On Error Rate Analysis for URLLC over Multiple Fading Channels

TL;DR

This paper derives an upper-bound on packet error rate for URLLC over multiple fading channels, providing a practical tool to ensure ultra-reliable low-latency communication performance.

Contribution

It introduces a tight upper-bound on error rate for finite-length codes in multi-fading channels, aiding in URLLC parameter design.

Findings

The bound is tight across various SNRs and bin counts.

The bound effectively guides URLLC reliability guarantees.

Simulation confirms the bound's accuracy.

Abstract

In this paper, we study ultra-reliable and low-latency communication (URLLC) under fading using multiple frequency or time bins. We investigate an approach to find an upper-bound on the packet error rate when a finite-length code is used. From simulation results, we find that the bound is reasonably tight for wide ranges of signal-to-noise ratio (SNR) and the number of multiple bins. Thus, the derived bound can be used to determine key parameters to guarantee the performance of URLLC in terms of the packet error rate.

Equations

$${\mathbf{r}}_l=h_l\mathbf{s}+{\mathbf{n}}_l,l\in \{1,\dots ,L\},$$

$$\mathbf{y}$$

$$=\sum _l|h_l{|}^2\mathbf{s}+h_l^{\ast }{\mathbf{n}}_l.$$

$$\rho =\frac{||\mathbf{h}|{|}^{2}P}{N_0},$$

$$R^{\ast }(n,ϵ)\approx {\log }_2(1+\rho )-\sqrt{\frac{V(\rho )}{n}}{\mathcal{Q}}^{-1}(ϵ)+\frac{{\log }_{2}n}{2n},$$

$$V(\rho )=\frac{\rho (2+\rho )}{(1+\rho {)}^2}({\log }_{2}e{)}^2,$$

$$\underline{R}(\rho ,n,ϵ)={\log }_2(1+\rho )-\sqrt{\frac{\bar{V}}{n}}{\mathcal{Q}}^{-1}(ϵ),$$

$${\mathbb{P}}_{\mathrm{err}}\le ϵ+(1-ϵ)\mathrm{Pr}(\rho <\tau (ϵ)),$$

$$\tau (ϵ)=2^{R+\sqrt{\frac{\bar{V}}{n}}{\mathcal{Q}}^{-1}(ϵ)}-1.$$

$${\mathbb{P}}_{\mathrm{err}}\le \mathrm{Pr}(\mathrm{err}|\rho \ge \tau )\mathrm{Pr}(\rho \ge \tau )+\mathrm{Pr}(\rho <\tau ),$$

$${\mathbb{P}}_{\mathrm{err}}\le ϵ\mathrm{Pr}(\rho \ge \tau )+\mathrm{Pr}(\rho <\tau ),$$

$$\frac{{\mathbb{P}}_{\mathrm{err}}\le {\bar{\mathbb{P}}}_{\mathrm{err}}\text{buildrel}△}{={\min }_{0\le ϵ\le 1}ϵ+(1-ϵ)\mathrm{Pr}(\rho <\tau (ϵ)).}$$

$${\mathbb{P}}_{\mathrm{err}}=\mathrm{Pr}({\log }_2(1+\rho )<R)=\mathrm{Pr}\left(\rho <\tau \right),$$

$$\mathbb{E}[h_l{h}_{l^{\prime }}^{\ast }]={\sigma }_h^2{\delta }_{l,l^{\prime }}.$$

$$|h_l{|}^2\sim \mathrm{Exp}({\sigma }_h^2)=\frac{1}{{\sigma }_h^2}\exp \left(-\frac{|h_l{|}^2}{{\sigma }_h^2}\right).$$

$$\frac{||\mathbf{h}|{|}^2}{{\sigma }_h^2}=\frac{{\chi }_{2L}^2}{2},$$

$$Z_L=\frac{{\chi }_{2L}^2}{2L}.$$

$$\mathrm{Pr}(\rho <\tau )=\mathrm{Pr}\left(Z_L<\frac{\tau }{\beta }\right)=\frac{\gamma \left(L,\frac{\tau L}{\beta }\right)}{(L-1)!},$$

$$\beta =\frac{LP{\sigma }_h^2}{N_0}.$$

$$\mathrm{Pr}(Z_L<z)$$

$$=e^{2Ltz}{\left(\frac{1}{1+2t}\right)}^L$$

$$={\left(\frac{e^{2tz}}{1+2t}\right)}^L,t\ge 0.$$

$$\frac{\mathrm{Pr}(Z_L<z)\le U_L(z)\text{buildrel}△}{=(z{e}^{1-z}{)}^L,z\in [0,1),}$$

$$B_L(z)=(c_{L}z{e}^{1-z{c}_L}{)}^L.$$

$$\underset{z\to 0}{\lim }\frac{B_L(z)}{\mathrm{Pr}(Z_L\le z)}=1$$

$$\gamma (s,x)=x^s\sum _{k=0}^{\infty }\frac{(-x{)}^k}{k!(s+k)},$$

$$\mathrm{Pr}(Z_L\le z)=\frac{1}{L!}((Lz{)}^L+O(z^{L+1})).$$

$$B_L(z)=(c_{L}ze{)}^L+O(z^{L+1}).$$

$$c_L=L{e}^{-1}{\left(L!\right)}^{-\frac{1}{L}}.$$

$$c_L<e^{-\frac{1}{L}}\le 1.$$

$$B_L(z)\le U_L(z),z\in [0,1).$$

$${\bar{\mathbb{P}}}_{\mathrm{err},B}=\underset{0\le ϵ\le 1}{\min }ϵ+(1-ϵ)B_L\left(\frac{\tau (ϵ)}{\beta }\right).$$

Full text

I Introduction

Ultra-reliable and low-latency communication (URLLC) has been considered for a number of real-time applications such as factory automation, autonomous driving, and remote surgery. In 5th generation (5G) cellular systems, URLLC is to be supported [1] [2]. For URLLC, in [3], resource allocation and hybrid automatic repeat request (HARQ) schemes are investigated. In [4], the notion of effective bandwidth [5] is studied in order to guarantee a certain latency with quality of service (QoS) exponent. URLLC in machine-type communication (MTC) [6] is studied with random access in [7].

In order to perform reliable transmissions, channel coding can be employed. Since the length of packets can be short it is necessary to consider finite-length codes and understand their impact on the performance [8]. In [9], URLLC is considered with finite-length codes. In [10], various existing channel codes are studied and compared with the theoretical performance obtained in [8].

As in [11] [12], multi-connectivity can be used to provide a diversity gain, which improves the reliability in transmissions over fading. In this paper, we also consider multichannel transmissions to exploit a high diversity gain so that one transmission (through multiple channels) is sufficient as its packet error rate can be low without re-transmission in URLLC. In order to guarantee a sufficient low packet error rate, it is necessary to decide key parameters in advance, which requires a good prediction of the packet error rate. To this end, in this paper, we focus on the derivation of an upper-bound on the packet error probability when finite-length codes are used in URLLC.

Notation

Matrices and vectors are denoted by upper- and lower-case boldface letters, respectively. The superscript ${\rm T}$ denotes the transpose. The 2-norm is denoted by $||\cdot||$. For a matrix ${\bf X}$, $[{\bf X}]_{m,n}$ represents the $(m,n)$th element. ${\mathbb{E}}[\cdot]$ and ${\rm Var}(\cdot)$ denote the statistical expectation and variance, respectively. ${\cal C}{\cal N}({\bf a},{\bf R})$ represents the distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean vector ${\bf a}$ and covariance matrix ${\bf R}$.

II System Model

Suppose that we have a set of $L$ frequency bins or blocks for multi-connectivity in URLLC [11]. A transmitter is to transmit the same coded packet through $L$ blocks. Then, the received signal at a receiver through the $l$th block is given by

[TABLE]

where $h_{l}$ represents the channel coefficient from the transmitter to the receiver through the $l$th block, ${\bf s}$ is a coded packet, and ${\bf n}_{l}\sim{\cal C}{\cal N}({\bf 0},N_{0}{\bf I})$ is the background noise vector.

To decode the signal, we can consider the maximal ratio combining (MRC) [13] [14] as follows:

[TABLE]

Then, the instantaneous signal-to-noise ratio (SNR) after MRC, which is referred to as MRC-SNR for convenience, is given by

[TABLE]

where ${\bf h}=[h_{1}\ \ldots\ h_{L}]^{\rm T}$ and ${\bf E}[{\bf s}{\bf s}^{\rm H}]=P{\bf I}$. Here, $P$ represents the signal transmit power and $\frac{P}{N_{0}}$ is referred to as the SNR. For a reliable communication, a high diversity gain with a sufficient SNR is required. In addition, if the packet error rate is sufficiently low with a large $L$, no re-transmission might be required, which can result in low-latency communication. Thus, it is expected to predict the packet error rate in terms of $L$ and SNR in URLLC.

Note that the outage probability of MRC is well-known [15]. However, when finite-length codes are used [8], the packet error rate cannot be directly expressed by the outage probability.

III Error Probability Analysis

In URLLC, it is necessary to decide key parameters (e.g., the signal transmit power, $P$, and the number of blocks, $L$) to provide a certain guaranteed performance. For example, we can consider the packet error rate. In this section, we find a closed-form expression for the packet error rate that helps decide the values of key parameters when finite-length codes are used.

III-A Error Probability of Finite Length Codes

For a given $\rho$, according to [8] [16], the achievable rate (for complex Gaussian channel [9]) is given by

[TABLE]

where $V(\rho)$ is the channel dispersion that is given by

[TABLE]

$n$ is the length of codeword when a codeword is transmitted within a block, and $\epsilon$ is the nominal111This becomes the error probability when $\rho$ is fixed. However, if $\rho$ is a random variable (due to fading), we also need to take into account the outage probability to find the error probability. From this reason, it is referred to as the nominal error probability. error probability. It can be shown that $\bar{V}>V(\rho)$, where $\bar{V}=\frac{1}{(\ln 2)^{2}}\approx 2.0814$ [4]. Thus, ignoring the term of $O\left(\frac{\log_{2}n}{n}\right)$, a lower-bound on the achievable rate can be obtained as follows:

[TABLE]

which might be tight for a sufficiently high MRC-SNR, $\rho$, because $V(\rho)\to\bar{V}$ as $\rho\to\infty$. The lower-bound in (7) allows tractable analysis, since the terms of $\epsilon$ and $\rho$ are decoupled.

Lemma 1

With a code rate $R$, the probability of unsuccessful decoding is upper-bounded as

[TABLE]

where

[TABLE]

Proof:

For convenience, let $\tau=\tau(\epsilon)$. From (7), the probability that the achievable rate is lower than $R$ is upper-bounded by $\Pr(\rho<\tau)$. To find an upper-bound on the probability of unsuccessful decoding we can assume that the decoding fails if $\rho<\tau$ with probability 1. Then, we have

[TABLE]

where $\Pr({\rm err}\,|\,\rho\geq\tau)$ is the conditional probability of decoding error for given $\rho\geq\tau$. Thus, $\Pr({\rm err}\,|\,\rho\geq\tau)$ is upper-bounded by $\epsilon$, and we have

[TABLE]

which becomes (8). ∎

By taking $\epsilon$ as a parameter to minimize the upper-bound in (8), we can have the following tight upper-bound:

[TABLE]

In (11), $\Pr(\rho<\tau(\epsilon))$ is referred to as the outage probability.

Note that if $n\to\infty$ and a capacity achieving code is used, the packet error rate can be simply expressed by the outage probability, i.e.,

[TABLE]

where $\tau=2^{R}-1$.

III-B Outage Probability

In this subsection, we consider a closed-form expression for the outage probability.

We assume independent Rayleigh fading channels and

[TABLE]

Thus, $|h_{l}|^{2}$ has the following exponential distribution:

[TABLE]

From (13), we can show that

[TABLE]

where $\chi_{n}^{2}$ represents a chi-squared random variable with $n$ degrees of freedom. For convenience, let

[TABLE]

The cdf of $\rho$ is given by

[TABLE]

where $\gamma(s,x)=\int_{0}^{x}t^{s-1}e^{-t}dt$ is the lower incomplete gamma function and

[TABLE]

In order to have a tractable expression for (15), an upper-bound on the tail probability of $Z_{d}$ can be considered. Using the Chernoff bound [17], it can be shown that

[TABLE]

Letting $z=\frac{1}{1+2t}$, we have

[TABLE]

which is reasonably tight. Note that $ze^{1-z}$ increases in $z\in[0,1)$. Thus, the upper-bound in (20) increases with $z$.

However, if a low outage probability is considered with $z\to 0$ for reliable communication in URLLC, the upper-bound may not be satisfactory. For a tight bound when $z\to 0$, a term can be introduced. In particular, we replace $z$ with $c_{L}z$ in (20), where $c_{L}$ is the correction term, and let

[TABLE]

The correction term $c_{L}$ can be decided to satisfy

[TABLE]

so that $B_{L}(z)$ can approach the outage probability, $\Pr(Z_{L}\leq z)$, when $z\to 0$. Since [18]

[TABLE]

from (15), we have

[TABLE]

In addition,

[TABLE]

Thus, to satisfy (22), from (24) and (25) we have

[TABLE]

Since $n!>e\left(\frac{n}{e}\right)^{n}$, it can be shown that

[TABLE]

From this and the fact that $U_{L}(z)$ increases with $z$, it follows that

[TABLE]

Consequently, we can see that $B_{L}(z)$ can be tighter than the upper-bound in (20) (i.e., $U_{L}(z)$) and $B_{L}(z)$ approaches $\Pr(Z_{L}<z)$ as $z\to 0$. However, we are unable to prove that $B_{L}(z)$ is an upper-bound on $\Pr(Z_{L}<z)$, although it seems that $B_{L}(z)$ is an upper-bound222In the rest of the paper, we conjecture that it is an upper-bound. based on numerical results as shown in Fig. 1.

With $B_{L}(z)$, (11) is modified as follows:

[TABLE]

Since $B_{L}(z)$ is a closed-form expression, a tight upper-bound can be found using a one-dimensional numerical search algorithm.

IV Simulation Results

In this section, we present simulation results under independent Rayleigh fading channels with $\sigma_{h}^{2}=1$. The bound in (28) is used to predict the performance in terms of the packet error rate. In URLLC, the packet error rate is to be $10^{-3}-10^{-5}$ [19] [2].

Fig. 2 shows the packet error rate as a function of rate, $R$, when $n=2^{12}$, $\frac{P}{N_{0}}=3$ dB, and $L=4$. We can see that the bound becomes tighter as $R$ increases.

In Fig. 3 we show the packet error rate as a function of the number of blocks, $L$, when $n=2^{12}$, $R=1/2$, and $\frac{LP}{N_{0}}=3$ dB for all $L\in\{1,10\}$. Although the total signal power remained unchanged when $L$ increases, we can see that the packet error rate decreases with $L$ thanks to the diversity gain.

The impact of the SNR on the packet error rate is shown in Fig. 4 when $n=2^{12}$, $R=1/2$, and $L=4$. As expected, the packet error rate decreases with the transmit power. We can also see that the upper-bound is reasonably tight.

Fig. 5 shows the packet error rate as a function of the codeword length, $n$, when $R=1/2$, $\frac{P}{N_{0}}=3$ dB, and $L=4$. It is shown that the bound becomes tight when the codeword length is sufficiently long. If $n$ is small ($\leq 1024$), the bound is not tight.

V Concluding Remarks

In this paper, we considered multichannel transmissions for URLLC and derived an upper-bound on the packet error rate when finite-length codes are used. In particular, to take into account fading, an upper-bound on the packet error rate was considered with the outage probability. To find a tight bound, an optimization problem was formulated and a correction term was introduced for the Chernoff bound on the outage probability. From simulation results, we found that the derived upper-bound is reasonably tight and can be used to determine key parameters in order to guarantee a sufficiently low packet error rate in URLLC.