Joint Coding of eMBB and URLLC in Vehicle-to-Everything (V2X) Communications
Homa Nikbakht^1, Eric Ruzomberka^1, Michèle Wigger^2, Shlomo Shamai (Shitz)^3, and H. Vincent Poor^1
^1Princeton University,   ^2LTCI,   Tlcom Paris, IP Paris,  ^3Technion,

{homa, er6214<EMAIL_ADDRESS><EMAIL_ADDRESS>
A point-to-point communication is considered where a roadside unite (RSU) wishes to simultaneously send messages of enhanced mobile broadband (eMBB) and ultra-reliable low-latency communication (URLLC) services to a vehicle. The eMBB message arrives at the beginning of a block and its transmission lasts over the entire block. During each eMBB transmission block, random arrivals of URLLC messages are assumed. To improve the reliability of the URLLC transmissions, the RSU reinforces their transmissions by mitigating the interference of eMBB transmission by means of dirty paper coding (DPC).   In the proposed coding scheme, the eMBB messages are decoded based on two approaches: treating interference as noise, and successive interference cancellation. Rigorous bounds are derived for the error probabilities of eMBB and URLLC transmissions achieved by our scheme. Numerical results illustrate that they are lower than bounds for standard time-sharing.

§ INTRODUCTION

Enhanced mobile broadband (eMBB) and ultra-reliable low-latency communication (URLLC) services enabled by  5G new radio (NR) are considered as key enablers of the vehicle-to-everything (V2X) technology [1, 2, 3, 4, 5, 6]. Particularly,  eMBB services aim to provide high data rate for content delivery and therefore improve the quality of experience (QoE) of in-vehicle entertainment applications. URLLC services, however, are key to guarantee the delivery of critical road safety information and thus enable fully autonomous driving of connected vehicles [7, 8].

Coexistence of eMBB and URLLC services in V2X communications  has been studied in the literature [9, 10, 11]. In [9], a novel URLLC and eMBB coexistence mechanism for the cellular V2X framework is proposed where at the begining of the transmission interval  eMBB users are associated with a V2X base station, whereas, URLLC users are allowed to puncture the eMBB transmissions upon arrival. The work in [10] formulates an optimization problem for joint scheduling of punctured eMBB and URLLC traffic to  maximize the  aggregate utility of the eMBB users subject to latency constraints for the URLLC users. Related to this work is [11], where   resources are allocated jointly between eMBB and URLLC messages  for a one-way highway vehicular network in which a vehicle receives an eMBB message from the nearest roadside unit (RSU) and URLLC messages from the nearest vehicle. During each eMBB transmission interval, random arrivals of URLLC messages are assumed. The eMBB time slot is thus divided into mini-slots and the newly arrived URLLC messages are immediately scheduled in the next mini-slot by puncturing the on-going eMBB transmissions. To guarantee the reliability of the URLLC transmission, guard zones are deployed around the vehicle and the eMBB transmissions are not allowed inside such zones.

In this work, the RSU wishes to transmit both eMBB and URLLC messages to a vehicle.

The eMBB message arrives at the beginning of a block and  its transmission lasts over the entire block. The eMBB blocklength is again divided into mini-slots and URLLC messages  arrive randomly at the beginning of these mini-slots. Specifically, at the beginning of each of these mini-slots a URLLC message arrives with probability $\rho \in [0,1]$ and the
RSU simultaneously sends the eMBB message as well as the newly arrived URLLC message over this mini-slot. With probability $1-\rho$  no URLLC message arrives at the beginning of the mini-slot and the RSU only sends the eMBB message. In our work, we do not use guard zones, but instead the RSU reinforces transmission of URLLC messages by mitigating the interference of eMBB transmission by means of  dirty paper coding [12, 13, 14]. After each mini-slot, the receiving vehicle attempts to decode a URLLC message, and after the  entire transmission interval  it decodes the eMBB message. Given that the URLLC transmissions interfere with the transmission of eMBB, we employ two different eMBB decoding approaches.  The first approach, known as treating interference as noise (TIN), is to treat the URLLC interference as noise.  The second approach, known as successive interference cancellation (SIC),  is to first subtract the decoded   URLLC message  and then decode the eMBB message based on the received signal.  Rigorous bounds are derived for achievable error probabilities of eMBB (in both approaches) and URLLC transmissions.
Numerical results illustrate that our proposed scheme significantly outperforms the standard time-sharing scheme.

§ PROBLEM SETUP

Consider a point-to-point setup with one RSU (transmitter) and one vehicle (receiver) communicating over a $\Ne$ uses of an AWGN channel. The  transmitter (Tx)  sends a single, so called eMBB-type message $\MkS$, over the entire blocklength $\Ne$, where $\MkS$  is uniformly distributed over a given set $\mathcal{M}^{(\e)} := \{1, \ldots, L_{\e}\}$. Message $\MkS$ is thus available at the Tx at time $t=1$ (and remains until time $\Ne$). Additionally, prior to each channel use in

: =    {1, 1+ Ν,  1+2Ν,…, 1 + (η-1 )Ν},

\begin{equation}
\eta := \left \lfloor \frac{\Ne}{\Nu}\right \rfloor,
\end{equation}

the Tx generates with probability $\rho$ an additional, so called, URLLC-type message that it wishes to convey to the Rx. With probability $1-\rho$ no URLLC-type message is generated.
For each $b\in [\eta]$, if a URLLC message is generated at time $t=(b-1)\Nu+1$, then we set $A_b=1$, and otherwise we set $A_b=0$. Denote  the time-instances from $(b-1)\cdot \Nu +1$ to $b\cdot \Nu$ by block $b$. If in block $b$ a message is generated we denote it by $M_{b}^{(\U)}$ and  assume that it is  uniformly distributed over the set $\mathcal{M}^{(\U)}:= \{1, \ldots, L_{\U}\}$.

During block $b$, the  Tx computes its  inputs  as:
X_t    =    f^()_t ( M_b^(), ),     if  A_b = 1,

f^()_t ( ),      if  A_b = 0,

for  $t=(b-1)\cdot \Nu+1,\ldots, b\cdot \Nu$ and some encoding functions $f^{(\U)}_t$ and  $f_t^{(\e)}$ on appropriate domains. After the last URLLC block, i.e. at times $t=\eta \Nu +1, \ldots, \Ne$, the Tx produces the inputs
X_t    =
f^()_t ( ),
t= ηΝ+1, …, .

The sequence of channel inputs $X_1,\ldots, X_{\Ne}$ has to satisfy
the average block-power constraint

\begin{equation}\label{eq:power}
\frac{1}{\Ne} \sum_{t=1}^{\Ne} X_{t}^2
\leq \P,
\qquad \textnormal{almost surely.}
\end{equation}

The input-output relation of the network is  described as

\begin{equation}\label{Eqn:Channel}
{Y}_{t} = h {X}_{t}+ {Z}_{t},
\end{equation}

where $\{Z_{t}\}$ are independent and identically distributed (i.i.d.) standard Gaussian for all  $t$ and independent of all messages; $h> 0$ is the fixed channel coefficient between the Tx and Rx.

After each URLLC block $b$ the receiver (Rx) decodes the transmitted URLLC message $M_b^{(\U)}$ if $A_b=1$. Moreover, at the end of the entire $\Ne$ channel uses it decodes the eMBB message $M^{(\e)}$.
Thus, if $A_b=1$ it produces

\begin{equation}
\hat M_b^{(\U)}={g^{(\Nu)}}\big( Y_{(b-1)\Nu +1}, \ldots, Y_{b\Nu} \big),
\end{equation}

for some decoding function $g^{(\Nu)}$ on appropriate domains. Otherwise, it sets $  \hat M_b^{(\U)} = 0$.
We define the average error probability for each message $M_b^{(\U)}$ as:
ϵ^()_b    :=   ρ[  M̂_b^() ≠M_b^() | A_b=1 ]

+(1-ρ) [  M̂_b^() ≠0 | A_b=0 ] .
At the end of the  $\Ne$ channel uses,
the Rx decodes its desired eMBB message as:

\begin{equation}\label{mhats}
\hat{{M}}^{(\e)}={\psi^{(\Ne)}}\left ( \vect Y^{\Ne} \right ),
\end{equation}

where $\vect Y^{\Ne}: = (Y_1, \ldots, Y_{\Ne})$ and  $\psi^{(\Ne)}$ is a decoding function on appropriate domains. We define the average error probability for  message $M^{(\e)}$ as

\begin{equation}
\epsilon^{(\e)} := \Pr\left [ \hat M^{(\e)} \neq M^{(\e)}\right].
\end{equation}

The goal is to propose a coding scheme that simultaneously has small error probabilities $\epsilon^{(\U)}_b$ and $\epsilon^{(\e)}$.

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[<->] (0,0.6)–(4.75,0.6);

[draw = none] at (2.5,0.7) $\Ne$;
[draw = none] at (0.5,0.4) $\Nu$;
[draw = none] at (0.5+1,0.4) $\Nu$;
[draw = none] at (0.5+2,0.4) $\Nu$;
[draw = none] at (0.5+3,0.4) $\Nu$;
[draw = none] at (0.5+3.88,0.4) $\Ne - \eta\Nu$;
[draw = none] at (0.5,-0.3) $ \vect X_{1}^{(\U)}\hspace{-0.15cm} + \vect X_{1}^{(\e,2)}$;
[draw = none] at (0.5+1,-0.3) $  \vect X_{2}^{(\e,1)}$;
[draw = none] at (0.5+2,-0.3) $ \vect X_{3}^{(\U)} \hspace{-0.15cm}+\hspace{-0.05cm} \vect X_{3}^{(\e,2)}$;
[draw = none] at (0.5+3,-0.3) $  \vect X_{4}^{(\e,1)}$;
[draw = none] at (0.5+3.9,-0.3) $  \vect X_{5}^{(\e,1)}$;

Example of the coding scheme with $\eta = 4$ and $\mbt = \{1,3\}$.

§ JOINT TRANSMISSION OF URLLC AND EMBB MESSAGES

§.§ Construction of Codebooks

Define rCl
: ={b ∈[η]: A_b = 1}.

Choose $\bu$ and $\bef \in [0,1]$ such that:

\begin{equation} \label{eq:10}
\bu + \bef = 1.
\end{equation}

Fix a value of $\alpha  \in [0,1]$. For each block $b\in [\eta]$, for each $j \in [ L_v]$ and each  realization  $m \in [ L_{\U}]$, generate  codewords $\vect V_b(m,j)$    by picking them uniformly over a centered $\Nu$-dimensional  sphere of radius $\sqrt{\Nu\vnorm  \P}$ independently of each other and  of all other codewords, for
: = + α^2 .

For each $\ell \in [L_{\e}]$ randomly draw a  codeword $\xkef(\ell)$ uniformly distributed  on the centered $\Nu$-dimensional sphere of radius $\sqrt{\Nu \bef \P}$ and a codeword $\xkes(\ell)$ uniformly distributed  on the centered $\Nu$-dimensional sphere of radius  $\sqrt{\Nu \Pb}$. All codewords are chosen independently of each other.

§.§ Encoding

§.§.§ Encoding at Blocks $b \in \mb$

In each block $b \in \mb$, the Tx has both an eMBB and an URLLC message to send.  It first picks the codeword $\xkef(\MkS)$ and then
employs DPC to encode $M^{(\U)}_b$  while precanceling the interference of its own eMBB codeword  $\xkef(\MkS)$.
Specifically, it chooses an index $j$ such that the

\begin{equation} \label{eq:x21}
\xku :  = \vect V_b (M^{(\U)}_b,j )- \alpha  \xkef
\end{equation}

lies in the set
𝒟_b := { x_b^(): Ν- δ_b ≤x_b^()^2 ≤Ν}
for a given $\delta_b> 0$.
If multiple  such codewords  exist, the index $j^\star$ is chosen at random from this set, and the Tx sends:

\begin{equation}
\vect{X}_b= \xku + \xkef.
\end{equation}
We also set $\abst=1$.

no appropriate codeword exists, the Tx  discards the arrived URLLC message by setting $\abst=0$ and sends only the eMBB message

\begin{equation}
\vect{X}_b=\xkes(\MkS)
\end{equation}

over this block.

:= {b ∈:  = 1},

where $\mbt \subseteq \mb$ and represents the set of blocks in which an URLLC message is sent. See Figure <ref>.

§.§.§ Encoding at Blocks $b \in [\eta] \backslash \mb$ and in Block $\eta+1$ when $\Ne > \eta \Nu$

In each Block $b \in [\eta] \backslash \mb$, the Tx sends only eMBB message $M^{(\e)}$:

\begin{equation}
\vect{X}_b=\vect X_{b,1}^{(\e)}(M^{(\e)}).
\end{equation}

Over Block $b$, the Tx  thus transmits
X_b = +     if   b ∈,

§.§ Decoding

After each block $b \in [\eta]$, the Rx
attempts to decode a URLLC message, and after the  entire block of $\Ne$ channel uses it decodes the transmitted  eMBB message. Given that the URLLC transmissions interfere with the transmission of eMBB, the Rx envisions  two different approaches to decode the eMBB message. The first approach, termed  TIN approach,  is to treat the URLLC interference as noise.  The second approach, termed SIC approach, is to first subtract the decoded   URLLC message  and then decode the eMBB message based on the received signal.

§.§.§ Decoding of URLLC Messages
At the end of each block $b \in [\eta]$, the Rx observes the following channel outputs $\vect Y_b: = \{Y_{(b-1)\Nu + 1}, \ldots, Y_{b \Nu} \}$:

Y_b =  h+ h+ Z_b       if  b ∈

h +  Z_b       o.w.

with $\vect Z_b \sim \mathcal N(0, I_{\Nu})$.

Define the information density metric between $\vect y_b$ and $\vect v_b$ by:

\begin{equation} \label{eq:ibU}
i^{(\U)}_b  (\vect v_b; \vect y_b ) := \ln \frac{f_{\vect Y_b| \vect V_b} (\vect y_b| \vect v_b)}{f_{\vect Y_b}(\vect y_b)}.
\end{equation}

After observing $\vect Y_b$,  the Rx  chooses  the pair

\begin{equation}
(m',j') =\text{arg} \max_{ m, j}  i^{(\U)}_b  (\vect v_b(m,j); \vect Y_b ) .
\end{equation}

If for this pair

\begin{equation}
i^{(\U)}_b  (\vect v_b(m',j'); \vect Y_b ) > \gamma^{(\U)}
\end{equation}

where $\gamma^{(\U)}$ is a threshold over which we optimize, the Rx chooses $(\hat M_b^{(\U)},\hat j)= (m',j')$ and sets $\abd = 1$. Otherwise the receiver declares that no URLLC message has been sent and indicates it by setting $\hat M_b^{(\U)}=0$ and $\abd = 0$.

:= {b ∈[η]: = 1}

that is the set of blocks in which an URLLC message is detected. A detection error happens if
$\mbd \neq \mbt$.

In each block $b \in \mbd$, set $\abc = 1$ if $(\hat M_b^{(\U)}, \hat j) = (M_b^{(\U)} , j)$, otherwise set $\abc = 0$. Define
: = {b ∈: = 1}
that is the set of blocks in which an URLLC message is decoded correctly.

§.§.§ Decoding the eMBB Message under the TIN approach

To decode its desired eMBB message under this approach, the Rx treats URLLC transmissions as noise. Therefore,
the decoding of the eMBB message depends on the detection of URLLC messages sent over the $\eta$ blocks.

Let $\bku$ be the realization of the set $\mbd$ defined in (<ref>).
Given $\bku$, the Rx decodes its desired eMBB message based on the outputs of the entire $\Ne$ channel uses by  looking for an index $m$ such that its corresponding codewords $\left \{ \{\vect x_{b}^{(\e,1)}(m)\}_{b \notin  \bku },  \{\vect x_{b}^{(\e,2)}(m)\}_{b \in \bku } \right \}$ maximize
i^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  y^| = )

: = ln∏_b∉ f_Y_b|  (y_b| x_b,1^())/f_Y_b(y_b) +  ln∏_b∈  f_Y_b|  (y_b| x_b,2^())/f_Y_b(y_b)
among all codewords
$ \{ \{\vect x_{b}^{(\e,1)}(m')\}_{b \notin  \bku },  \{\vect x_{b}^{(\e,2)}(m')\}_{b \in \bku } \}$.

§.§.§ Decoding the eMBB Message under the SIC approach

Under this approach, before decoding the desired eMBB message, the Rx mitigates the interference of the correctly decoded URLLC messages from its observed output signal. Therefore, the decoding of the eMBB message depends not only on the  detection of the sent URLLC messages but also on the  decoding of such messages.

For each Block $b \in \mbd$, we define $\abc = 1$ if $(\hat M_b^{(\U)}, \hat j) = (M_b^{(\U)} , j)$, otherwise set $\abc = 0$. Define the set of blocks in which an URLLC message is decoded correctly:
: = {b ∈: = 1}.

Let  $\bku$ be a realization of the set $\mbd$ and $\bkut$ be a realization of the set $\mbc$.
After observing the channel outputs of the entire $\Ne$ channel uses, the Rx decodes its desired eMBB message  by  looking for an index $m$ such that its corresponding codewords $\left \{ \{\vect x_{b}^{(\e,1)}(m)\}_{b \notin \bku},  \{\vect x_{b}^{(\e,2)}(m)\}_{b \in \bku } \right \}$ maximize
i^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^| , , {V_b}_b ∈)

: = ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/f_Y_b(y_b) +  ln∏_b∈\  f_Y_b|  (y_b| x_b^(,2))/f_Y_b(y_b)

+  ln∏_b∈  f_Y_b| , V_b (y_b| x_b^(,2), v_b)/f_Y_b| V_b(y_b| v_b)

among all codewords $\{ \{\vect x_{b}^{(\e,1)}(m')\}_{b \notin \bku  },  \{\vect x_{b}^{(\e,2)}(m')\}_{b \in \bku } \}$.

J_     :=   ( π2^Ν+1/2e^-h^2 Ν/2 √()/9h^2 (1- α)^Ν-1 (+ (1- α)^2 ) )^k  ·( √(8 (1 + 2 h^2))/27√(π) (1+ h^2 ) )^η-k

J̃_     :=    ( π2^Ν+1/2e^-h^2 Ν/2 √()/9h^2 (1- α)^Ν-1 (+ (1- α)^2 ) )^ k-k̃ ·( √(8 (1 + 2 h^2))/27√(π) (1+ h^2 ) )^η-k  ·( √(8 (1 + 2 h^2(1-α)^2 ))/27√(π) (1+ h^2(1-α)^2) )^k̃

ζ   : =    1/√(π)Γ(Ν/2)/Γ(Ν-1/2) (κ_Ν-3/2 (α√(/) + δ_b/(2 αΝ√())  )  -  κ_Ν-3/2 (α√(/)) )

μ_    :=     2/h^2(-) (Ν/2ln/ - γ^() + lnJ_ )  + /-  ( Ν(√() - √())^2 - δ_b) -  Ν(1-α)^2/-

μ̃_     :=     2/h^2(-) (Ν/2ln/ - γ^() + lnJ̃_ )  + /-  ( Ν(√() +√())^2 )  -  Ν(1-α)^2/-

μ   :=     /2 ln- kΝ/2 ln- η-k/2 Ν+   k/2Ν- k/2(√() + (1- α)√() )^2Ν-γ^() + lnJ_

μ̃   : =     /2 ln+ Ν(k - k̃/2(   / - (√() + (1- α)√() )^2 /   -ln/ ) + k̃/2 ln/  - η-k/2  - k̃(1- α)^2 /2 )  + lne^-γ̃^() J̃_

T    :=      (- k Ν)(-1)/2μ +  (η+1 - k)√(Ν)/μ √(2) Γ(Ν+1/2)/Γ(Ν/2) + kτ/μ√(2) Γ(Ν+1/2)/Γ(Ν/2) +  k Ν(-)/2μ + (L_e -1) e^-γ^()

ν    :=    k̃/μ̃ ( √(2) Γ(Ν+1/2)/Γ(Ν/2)  (τ-(1-α)√(Ν)/) + Ν( -/2 - -1/2) )  + (L_e -1) ( μ/μ̃ e^-γ̃^()+ e^-γ̃^() )

§ MAIN RESULTS

Define $\sy := h^2  \Pb + 1$, $\syx := h^ 2\vnorm \P+ 1$, $\syv := h^ 2(1 - \alpha)^2 \bef \Pb+ 1$ and

λ(x)     :=    x/2 + u^2/4 - u/2 √(x+ u^2/4 ),

λ̃(x)    :=    x/2 + u^2/4 + u/2 √(x + u^2/4 ),

u     :=       2√(Ν) ( (√() + √()) + √()(1- α))/h (- ),

τ   : =    √(Ν) (√() (+ ) + (1- α) √())/,

and for all integer values $n=1,2,\ldots$:
κ_n(x)    :=    x(1-x^2)^n/2n+1 + 2n/2n+1 κ_n-1(x)

where $\kappa_0(x) := x$.
By employing the scheme proposed in Section <ref>, we have the following theorem on the upper bounds on the URLLC and eMBB error probabilities $\epsilon^{(\U)}_b$, $\epsilon^{(\e)}_{\text{TIN}}$, and $\epsilon^{(\e)}_{\text{SIC}}$.

For fixed $\bef$ , $\bu \in [0,1] $ and message set sizes $L_{\U}$ and $L_{\e}$, the average error probabilities $\epsilon^{(\U)}_b$, $\epsilon^{(\e)}_{\text{TIN}}$, and $\epsilon^{(\e)}_{\text{SIC}}$ are bounded by
ϵ^()_b     ≤    ρ(( 1-  ζ)^L_v  + q + 1- q_2 ) + (1- ρ)q_1

ϵ^()_TIN     ≤    ∑_k = 0^ηηk q_3^k (1-q_2)^η-k (1- Δ+ T )

ϵ^()_SIC     ≤   ∑_k= 0^ηηk q_4^k (1-q_2)^η-k

·(1- Δ+ ∑_k̃=0^k k k̃ q^k̃ (1- q)^k - k̃  (μT /μ̃ -ν) ),

where  $\gamma^{(\U)}, \gamma^{(\e)},  \tilde{\gamma}^{(\e)}$ are arbitrary positive parameters, $G(\cdot, \cdot)$  denotes the regularized gamma function,  $k:=|\bku|$, $\tilde k = |\bkut|$, $\ru := \rho\left(1- (1-\zeta)^{L_v}\right)$, $q_3: = \ru q_4 + (1- \ru)q_1$, and

q   : =    √(1-q_2)  + (L_vL_ -1) e^-γ^(),

q_1    :=    1- (1- e^-γ^() ) ^L_vL_,

q_2    :=    1 - (1- G(Ν/2, λ(μ_)) + G(Ν/2, λ̃(μ_)) ) ^L_vL_

q_4    :=    1 - (1- G(Ν/2, λ̃(μ̃_)) + G(Ν/2,  λ(μ̃_)) ) ^L_vL_

Δ   :=    ^k(1-)^η-k q_2^k  (1-q_1)^η-k/(·q_3 + (1- )·q_1)^k  (1-·q_2)^η- k

J_     :=      π√( )  2^Ν+1/2e^-h^2(1-α)^2Ν/2/9h^2(1-α) (+ (1-α)^2 ),

J̃_     :=   27 √(π) (1+h^2(1-α)^2)e^Νh^2 (+ (1-α)^2 )/2(h^2(1-α))^Ν-2√(8 (1+2h^2(1-α)^2) .

and $J_e, \tilde J_e, \zeta, \mu_{\U}, \tilde \mu_{\U},  \mu, \tilde \mu, T$ and $\nu$ are defined in (<ref>).

See Section <ref>.

§ NUMERICAL ANALYSIS

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Time Sharing,  $\epsilon_{b}^{(\U)}$, $\epsilon^{(\e)}_{\text{TIN}}$, $\epsilon^{(\e)}_{\text{SIC}}$

Upper bounds on $\epsilon_{\text{TIN}}^{(\e)}, \epsilon_{\text{SIC}}^{(\e)}$ for $\Pb = 5, \Ne = 600$ and $\Nu = 200$  and for maximum value of $\epsilon_{b}^{(\U)}$ fixed at $10^{-5}$.

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[ color=black, very thick,  dashed] coordinates  (1e-05,0.0135448415730991)(1.175194363069249e-04,0.042694161232047 )(7.230934120238305e-04,0.069082011213920)(0.004650839910841,0.108666672622638)(0.032956352334015,0.191516289054611);

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[ color=red, very thick,  mark=otimes] coordinates  (1e-5,7.41089720909776e-11)(1.175194363069249e-04, 2.723680881461321e-09)(7.230934120238305e-04,5.178313428907042e-08)(0.004650839910841,1.956009996868224e-06)(0.032956352334015,0.0001145485767219);

TS, $\rho = 0.8$, TS, $\rho = 0.2$, TIN, $\rho = 0.8$,   TIN, $\rho = 0.2$,SIC, $\rho = 0.8$,SIC, $\rho = 0.2$

Upper bounds on $\epsilon_{\text{TIN}}^{(\e)}, \epsilon_{\text{SIC}}^{(\e)}$, $\epsilon_{b}^{(\U)}$ for $\P = 5$ and $\Nu = 20\cdot b$ and $\Ne = 3\Nu$ for values of  $b$ in $\{ 10,8,6,4,2\}$.

In Figure <ref>, we numerically compare the bounds in Theorem <ref> with the time-sharing scheme where URLLC transmissions puncture the eMBB transmission upon arrival. In this figure,  we set the maximum error probability of URLLC transmission to be equal to $10^{-5}$. For each value of $\rho \in \{0.2,0.4,0.6,0.8,1\}$, we then  optimize the parameters $\alpha$, $\bef$ and $\bu$ to minimize the eMBB error probability under both TIN and SIC approaches. As can be seem from this figure, our schemes outperform the time-sharing scheme specifically for large values of $\rho$, i.e., in regions with dense URLLC arrivals.

In Figure <ref>,  we numerically compare the bounds in Theorem <ref> for $\rho = 0.2$ and $\rho = 0.8$. In this plot, $\Nu = 20 \cdot b$ and $\Ne = 3\Nu$ and the value of $b$ varies from $10$ to $2$ with step size $2$. The values of $\alpha$, $\bef$ and $\bu$ are optimized to minimize  $\epsilon_{\text{TIN}}^{(\e)}$ and $ \epsilon_{\text{SIC}}^{(\e)}$ for a given maximum $\epsilon_{b}^{(\U)}$.  As can be seen from this figure, when $\rho$ is high, the TIN scheme outperforms the SIC and the time-sharing schemes. For low values of $\rho$, however, the SIC scheme outperforms the other two schemes. The reason is that for high values of  $\rho$, more subtracted URLLC interference will be wrong which introduces error in the eMBB decoding under the SIC scheme.

§ PROOF OF THEOREM <REF>

§.§ Bounding $\epsilon_b^{(\U)}$

Recall the definition of the sets $\mb$, $\mbt$ and $\mbd$ from (<ref>), (<ref>) and (<ref>), respectively.
Given that URLLC message $M_b^{(\U)}$ arrives at the beginning of Block $b$, i.e., $b \in \mb$,  we have the following error events:
ℰ_,1    : =    {b ∉}

ℰ_,2    :=    { b ∉}

ℰ_,3    : =    { (M̂_b^(), ĵ ) ≠(M_b^(), j ) }.

Given that no URLLC message is sent over  Block $b$, i.e., $b \notin \mbt$,  we have the following error event:
ℰ_,4    : =     {b ∈} .

The  error probability of decoding URLLC message $M_b^{(\U)}$ of Block $b$  thus is bounded by
ϵ_b^()    ≤     [b ∈] [ℰ_,1 | b ∈]

+ [b ∈] [ℰ_,2  | ℰ_,1^c, b ∈]

+  [b ∈] [ℰ_,3| ℰ_,2^c, ℰ_,1^c, b ∈]

+  [b ∉] [ℰ_,4 | b ∉] .

§.§.§ Analyzing $\Pr [\mathcal E_{\U,1} | b \in \mb]$

From (<ref>) we notice  that $\Big(\vect V_b - \alpha  \xkef \Big)\in \mathcal D_b$ if and only if
Ν- δ_b ≤||V_b - α||^2  ≤Ν.

Recall that  $||\vect V_k||^2  = \Nu \vnorm \Pb$ almost surely.

We can prove that
[(V_b - α) ∈𝒟_b ] = ζ
where $\zeta$ is defined in (<ref>).

see Appendix <ref>.

Since the $L_v$ codewords are generated independently:
[ℰ_,1 | b ∈] = ( 1-  ζ)^L_v.

To analyze the remaining error events, we employ the following lemma.

For any $\gamma^{(\U)}>0$:
[i_b^()(V_b(m,j); Y_b) ≤γ^() ]

≤    1-G(Ν/2, λ(μ_ )) + G(Ν/2, λ̃(μ_ )),

where $G(\cdot,\cdot)$ is the regularized gamma function
and $\lambda(\cdot)$ and $ \tilde \lambda(\cdot)$ are defined in (<ref>) and $\mu_{\U}$ is defined in (<ref>).

See Appendix <ref>.

§.§.§ Analyzing $\Pr [\mathcal E_{\U,2}  | \mathcal E_{\U,1}^c, b \in \mb]$

This error event is equivalent to the probability that for all $j \in [L_v]$ and for all $m \in [L_{\U}]$ there is no codeword $V_b(m,i)$ such that $i(\vect V_b(m,i); \vect Y_b) > \gamma^{(\U)}$. Therefore,
[ℰ_,2  | ℰ_,1^c, b ∈]

=   ( [i(V_b(m,j); Y_b) ≤γ^()])^L_vL_

≤    (1- G(Ν/2, λ(μ_) ) + G(Ν/2, λ̃(μ_)) ) ^L_vL_

where the last inequality holds by Lemma <ref>.

§.§.§ Analyzing $\Pr [\mathcal E_{\U,3}| \mathcal E_{\U,2}^c, \mathcal E_{\U,1}^c, b \in \mb ]$

To evaluate this probability, we use the threshold bound for maximum-metric decoding. For any given threshold $\gamma^{(\U)} $:
[ℰ_,3| ℰ_,2^c, ℰ_,1^c, b ∈]

≤    [i(V_b(M_b^(),  j); Y_b) ≤γ^()]

+ (L_vL_ -1) ℙ[i(V̅_b(m', j'); Y_b)> γ^]
where  $m' \in \{1, \ldots, L_{\U}\}$, $j' \in \{1, \ldots, L_v\}$, $(M_b^{(\U)}, j) \neq (m',j')$,  $\bar{\vect V}_b \sim f_{\vect V_b}$ and is  independent of $(\vect V_b, \vect Y_b)$.

For any  $\gamma^{(\U)}>0$:
[i(V̅_b; Y_b)> γ^()] ≤e^-γ^().

See Appendix <ref>.

By Lemmas <ref> and <ref>, we have
[ℰ_,3| ℰ_,2^c, ℰ_,1^c, b ∈]

≤   1- G(Ν/2, λ(μ_) ) + G(Ν/2, λ̃(μ_))    + (L_vL_ -1) e^-γ^() .

§.§.§ Analyzing $  \Pr [\mathcal E_{\U,4} | b \notin \mb]  $

This error event is equivalent to the probability that given no URLLC is arrived, there exists at least one codeword $V_b(m,i)$ with $ m \in [L_{\U}]$ and $j \in [ L_v]$ such that $i(\vect V_b(m,j); \vect Y_b) > \gamma^{(\U)}$. Therefore,
[ℰ_,4 | b ∉]

=    1 - ([i(V_b(m,j); Y_b) ≤γ^() ])^L_vL_

≤    1- (1- e^-γ^() )^L_vL_.

where the last inequality follows by Lemma <ref>.

By combining (<ref>), (<ref>), (<ref>) and (<ref>) we prove bound (<ref>).

§.§ Bounding $\epsilon^{(\e)}_{\text{TIN}}$

:=     [b ∈],

:=    [b ∈| b ∈],

:=    [b ∈| b ∉].

We prove that

=    ρ(1- (1-ζ)^L_v),   ≤q_1,   q_2 ≤≤q_3,

where $q_1$, $q_2$ and $q_3$ are defined in (<ref>) and $\zeta$  in (<ref>).

See Appendix <ref>.

Given  $\mbd=\bku$, we have the following two error events:
ℰ_TIN,1    =    {≠}

ℰ_TIN,2    =    {M̂^() ≠M^() } .

The  eMBB decoding error probability under the TIN approach thus is bounded by
ϵ_^TIN ≤∑_ [= ]

·( [ℰ_TIN,1| = ] +[ ℰ_TIN,2| = , ℰ_TIN,1^c]).

§.§.§ Analyzing $ \Pr [\mbd = \bku]$

:=    [b ∈, b ∈] + [ b ∈, b ∉]

=     + (1- ) ,
where $\ru$, $\rdz$ and $\rdo$ are defined in (<ref>). By Lemma <ref>:
·q_2 ≤≤·q_3 + (1- )·q_1,

and thus by the independence of the blocks:
[= ]

=    ^||  (1 - )^η- ||

≤    (·q_3 + (1- )·q_1)^||  (1-·q_2)^η- ||

§.§.§ Analyzing $ \Pr [\mathcal E_{\text{TIN},1}| \mbd = \bku ]$

Notice that the values of $\ru, \rdz$ and $\rdo$   stay the same for all blocks in $[\eta]$. Thus
[≠| = ]

=    1- [= | = ]

=     1 - [= , = ]/[= ]

=     1- [= ] [= | = ]/^|| (1 - )^η- ||

=     1- ^||(1-)^η-|| ^|| (1-)^η-||/^|| (1 - )^η- ||

≤    1 -  ^||(1-)^η-|| q_2^||  (1-q_1)^η-||/(·q_3 + (1- )·q_1)^||  (1-·q_2)^η- ||

where $\ru, q_1,q_2$ and $q_3$ are defined in (<ref>).

The inequality in (<ref>) follows by Lemma <ref>.

§.§.§ Analyzing $\Pr[ \mathcal E_{\text{TIN},2}| \mbd = \bku, \mathcal E_{\text{TIN},1}^c]$

To bound $\Pr[\hat M^{(\e)} \neq M^{(\e)} |\mbd = \bku,  \mathcal E_{\text{TIN},1}^c ]$, we use the threshold bound for maximum-metric decoding. For any given threshold $\gamma^{(\e)}$:
[M̂^() ≠M^() | = ,  ℰ_TIN,1^c ]

≤[i^()_TIN ( {}_b ∉,  {}_b ∈; Y^ | ) < γ^()]

+      [i^()_TIN ( {X̅_b^(,1) }_b ∉,  {X̅_b^(,2)}_b ∈;  Y^| ) ≥γ^()]

where for each $b$, $\bar {\vect X}_{b}^{(\e,1)} \sim f_{\xkes}$ and  $\bar {\vect X}_{b}^{(\e,2)} \sim f_{\xkef}$ and are independent of $(\xkes, \xkef, \vect Y_b)$. We use the following two lemmas to bound the above two probability terms.

For any $\gamma^{(\e)} >0$:
[i^()_TIN ( {}_b ∉,  {}_b ∈;   Y^ | ) < γ^()]

≤T - (L_v-1)e^-γ^()

where $T$ is defined in (<ref>).

See Appendix <ref>.

For any $\gamma^{(\e)}>0$:
[i^()_TIN ( {X̅_b^(,1) }_b ∉,  {X̅_b^(,2)}_b ∈;  {Y_b}_b = 1^η+1 | ) ≥γ^()]

The proof is similar to the proof of  Lemma <ref> and omitted.

Combining Lemmas <ref> and <ref> with (<ref>) and defining $k:=|\bku|$ proves the bound in (<ref>).

§.§ Bounding $\epsilon^{(\e)}_{\text{SIC}}$

Recall the definition of the sets $\mb$, $\mbt$, $\mbd$ and $\mbc$ from (<ref>), (<ref>), (<ref>), and (<ref>), respectively. Let $\bku$ be a realization of the set $\mbd$, and $\bkut$ be a realization of the set $\mbc$. We have the following two error events:
ℰ_SIC,1    =    {≠}

ℰ_SIC,2    =    {M̂^() ≠M^() }

The  eMBB decoding error probability under the SIC approach  thus is given by
≤∑_ [= ]

( [ℰ_SIC,1| = ]

+∑_ [= | ℰ_SIC,1^c, = ]

·[ ℰ_SIC,2| = , = , ℰ_SIC,1^c]).

§.§.§ Analyzing $\Pr[\mbc = \bkut| \mathcal E_{\text{SIC},1}^c, \mbd = \bku]$

For any subset $B_{c} \subseteq B_d$ we have:
[= | = = ]

=     ∏_b ∈ [M̂_b^() = M_b^()| = = ]

·∏_b∈\ (1- [M̂_b^() = M_b^()| = =])

≤    q^|| (1- q)^|| - ||

where $q$ is defined in (<ref>).

Inequality (<ref>) holds by (<ref>) and by the independence of the blocks.

§.§.§ Analyzing $\Pr[ \mathcal E_{\text{SIC},2}| \mbd = \bku, \mbc = \bkut, \mathcal E_{\text{SIC},1}^c]$

To bound this probability, we use the threshold bound for maximum-metric decoding. For any given threshold $\tilde \gamma^{(\e)}$:
[M̂^() ≠M^() | = , = , ℰ_SIC,1^c ]

≤    [i^()_SIC  ( {}_b ∉,  {}_b ∈;

Y^| , ,   {V_b}_b ∈ ) < γ̃^()]

+ (L_-1)    [i^()_SIC  ( {X̅_b,1^() }_b ∉,  {X̅_b,2^()}_b ∈;

Y_b^ | , ,  {V_b}_b ∈ ) ≥γ̃^()]
where for each $b$, $\bar {\vect X}_{b}^{(\e,1)} \sim f_{\xkes}$ and  $\bar {\vect X}_{b}^{(\e,2)} \sim f_{\xkef}$ and are independent of $(\xkes, \xkef, \vect Y^{\Ne})$. We use the following two lemmas to bound the above two probability terms.

Given $\tilde \gamma^{(\e)}$, we prove that
[i^()_SIC  ( {}_b ∉,  {}_b ∈; {Y_b}_b = 1^η+1

| ,  {V_b}_b ∈ ) < γ̃^()]

≤μT /μ̃ -ν
where $T$, $\nu, \mu$ and $\tilde \mu$ are defined in (<ref>).

See Appendix <ref>.

We can prove that
[i^()_SIC  ( {X̅_b,1^() }_b ∉,  {X̅_b,2^()}_b ∈;

{Y_b}_b = 1^η+1 | ,  {V_b}_b ∈ ) ≥γ̃^()]≤e^-γ̃^().

The proof is based on the argument provided in the proof of  Lemma <ref>.

Combining Lemmas <ref> and <ref> with  (<ref>) and defining $\tilde k = |\bkut|$  proves the bound in (<ref>).

§ CONCLUSIONS

We considered a point-to-point scenario where a roadside unite (RSU) wishes to simultaneously send eMBB and URLLC messages to a vehicle.  During each eMBB transmission interval, random arrivals of URLLC messages are assumed. To improve the reliability of the URLLC transmissions, we proposed a coding scheme that mitigates the interference of eMBB transmission by means of dirty paper coding (DPC).

We derived rigorous upper bounds on the error probabilities of eMBB and URLLC transmissions achieved by our scheme. Our numerical analysis shows that the proposed scheme significantly improves over
the standard time-sharing.

§ ACKNOWLEDGMENT

The work of   H. V. Poor has been supported by the U.S. National Science Foundation (NSF) within the Israel-US Binational program
under grant CCF-1908308.
The work of S. Shamai (Shitz) has been supported
by the US-Israel Binational Science Foundation
(BSF) under grant BSF-2018710.

§ PROOF OF LEMMA <REF>

By (<ref>) and since  $ \xkef$ and $\vect{V}_b$ are  drawn uniformly on the  $\Nu$-dimensional spheres of radii $\sqrt{\Nu \beta_{\e}\P}$ and $\sqrt{\Nu (\beta_{\U}+\alpha^2 \beta_{\e})\P}$, the error event $\mathcal{E}_{b,v}$ holds whenever the following condition is violated:
αΝ≤⟨V_b,  ⟩≤αΝ+ δ_b/2 α.
The distribution of  $\langle\vect V_b,   \xkef \rangle$  depends on $\vect V_b$ only through its magnitude, because $\xkef$ is uniform over a sphere and applying an orthogonal transformation to $\vect V_b$ and $\xkef$ does neither change  the inner product of the two vectors nor the distribution of $\xkef$.
In the following we therefore assume that $\vect V_b = (||\vect V_b||, 0, \ldots, 0)$, in which case (<ref>) is equivalent to:
αΝ/√(Ν) ≤X_b,2,1^() ≤αΝ/√(Ν) + δ_b/2 α√(Ν)

where $X_{b,2,1}^{(\e)}$ is the first entry of the vector $\xkef$.

The distribution of a given symbol in a length-$\Nu$ random sequence distributed uniformly on the sphere is [15]
f_X_b,2,1^()(x_b,2,1^())   =    1/√(πΝ)Γ(Ν/2)/Γ(Ν-1/2) (1 - (x_b,2,1^())^2/Ν )^Ν-3/2

×1{(x_b,2,1^())^2 ≤Ν}.

[V_b - α∈𝒟_k ]

=    ∫_αΝ/√(Ν)^αΝ/√(Ν) + δ_b/2 α√(Ν)   f_X_b,2,1^()(x_b,2,1^() ) dx_b,2,1^()

=     1/√(π)Γ(Ν/2)/Γ(Ν-1/2) κ_Ν-3/2 ( 2 α^2 Ν+ δ_b/2 αΝ√() )

- 1/√(π)Γ(Ν/2)/Γ(Ν-1/2) κ_Ν-3/2 (  α√(/)) ,

κ_n(x) = x(1-x^2)^n/2n+1 + 2n/2n+1 κ_n-1(x)

with $\kappa_0(x) = x$.
This concludes the proof.

§ PROOF OF LEMMA <REF>

Note that $\vect Y_b$  and $\vect Y_b| \vect V_b$ do not follow a Gaussian distribution.
Q_Y_b (y_b)    =    𝒩(y_k,1; 0, I_Ν )

Q_Y_b| V_b (y_b| v_b)    =    𝒩(y_h; h V_b, I_Ν )

with $\sy = h^2  \Pb + 1$ and $\syv = h^ 2(1 - \alpha)^2 \bef \Pb+ 1$.

ĩ_b^()(v_b; y_b ) := lnQ_Y_b| V_b (y_b| v_b) /Q_Y_b (y_b)  .

We can prove that
i_b^()(v_b; y_b ) ≥ĩ_b^()(v_b; y_b ) + lnJ_,

J_     :=     π√( )  2^Ν+1/2e^-h^2(1-α)^2Ν/2/9h^2(1-α) (+ (1-α)^2 )

By <cit.>:
f_Y_b (y_b)/Q_Y_b (y_b) ≤9((1-α)h)^Ν/2 π√(2) + (1-α)^2 /(1-α) √( ).

By <cit.>:
f_Y_b| V_b (y_b| v_b)/Q_Y_b| V_b (y_b| v_b) ≥2^Ν-2/2(h(1-α))^Ν-2 e^-h^2(1-α)^2Ν/2
Combining the two bounds concludes the proof.

As a result, we have
[i_b^()(V_b; Y_b) ≤γ^() ]

≤    [ĩ(V_b; Y_b ) ≤γ^()  - lnJ_]

=        [lnQ_Y_b| V_b(Y_b| V_b)/Q_Y_b(Y_b)   ≤γ^()  - lnJ_ ]

=     [ ln1/(√(2π))^Νexp(- || Y_b - hV_b||^2/2)/1/(√(2π))^Νexp(- || Y_b||^2/2 ) ≤γ^()  - lnJ_]

=     [ Ν/2 ln/ + || Y_b||^2/2 - || Y_b - hV_b||^2/2 ≤γ^()  - lnJ_]

=     [h^2/2|| ||^2 + h^2/2(1/ - (1-α)^2/) ||||^2

+h^2/2(1/ - 1/) ||Z_b||^2+ h/ ⟨, ⟩

+ h/ ⟨, Z_b ⟩+ (h/ + h(1-α)/ ) ⟨, Z_b ⟩

≤γ^()  - lnJ_ - Ν/2ln/ ]

≤    [h^2(Ν- δ_b)/2 + h^2Ν/2(1/ - (1-α)^2/)

+h^2/2(1/ - 1/) ||Z_b||^2 -h Ν√()/

- h √(Ν) (√()/  + √()/ + √() (1-α)/ ) || Z_b||

≤γ^()  - lnJ_ - Ν/2ln/ ]

=     [ ||Z_b||^2 + u ||Z_b|| ≥μ_]

=     [ (||Z_b|| + u/2 ) ^2  ≥μ_ + u^2/4]

=     1- F(√(μ_ + u^2/4) - u/2) + F (-√(μ_ + u^2/4) - u/2 )

μ_     :=     2/h^2(-) (Ν/2ln/ - γ^()  + lnJ_ )

+ /-  ( Ν(√() - √())^2 - δ_b)

-  Ν(1-α)^2/-

u     :=       2√(Ν) ( (√() + √()) + √()(1- α))/h (- )

Notice that in (<ref>) we use the fact that $||\vect Z_b||$ follows a chi-distribution with degree $\Nu$ and $F(\cdot)$ represents its CDF.

§ PROOF OF LEMMA <REF>

By Bayes' rule we have
f_V_b(v̅_b)    =    f_Y_b(y_b)f_V_b | Y_b (v̅_b | y_b) /f_Y_b| V_b (y_b| v̅_b)

=    f_V_b | Y_b (v̅_b | y_b) exp( - i(v̅_b, y_b ) ).

By multiplying both sides of the above equation by $\mathbbm {1} \{i(\bar{\vect v}_b, \vect y_b )> \gamma\}$ and integrating over all $\bar{\vect v}_b$, we have
∫_v̅_b 1 {i(v̅_b, y_b )> γ} f_V_b(v̅_b) d v̅_b =

∫_v̅_b 1 {i(v̅_b, y_b )> γ}  e^  - i(v̅_b, y_b ) f_V_b| Y_b (v̅_b | y_b) d v̅_b.
Note that the left-hand side of (<ref>) is equivalent to $\Pr [i(\bar{\vect v}_b, \vect y_b)> \gamma| \vect Y_b = \vect y_b ] $. Thus
[i(v̅_b, y_b )> γ| Y_b = y_b ]

=     ∫_v̅_b 1 {i(v̅_b, y_b )> γ}

×exp( - i(v̅_b, y_b ) ) f_V_b | Y_b (v̅_b | y_b) d v̅_b

=     ∫_v̅_b 1 {f_Y_b| V_b (y_b| v̅_b)/f_Y_b(y_b) e^-γ>1 }

×exp( - i(v̅_b, y_b ) ) f_V_b | Y_b (v̅_b | y_b) d v̅_b

≤    ∫_v̅_b f_Y_b| V_b  (y_b| v̅_b)/f_Y_b(y_b) e^-γ

×exp( - i(v̅_b, y_b ) ) f_V_b | Y_b (v̅_b | y_b) d v̅_b

=     ∫_v̅_b e^-γ f_V_b | Y_b (v̅_b | y_b) d v̅_b

=      e^-γ.

§ PROOF OF LEMMA <REF>

We start by analyzing the quantities in $\ru$, $\rdz$ and $\rdo$ defined in (<ref>), (<ref>) and (<ref>).

§.§.§ Analyzing $\ru$

=    ρ·[ ∃   j ∈[L_v]   s.t.    (V_b(M_b^(),j) ) ∈𝒟_b ]

=     ρ(1- (1-ζ)^L_v)

where the last equality is by (<ref>).

§.§.§ Bounding $\rdz$

=     [b ∈| b ∈]

=     1- [∀m, ∀j: i^()_b  (V_b(m,j); Y_b ) ≤γ^()| b ∈]

≥    1 - (1- G(Ν/2, λ(μ_) ) + G(Ν/2, λ̃(μ_))  ) ^L_vL_

where (<ref>) is by (<ref>).

For any $\gamma^{(\U)}>0$:
[i_b^()(V_b(m,j); Y_b) ≤γ^() ]

≥    1- G(Ν/2, λ̃(μ̃_)) + G(Ν/2,  λ(μ̃_))

where $G(\cdot,\cdot)$ is the regularized gamma function, $\lambda (\cdot)$ and $\tilde \lambda (\cdot)$ are defined in (<ref>) and $\tilde \mu_{\U}$ is defined in (<ref>).

The proof is similar to the proof of Lemma <ref>. We present a sketch of the proof.

We start by upper bounding
i_b^()(v_b; y_b )  ≤ĩ_b^()(v_b; y_b ) +  lnJ̃_,

where by <cit.> and <cit.> we can prove that
J̃_     :=   27 √(π) (1+h^2(1-α)^2)e^Νh^2 (+ (1-α)^2 )/2(h^2(1-α))^Ν-2√(8 (1+2h^2(1-α)^2) .

[i_b^()(V_b; Y_b) ≤γ^() ]

≥    [ĩ(V_b; Y_b ) ≤γ^() - lnJ̃_]

=     [ ||Z_b||^2 - u ||Z_b|| ≥μ̃_]

=     [ (||Z_b|| - u/2 ) ^2  ≥μ̃_ + u^2/4]

=     1- F(√(μ̃_ + u^2/4) + u/2) + F (-√(μ_ + u^2/4) + u/2 )

μ̃_     :=     2/h^2(-) (Ν/2ln/ - γ^() + lnJ̃_ )

+ /-  ( Ν(√() +√())^2 )

-  Ν(1-α)^2/-

By Lemma <ref>:
≤1 - (1- G(Ν/2, λ̃_1) + G (Ν/2, λ̃_2 ) ) ^L_vL_.

§.§.§ Upper  Bounding $\rdo$

=     [b ∈| b ∉]

=     [∃m ∈[L_], j∈[L_v]: i^()_b  (V_b(m,j); Y_b ) ≥γ^()| b ∉]

=     1- [∀m, ∀j: i^()_b  (V_b(m,j); Y_b ) ≤γ^()| b ∈]

≤    1 - (1- e^-γ^() ) ^L_vL_

where (<ref>) is by (<ref>).

§ PROOF OF LEMMA <REF>

Notice that for each $b \in [1:\eta+1]$, $\vect Y_b$  and for $b \in \bku$, $\vect Y_b| \xkef$  do not follow a Gaussian distribution.
Define $Q_{\vect Y_b}(\vect y_b)$ as in (<ref>) and

Q_Y_b|  (y_b| x_b^(,2))    =    𝒩(y_b; h (1-α), I_Ν )
with  $\syx = h^ 2\vnorm \P+ 1$.

ĩ^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  {y_b}_b = 1^η+1|)

: = ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b) +  ln∏_b∈  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)

We can prove that
i^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  {y_b}_b = 1^η+1 | )

≥    ĩ^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  {y_b}_b = 1^η+1 | )   + lnJ_,
J_     :=    ( π2^Ν+1/2e^-h^2 Ν/2 √()/9h^2 (1- α)^Ν-1 (+ (1- α)^2 ) )^k

·( √(8 (1 + 2 h^2))/27√(π) (1+ h^2 ) )^η-k

similar to the proof of Lemma <ref> and by <cit.>, for $b \notin \bku$:
f_Y_b (y_b)/Q_Y_b (y_b) ≤27√(π) (1+h^2 )/√(8 (1 + 2h^2 )).

As a result, we have
[i^()_TIN ( {}_b ∉,  {}_b ∈;  Y^ |)  < γ^() ]

≤    [ĩ^()_TIN ( {}_b ∉,  {}_b ∈;  Y^ |)

<γ^() - lnJ_ ]

=    [ ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b)

+  ln∏_b∈  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)<γ^() - lnJ_ ]

=   [ ln∏_b∉\η+1   1/(√(2π))^Ν e^-||Z_b||^2/2/1/(√(2π))^Ν e^-||+ Z_b||^2/2

+  ln∏_b∈  1/(√(2π))^Ν e^-||V_b + Z_b||^2/2/1/(√(2π))^Ν e^-||+ + Z_b||^2/2

+ ln1/(√(2π))^-ηΝ e^-|| Z_η+1||^2/2/1/(√(2π))^-ηΝ e^-|| X_η+1,1^() + Z_η+1||^2/2 <γ^() - lnJ_ ]

=   [  1/2 ∑_b ∉ ||Z_b||^2 - 1/2 ||+ Z_b||^2

+  ∑_b ∈||V_b + Z_b||^2/2  -   ||V_b + (1- α)+ Z_b||^2/2

>-γ^() +lnJ_ +/2 ln-Νk/2  ln]

≤     [ -1/2 ∑_b ∉ ||Z_b||^2 +   √(Ν)/∑_b ∉ ||Z_b||

+   τ∑_b ∈ ||Z_b||  +  -/2∑_b ∈ ||Z_b||^2  > μ]

(a)=    [ -1/2 Z̃_1 +   √(Ν)/∑_b ∉ ||Z_b||

+   τ∑_b ∈ ||Z_b||  +  -/2 Z̃_2  > μ]

(b)≤    𝔼 [ -1/2 Z̃_1 +   √(Ν)/∑_b ∉ ||Z_b|| ]/μ

+ 𝔼 [  τ∑_b ∈ ||Z_b||  +  -/2 Z̃_2 ]/μ

=    (- k Ν)(-1)/2μ +  (η+1 - k)√(Ν)/μ √(2) Γ(Ν+1/2)/Γ(Ν/2)

+ kτ/μ√(2) Γ(Ν+1/2)/Γ(Ν/2) +  k Ν(-)/2μ

τ   : =    √(Ν) (√() (+ ) + (1- α) √())/

μ   :=    -γ^() + lnJ_+ /2 ln- kΝ/2 ln- η+1-k/2 Ν

+   k/2Ν- k/2(√() + (1- α)√() )^2Ν
In step $(a)$, we define
Z̃_1    :=    ∑_b ∉ ||Z_b||^2 ∼𝒳^2(- kΝ)

Z̃_2    :=    ∑_b ∈ ||Z_b||^2 ∼𝒳^2(kΝ)

where $\mathcal X^2(n)$ represents chi-squared distribution of degree $n$.
In step $(b)$, we use the following  Markov's inequality:
[X > a] ≤𝔼[X]/a.

In step $(c)$:
𝔼 [Z̃_1]    =    - kΝ,

𝔼 [Z̃_2]    =    kΝ,

𝔼 [||Z_b||]    =    √(2) Γ(Ν+1/2)/Γ(Ν/2).

§ PROOF OF LEMMA <REF>

Define $Q_{\vect Y_b} (\vect y_b)$ as in (<ref>), $Q_{\vect Y_b |\vect V_b} (\vect y_b| \vect v_b)$ as in (<ref>) and $Q_{\vect Y_b| \xkef} (\vect y_b| \vect x_{b}^{(\e,2)})$ as in (<ref>).

ĩ^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^ | , , {V_b}_b ∈)

: = ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b) +  ln∏_b∈\  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)

+  ln∏_b∈  f_Y_b| , V_b (y_b| x_b^(,2), v_b)/Q_Y_b| V_b(y_b| v_b)

We can prove that
i^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^ | , , {v_b}_b ∈)

≥   ĩ^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^ | , ,  {v_b}_b ∈)

+ lnJ̃_,

J̃_     :=    ( π2^Ν+1/2e^-h^2 Ν/2 √()/9h^2 (1- α)^Ν-1 (+ (1- α)^2 ) )^ k-k̃

·( √(8 (1 + 2 h^2))/27√(π) (1+ h^2 ) )^η-k

·( √(8 (1 + 2 h^2(1-α)^2 ))/27√(π) (1+ h^2(1-α)^2) )^k̃

similar to the proof of Lemmas <ref> and <ref>.

As a result, we have
[i^()_SIC ( {}_b ∉,  {}_b ∈;

Y^ | , , {V_b}_b ∈)  ≤γ̃^() ]

≤    [ĩ^()_SIC ( {}_b ∉,  {}_b ∈;

Y^ | , ,  {V_b}_b ∈) < γ̃^() - lnJ̃_ ]

=    [ ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b)

+  ln∏_b∈\  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)

+  ln∏_b∈  f_Y_b| , V_b (y_b| x_b^(,2), v_b)/Q_Y_b|V_b(y_b| y_b)< γ̃^() - lnJ̃_ ]

=   [ ln∏_b∉\η+1   1/(√(2π))^Ν e^-||Z_b||^2/2/1/(√(2π))^Ν e^-||+ Z_b||^2/2

+  ln∏_b∈\  1/(√(2π))^Ν e^-||V_b + Z_b||^2/2/1/(√(2π))^Ν e^-||+ + Z_b||^2/2

+  ln∏_b∈  1/(√(2π))^Ν e^-|| Z_b||^2/2/1/(√(2π))^Ν e^-||(1-α) + Z_b||^2/2

+ ln1/(√(2π))^-ηΝ e^-|| Z_η+1||^2/2/1/(√(2π))^-ηΝ e^-|| X_η+1,1^() + Z_η+1||^2/2 < γ̃^() - lnJ̃_ ]

=   [  1/2 ∑_b ∉ ||Z_b||^2 - 1/2 ||+ Z_b||^2

+  ∑_b ∈\ (||V_b + Z_b||^2/2

-   ||V_b + (1- α)+ Z_b||^2/2 )

+  ∑_b ∈|| Z_b||^2/2  -   || (1- α)+ Z_b||^2/2

> -γ̃^() + lnJ̃_+- k Ν/2 ln

+(k-k̃)Ν/2 ln/+Νk̃/2  ln]

(a)≤     [ -1/2 ∑_b ∉ ||Z_b||^2 +   √(Ν)/∑_b ∉ ||Z_b||

+   τ∑_b ∈\ ||Z_b||

+  -/2∑_b ∈\ ||Z_b||^2

+   (1-α)√(Ν)/ ∑_b ∈ ||Z_b||

+  -1/2∑_b ∈ ||Z_b||^2  > μ̃]

≤    𝔼 [ -1/2 ∑_b ∉ ||Z_b||^2 +   √(Ν)/∑_b ∉ ||Z_b|| ]/μ̃

+ τ𝔼 [∑_b ∈\ ||Z_b|| ]/μ̃

+ 𝔼 [ -/2∑_b ∈\ ||Z_b||^2 ]/μ̃

+  𝔼 [(1-α)√(Ν)/ ∑_b ∈ ||Z_b|| ]/μ̃

+ 𝔼 [ -1/2∑_b ∈ ||Z_b||^2 ]/ μ̃

=     (- k Ν)(-1)/2μ̃ +  (η+1 - k)√(Ν)/μ̃ √(2) Γ(Ν+1/2)/Γ(Ν/2)

+ kτ/μ̃√(2) Γ(Ν+1/2)/Γ(Ν/2) +  k Ν(-)/2μ̃

- k̃/μ̃  √(2) Γ(Ν+1/2)/Γ(Ν/2)  (τ-(1-α)√(Ν)/)

- Νk̃/μ̃ ( -/2 - -1/2)

μ̃   : =     - kΝ/2 ln+ (k - k̃)Ν/2 ln/+ k̃Ν/2 ln

- η-k/2 Ν+   k- k̃/2Ν- k̃(1- α)^2 Ν/2

- k-k̃/2(√() + (1- α)√() )^2Ν- γ̃^() + lnJ̃_.

This concludes the proof.

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§ PROOF OF LEMMA <REF>

By (<ref>) and since  $ \xkef$ and $\vect{V}_b$ are  drawn uniformly on the  $\Nu$-dimensional spheres of radii $\sqrt{\Nu \beta_{\e}\P}$ and $\sqrt{\Nu (\beta_{\U}+\alpha^2 \beta_{\e})\P}$, the error event $\mathcal{E}_{b,v}$ holds whenever the following condition is violated:
αΝ≤⟨V_b,  ⟩≤αΝ+ δ_b/2 α.
The distribution of  $\langle\vect V_b,   \xkef \rangle$  depends on $\vect V_b$ only through its magnitude, because $\xkef$ is uniform over a sphere and applying an orthogonal transformation to $\vect V_b$ and $\xkef$ does neither change  the inner product of the two vectors nor the distribution of $\xkef$.
In the following we therefore assume that $\vect V_b = (||\vect V_b||, 0, \ldots, 0)$, in which case (<ref>) is equivalent to:
αΝ/√(Ν) ≤X_b,2,1^() ≤αΝ/√(Ν) + δ_b/2 α√(Ν)

where $X_{b,2,1}^{(\e)}$ is the first entry of the vector $\xkef$.

The distribution of a given symbol in a length-$\Nu$ random sequence distributed uniformly on the sphere is [15]
f_X_b,2,1^()(x_b,2,1^())   =    1/√(πΝ)Γ(Ν/2)/Γ(Ν-1/2) (1 - (x_b,2,1^())^2/Ν )^Ν-3/2

×1{(x_b,2,1^())^2 ≤Ν}.

[V_b - α∈𝒟_k ]

=    ∫_αΝ/√(Ν)^αΝ/√(Ν) + δ_b/2 α√(Ν)   f_X_b,2,1^()(x_b,2,1^() ) dx_b,2,1^()

=     1/√(π)Γ(Ν/2)/Γ(Ν-1/2) κ_Ν-3/2 ( 2 α^2 Ν+ δ_b/2 αΝ√() )

- 1/√(π)Γ(Ν/2)/Γ(Ν-1/2) κ_Ν-3/2 (  α√(/)) ,

κ_n(x) = x(1-x^2)^n/2n+1 + 2n/2n+1 κ_n-1(x)

with $\kappa_0(x) = x$.
This concludes the proof.

§ PROOF OF LEMMA <REF>

Note that $\vect Y_b$  and $\vect Y_b| \vect V_b$ do not follow a Gaussian distribution.
Q_Y_b (y_b)    =    𝒩(y_k,1; 0, I_Ν )

Q_Y_b| V_b (y_b| v_b)    =    𝒩(y_h; h V_b, I_Ν )

with $\sy = h^2  \Pb + 1$ and $\syv = h^ 2(1 - \alpha)^2 \bef \Pb+ 1$.

ĩ_b^()(v_b; y_b ) := lnQ_Y_b| V_b (y_b| v_b) /Q_Y_b (y_b)  .

We can prove that
i_b^()(v_b; y_b )/ĩ_b^()(v_b; y_b ) ≥J_,

J_     :=    ln π√( )  2^Ν+1/2e^-h^2(1-α)^2Ν/2/9h^2(1-α) (+ (1-α)^2 )

By <cit.>:
f_Y_b (y_b)/Q_Y_b (y_b) ≤9((1-α)h)^Ν/2 π√(2) + (1-α)^2 /(1-α) √( ).

By <cit.>:
f_Y_b| V_b (y_b| v_b)/Q_Y_b| V_b (y_b| v_b) ≥2^Ν-2/2(h(1-α))^Ν-2 e^-h^2(1-α)^2Ν/2
Combining the two bounds concludes the proof.

As a result, we have
[i_b^()(V_b; Y_b) ≤γ^() ]

≤    [ĩ(V_b; Y_b ) ≤γ^() /J_]

=        [lnQ_Y_b| V_b(Y_b| V_b)/Q_Y_b(Y_b)   ≤γ_ /J_ ]

=     [ ln1/(√(2π))^Νexp(- || Y_b - hV_b||^2/2)/1/(√(2π))^Νexp(- || Y_b||^2/2 ) ≤γ^() /J_]

=     [ Ν/2 ln/ + || Y_b||^2/2 - || Y_b - hV_b||^2/2 ≤γ^() /J_]

=     [h^2/2|| ||^2 + h^2/2(1/ - (1-α)^2/) ||||^2

+h^2/2(1/ - 1/) ||Z_b||^2+ h/ ⟨, ⟩

+ h/ ⟨, Z_b ⟩+ (h/ + h(1-α)/ ) ⟨, Z_b ⟩

≤γ^() /J_ - Ν/2ln/ ]

≤    [h^2(Ν- δ_b)/2 + h^2Ν/2(1/ - (1-α)^2/)

+h^2/2(1/ - 1/) ||Z_b||^2 -h Ν√()/

- h √(Ν) (√()/  + √()/ + √() (1-α)/ ) || Z_b||

≤γ^() /J_ - Ν/2ln/ ]

=     [ ||Z_b||^2 + u ||Z_b|| ≥μ_]

=     [ (||Z_b|| + u/2 ) ^2  ≥μ_ + u^2/4]

=     1- F(√(μ_ + u^2/4) - u/2) + F (-√(μ_ + u^2/4) - u/2 )

μ_     :=     2/h^2(-) (Ν/2ln/ - γ^() /J_ )

+ /-  ( Ν(√() - √())^2 - δ_b)

-  Ν(1-α)^2/-

u     :=       2√(Ν) ( (√() + √()) + √()(1- α))/h (- )

Notice that in (<ref>) we use the fact that $||\vect Z_b||$ follows a chi-distribution with degree $\Nu$ and $F(\cdot)$ represents its CDF.

§ PROOF OF LEMMA <REF>

By Bayes' rule we have
f_V_b(v̅_b)    =    f_Y_b(y_b)f_V_b | Y_b (v̅_b | y_b) /f_Y_b| V_b (y_b| v̅_b)

=    f_V_b | Y_b (v̅_b | y_b) exp( - i(v̅_b, y_b ) ).

By multiplying both sides of the above equation by $\mathbbm {1} \{i(\bar{\vect v}_b, \vect y_b )> \gamma\}$ and integrating over all $\bar{\vect v}_b$, we have
∫_v̅_b 1 {i(v̅_b, y_b )> γ} f_V_b(v̅_b) d v̅_b =

∫_v̅_b 1 {i(v̅_b, y_b )> γ}  e^  - i(v̅_b, y_b ) f_V_b| Y_b (v̅_b | y_b) d v̅_b.
Note that the left-hand side of (<ref>) is equivalent to $\Pr [i(\bar{\vect v}_b, \vect y_b)> \gamma| \vect Y_b = \vect y_b ] $. Thus
[i(v̅_b, y_b )> γ| Y_b = y_b ]

=     ∫_v̅_b 1 {i(v̅_b, y_b )> γ}

×exp( - i(v̅_b, y_b ) ) f_V_b | Y_b (v̅_b | y_b) d v̅_b

=     ∫_v̅_b 1 {f_Y_b| V_b (y_b| v̅_b)/f_Y_b(y_b) e^-γ>1 }

×exp( - i(v̅_b, y_b ) ) f_V_b | Y_b (v̅_b | y_b) d v̅_b

≤    ∫_v̅_b f_Y_b| V_b  (y_b| v̅_b)/f_Y_b(y_b) e^-γ

×exp( - i(v̅_b, y_b ) ) f_V_b | Y_b (v̅_b | y_b) d v̅_b

=     ∫_v̅_b e^-γ f_V_b | Y_b (v̅_b | y_b) d v̅_b

=      e^-γ.

§ PROOF OF LEMMA <REF>

We start by analyzing the quantities in $\ru$, $\rdz$ and $\rdo$ defined in (<ref>), (<ref>) and (<ref>).

§.§.§ Analyzing $\ru$

=    ρ·[ ∃   j ∈[L_v]   s.t.    (V_b(M_b^(),j) ) ∈𝒟_b ]

=     ρ(1- (1-ζ)^L_v)

where the last equality is by (<ref>).

§.§.§ Bounding $\rdz$

=     [b ∈| b ∈]

=     1- [∀m, ∀j: i^()_b  (V_b(m,j); Y_b ) ≤γ^()| b ∈]

≥    1 - (1- G(Ν/2, λ(μ_) ) + G(Ν/2, λ̃(μ_))  ) ^L_vL_

where (<ref>) is by (<ref>).

For any $\gamma^{(\U)}>0$:
[i_b^()(V_b(m,j); Y_b) ≤γ^() ]

≥    1- G(Ν/2, λ̃(μ̃_)) + G(Ν/2,  λ(μ̃_))

where $G(\cdot,\cdot)$ is the regularized gamma function, $\lambda (\cdot)$ and $\tilde \lambda (\cdot)$ are defined in (<ref>) and $\tilde \mu_{\U}$ is defined in (<ref>).

The proof is similar to the proof of Lemma <ref>. We present a sketch of the proof.

We start by upper bounding
i_b^()(v_b; y_b )/ĩ_b^()(v_b; y_b ) ≤J̃_,

where by <cit.> and <cit.> we can prove that
J̃_     :=   27 √(π) (1+h^2(1-α)^2)e^Νh^2 (+ (1-α)^2 )/2(h^2(1-α))^Ν-2√(8 (1+2h^2(1-α)^2) .

[i_b^()(V_b; Y_b) ≤γ^() ]

≥    [ĩ(V_b; Y_b ) ≤γ^() /J̃_]

=     [ ||Z_b||^2 - u ||Z_b|| ≥μ̃_]

=     [ (||Z_b|| - u/2 ) ^2  ≥μ̃_ + u^2/4]

=     1- F(√(μ̃_ + u^2/4) + u/2) + F (-√(μ_ + u^2/4) + u/2 )

μ̃_     :=     2/h^2(-) (Ν/2ln/ - γ^() /J̃_ )

+ /-  ( Ν(√() +√())^2 )

-  Ν(1-α)^2/-

By Lemma <ref>:
≤1 - (1- G(Ν/2, λ̃_1) + G (Ν/2, λ̃_2 ) ) ^L_vL_.

§.§.§ Upper  Bounding $\rdo$

=     [b ∈| b ∉]

=     [∃m ∈[L_], j∈[L_v]: i^()_b  (V_b(m,j); Y_b ) ≥γ^()| b ∉]

=     1- [∀m, ∀j: i^()_b  (V_b(m,j); Y_b ) ≤γ^()| b ∈]

≤    1 - (1- e^-γ^() ) ^L_vL_

where (<ref>) is by (<ref>).

§ PROOF OF LEMMA <REF>

Notice that for each $b \in [1:\eta+1]$, $\vect Y_b$  and for $b \in \bku$, $\vect Y_b| \xkef$  do not follow a Gaussian distribution.
Define $Q_{\vect Y_b}(\vect y_b)$ as in (<ref>) and

Q_Y_b|  (y_b| x_b^(,2))    =    𝒩(y_b; h (1-α), I_Ν )
with  $\syx = h^ 2\vnorm \P+ 1$.

ĩ^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  {y_b}_b = 1^η+1|)

: = ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b) +  ln∏_b∈  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)

We can prove that
i^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  {y_b}_b = 1^η+1 | )/ĩ^()_TIN ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈;  {y_b}_b = 1^η+1 | )  ≥J_,
J_     :=    k lnπ2^Ν+1/2e^-h^2 Ν/2 √()/9h^2 (1- α)^Ν-1 (+ (1- α)^2 )

- (η-k) ln27√(π) (1+ h^2 )/√(8 (1 + 2 h^2))

similar to the proof of Lemma <ref> and by <cit.>, for $b \notin \bku$:
f_Y_b (y_b)/Q_Y_b (y_b) ≤27√(π) (1+h^2 )/√(8 (1 + 2h^2 )).

As a result, we have
[i^()_TIN ( {}_b ∉,  {}_b ∈;  Y^ |)  < γ^() ]

≤    [ĩ^()_TIN ( {}_b ∉,  {}_b ∈;  Y^ |)  < γ^()/J_ ]

=    [ ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b)

+  ln∏_b∈  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)< γ^()/J_ ]

=   [ ln∏_b∉\η+1   1/(√(2π))^Ν e^-||Z_b||^2/2/1/(√(2π))^Ν e^-||+ Z_b||^2/2

+  ln∏_b∈  1/(√(2π))^Ν e^-||V_b + Z_b||^2/2/1/(√(2π))^Ν e^-||+ + Z_b||^2/2

+ ln1/(√(2π))^-ηΝ e^-|| Z_η+1||^2/2/1/(√(2π))^-ηΝ e^-|| X_η+1,1^() + Z_η+1||^2/2 < γ^()/J_ ]

=   [  1/2 ∑_b ∉ ||Z_b||^2 - 1/2 ||+ Z_b||^2

+  ∑_b ∈||V_b + Z_b||^2/2  -   ||V_b + (1- α)+ Z_b||^2/2

>- γ^()/J_ +/2 ln-Νk/2  ln]

≤     [ -1/2 ∑_b ∉ ||Z_b||^2 +   √(Ν)/∑_b ∉ ||Z_b||

+   τ∑_b ∈ ||Z_b||  +  -/2∑_b ∈ ||Z_b||^2  > γ]

(a)=    [ -1/2 Z̃_1 +   √(Ν)/∑_b ∉ ||Z_b||

+   τ∑_b ∈ ||Z_b||  +  -/2 Z̃_2  > γ]

(b)≤    𝔼 [ -1/2 Z̃_1 +   √(Ν)/∑_b ∉ ||Z_b|| ]/γ

+ 𝔼 [  τ∑_b ∈ ||Z_b||  +  -/2 Z̃_2 ]/γ

=    (- k Ν)(-1)/2γ +  (η+1 - k)√(Ν)/γ √(2) Γ(Ν+1/2)/Γ(Ν/2)

+ kτ/γ√(2) Γ(Ν+1/2)/Γ(Ν/2) +  k Ν(-)/2γ

τ   : =    √(Ν) (√() (+ ) + (1- α) √())/

μ   :=    -γ^()/J_ + /2 ln- kΝ/2 ln- η+1-k/2 Ν

+   k/2Ν- k/2(√() + (1- α)√() )^2Ν
In step $(a)$, we define
Z̃_1    :=    ∑_b ∉ ||Z_b||^2 ∼𝒳^2(- kΝ)

Z̃_2    :=    ∑_b ∈ ||Z_b||^2 ∼𝒳^2(kΝ)

where $\mathcal X^2(n)$ represents chi-squared distribution of degree $n$.
In step $(b)$, we use the following  Markov's inequality:
[X > a] ≤𝔼[X]/a.

In step $(c)$:
𝔼 [Z̃_1]    =    - kΝ,

𝔼 [Z̃_2]    =    kΝ,

𝔼 [||Z_b||]    =    √(2) Γ(Ν+1/2)/Γ(Ν/2).

§ PROOF OF LEMMA <REF>

Define $Q_{\vect Y_b} (\vect y_b)$ as in (<ref>), $Q_{\vect Y_b |\vect V_b} (\vect y_b| \vect v_b)$ as in (<ref>) and $Q_{\vect Y_b| \xkef} (\vect y_b| \vect x_{b}^{(\e,2)})$ as in (<ref>).

ĩ^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^ | , , {V_b}_b ∈)

: = ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b) +  ln∏_b∈\  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)

+  ln∏_b∈  f_Y_b| , V_b (y_b| x_b^(,2), v_b)/Q_Y_b| V_b(y_b| v_b)

We can prove that
i^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^ | , , {v_b}_b ∈)/ĩ^()_SIC ( {x_b^(,1)}_b ∉,  {x_b^(,2)}_b ∈; y^ | , ,  {v_b}_b ∈)

J̃_     :=    (k-k̃)  lnπ2^Ν+1/2e^-h^2 Ν/2 √()/9h^2 (1- α)^Ν-1 (+ (1- α)^2 )

- (η-k) ln27√(π) (1+ h^2 )/√(8 (1 + 2 h^2))

- k̃ ln27√(π) (1+ h^2(1-α)^2)/√(8 (1 + 2 h^2(1-α)^2 ))

similar to the proof of Lemmas <ref> and <ref>.

As a result, we have
[i^()_SIC ( {}_b ∉,  {}_b ∈;

Y^ | , , {V_b}_b ∈)  ≤γ̃^() ]

≤    [ĩ^()_SIC ( {}_b ∉,  {}_b ∈;

Y^ | , ,  {V_b}_b ∈) < γ̃^()/J̃_ ]

=    [ ln∏_b∉ f_Y_b|  (y_b| x_b^(,1))/Q_Y_b(y_b)

+  ln∏_b∈\  Q_Y_b|  (y_b| x_b^(,2))/Q_Y_b(y_b)

+  ln∏_b∈  f_Y_b| , V_b (y_b| x_b^(,2), v_b)/Q_Y_b|V_b(y_b| y_b)< γ̃^()/J̃_ ]

=   [ ln∏_b∉\η+1   1/(√(2π))^Ν e^-||Z_b||^2/2/1/(√(2π))^Ν e^-||+ Z_b||^2/2

+  ln∏_b∈\  1/(√(2π))^Ν e^-||V_b + Z_b||^2/2/1/(√(2π))^Ν e^-||+ + Z_b||^2/2

+  ln∏_b∈  1/(√(2π))^Ν e^-|| Z_b||^2/2/1/(√(2π))^Ν e^-||(1-α) + Z_b||^2/2

+ ln1/(√(2π))^-ηΝ e^-|| Z_η+1||^2/2/1/(√(2π))^-ηΝ e^-|| X_η+1,1^() + Z_η+1||^2/2 < γ̃^()/J̃_ ]

=   [  1/2 ∑_b ∉ ||Z_b||^2 - 1/2 ||+ Z_b||^2

+  ∑_b ∈\ (||V_b + Z_b||^2/2

-   ||V_b + (1- α)+ Z_b||^2/2 )

+  ∑_b ∈|| Z_b||^2/2  -   || (1- α)+ Z_b||^2/2

>-γ̃^()/J̃_ +- k Ν/2 ln

+(k-k̃)Ν/2 ln/+Νk̃/2  ln]

(a)≤     [ -1/2 ∑_b ∉ ||Z_b||^2 +   √(Ν)/∑_b ∉ ||Z_b||

+   τ∑_b ∈\ ||Z_b||

+  -/2∑_b ∈\ ||Z_b||^2

+   (1-α)√(Ν)/ ∑_b ∈ ||Z_b||

+  -1/2∑_b ∈ ||Z_b||^2  > γ̃]

≤    𝔼 [ -1/2 ∑_b ∉ ||Z_b||^2 +   √(Ν)/∑_b ∉ ||Z_b|| ]/γ̃

+ τ𝔼 [∑_b ∈\ ||Z_b|| ]/γ̃

+ 𝔼 [ -/2∑_b ∈\ ||Z_b||^2 ]/γ̃

+  𝔼 [(1-α)√(Ν)/ ∑_b ∈ ||Z_b|| ]/γ̃

+ 𝔼 [ -1/2∑_b ∈ ||Z_b||^2 ]/ γ̃

=     (- k Ν)(-1)/2γ̃ +  (η+1 - k)√(Ν)/γ̃ √(2) Γ(Ν+1/2)/Γ(Ν/2)

+ kτ/γ̃√(2) Γ(Ν+1/2)/Γ(Ν/2) +  k Ν(-)/2γ̃

- k̃/γ̃  √(2) Γ(Ν+1/2)/Γ(Ν/2)  (τ-(1-α)√(Ν)/)

- Νk̃/γ̃ ( -/2 - -1/2)

μ̃   : =     - kΝ/2 ln+ (k - k̃)Ν/2 ln/+ k̃Ν/2 ln

- η-k/2 Ν+   k- k̃/2Ν- k̃(1- α)^2 Ν/2

- k-k̃/2(√() + (1- α)√() )^2Ν-γ̃^()/J̃_

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ℱ_k := { y_k,3 ∈ℝ^Ν̃: 1/Ν̃ ||ỹ_k,3||^2  ∈[   σ_y_k,3^2 - δ_y, σ_y_k,3^2 + δ_y] }.

for a fixed $\delta_y > 0$ and where
σ_y_k,3^2 := h_k,k^2(β_3,1 + β_3,2)P + h_k-1,k^2 β_3 P + 1.

By Cramer's theorem in <cit.>, we have

\begin{equation}
\mathbb P [ {\vect y}_{k,3} \notin \mathcal F_k]   <  \exp(- \tilde n_{u}\kappa \delta_y^2)
\end{equation}

for some constant $\kappa > 0$.

We then rewrite (<ref>) by
[ℰ_k,u,4| ℰ_k,u,1^c, ℰ_k,u,2^c, ℰ_k,u,3^c]    ≤    ℙ [ y_k,3 ∈ℱ_k][i(u_k, y_k,3 ) ≤γ_| y_k,3 ∈ℱ_k]

+ M_u L_k ℙ [ y_k,3 ∈ℱ_k] ·ℙ[i(u̅_k, y_k,3 )> γ_| y_k,3 ∈ℱ_k]  + ℙ [ y_k,3 ∉ℱ_k].

By <cit.>, if we define an output $\vect Y^*_{k,3} \sim \mathcal N (0, \sigma_y^2 )$, then  we have
min_y_k,3 ∈ℱ_k f_Y_k,3 (y_k,3)/f_Y^*_k,3 ( y_k,3) ≤J_

for a finite constant $J$. To calculate $ f_{\vect Y_{k,3}| \vect U_k}(\vect y_{k,3}| \vect u_k)$, we define  $\vect U_k^*$ that  follows the distribution  $\mathcal N(0,  I_{\tilde \Nu} (\beta_{3,1} P + \alpha_{k,1}^2 \mathrm P_{X_k} + \alpha_{k,2}^2 \mathrm P_{\hat X_{k-1}}))$. Then by <cit.>
and also by <cit.>:

\begin{equation} \label{eq:87}
\min_{\vect u_k: \vect u_k - \alpha_{k,1} \vect x^{(e)}_{k,3} - \alpha_{k,2} \hat{\vect x}^{(e)}_{k-1,3} \in \mathcal D_k} \frac{f_{\vect U_k }(\vect u_k)}{f_{\vect U_k^* } (\vect u_k)} \le J_2
\end{equation}

where $J_{\U} \le 1$ is a constant.  Define $Q_{\vect Y^*_{k,3}, \vect U_k^*} (\vect y_{k,3}, \vect u_k^*)$ as the joint distribution of  $\vect Y^*_{k,3}$ and $\vect U_k^*$, and
f_Y_k,3, U_k(y_k,3, u_k)/Q_Y^*_k,3, U_k^* (y_k,3, u_k) = D_f,Q(y_k,3, u_k).

One can prove that
D_f,Q(y_k,3, u_k) ≥J_3

where $J_3$ is a constant.

By combining (<ref>), (<ref>), (<ref>) and (<ref>), we have
f_Y_k,3| U_k(y_k,3| u_k)/f_Y_k,3 (y_k,3)    =   f_U_k| Y_k,3(u_k| y_k,3)/f_U_k (u_k)

≥   f_U_k| Y_k,3(u_k| y_k,3)/J_2f_U^*_k (u_k)

=    f_U_k, Y_k,3(u_k, y_k,3)/J_2f_U^*_k (u_k)f_Y_k,3 (y_k,3)

≥   f_U_k, Y_k,3(u_k, y_k,3)/J_1J_2f_U^*_k (u_k)f_Y^*_k,3 (y_k,3)

=     Q_Y^*_k,3, U_k^* (y_k,3, u_k)·D_f,Q(y_k,3, u_k)/J_1J_2f_U^*_k (u_k)f_Y^*_k,3 (y_k,3)

≥     J_3 Q_Y^*_k,3, U_k^* (y_k,3, u_k)/J_1J_2f_U^*_k (u_k)f_Y^*_k,3 (y_k,3)

=     J_k   f_Y^*_k,3| U_k^* (y_k,3| u_k)/f_Y^*_k,3 (y_k,3)

where $J_k := \frac{J_3}{J_1J_2}$.
As a result
[Y_k,3 ∈ℱ_k] [lnf_Y_k,3 | U_k (y_k,3| u_k) /f_Y_k,3 (y_k,3) ≤γ_ | y_k,3 ∈ℱ_k ]

≤[lnf_Y^*_k,3 | U^*_k (y_k,3| u_k) /f_Y^*_k,3 (y_k,3) ≤γ_ - lnJ_k ]

=      [  ln1/(√(2πσ_y|u,3^2))^Ν̃exp(- || y_k,3 - h_k,ku_k||^2/2σ_y|u,3)/1/(√(2πσ_y_k,3^2))^Ν̃exp(- || y_k,3||^2/2 σ_y^2) ≤γ_ - lnJ_k ]

=    [ Ν̃/2 ln(σ_y_k,3^2) - Ν̃/2  ln(σ_y|u,3^2)  - || y_k,3 - h_k,ku_k||^2/2σ_y|u,3 +  || y_k,3||^2/2 σ_y_k,3^2 ≤γ_ - lnJ_k ]

=     [  || y_k,3/σ_y_k,3||^2 - || y_k,3 - h_k,ku_k/σ_y|u,3||^2   ≤γ̃_ ]

σ_y_k,3^2     =    1 + P ( β_3,2 h_k,k^2 ( 1- α_k,1)^2 + β_3 (h_k-1,k^2 + h_k,k^2 α_k,2^2)

γ̃_    =    2 γ_ - 2 ln(J_k) -Ν̃ln(σ_y_k,3^2) + Ν̃ln(σ_y|u,3^2)

Note that $\frac{\vect y_{k,3}}{\sigma_{y_{k,3}}} \sim \mathcal N (0, I_{\tilde \Nu })$ and  $\frac{ \vect y_{k,3} - h_{k,k}\vect u_{k}}{\sigma_{y|u,3}} \sim \mathcal N (0, I_{\tilde \Nu})$.  Define
v_1    :=     || y_k,3/σ_y_k,3||^2:       v_1 ∼𝒳^2 (Ν̃)

v_2    :=    || y_k,3 - h_k,ku_k/σ_y|u,3||^2:      v_2 ∼𝒳^2 (Ν̃)

where $\mathcal X^2(s)$ is central chi-squared distribution of degree $s$. We define

\begin{equation}
Q_u := v_1 - v_2
\end{equation}

[Y_k,3 ∈ℱ_k] [lnf_Y_k,3 | U_k (y_k,3| u_k) /f_Y_k,3 (y_k,3) ≤γ_ | y_k,3 ∈ℱ_k ]
≤[ Q_u ≤γ̃_] =F_Q_u (γ̃_)

where $F_{Q_u}$ is the CDF of $Q_u$. To calculate this CDF we use the following theorem that is on the CDF of linear combination of random chi-squared variables.

\begin{equation}
Q = \sum_{i= 1}^p c_i v_i
\end{equation}

with  $v_i \sim \mathcal X^2 (n_i)$ and $c_i$s being real non-zero constants. Then
{Q ≤y}=  F_Q(y) = ( ∏_i = 2 ^p b_i ) ∑_j = 0 ^ ∞ γ(s + j, y/2c_1)/Γ(s+j) a_j
where  $ \Gamma (\cdot)$ is the gamma function, $\gamma(\cdot, \cdot)$ is the lower incomplete gamma function, and

b_i    :=     ( c_1/c_i ) ^n_i/2,

s     :=    ∑_i = 1 ^p n_i/2,

a_j     :=     A_j^(p),

A_j^(i)     :=     ∑_k = 0^j A_k^(i-1) A (c_i, j-k),

A (c_i, r)     :=      (n_i/2)_r ( 1 - c_1/c_i)^r / r! ,

\begin{equation}
\left (\frac{n_i}{2}\right)_r  = \begin{cases} 1,& \text{if} \; \;  r = 0\\
\frac{n_i}{2}(\frac{n_i}{2}+1)\ldots(\frac{n_i}{2}+r-1), &\; \text{O.W}.
\end{cases}
\end{equation}

To sum up
ϵ_u,k    ≤      Γ(Ν̃/2, Ν̃Π_1/2β_3P )/Γ(Ν̃/2) +  Γ(Ν̃/2, Ν̃Π_2/2β_3,2P )/Γ(Ν̃/2) + ( 1- B_k )^L_k

+ M_u L_k (1- exp(- ñ_uκδ_y^2)) + exp(- ñ_uκδ_y^2) + F_Q_u (γ̃_) + ϵ_T,k,1 + ϵ_T,k,2

§.§ Bounding $\epsilon_{e,k}$

§.§.§ Bounding $\epsilon_{e,k}$ at Rx $k \in \Tu$

Recall the definition of the error event $\mathcal E^{(e)}_{k,1}$ from (<ref>). We bound $\epsilon_{e,k}$ as
ϵ_e,k ≤[ℰ^(e)_k,1] + [||X_k,3^(e)||^2 > Ν̃β_3,2 P] + [||X̂_k-1,3^(e)||^2 > Ν̃β_3 P].

Analyzing  $\Pr [\mathcal E^{(e)}_{k,1}]$: To evaluate this error event, we use the threshold bound for maximum-metric decoding. I.e.
[ℰ^(e)_k,1]    ≤    [i(x_k,3^(e), x_k,4^(e), y_k,3, y_k,4 ) ≤γ_2]  + M_e ·ℙ[i(x̅_k,3^(e), x̅_k,4^(e), y_k,3, y_k,4 )> γ_2]

for any $\gamma_2$, where $\bar{\vect x}_{k,3}^{(e)} \sim f_{\vect X_{k,3}^{(e)}}$ and $ \bar{\vect x}_{k,4}^{(e)}\sim f_{\vect X_{k,4}^{(e)}}$ and are  independent of $(\vect x_{k,3}^{(e)}, \vect x_{k,4}^{(e)}, \vect y_{k,3}, \vect y_{k,4} )$.

§ ACKNOWLEDGMENT

The works of M. Wigger and η. have been supported by the
European Union's Horizon 2020 Research And Innovation Programme,
grant agreements no. 715111 for M. Wigger and no. 694630 for η. . The work of H. Nikbakht and JM Gorce have been supported by the Nokia Bell Labs - Inria common lab, grant agreement “Network Information Theory”.
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