}(\hat{l}),{\mathbf{y}}_{2})\in A_{\epsilon}^{(n)}(V,Y_{2})$ ). If $\hat{k},\hat{l}$ are not unique or does not exist, declare an error, i.e. ${\hat{W}}_{1}=0$ and/or ${\hat{W}}_{2}=0$ . Else, decide ${\hat{W}}_{1}\in{\mathcal{W}}_{1}$ (resp. ${\hat{W}}_{2}\in{\mathcal{W}}_{2}$ ) at mobile unit one (resp two), such that $\hat{k}\in B_{{\hat{W}}_{1}}$ (resp. $\hat{l}\in C_{{\hat{W}}_{2}}$ ).

Encoding at MAC part:

i)

Generation of the codebook(Nested codebook structure): Fix $p(q_{1}),p(q_{2})$ . Let $p(q_{1},q_{2})=p(q_{1})p(q_{2})$ . Generate the $w_{i}$ -th codebook $\mathcal{C}_{MAC}^{w_{i}}\in{\mathcal{Q}}_{i}^{2^{nR_{i}^{Data}}}\times{\mathcal{Q}}_{i}^{n}$ such that $(j,k)$ -th element is $q_{w_{i},k}(j)$ and $q_{w_{i},k}(j)$ s are i.i.d. realizations of $Q_{i}$ of which distribution is $p(q_{i})$ for all $j\in\{1,\ldots,2^{nR_{i}^{Data}}\}$ , $k\in\{1,\ldots,n\}$ and $i\in\{1,2\}$ .
ii)

Choose a message $M_{i}\in{\mathcal{M}}_{i}^{{\hat{W}}_{i}}$ uniformly for the ${\hat{W}}_{i}$ decided at the BCC part, i.e. $\Pr(M_{i}=m_{i})=\frac{1}{2^{nR_{i}^{Data}}}$ , for all $m_{i}\in{\mathcal{M}}_{i}^{{\hat{W}}_{i}}$ and for $i\in\{1,2\}$ . In order to send the message $m_{i}$ , pick the corresponding codeword ${\mathbf{q}}_{{\hat{W}}_{i}}^{n}(m_{i})$ of $\mathcal{C}_{MAC}^{{\hat{W}}_{i}}$ and send over the imperfection channel $p({\hat{q}}_{i}|q_{{\hat{W}}_{i}})$ resulting in ${\hat{\mathbf{q}}}_{i}^{n}$ for $i\in\{1,2\}$ . The pair of $({\hat{\mathbf{q}}}_{1},{\hat{\mathbf{q}}}_{2})$ is the input to the MAC part, $p(s|{\hat{q}}_{1},{\hat{q}}_{2})$ .

Decoding at MAC part:

Find the pair of indexes $\left({\hat{M}}_{1},{\hat{M}}_{2}\right)\in{\mathcal{M}}_{1}^{w_{1}}\times{\mathcal{M}}_{2}^{w_{2}}$ such that $({\mathbf{q}}_{w_{1}}^{n}({\hat{M}}_{1}),{\mathbf{q}}_{w_{2}}^{n}({\hat{M}}_{2}),{\mathbf{s}}^{n})\in A_{\epsilon}^{(n)}(Q_{1},Q_{2},S)$ , where $A_{\epsilon}^{(n)}(Q_{1},Q_{2},S)$ is the ${\epsilon}$ -typical set with respect to distribution

$\displaystyle p(q_{1},q_{2},s)$	$\displaystyle=$	$\displaystyle\sum_{{\hat{q}}_{1},{\hat{q}}_{2}}p(s\|{\hat{q}}_{1},{\hat{q}}_{2},q_{1},q_{2})p({\hat{q}}_{1},{\hat{q}}_{2}\|q_{1},q_{2})p(q_{1})p(q_{2}),$	(13)
	$\displaystyle=$	$\displaystyle\sum_{{\hat{q}}_{1},{\hat{q}}_{2}}p(s\|{\hat{q}}_{1},{\hat{q}}_{2})p({\hat{q}}_{1},{\hat{q}}_{2}\|q_{1},q_{2})p(q_{1})p(q_{2}),$	(14)
	$\displaystyle=$	$\displaystyle\sum_{{\hat{q}}_{1},{\hat{q}}_{2}}p(s\|{\hat{q}}_{1},{\hat{q}}_{2})p({\hat{q}}_{1}\|q_{1})p({\hat{q}}_{2}\|q_{2})p(q_{1})p(q_{2}),$	(15)

where (13) follows since $p(q_{1},q_{2})=p(q_{1})p(q_{2})$ (cf. the codebook generation of MAC part), (14) follows since MAC channel depends on only $({\hat{q}}_{1},{\hat{q}}_{2})$ and (15) follows since imperfection channels are independent and depends on only $q_{1}$ and $q_{2}$ , respectively.

If such a $\left({\hat{M}}_{1},{\hat{M}}_{2}\right)$ pair does not exist or is not unique, then declare an error, i.e. ${\hat{M}}_{1}=0$ and/or ${\hat{M}}_{2}=0$ ; otherwise decide $\left({\hat{M}}_{1},{\hat{M}}_{2}\right)$ .

Analysis of Probability of Error:
We begin with BCC part. By defining the error event as $\mathcal{E}^{BCC}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{({\hat{W}}_{1}({\mathbf{Y}}_{1}^{n}),{\hat{W}}_{2}({\mathbf{Y}}_{2}^{n}))\neq(W_{1},W_{2})\right\}$ , we have the following expression for the average probability of error averaged over all messages, $(w_{1},w_{2})$ , and codebooks, ${\mathcal{C}}_{BCC}$

	$\displaystyle P_{e,BCC}^{(n)}$	$\displaystyle=$	$\displaystyle\Pr\left(\mathcal{E}^{BCC}\right),$		(16)
		$\displaystyle=$	$\displaystyle\Pr\left(\mathcal{E}^{BCC}\|(W_{1},W_{2})=(1,1)\right),$		(16)

where (16) follows by noting the equality of arithmetic average probability of error and the average probability of error given in (8) and the symmetry of the codebook construction at the BCC part.

Next, we define following type of error events:

$\displaystyle\mathcal{E}_{1}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{\nexists({\mathbf{u}}^{n}(k),{\mathbf{v}}^{n}(l))\in(B_{1}\times C_{1})\cap A_{\epsilon}^{(n)}(U,V)\right\},$	(17)
$\displaystyle\mathcal{E}_{2}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{({\mathbf{u}}^{n}(k),{\mathbf{v}}^{n}(l),{\mathbf{x}}^{n}(w_{1},w_{2}),{\mathbf{y}}_{1}^{n},{\mathbf{y}}_{2}^{n})\not\in A_{\epsilon}^{(n)}(U,V,X,Y_{1},Y_{2})\right\},$	(18)
$\displaystyle\mathcal{E}_{3}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{\exists\hat{k}\neq k,\textrm{ s.t. }({\mathbf{u}}^{n}(\hat{k}),{\mathbf{y}}_{1}^{n})\in A_{\epsilon}^{(n)}(U,Y_{1})\right\},$	(19)
$\displaystyle\mathcal{E}_{4}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{\exists\hat{l}\neq l,\textrm{ s.t. }({\mathbf{v}}^{n}(\hat{l}),{\mathbf{y}}_{2}^{n})\in A_{\epsilon}^{(n)}(V,Y_{2})\right\},$	(20)

where (17) corresponds to the failure of the encoding, (19) (resp. (20)) corresponds to the failure of the decoding at mobile unit one (resp. mobile unit two).

Using typicality arguments, it can be shown that $\Pr\left(\mathcal{E}_{i}^{BCC}\right)\leq\epsilon/4$ for $i\in\{2,3,4\}$ and Lemma 1 of [2] also guarantees that $\Pr\left(\mathcal{E}_{1}^{BCC}\right)\leq\epsilon/4$ . Using these facts and the union bound, we conclude that

P_{e,BCC}^{(n)}=\Pr(\mathcal{E}^{BCC})=\Pr(\mathcal{E}^{BCC}|(W_{1},W_{2})=(1,1))\leq{\epsilon},

(21)

for any ${\epsilon}>0$ , for sufficiently large $n$ ; provided that $I(U;Y_{1})>R_{1}^{ID}+{\epsilon}$ , $I(V;Y_{2})>R_{2}^{ID}+{\epsilon}$ , $I(U;Y_{1})+I(V;Y_{2})-I(U;V)>R_{1}^{ID}+R_{2}^{ID}+2{\epsilon}+\delta({\epsilon})$ , such that $\delta({\epsilon})\rightarrow 0$ as ${\epsilon}\rightarrow 0$ .

Further, using standard arguments for finding a code with negligible maximal probability of error (cf. [1] pp. 203-204) from the one with $P_{e,BCC}^{(n)}\leq{\epsilon}$ we conclude that we have

\lambda_{BCC}^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\max_{w_{1},w_{2}}\lambda_{BCC}^{w_{1},w_{2}}\leq 2{\epsilon},

(22)

for any ${\epsilon}>0$ and for sufficiently large $n$ , which concludes the BCC part.

By defining the error event as $\mathcal{E}^{MAC}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{\left({\hat{M}}_{1}({\mathbf{S}}^{n}),{\hat{M}}_{2}({\mathbf{S}}^{n})\right)\neq(M_{1},M_{2})|\left({\hat{W}}_{1},{\hat{W}}_{2}\right)=\left(w_{1},w_{2}\right)\right\}$ , we have the following expression for the average probability of error averaged over all messages, $(m_{1},m_{2})$ , and codebooks corresponding to the messages, ${\mathcal{C}}_{MAC}^{w_{1}}$ and ${\mathcal{C}}_{MAC}^{w_{2}}$

	$\displaystyle P_{e,MAC}^{(n)}$	$\displaystyle=$	$\displaystyle\Pr\left(\mathcal{E}^{MAC}\right),$		(23)
		$\displaystyle=$	$\displaystyle\Pr\left(\mathcal{E}^{MAC}\|(M_{1},M_{2})=(1,1)\right),$		(23)

where (23) follows by noting the equality of arithmetic average probability of error and the average probability of error given in (10) and the symmetry of the nested codebook construction at the MAC part.

Next, we define the following events

\mathcal{E}_{ij}^{MAC}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{({\mathbf{q}}_{w_{1}}^{n}(i),{\mathbf{q}}_{w_{2}}^{n}(j),{\mathbf{s}}^{n})\in A_{\epsilon}^{(n)}(Q_{1},Q_{2},S)\right\},

(24)

Using union bound and appropriately bounding each error event by exploiting typicality arguments, one can show that

P_{e,MAC}^{(n)}=\Pr\left(\mathcal{E}^{MAC}\right)=\Pr\left(\mathcal{E}^{MAC}|(M_{1},M_{2})=(1,1)\right)\leq{\epsilon},

(25)

for any ${\epsilon}>0$ and sufficiently large $n$ ; provided that $I(Q_{1};S|Q_{2})-R_{1}^{Data}>3{\epsilon}$ , $I(Q_{2};S|Q_{1})-R_{2}^{Data}>3{\epsilon}$ and $I(Q_{1},Q_{2};S)-(R_{1}^{Data}+R_{2}^{Data})>4{\epsilon}$ .

Further, using standard arguments for finding a code with negligible maximal probability of error (cf. [1] pp. 203-204) from the one with $P_{e,MAC}^{(n)}\leq{\epsilon}$ we conclude that we have

\lambda_{MAC}^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\max_{m_{1},m_{2}}\lambda_{MAC}^{m_{1},m_{2}}\leq 2{\epsilon},

(26)

for any ${\epsilon}>0$ and for sufficiently large $n$ , which concludes the MAC part.

Next, we sum up things and conclude the proof in the following manner.

First, by plugging (11) in (6), we have

\lambda^{(n)}=\max_{\lambda_{BCC}^{w_{1},w_{2}},\lambda_{MAC}^{m_{1},m_{2}}}\lambda_{BCC}^{w_{1},w_{2}}+\lambda_{MAC}^{m_{1},m_{2}}-\lambda_{BCC}^{w_{1},w_{2}}\lambda_{MAC}^{m_{1},m_{2}}.

(27)

Further, using the fact that the cost function in (27) is monotonic increasing in both $\lambda_{BCC}^{w_{1},w_{2}}$ and $\lambda_{MAC}^{m_{1},m_{2}}$ , we conclude that (cf. (22) and (26))

\lambda^{(n)}\leq 4{\epsilon}-4{\epsilon}^{2},

(28)

for any $0<{\epsilon}<1$ and sufficiently large $n$ . Since ${\epsilon}$ may be arbitrarily small, (28) concludes the proof. ∎

IV Power Constrained Gaussian Case

IV-A Problem Statement

In this section, we generalize the communication problem stated in Section II-B to continuous random variables under the assumption of Gaussian noise and power constraint on the codebooks. To be more precise we have the problem depicted in Figure 2, with the power constraints:

$\displaystyle\textrm{E}\left[X^{2}\right]$	$\displaystyle\leq$	$\displaystyle P,$	(29)
$\displaystyle\textrm{E}\left[(Q_{1,{\hat{W}}_{1}})^{2}\right]$	$\displaystyle\leq$	$\displaystyle\alpha_{1}P_{1},$	(30)
$\displaystyle\textrm{E}\left[(Q_{1,{\hat{W}}_{2}})^{2}\right]$	$\displaystyle\leq$	$\displaystyle\alpha_{2}P_{2},$	(31)

such that $\alpha_{1},\alpha_{2}<1$ and $P_{1}+P_{2}\leq P$ , where $P_{1}$ (resp. $P_{2}$ ) is the power delivered to mobile unit one (resp. two) and w.l.o.g. we assume that $N_{1}<N_{2}$ .

Figure 2: Block Diagram Representation of the multiuser communication system under Gaussian noise assumption.

Note that both Definition II.1 (excluding imperfection channels, which are irrelevant for this case) and Definition II.2 are valid for this case, with ${\mathcal{X}}={\mathcal{Q}}_{1}={\mathcal{Q}}_{2}={\mathcal{S}}={\mathbb{R}}$ .

Remark IV.1

(i)

Observe that, we model the “imperfection channel” of discrete case as an additional power constraint for the Gaussian case.
(ii)

BCC part for the Gaussian case at hand is equivalent to “degraded BCC”, which enables us to state the capacity region instead of characterizing achievable region only.

IV-B Capacity Region for Gaussian Case

In this section, we state the capacity region of the communication system given in Section IV-A. Note that throughout the section, all the logarithms are base $e$ , in other words the unit of information is “nats”.

Theorem IV.1

The capacity region, ${\mathcal{R}}_{1}\subset{\mathbb{R}}^{4}$ , of the system shown in Figure 2 is given by

	$\displaystyle{\mathcal{R}}_{1}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data})\;:\;R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data}\geq 0,\;R_{1}^{ID}<\frac{1}{2}\log\left(1+\frac{\alpha P}{N_{1}}\right),\right.$
	$\displaystyle R_{2}^{ID}<\frac{1}{2}\log\left(1+\frac{(1-\alpha)P}{N_{2}+\alpha P}\right),\;R_{1}^{Data}<\frac{1}{2}\log\left(1+\frac{\alpha\alpha_{1}P}{N_{3}}\right),\;R_{2}^{Data}<\frac{1}{2}\log\left(1+\frac{(1-\alpha)\alpha_{2}P}{N_{3}}\right),$
	$\displaystyle\left.R_{1}^{Data}+R_{2}^{Data}<\frac{1}{2}\log\left(1+\frac{\alpha\alpha_{1}P+(1-\alpha)\alpha_{2}P}{N_{3}}\right),\textrm{ s. t. }0\leq\alpha\leq 1,\;0\leq\alpha_{1},\alpha_{2}\leq 1\right\},$		(32)

where $\alpha$ may be chosen arbitrarily in the given range and $\alpha_{1}$ and $\alpha_{2}$ are system parameters.

IV-B1 Achievability

In section, we prove the forward part of Theorem IV.1, in other words following theorem:

Theorem IV.2

Any rate quadruple $(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data})\in{\mathbb{R}}^{4}$ , there exists a sequence of
$\left(2^{nR_{1}^{ID}},2^{nR_{2}^{ID}},2^{nR_{1}^{Data}},2^{nR_{2}^{Data}},n\right)$ codes with arbitrarily small probability of error for sufficiently large $n$ , provided that

$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha P}{N_{1}}\right)$	$\displaystyle>$	$\displaystyle R_{1}^{ID}+{\epsilon},$	(33)
$\displaystyle\frac{1}{2}\log\left(1+\frac{(1-\alpha)P}{\alpha P+N_{2}}\right)$	$\displaystyle>$	$\displaystyle R_{2}^{ID}+{\epsilon},$	(34)
$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha_{1}\alpha P}{N_{3}}\right)$	$\displaystyle>$	$\displaystyle R_{1}^{Data}+3{\epsilon},$	(35)
$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha_{2}(1-\alpha)P}{N_{3}}\right)$	$\displaystyle>$	$\displaystyle R_{2}^{Data}+3{\epsilon},$	(36)
$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha_{1}\alpha P+\alpha_{2}(1-\alpha)P}{N_{3}}\right)$	$\displaystyle>$	$\displaystyle R_{1}^{Data}+R_{2}^{Data}+4{\epsilon},$	(37)

for any ${\epsilon}>0$ , $0\leq\alpha\leq 1$ and $0\leq\alpha_{1},\alpha_{2}\leq 1$ .

Proof:

In order to prove the theorem, we use superposition coding [1] at BCC part and standard random coding at MAC part. W.l.o.g. suppose $\epsilon\in(0,13/84)$ . ⁴⁴4Since we want to show that $\lambda^{(n)}\rightarrow 0$ as $n\rightarrow\infty$ , this will suffice. To see this, observe that in the proof of the theorem, we show that for any sufficiently large $n$ and for any $\epsilon\in(0,13/84)$ , $\lambda^{(n)}\leq\epsilon$ , which directly implies $\lambda^{(n)}\leq\epsilon^{\prime}$ for any $\epsilon^{\prime}\geq 13/84$ .

Encoding at BCC part:

i)

Generation of the codebook:(Superposition Coding) Generate codebook, ${\mathcal{C}}_{BCC}^{1}$ (resp. ${\mathcal{C}}_{BCC}^{2}$ ) with corresponding rate $R_{1}^{ID}$ (resp. $R_{2}^{ID}$ ) such that both $R_{1}^{ID}$ and $R_{2}^{ID}$ satisfy the conditions (33), (34) and (35) where

${\mathcal{C}}_{BCC}^{1}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left[x_{1,i}(w_{1})\right],$ (38)

such that each $x_{1,i}(w_{1})$ are i.i.d. realizations of $X_{1}\sim{\mathcal{N}}(0,\alpha P-{\epsilon}/2)$ and

${\mathcal{C}}_{BCC}^{2}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left[x_{2,i}(w_{2})\right],$ (39)

such that each $x_{2,i}(w_{2})$ are i.i.d. realizations of $X_{2}\sim{\mathcal{N}}(0,(1-\alpha)P-{\epsilon}/2)$ . Reveal both ${\mathcal{C}}_{BCC}^{1}$ and ${\mathcal{C}}_{BCC}^{2}$ to each mobile unit.
ii)

Choose a message pair $(w_{1},w_{2})\in{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ , uniformly over ${\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ , i.e. $\Pr(W_{1}=w_{1},W_{2}=w_{2})=1/2^{n(R_{1}^{ID}+R_{2}^{ID})}$ , for all $(w_{1},w_{2})\in{\mathcal{W}}_{1}\times{\mathcal{W}}_{2}$ .
iii)

In order to send message $(w_{1},w_{2})$ , take ${\mathbf{x}}_{1}^{n}(w_{1})$ from ${\mathcal{C}}_{BCC}^{1}$ and ${\mathbf{x}}_{2}^{n}(w_{2})$ from ${\mathcal{C}}_{BCC}^{2}$ and send ${\mathbf{x}}^{n}(w_{1},w_{2})\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}{\mathbf{x}}_{1}^{n}(w_{1})+{\mathbf{x}}_{2}^{n}(w_{2})$ over the BCC to both sides, yielding $Y_{1}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}{\mathbf{x}}^{n}(w_{1},w_{2})+Z_{1}$ at mobile unit one and $Y_{2}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}{\mathbf{x}}^{n}(w_{1},w_{2})+Z_{2}$ at mobile unit two, where $Z_{1}$ and $Z_{2}$ are arbitrarily correlated with following marginal distributions $Z_{1}\sim{\mathcal{N}}(0,N_{1})$ , $Z_{2}\sim{\mathcal{N}}(0,N_{2})$ . Note that law of large numbers ensures ${\mathbf{x}}^{n}(w_{1},w_{2})$ satisfies the power constraint of (29).

Decoding at BCC part:

i)

Upon receiving ${\mathbf{y}}_{2}^{n}$ , second mobile unit performs jointly typical decoding, i.e. decides the unique ${\hat{W}}_{2}\in{\mathcal{W}}_{2}$ such that $\left({\mathbf{y}}_{2}^{n},{\mathbf{x}}_{2}^{n}({\hat{W}}_{2})\right)\in A_{\epsilon}^{(n)}(X_{2},Y_{2})$ . If such a ${\hat{W}}_{2}\in{\mathcal{W}}_{2}$ does not exist or is not unique, then declares an error, i.e. ${\mathcal{W}}_{2}=0$ .

Mobile unit one also performs the same jointly typical decoding first with ${\mathbf{y}}_{1}^{n}$ in order to decide the unique ${\hat{W}}_{2}\in{\mathcal{W}}_{2}$ such that $\left({\mathbf{y}}_{1}^{n},{\mathbf{x}}_{1}^{n}({\hat{W}}_{2})\right)\in A_{\epsilon}^{(n)}(X_{2},Y)$ . If such ${\hat{W}}_{2}\in{\mathcal{W}}_{2}$ does not exist or is not unique, then declares an error, i.e. ${\mathcal{W}}_{2}=0$ . After deciding on ${\hat{W}}_{2}$ , mobile unit one calculates the corresponding ${\mathbf{y}}^{n}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}{\mathbf{y}}_{1}^{n}-{\mathbf{x}}_{2}^{n}({\hat{W}}_{2})$ and then performs jointly typical decoding, i.e. decides the unique ${\hat{W}}_{1}\in{\mathcal{W}}_{1}$ such that $\left({\mathbf{y}}^{n},{\mathbf{x}}_{1}^{n}({\hat{W}}_{1})\right)\in A_{\epsilon}^{(n)}(X_{1},Y)$ . If such a ${\hat{W}}_{1}\in{\mathcal{W}}_{1}$ does not exist or is not unique, then declares an error, i.e. ${\hat{W}}_{1}=0$ .

Encoding at MAC part:

i)

Generation of Codebook (Nested Codebook Structure): Fix $f(q_{1}),f(q_{2})$ . Let $f(q_{1},q_{2})=f(q_{1})f(q_{2})$ . Generate the $w_{1}$ -th (resp. $w_{2}$ -th) codebook as $\mathcal{C}_{MAC}^{w_{1}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left[q_{w_{1},j}(m_{1})\right]$ (resp. $\mathcal{C}_{MAC}^{w_{2}}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left[q_{w_{2},j}(m_{2})\right]$ ), such that $q_{w_{1},j}(m_{1})$ (resp. $q_{w_{2},j}(m_{2})$ ) are i.i.d. realizations of $Q_{1}\sim{\mathcal{N}}(0,\alpha_{1}\alpha P-{\epsilon})$ (resp. $Q_{2}\sim{\mathcal{N}}(0,\alpha_{2}(1-\alpha)P-{\epsilon})$ ) for all $w_{1}\in\{1,\ldots,2^{nR_{1}^{ID}}\}$ (resp. $w_{2}\in\{1,\ldots,2^{nR_{2}^{ID}}\}$ ), $m_{1}\in\{1,\ldots,2^{nR_{1}^{Data}}\}$ (resp. $m_{2}\in\{1,\ldots,2^{nR_{2}^{Data}}\}$ ) and $j\in\{1,\ldots,n\}$ .
ii)

Choose a message $M_{i}\in{\mathcal{M}}_{i}^{{\hat{W}}_{i}}$ uniformly, i.e. $\Pr(M_{i}=m_{i})=1/2^{nR_{i}^{Data}}$ , for all $m_{i}\in{\mathcal{M}}_{i}^{{\hat{W}}_{i}}$ and for $i\in\{1,2\}$ . In order to send a message $m_{i}$ , take the corresponding codeword ${\mathbf{q}}_{{\hat{W}}_{i}}^{n}$ of ${\mathcal{C}}_{MAC}^{{\hat{W}}_{i}}$ and send over the MAC, for $i\in\{1,2\}$ , resulting in ${\mathbf{S}}^{n}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}{\mathbf{q}}_{{\hat{W}}_{1}}^{n}+{\mathbf{q}}_{{\hat{W}}_{2}}^{n}+{\mathbf{Z}}_{3}^{n}$ .

Decoding at MAC part:

i)

Find the pair of indexes $({\hat{M}}_{1},{\hat{M}}_{2})\in{\mathcal{M}}_{1}^{w_{1}}\times{\mathcal{M}}_{2}^{w_{2}}$ such that $({\mathbf{q}}_{w_{1}}({\hat{M}}_{1}),{\mathbf{q}}_{w_{2}}({\hat{M}}_{2}),{\mathbf{s}}^{n})\in A_{\epsilon}^{(n)}(Q_{1},Q_{2},S)$ . If such a pair does not exist or is not unique, then declare an error, i.e. ${\hat{M}}_{1}=0$ and/or ${\hat{M}}_{2}=0$ ; otherwise decide $({\hat{M}}_{1},{\hat{M}}_{2})$ .

Analysis of Probability of Error: We begin with the BCC part. First, note that (16) is still valid as well as the error event definition. Next, we define following type of error events

$\displaystyle\mathcal{E}_{0}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{\frac{1}{n}\sum_{j=1}^{n}x_{j}^{2}(1,1)>P\right\},$	(40)
$\displaystyle\mathcal{E}_{1,i}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{({\mathbf{x}}_{2}^{n}(i),{\mathbf{y}}_{1}^{n})\in A_{\epsilon}^{(n)}(X_{2},Y_{1}),\textrm{ s.t. }i\neq 1\right\},$	(41)
$\displaystyle\mathcal{E}_{2,j}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{({\mathbf{x}}_{1}^{n}(j),{\mathbf{y}}^{n})\in A_{\epsilon}^{(n)}(X_{1},Y),\textrm{ s.t. }j\neq 1\right\},$	(42)
$\displaystyle\mathcal{E}_{3,k}^{BCC}$	$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$	$\displaystyle\left\{({\mathbf{x}}_{2}^{n}(k),{\mathbf{y}}_{2}^{n})\in A_{\epsilon}^{(n)}(X_{2},Y_{2}),\textrm{ s.t. }k\neq 1\right\},$	(43)

where (40) corresponds to the violation of the power constraint, (41) corresponds to the failure of the first step of the decoding at the mobile unit one, (42) corresponds to the failure of the second step of the decoding at the mobile unit one, (43) corresponds to the failure of the decoding at the mobile unit two.

Using union bound and appropriately bounding the probability of each error event term by using arguments of typicality (except for the power constraint, which follows from law of large numbers), one can show that

P_{e,BCC}^{(n)}=\Pr\left(\mathcal{E}^{BCC}\right)=\Pr\left(\mathcal{E}^{BCC}|(W_{1},W_{2})=(1,1)\right)\leq 7{\epsilon},

(44)

for any ${\epsilon}>0$ and sufficiently large $n$ , provided that $\frac{1}{2}\log\left(1+\frac{\alpha P}{N_{1}}\right)-R_{1}^{ID}>{\epsilon}$ (cf. (33)), $\frac{1}{2}\log\left(1+\frac{(1-\alpha)P}{\alpha P+N_{2}}\right)-R_{2}^{ID}>{\epsilon}$ (cf. (34)) and $\frac{1}{2}\log\left(1+\frac{(1-\alpha)P}{\alpha+N_{1}}\right)-R_{2}^{ID}>{\epsilon}$ ( which is guaranteed by recalling $N_{1}<N_{2}$ and (33).

Further, using standard arguments for finding a code with negligible maximal probability of error (cf. [1] pp. 203-204) from the one with $P_{e,BCC}^{(n)}\leq 7{\epsilon}$ we conclude that we have

\lambda_{BCC}^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\max_{w_{1},w_{2}}\lambda_{BCC}^{w_{1},w_{2}}\leq 14{\epsilon},

(45)

for any ${\epsilon}>0$ and sufficiently large $n$ , provided that (33) and (34) hold, which concludes the BCC part.

Now, we continue with the MAC part and note that (23) is still valid as well as the error event definition. We additionally include the following type of error event, which deals with the power constraints

\mathcal{E}^{MAC}_{0,i}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\left\{\frac{1}{n}\sum_{j=1}^{n}q_{w_{i},j}^{2}(1)>\alpha_{i}P_{i}\right\},

(46)

for $i\in\{1,2\}$ , such that $P_{1}=\alpha P$ and $P_{2}=(1-\alpha)P$ and $\alpha$ is the same as the one given in BCC case.

Using union bound and appropriately bounding the probability of each error event term by using arguments of typicality (except for the power constraint related terms, which follow from law of large numbers), one can show that

P_{e,MAC}^{(n)}=\Pr\left(\mathcal{E}^{MAC}\right)=\Pr\left(\mathcal{E}^{MAC}|(M_{1},M_{2})=(1,1)\right)\leq 6{\epsilon},

(47)

for any ${\epsilon}>0$ and sufficiently large $n$ , provided that $\frac{1}{2}\log\left(1+\frac{\alpha_{1}\alpha P}{N_{3}}\right)>R_{1}^{Data}+3{\epsilon}$ , $\frac{1}{2}\log\left(1+\frac{\alpha_{2}(1-\alpha)P}{N_{3}}\right)>R_{2}^{Data}+3{\epsilon}$ , $\frac{1}{2}\log\left(1+\frac{\alpha_{1}\alpha P+\alpha_{2}(1-\alpha)P}{N_{3}}\right)>R_{1}^{Data}+R_{2}^{Data}+4{\epsilon}$ .

Further, using standard arguments for finding a code with negligible maximal probability of error (cf. [1] pp. 203-204) from the one with $P_{e,MAC}^{(n)}\leq 6{\epsilon}$ we conclude that we have

\lambda_{MAC}^{(n)}\mbox{$\>\stackrel{{\scriptstyle\triangle}}{{=}}\>$}\max_{m_{1},m_{2}}\lambda_{MAC}^{m_{1},m_{2}}\leq 12{\epsilon},

(48)

for any ${\epsilon}>0$ and sufficiently large $n$ , provided that (35), (36) and (37) hold, which concludes the MAC part.

Following similar arguments as in Section III-A and using (45) and (48), we conclude that

\lambda^{(n)}\leq{\epsilon}(26-168{\epsilon}),

(49)

for any $0<{\epsilon}<\frac{13}{84}$ , where $\lambda^{(n)}$ is as defined in (6). Since ${\epsilon}$ may be arbitrarily small, (49) concludes the proof. ∎

IV-B2 Converse

In this section, we prove the converse part of Theorem IV.1, in other words we have the following theorem:

Theorem IV.3

For any sequence of $\left(2^{nR_{1}^{ID}},2^{nR_{2}^{ID}},2^{nR_{1}^{Data}},2^{nR_{2}^{Data}},n\right)$ -RFID codes with $P_{e}^{(n)}<{\epsilon}$ , for any ${\epsilon}>0$ , we have $\left(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data}\right)\in{\mathcal{R}}_{1}$ .

Proof:

Proof relies on ideas from [3] for BCC part and [1] for MAC part.

First of all, we have following

	$\displaystyle P_{e}^{(n)}$	$\displaystyle=$	$\displaystyle 1-\Pr\left(\left[({\hat{W}}_{1},{\hat{W}}_{2})=(W_{1},W_{2})\right]\wedge\left[({\hat{M}}_{1},hM_{2})=(M_{1},M_{2})\right]\right),$		(50)
		$\displaystyle=$	$\displaystyle 1-\Pr\left(({\hat{W}}_{1},{\hat{W}}_{2})=(W_{1},W_{2})\right)\Pr\left(({\hat{M}}_{1},{\hat{M}}_{2})=(M_{1},M_{2})\|({\hat{W}}_{1},{\hat{W}}_{2})=(W_{1},W_{2})\right).$		(50)

Using (50) and noting that $P_{e}^{(n)}\leq{\epsilon}$ , we have

\left(1-\Pr\left(({\hat{W}}_{1},{\hat{W}}_{2})\neq(W_{1},W_{2})\right)\right)\left(\Pr\left(({\hat{M}}_{1},{\hat{M}}_{2})\neq(M_{1},M_{2})|({\hat{W}}_{1},{\hat{W}}_{2})=(W_{1},W_{2})\right)\right),

which implies

P_{e,BCC}^{(n)}=\Pr\left(({\hat{W}}_{1},{\hat{W}}_{2})\neq(W_{1},W_{2})\right)\leq{\epsilon},

(51)

and

P_{e,MAC}^{(n)}=\Pr\left(({\hat{M}}_{1},{\hat{M}}_{2})\neq(M_{1},M_{2})|({\hat{W}}_{1},{\hat{W}}_{2})=(W_{1},W_{2})\right)\leq{\epsilon},

(52)

Next, (51) enables us to use the result of [3] for BCC case, hence we state that

	$\displaystyle R_{1}^{ID}$	$\displaystyle\leq$	$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha P}{N_{1}}\right),$		(53)
	$\displaystyle R_{2}^{ID}$	$\displaystyle\leq$	$\displaystyle\frac{1}{2}\log\left(1+\frac{(1-\alpha)P}{\alpha P+N_{2}}\right),$		(54)

for any $0\leq\alpha\leq 1$ .

Further, (52) enables us to use the result of [1] for MAC case, hence we state that

$\displaystyle R_{1}^{Data}$	$\displaystyle\leq$	$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha_{1}\alpha P}{N_{3}}\right),$	(55)
$\displaystyle R_{2}^{Data}$	$\displaystyle\leq$	$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha_{2}(1-\alpha)P}{N_{3}}\right),$	(56)
$\displaystyle R_{1}^{Data}+R_{2}^{Data}$	$\displaystyle\leq$	$\displaystyle\frac{1}{2}\log\left(1+\frac{\alpha_{1}\alpha P+\alpha_{2}(1-\alpha)P}{N_{3}}\right).$	(57)

Combining (53), (54), (55), (56) and (57) we conclude that for any $\left(2^{nR_{1}^{ID}},2^{nR_{2}^{ID}},2^{nR_{1}^{Data}},2^{nR_{2}^{Data}},n\right)$ -RFID codes with $P_{e}^{n}$ , we have $\left(R_{1}^{ID},R_{2}^{ID},R_{1}^{Data},R_{2}^{Data}\right)\in{\mathcal{R}}_{1}$ , which concludes the proof. ∎

V Conclusion

In this paper, we studied the RFID capacity problem by modeling the underlying structure as a specific multiuser communication system that is represented by a cascade of a BCC and a MAC. The BCC and MAC parts are used to model communication between the RFID reader and the mobile units, and between the mobile units and the RFID reader, respectively. To connect the BCC and MAC parts, we used a “nested codebook” structure. We further introduced imperfection channels for discrete alphabet case as well as additional power limitations for continuous alphabet additive Gaussian noise case to accurately model the physical medium of the RFID system. We provided the achievable rate region in the discrete alphabet case and the capacity region for the continuous alphabet additive Gaussian noise case. Hence, overall, we characterized the maximal achievable error free communication rates for any RFID protocol for the latter case.

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