On the Capacity of the Slotted Strongly Asynchronous Channel with a Bursty User

TL;DR

This paper analyzes the fundamental limits of asynchronous communication in slotted channels with bursty users, deriving bounds on the trade-offs between transmission rate, asynchronism, and burstiness in the asymptotic regime.

Contribution

It provides the first comprehensive bounds on the capacity of strongly asynchronous channels with bursty users, including models with random access and matching bounds in special cases.

Findings

Achievability and converse bounds on the $(R, u, ext{or } eta)$ trade-offs.

Bounds coincide for $ u=0$ and $R=0$, and for $eta= ext{or } u$ and $R=0$.

Characterization of the capacity region in asymptotic regimes.

Abstract

In this paper, the trade-off between the number of transmissions (or burstiness) $K_n=e^{n\nu}$ of a user, the asynchronism level $A_n=e^{n\alpha}$ in a slotted strongly asynchronous channel, and the ability to distinguish $M_n=e^{nR}$ messages per transmission with vanishingly error probability is investigated in the asymptotic regime as blocklength $n$ goes to infinity. The receiver must locate and decode, with vanishing error probability in $n$, all of the transmitted messages. Achievability and converse bounds on the trade-off among $(R,\alpha,\nu)$ is derived. For cases where $\nu=0$ and $ R=0$, achievability and converse bounds coincide. A second model for a bursty user with random access in which the user may access and transmit a message in each block with probability $e^{-n\beta}$ in then considered. Achievability and converse bounds on the trade-off between $(R, \alpha, \beta)$ is also characterized. For cases where $\beta =\alpha$ and $R=0$, the achievability and converse bounds match.

Equations

$$[a{]}^{+}:=\{\begin{array}{cc}a,& a>0\\ 0,& a\le 0\end{array}.$$

$${\hat{P}}_{x^n}(a):=\frac{1}{n}\mathcal{N}(a|x^n)=\frac{1}{n}\sum _{i=1}^n\text{mathbbm}{1}_{\{x_i=a\}},\forall a\in \mathcal{X},$$

$$T(P)$$

$$T_V(x^n)$$

$$g({\mathcal{Y}}^{nA})=\left(({\hat{\nu }}_1,\widehat{m_1}),\dots ,({\hat{\nu }}_K,\widehat{m_K})\right),$$

$$P_e^{(n)}:=\frac{1}{M^K\left(\genfrac{}{}{0ex}{}{A}{K}\right)}\sum _{({\nu }_1,m_1),\dots ,({\nu }_K,m_K)}\mathbb{P}[g(y^{nA})\ne \left(({\nu }_1,m_1),\dots ,({\nu }_K,m_K)\right)|H_{\left(({\nu }_1,m_1),\dots ,({\nu }_K,m_K)\right)}],$$

$${\mathcal{R}}^{in}:=\bigcup _{\lambda \in [0,1],P\in {\mathcal{P}}_{\mathcal{X}}}$$

$${\mathcal{R}}^{out}:=\bigcup _{\lambda \in [0,1],P\in {\mathcal{P}}_{\mathcal{X}}}$$

$$Q_{\lambda }(.|x)$$

$$P_e^{(n)}$$

$$T:-D(Q_{\star }\parallel Q|P)\le T\le D(Q\parallel Q_{\star }|P),$$

$$L\left(y_j^n,x_j^n(m)\right):=\frac{1}{n}\log \frac{Q\left(y_j^n|x_j^n(m)\right)}{Q_{{\star }^n}(y_j^n)}\ge T,$$

$$H_1:=H_{\left((1,1),\dots ,(K_n,1)\right)}$$

$$\mathbb{P}\left[\text{synch error}|H_1\right]$$

$$\le \text{mathbbm}P\left[\bigcup _{j=1}^{K_n}\bigcap _{m=1}^{M_n}L\left(Y_j^n,x_j^n(m)\right)<T|H_1\right]$$

$$+\text{mathbbm}P\left[\bigcup _{j=K_n+1}^{A_n}\bigcup _{m=1}^{M_n}L\left(Y_j^n,x_j^n(m)\right)\ge T|H_1\right]$$

$$\le \sum _{j=1}^{K_n}\mathbb{P}\left[L\left(Y_j^n,x_j^n(1)\right)<T|H_1\right]+e^{nR}\sum _{j=K_n+1}^{A_n}\mathbb{P}\left[L\left(Y_j^n,x_j^n(1)\right)\ge T|H_1\right]$$

$$\le e^{n\nu }\sum _{\begin{array}{c}\hat{Q}:\\ D(\hat{Q}||Q_{\star }|P)-D(\hat{Q}||Q|P)<T\end{array}}\mathbb{P}\left[Y^n\in T_{\hat{Q}}\left(x^n(1)\right)|H_1\right]$$

$$+e^{n\left(R+\alpha \right)}\sum _{\begin{array}{c}\hat{Q}:\\ D(\hat{Q}||Q_{\star }|P)-D(\hat{Q}||Q|P)\ge T\end{array}}\mathbb{P}\left[Y^n\in T_{\hat{Q}}(x^n(1))|H_1\right]$$

$$\le e^{n\nu }{e}^{-nD(Q_{\lambda }\parallel Q|P)}+e^{n(\alpha +R)}{e}^{-nD(Q_{\lambda }\parallel Q_{\star }|P)},$$

$$\lambda :D(Q_{\lambda }\parallel Q_{\star }|P)-D(Q_{\lambda }\parallel Q|P)=T.$$

$$\nu$$

$$\alpha +R$$

$$\left\{\text{error}|H_1\right\}=\bigcup _{\begin{array}{c}\left((l_1,{\tilde{m}}_1)\dots ,(l_{K_n},{\tilde{m}}_{K_n})\right)\\ \ne \left((1,m_1),\dots ,(K_n,m_{K_n})\right)\end{array}}\left\{\sum _{i=1}^{K_n}L\left(Y_i^n,x_i^n(m_i)\right)\le \sum _{i=1}^{K_n}L\left(Y_{l_i}^n,x_{l_i}^n({\tilde{m}}_i)\right)\right\},$$

$$\left((l_1,{\tilde{m}}_1)\dots ,(l_{K_n},{\tilde{m}}_{K_n})\right)\ne \left((1,m_1),\dots ,(K_n,m_{K_n})\right),$$

$$\displaystyle\left\{\text{error}|H_{1}\right\}\supseteq\bigcup_{\begin{subarray}{c}i\in[K_{n}]\\ j\in[K_{n}+1:A_{n}]\\ {m}\in[M_{n}]\end{subarray}}\big{\{}L\left(Y_{i}^{n},x_{i}^{n}(m_{i})\right)\leq L\left(Y_{j}^{n},x_{j}^{n}({m})\right)\big{\}}$$

$$\supseteq \left\{\bigcup _{i\in [K_n]}\left\{L\left(Y_i^n,x_i^n(m_i)\right)\le T\right\}\right\}\bigcap \left\{\bigcup _{\begin{array}{c}j\in [K_n+1:A_n]\\ m\in [M_n]\end{array}}\left\{L\left(Y_j^n,x_j^n(m)\right)\ge T\right\}\right\}.$$

$$T\in [-D(Q_{\star }\parallel Q|P_{i^{\star }}),D(Q\parallel Q_{\star }|P_{i^{\star }})],$$

$$i^{\star }:=\arg \underset{\begin{array}{c}i,{\lambda }_i:\\ D(Q_{{\lambda }_i}\parallel Q_{\star }|P_i)-D(Q_{{\lambda }_i}\parallel Q|P_i)=T\end{array}}{\max }D(Q_{{\lambda }_i}\parallel Q|P_i).$$

$$\displaystyle\mathbbm{P}\left[\text{error}\Big{|}H_{1}\right]$$

$$\ge \text{mathbbm}P\left[\bigcup _{i\in [K_n]}L\left(Y_i^n,x_i^n(m)\right)\le T|H_1\right]\cdot \text{mathbbm}P\left[\bigcup _{\begin{array}{c}j\in [K_n+1:A_n]\\ m\in [M_n]\end{array}}L\left(Y_j^n,x_j^n(m)\right)\ge T|H_1\right]$$

$$\ge \left(1-e^{-n\left[\nu -D\left(Q_{{\lambda }_{i^{\star }}}\parallel Q|P_{i^{\star }}\right)\right]}\right)$$

$$\cdot \left(1-e^{-n\left[\alpha +R\text{mathbbm}{1}_{\left\{R<I(P,Q_{{\lambda }_{i^{\ast }}})\right\}}-D(Q_{{\lambda }_{i^{\ast }}}\parallel Q_{\star }|P_{i^{\ast }})\right]}\right),$$

$$\nu$$

$$\alpha +R\text{mathbbm}{1}_{\left\{R<I(P,Q_{{\lambda }_{i^{\ast }}})\right\}}$$

$$\nu$$

$$\alpha$$

Full text

I Introduction

It is widely believed that Machine-type Communications and Internet of Things are going to be the next dominant paradigm in wireless technology. The traffic pattern imposed by the devices within these networks have unique features different from the ones in human-type communication networks. The communications that take place within these networks are often sporadic and bursty, but must nonetheless be reliably detected and decoded. For example, each sensor node may want to transmit a signal to the base station only when some incident has taken place.

In this paper, we consider the problem of both detecting and decoding asynchronous data bursts of a single user. This extends work in [1]. In conventional methods the user transmits a pilot signal at the beginning of each data burst to notify the decoder of the upcoming data; the decoding phase may be performed using any synchronized decoding method. The loss in this approach is negligible when synchronization is done once and the cost of acquiring synchronization is absorbed into the lengthy data stream that follows. For sparse / bursty transmission, as in the problem considered here, this approach is not suitable as the training based schemes are known to be sub-optimal [2]. In this work we do not enforce the usage of pilot symbols, and the codebook serves the dual purpose of synchronization and data transfer. This paper’s central goal is to characterize the trade-off between the reliable transmission rate between one transmitter and one receiver, the burstiness of that transmitter, and the level of asynchronism.

I-A * Past Work*

In this paper, the trade-off between the number of transmissions (or burstiness) $K_{n}=e^{n\nu}$ of a user, the asynchronism level $A_{n}=e^{n\alpha}$ in a slotted strongly asynchronous channel, and the ability to distinguish $M_{n}=e^{nR}$ messages per transmission with vanishingly error probability is investigated in the asymptotic regime as blocklength $n$ goes to infinity. The problem considered here generalizes the one in [3]. In [3], the authors considered a user who transmits only once within an strong asynchronous window. The goal was to locate and decode the user transmission time and message. In our work, the user transmits exponentially many times in blocklength (or arbitrary number of times in our second model). Moreover our error metric is the global / joint probability of error (i.e., an error is declared if any of the user’s transmissions is in error, and we have an exponential number of transmissions) and we require the exact recovery of the transmission time and codeword in all transmissions. The approach in [3] does not extend to the global probability of error criterion for exponential number of transmission in blocklength $n$ (where the number of transmissions is equal to $K_{n}=e^{n\nu},\nu>0$). This is due to the fact that their achievability relies on the typicality decoder and the derived error bounds do not decay fast enough with blocklength $n$.

In [4], the authors considered the special case of the problem considered here where a user transmits one synchronization pattern of length $n$ (hence the rate $R=0$) only once (hence $\nu=0$) in a window of length $A_{n}=e^{n\alpha}$ of $n$ channel uses each. They showed that for any $\alpha$ below the synchronization threshold $\alpha_{0}$, the user can detect the location of the synchronization pattern. In addition they showed that a synchronization pattern consisting of the repetition of a single symbol which induces an output distribution with the maximum divergence from the noise distribution, suffices. The typicality decoder introduced in [4] however, even in a slotted channel model, only retrieves one of the trade-off points that we obtain in this paper that corresponds to a sub-exponential number of transmissions. We propose new achievability and converse techniques to support an exponential number of transmissions ($K_{n}=e^{n\nu},\nu>0$). Interestingly, we show that the symbol used for synchronization may change for different values of $\alpha$ and $\nu$.

The single user strongly asynchronous channel was also considered in [5], where it was shown that the exact transmission time recovery, as opposed to the error criterion in [3] which allows a sub-exponential delay in $n$, does not change the capacity.

Recently, the synchronous Gaussian massive multiple access channel with random access has been modeled in [6] where the number of users is let to grow linearly in the code blocklength and any random subset of users may try to access the channel. In [6], the authors took advantage of the Gaussian channel structure to exactly derive matching upper and lower bounds on the capacity. Since then, other versions of “massive number of users” have been proposed in [7], [8]. In [9], we studied a multi-user version of the slotted strongly asynchronous model for a discrete memoryless channel where we assumed that $K_{n}=e^{n\nu}$ different users transmit a message among $M_{n}^{(i)}=e^{nR_{i}},i\in\{1,\ldots,K_{n}\}$ of them only once in an asynchronous window of length $A_{n}=e^{n\alpha}$ blocks of $n$ channel uses each. Inner bounds on the trade-off between $(R_{i},\nu,\alpha),i\in\{1,\ldots,K_{n}\}$ were derived, but these were not shown to be tight. What renders the presented version of the problem – a single user transmitting multiple times rather than multiple users transmitting once each – more tractable is that one is guaranteed that in each block there is at most one transmitted message and we do not need to detect the user’s identity.

I-B * Contributions*

In this paper, we bridge the bursty random access channel model with the asynchronous communication and study the trade-off between the number of transmissions (or burstiness) $K_{n}=e^{n\nu}$ of a user, the asynchronism level $A_{n}=e^{n\alpha}$ in a slotted bursty and strongly asynchronous channel, and the ability to distinguish $M_{n}=e^{nR}$ messages per transmission with vanishingly error probability as blocklength $n$ goes to infinity. The slotted assumption restricts the transmission times to be integer multiples of the blocklength $n$; this assumption simplifies the error analysis yet captures the essence of the problem. We show:

For synchronization and data transmission ($R>0$), we find converse and achievability bounds on the capacity region of $(R,\alpha,\nu)$ that match for $\nu=0$. 2. 2.

For synchronization only ($R=0$), our proposed sequential decoder achieves the optimal trade-off. Surprisingly, we show that the optimal synchronization pattern is not fixed and may depend on the asynchronism level $\alpha$. 3. 3.

For certain values of $R$, which are small enough, the achievability and converse bounds match.

We also consider a slotted bursty and strongly asynchronous random access channel with asynchronous level $A_{n}=e^{n\alpha}$ where the number of transmissions of the user is not fixed and the user may randomly with probability $p_{n}=e^{-n\beta}$ transmit a message, among $M_{n}=e^{nR}$ possible ones, within each block of $n$ channel uses. In this case, we show:

The achievability and converse bounds on the capacity region $(R,\alpha,\beta)$ is derived. Our achievability result shows that the asynchronous window length $A_{n}=e^{n\alpha}$ will increases with the increase of $\beta$ since the number of transmissions to be detected decreases. Moreover, for $\beta=\alpha$ and $R=0$ the achievability and converse bounds match.

I-C Paper organization

The rest of this paper is organized as follows. In Section II we introduce the slotted bursty and strongly asynchronous channel model with fixed number of transmissions and analyze its capacity region by providing upper and lower bounds on its capacity. We also find an equivalent capacity region expression for the special case with zero rate (synchronization only). In Section III we introduce a model for slotted bursty and strongly asynchronous channel with random number of transmissions and find upper and lower bounds on its capacity region. Section IV concludes the paper.

I-D * Notation*

The notation $a_{n}\doteq e^{nb}$ means $\lim_{n\to\infty}\frac{\log a_{n}}{n}=b$. We write $[M:N]$, where $M,N\in\mathbb{Z},M\leq N$, to denote the set $\{M,M+1,\ldots,N\}$, and $[K]:=[1:K]$. In addition we use the notation

[TABLE]

Capital letters represent random variables that take on lower case letter values in calligraphic letter alphabets. A stochastic kernel / transition probability from $\mathcal{X}$ to $\mathcal{Y}$ is denoted by $Q(y|x),\forall(x,y)\in\mathcal{X}\times\mathcal{Y}$, and the output marginal distribution induced by $P\in\mathcal{P}_{\mathcal{X}}$ through the channel $Q$ as $[PQ](y):=\sum_{x}P(x)Q(y|x),\forall y\in\mathcal{Y}$ where $\mathcal{P}_{\mathcal{X}}$ is the space of all distributions on $\mathcal{X}$. As a shorthand notation, we also define $Q_{x^{n}}(.):=Q(.|x^{n})$. We use $y_{j}^{n}:=[y_{j,1},...,y_{j,n}]$, and simply $y^{n}$ instead of $y_{1}^{n}$. The empirical distribution of a sequence $x^{n}$ is

[TABLE]

where $\mathbbm{1}_{\{A\}}$ is the indicator function of the event $A$ and where $\mathcal{N}(a|x^{n})$ denotes the number of occurrences of letter $a\in\mathcal{X}$ in the sequence $x^{n}$; when using (1) the target sequence $x^{n}$ is usually clear from the context so we may drop the subscript $x^{n}$. The $P$-type set and the $V$-shell of the sequence $x^{n}$ are respectively defined as

[TABLE]

where $\mathcal{N}(a,b|x^{n},y^{n})=\sum_{i=1}^{n}\mathbbm{1}_{\left\{\begin{subarray}{c}x_{i}=a\\ y_{i}=b\end{subarray}\right\}}$ is the number of joint occurrences of $(a,b)$ in the pair of sequences $(x^{n},y^{n})$. We also use $I(P,Q)$ to denote the mutual information between random variable $(X,Y)\sim(P,[PQ])$ coupled via $P_{Y|X}(y|x)=Q(y|x)$, $D(P_{1}\parallel P_{2})$ for the Kullback Leibler divergence between distribution $P_{1}$ and $P_{2}$, and $D(Q_{1}\parallel Q_{2}|P):=\sum_{x,y\in\mathcal{X}\times\mathcal{Y}}P(x)Q_{1}(y|x)\log\frac{Q_{1}(y|x)}{Q_{2}(y|x)}$ for conditional Kullback Leibler divergence.

II System model for fixed number of transmissions and main results

We consider a discrete memoryless channel with transition probability matrix $Q(y|x)$ defined over all $(x,y)$ in the finite input and output alphabets $(\mathcal{X},\mathcal{Y})$. We also define a noise symbol $\star\in\mathcal{X}$ for which $Q_{\star}(y)>0,\ \forall y\in\mathcal{Y}$.

An $(M,A,K,n,\epsilon)$ code for the slotted bursty and strongly asynchronous discrete memoryless channel with transition probability matrix $Q(y|x)$ with fixed number of transmissions consists of:

• •

A message set $[M]$, from which messages are selected uniformly at random.

• •

Encoding functions $f_{i}:[M]\to\mathcal{X}^{n},\ i\in[A]$, where we define $x_{i}^{n}(m):=f_{i}(m)$. The transmitter chooses uniformly at random one set of $K$ blocks for transmission out of the ${A\choose K}$ possible ones, and a set of $K$ messages from $M^{K}$ possible ones, also uniformly at random, and sends $x_{\nu_{i}}^{n}(m_{i})$ in block $\nu_{i}$ for $i\in[K]$ and $\star^{n}$ in every other block. We denote the chosen blocks and messages as $\left((\nu_{1},m_{1}),\ldots,(\nu_{K},m_{K})\right)$.

• •

A destination decoder function

[TABLE]

such that the average probability of error associated to it, given by

[TABLE]

satisfies $P_{e}^{(n)}\leq\epsilon,$ where $H_{\left((\nu_{1},m_{1}),\ldots,(\nu_{K},m_{K})\right)}$ is the hypothesis that user transmits message $m_{i}$ at block $\nu_{i}$ with the codebook $x^{n}_{{\nu_{i}}}(m_{i})$, for all $i\in[K]$.

A tuple $(R,\alpha,\nu)$ is said to be achievable if there exists a sequence of codes $(e^{nR},e^{n\alpha},e^{n\nu},n,\epsilon_{n})$ with $\epsilon_{n}$ going to zero as $n$ goes to infinity. The capacity region is the set of all possible achievable $(R,\alpha,\nu)$ triplets.

We now introduce our main result. In Theorem 1 we show that an exponential number of transmissions for a single user is possible at the expense of a reduced rate and/or reduced asynchronous window length compared to the case of only one transmission $K_{n}=1$ (or more generally $\nu=0$).

Theorem 1**.**

Achievable and impermissible regions for the capacity region of a slotted bursty and strongly asynchronous discrete memoryless channel with transition probability matrix $Q(y|x)$ are given by

[TABLE]

and

[TABLE]

where

[TABLE]

Proof:

Achievability. Codebook generation. The user generates $A_{n}$ constant composition codebooks, of rate $R$ and blocklength $n$, by selecting each message’s codeword uniformly and independently from the $P$-type set of sequences in $\mathcal{X}^{n}$, one codebook for each available block.

Decoder. We perform a two-stage decoding. First, the decoder finds the location of the transmitted codewords (first stage, the synchronization stage) and it decodes the messages (second stage, the decoding stage). The probability of error for this two-stage decoder is given by

[TABLE]

For the first stage, fix

[TABLE]

which can be changed for different trade-off points. At each block $j\in[A_{n}]$, if there exists any message $m\in[M_{n}]$ such that the Log Likelihood Ratio (LLR)

[TABLE]

declare a codeword transmission block and a noise block otherwise. Given the hypothesis

[TABLE]

the probability of the synchronization error in the first stage is given by

[TABLE]

where $Q_{\lambda}$ is defined in (4) and

[TABLE]

The expression in (7) is the result of finding the minimum exponent in (6) using the Lagrangian method as in [10, Sec. 11.7].

By (7), the probability of error in the synchronization goes to zero as $n$ goes to infinity when

[TABLE]

Conditioning on the ‘no synchronization error’ and having found all $K_{n}$ ‘not noisy’ blocks, we can use a Maximum Likelihood (ML) decoder for random constant composition codes, introduced and analyzed in [11], on the super-block of length $nK_{n}$ to distinguish among $e^{nK_{n}R}$ different message combinations. If $R<I(P,Q)$, the probability of the error of the second stage also vanishes as $n\to\infty$.

Converse. The main technical difficulty and innovation in the proof relies on analyzing the probability of error in a ML decoder. In this regard, we boil down the problem to finding an exponentially decaying ‘lower’ bounds on the probability of the missed detection (where the likelihood ratio defined in (5) of an active block is less than a threshold) and false alarm (where the likelihood ratio defined in (5) of an idle block is larger than the threshold) error events. By the type counting argument and the fact that we have polynomially many types in blocklength at the expense of a small reduce in rate [12] we can restrict our attention to constant composition codes. In other words, we assume the use of codewords $x_{i}^{n}(.)$ with constant compositions $P_{i}$ in each block $i\in[A_{n}]$. Given the hypothesis $H_{1}:=H_{((1,1)\ldots(K_{n},1))}$, with a ML decoder (which achieves the minimum average probability of error) and for any $T\in\mathbb{R}$, the error events are given by

[TABLE]

where (9) the union of the events that the sum of the LLRs of the true hypothesis $\left((1,{m}_{1}),\ldots,(K_{n},{m}_{K_{n}})\right)$ is less than the sum of the LLRs of the wrong hypothesis (with arbitrary number of incorrect synchronization or decoding errors)

[TABLE]

where wrong means that we have at least one decoding or one synchronization error. We now focus our attention on a subset of these events which have a single synchronization error. i.e.,

[TABLE]

In other words, (10) is the union over the events that (any message, any noisy block) is selected instead of one of the *(correct message, correct block)*s; with the underlying assumption that the rest of the blocks are chosen correctly. We also further restrict

[TABLE]

where $i^{\star}$ is chosen such that

[TABLE]

The reason for this choice of $i^{*}$ will be become clear later (see (14) and (15)). By (11) we have

[TABLE]

where (13) is due to the independence of $Y_{j}^{n},j\in[A_{n}]$ and where (14) and (15) are proved in Appendix -A and -B, respectively.

The lower bound on the probability of error given in (14) and (15), would be bounded away from zero if

[TABLE]

which can be equivalently be written as

[TABLE]

and hence this region is impermissible.

Any asynchronous channel can be reduced to a synchronous channel by providing the decoder with side information about the transmission time. Hence, the same bound on the rate of a synchronous channel, i.e. $R<I(P_{i^{\star}},Q)$ also applies to the asynchronous channel. By the symmetry of the hypothesis, the same lower bound on probability of error holds for the average probability of error and hence we retrieve the bounds given in (3). ∎

Remark 1**.**

*We note that for the set of $\lambda:R<I(P,Q)$, the achievability and converse bounds match. As the result, there is a region for $R$ small enough, that our achievability and converse bounds coincide. *

The main novelty in this problem is to find exponentially decaying upper and lower bounds on the probability of error. The achievability scheme analysis is easier as we can easily pose it as a hypothesis testing problem. However, in the converse, we have to deal with the optimal ML decoder. As a first step in reducing the complexity of the ML decoder, we considered a set of error events with single synchronization errors (which we believe is the major error set and many other events are its subsets). Next, we had to find the probability that the LLR’s of the active blocks are smaller than a threshold. This again, would be easy to calculate for a single LLR; its probability is a function of the (imaginary) channel $Q_{\lambda}$ defined in (4). However, we have to deal with unions of such events as in (13). Calculation of these unions is also easy for $\nu=0$. In this case the optimal $\lambda=1$ and hence $Q_{\lambda=1}=Q$ and one can leverage the fact that the probability of decoding error for channel $Q$ is small to transform the union into a summation. If however $\nu\neq 0$ and hence $\lambda\neq 1$, probability of error for channel $Q_{\lambda}$ (for the same code as channel $Q$) would be dependent on the rate $R$. Transformation of a union to a summation is not straightforward anymore and hence we had to provide several additional steps (in Appendix -B and -C) to do so.

For a fixed $\lambda$, a comparison between the bounds given by (17b) and (8b) is shown in Fig. 1. It is easy to see that the bounds given in (2) and (3) will coincide (i.e., complement one another) for the case $\nu=0$ ($\lambda=1$) and retrieve the capacity region previously derived in [3].

Remark 2**.**

It is worth noting that the region specified in 1, need not be convex since $\alpha$ is a channel parameter and can not be chosen by user.

We now concentrate our attention to the synchronization case only.

Remark 3**.**

By specializing Theorem 1 for $R=0$, we can see that $\mathcal{R}^{in}|_{R=0}=\mathcal{R}^{out}|_{R=0}=\mathcal{R}|_{R=0}$.

It can be easily seen in Fig. 2 that by taking the union over $\lambda\in[0,1]$, the achievability and converse regions match for $R=0$.

In the following example, we consider a Binary Symmetric Channel and plot its achievable region.

Exmple 1**.**

To illustrate the capacity region in Theorem 1, we consider a Binary Symmetric Channel (BSC) $Q$ with cross over probability $\delta$ as it is shown in Fig. 3. We also assume $\star=0$. For the channel $Q_{\lambda}$ in (4) we have

[TABLE]

By changing $p=\mathbb{P}[X=0]\in[0,\frac{1}{2}]$ and $\lambda\in[0,1]$, we obtain the achievability region shown in Fig. 4(a). In addition, the (optimal) trade off for $(R,\alpha,\nu=0)$ can be seen in Fig. 4(b) which resembles the one in [5, Fig. 1]. The trade off between $(\alpha,\nu)$ can be seen in Fig. 4(c) which has the curvature we expect to see, like the one in Fig. 7 in the Appendix.

Theorem 2 provides another form for the trade-off between $(R=0,\alpha,\nu)$ which implies that using a repetition pattern for synchronization pattern is optimal.

Theorem 2**.**

For $R=0$, the capacity region $\mathcal{R}|_{R=0}$ in Remark 3 is equivalent to

[TABLE]

Proof:

$\mathcal{R}^{\text{synch}}\subseteq\mathcal{R}|_{R=0}$ is trivial since we can restrict the set of distributions $P\in\mathcal{P}_{\mathcal{X}}$ in $\mathcal{R}|_{R=0}$ to the distributions with weight one on a single symbol $x$ and zero weight on all other symbols.

We also prove $\mathcal{R}|_{R=0}\subseteq\mathcal{R}^{\text{synch}}$ by contradiction and by means of the following Lemma proved in Appendix -D.

Lemma 1**.**

The curve $\left(D(Q_{\lambda}\parallel Q_{\star}|P),D(Q_{\lambda}\parallel Q|P)\right)$ characterized by $\lambda\in[0,1]$ is the lower envelope of the set of curves

[TABLE]

which are each characterized by $\lambda_{x}\in[0,1]$.

We continue the proof by assuming $\mathcal{R}|_{R=0}\not\subseteq\mathcal{R}^{\text{synch}}$. Then there exists an element

[TABLE]

that is, which lies above all the $\left\{D(Q_{\lambda}\parallel Q_{\star}),D(Q_{\lambda}\parallel Q_{x})\right\}$ curves for all $x\in\mathcal{X}$. Hence, for any $x\in\mathcal{X}$, there exists a $\lambda_{x}$ such that

[TABLE]

As a result

[TABLE]

which contradicts Lemma 1 that $\left(D(Q_{\lambda}\parallel Q|P),D(Q_{\lambda}\parallel Q|P)\right)$ is the lower envelope of the set of $\bigcup_{x\in\mathcal{X}}\left\{\left(D(Q_{\lambda_{x}}\parallel Q_{\star}|P),D(Q_{\lambda_{x}}\parallel Q|P)\right)\right\}$ curves and hence the initial assumption that $\mathcal{R}|_{R=0}\not\subseteq\mathcal{R}^{\text{synch}}$ is not feasible. ∎

Note that by adapting the achievability scheme to synchronize only ($R=0$), we do not need a different synchronization pattern for each block. Using the same synchronization pattern in every block suffices to drive the probability of error in the synchronization stage to zero and since it matches the converse, it is optimal.

Theorem 2 also implies that depending on the value of $\alpha$ and $\nu$, using a repetition synchronization pattern with a single symbol is optimal. This symbol may change depending on the considered value of $\alpha$ and $\nu$. For the ternary channel in Fig. 5, for example, the resulting curves by using symbol $x=1$ and $x=2$ are shown in Fig. 5. As it is clear, for the regime $\alpha>0.356$, symbol $x=1$ has to be used whereas in the regime $\alpha\leq 0.356$ symbol $x=2$ has to be used in the synchronization pattern.

III System model for random transmissions and main result

We consider again a discrete memoryless channel with transition probability matrix $Q(y|x)$ defined over all $(x,y)$ in the finite input and output alphabets $(\mathcal{X},\mathcal{Y})$. We also define a noise symbol $\star\in\mathcal{X}$ for which $Q_{\star}(y)>0,\ \forall y\in\mathcal{Y}$.

An $(M,A,p,n,\epsilon)$ code for the slotted bursty and strongly asynchronous discrete memoryless channel with transition probability matrix $Q(y|x)$ with random access is defined as follows.

• •

A message set $[M]$, from which messages are selected uniformly at random.

• •

Encoding functions $f_{i}:[M]\to\mathcal{X}^{n},\ i\in[A]$, where we define $x_{i}^{n}(m):=f_{i}(m)$. For each block $i\in[A]$, the transmitter chooses a message among $M$ possible ones and transmit $x_{i}^{n}(m_{i})$ through the channel with probability $p$ or remains idle and transmits $\star^{n}$ with probability $1-p$.

• •

A destination decoder function

[TABLE]

such that the average probability of error associated to it, given by

[TABLE]

satisfies $P_{e}^{(n)}\leq\epsilon,$ where $H_{\left((\nu_{1},m_{1}),\ldots,(\nu_{k},m_{k})\right)}$ is the hypothesis that user transmits message $m_{i}$ at block $\nu_{i}$ with the codebook $x^{n}_{{\nu_{i}}}$, for all $i\in[k]$.

A tuple $(R,\alpha,\beta)$ is said to be achievable if there exists a sequence of codes $(e^{nR},e^{n\alpha},e^{-n\beta},n,\epsilon_{n})$ with $\epsilon_{n}\to 0$ as $n\to\infty$. The capacity region is the set of all possible achievable $(R,\alpha,\beta)$ triplets.

Theorem 3**.**

Achievable and impermissible regions for the capacity region of a slotted bursty and strongly asynchronous random access channel with transition probability matrix $Q(y|x)$ are given by

[TABLE]

and

[TABLE]

Proof:

**Achievability. ** The encoder and decoder are the same as the one given for the achievability proof of Theorem 1, except that the number of active blocks is not fixed. We denote $p_{n}:=e^{-n\beta}$ and $\hat{H}_{k}$ to be the hypothesis that the user is active in $k$ blocks. By the symmetry of the probability of error among hypotheses with the same number of occupied blocks, we can write

[TABLE]

With similar steps as those in the proof of Theorem 1, we obtain

[TABLE]

where

[TABLE]

By (22), we can upper bound (21) as

[TABLE]

which goes to zero for

[TABLE]

For the decoding stage, with the same strategy as the one in Theorem 1 we obtain the third bound in (19).

**Converse. ** The converse argument is also similar to the converse proof of Theorem 1. It can be shown that

[TABLE]

Hence

[TABLE]

where

[TABLE]

and where (23) is proved in Appendix -E. This retrieves the first two bounds in (20). The third bound in (20) is by the usual bound on the reliable rate of a synchronous channel. ∎

It is easy to see that (19) and (20) match for the cases that $R=0$ or $\beta=\alpha$. The latter case corresponds to $\lambda=1$.

Exmple 2**.**

We consider the same BSC channel defined in Example 1 and illustrate its achievability region for the slotted bursty and strongly asynchronous channel with random access in Fig. 6(a). For values of $\beta>D(Q\parallel Q_{\star}|P)=2.3527$, the achievable region is similar to the to the case $\beta=2.3527$ and the surface remains unchanged. This is also apparent in Fig. 6(b) where the trade-off between $(\alpha,\beta)$ is depicted. This is in fact obvious in Theorem 3 since for values of $\beta>D(Q\parallel Q_{\star}|P)$ the achievability (19) and converse bound (20) match and are equal to the capacity region for one for only one transmission as the one in [5, Fig. 1].

IV Conclusion

In this paper we study a slotted bursty and strongly asynchronous discrete memoryless channel where a user transmits a randomly selected message among $M_{n}=e^{nR}$ messages in each one of the $K_{n}=e^{n\nu}$ randomly selected blocks of the available $A_{n}=e^{n\alpha}$ blocks. We derive the upper and lower bounds on the trade-off among $(R,\alpha,\nu)$ by finding achievability and converse bounds where we analyze an optimal Maximum Likelihood decoder in the converse. For the case that the number of transmissions of the user is not fixed and the user may access the channel with probability $e^{-n\beta}$, we again provide upper and lower bounds on the trade-off between $(R,\alpha,\beta)$.

V Acknowledgement

The work of the authors was partially funded by NSF under award 1422511. The contents of this article are solely the responsibility of the authors and do not necessarily represent the official views of the NSF.

-A Proof of (14)

The main trick in the proof of (14) is to find an equivalent event and lower bound the probability of that event instead. In this regard we have

[TABLE]

where we define

[TABLE]

The equality in (25) is due to the equivalence of the events to the ones in (24) and  (26) is by Chebyshev’s inequlity. The inequality in (27) is by the choice of $i^{\star}$ in (12) and finally (28) is true because of the special choice of $T=D(Q_{\lambda_{i}}\parallel Q_{\star}|P_{i})-D(Q_{\lambda_{i}}\parallel Q|P_{i})$.

-B Proof of (15)

To find a lower bound on the term in (13), we proceed as before by writing

[TABLE]

where we have defined

[TABLE]

The equality in (30) is true because the two events in the probabilities are the same and the first inequality in (31) is by the Chebyshev inequality. The inequality in (33) is proved in Appendix -C. We should note that $\zeta_{j},j\in[K_{n}+1:A_{n}],$ are independent since $Y_{j}^{n},j\in[K_{n}+1:A_{n}]$ are independent.

-C Lower bound in (33).

We first define a new typical set $T_{Q_{\lambda+\epsilon}^{n}}^{\delta}$ as follows.

Definition 1**.**

For $\epsilon$ and $\delta$ define

[TABLE]

The new constraint

[TABLE]

that we included in the typical set definition ensures that all the sequences $y^{n}$ that belong to $T_{Q_{\lambda+\epsilon}^{n}}^{\delta}$ will also satisfy

[TABLE]

In addition, define

[TABLE]

where $\Delta>0$ since

[TABLE]

is decreasing in $\lambda$ [13]. By the Law of Large Numbers

[TABLE]

and

[TABLE]

and hence for any $\delta_{1}>0$ there exists $n_{1}$ such that for all $n\geq n_{1}$ we have

[TABLE]

Moreover, assume that $D_{Q^{n}_{\lambda+\epsilon}}(m)$ is the optimal (and disjoint) decoding region for message $m$, whose codeword is passed through the channel $Q^{n}_{\lambda+\epsilon}$. We also denote the average probability of decoding error associated with channel $Q^{n}_{\lambda+\epsilon}$ to be

[TABLE]

Now, if we drop half of the codewords in $\left(x^{n}(1),\ldots,x^{n}(M_{n})\right)$ with the largest probability of the error, the remaining half must must all satisfy

[TABLE]

otherwise, the average probability of error for the decoding regions $D_{Q_{\lambda+\epsilon}^{n}}(m)$ will be larger than $P_{e}^{(n)}(Q_{\lambda+\epsilon})$ and we reach a contradiction. Henceforth we restrict our attention to this half of the codebook (which without loss of generality we assume is the first $\frac{M_{n}}{2}$ codewords).

As the result for the optimal decoding regions $D_{Q_{\lambda+\epsilon}^{n}}(m)$ of channel channel $Q_{\lambda+\epsilon}^{n}$ and by (34) and (35) we have

[TABLE]

In addition, we can conclude from [5, Lemma 10] that for any two distributions $P_{1}^{n},P_{2}^{n}$ and any event $A$ such that

[TABLE]

we have

[TABLE]

In case the lower bound given in (36), i.e. $1-\delta_{1}-P_{e}^{(n)}(Q_{\lambda+\epsilon})$, is positive (which we discuss shortly) and by (37) we can write

[TABLE]

In addition, due to continuity of the divergence, as $\epsilon\to 0$, we have

[TABLE]

We now discuss the case that $1-\delta_{1}-P_{e}^{(n)}(Q_{\lambda+\epsilon})$ is positive. A sufficient condition for $1-\delta_{1}-P_{e}^{(n)}(Q_{\lambda+\epsilon})$ to be positive is that $P_{e}^{(n)}(Q_{\lambda+\epsilon})$ vanishes as $n\to\infty$. This is true if

[TABLE]

If, on the other hand $R\geq I(P,Q_{\lambda+\epsilon})$, we still can lower bound (32) by

[TABLE]

-D Proof of Lemma 1

We provide the proof for a binary alphabet $\mathcal{X}=\{a,b\}$ in a proof by contradiction. The proof for the general $|\mathcal{X}|>2$ is a straightforward generalization. For $x=a,b$ define

[TABLE]

Assume that the claim of the Lemma 1 is not valid and hence there exists $(\lambda_{a},\lambda_{b},{\widetilde{\lambda}})\in[0,1]^{3}$ such that

[TABLE]

or equivalently

[TABLE]

By [13, Theorem 2] we can exclude the cases where $\lambda_{a},\lambda_{b}<{\widetilde{\lambda}}$ and $\lambda_{a},\lambda_{b}>{\widetilde{\lambda}}$ and assume $\lambda_{a}<{\widetilde{\lambda}}<\lambda_{b}$, which implies

[TABLE]

for $x\in\{a,b\}$. Hence, by rearranging (39) and by dividing the two equations, we get

[TABLE]

Note since the $\left(E_{0}^{(x)}(\lambda),E_{1}^{(x)}(\lambda)\right)$ curve is convex and strictly decreasing, we have

[TABLE]

where $\frac{\partial E_{1}^{(x)}\left(E_{0}^{(x)}(\lambda)\right)}{\partial\lambda}$ is the slope of the $\left(E_{0}^{(x)}(\lambda),E_{1}^{(x)}(\lambda)\right)$, which can be visually seen in Fig. 7. However, according to [13, Theorem 6], the slope of the $\left(E_{0}^{(x)}(\lambda),E_{1}^{(x)}(\lambda)\right)$ curve at $\lambda={\widetilde{\lambda}}$ is equal to $\frac{{\widetilde{\lambda}}-1}{{\widetilde{\lambda}}}$ and is independent of $x$.

Putting (40), (41) and (42) together, we reach a contradiction and the proof is complete.

-E Proof of (23)

Note that

[TABLE]

and similarly

[TABLE]