## 4 A Heuristic Algorithm to Obtain the Hard Delay Constraint Policy
### Cross-Layer Optimization Problem  
To design a truncated channel inversion policy, we formulate an optimization problem that aims to minimize the average power while satisfying the hard delay constraint. The problem is expressed as  
\[\min_{\{\mathbf{\pi}_{q},\delta_{\text{th}}(\zeta,k,l)\}} \frac{\sum\limits_{\zeta\in\mathcal{Q}}\sum\limits_{k=1}^{K}\sum\limits_{l=0 }^{L(\zeta)}\pi^{\mathbf{\mathcal{F}}}(\zeta)p(l|k,\zeta)\bar{P}(g_{th}(\zeta,k,l))}{ \sum\limits_{\zeta\in\mathcal{Q}}\pi^{\mathbf{\mathcal{F}}}(\zeta)\zeta} \tag{50a}\] \[s.t.\quad\mathbf{H}\pi^{\mathbf{\mathcal{F}}}=\pi^{\mathbf{\mathcal{F}}},\] (50b) \[\quad\mathbf{1}^{T}\pi^{\mathbf{\mathcal{F}}}=1,\] (50c) \[\quad\pi^{\mathbf{\mathcal{F}}}(\zeta)\geq 0,\;\forall\zeta, \tag{50d}\]  
where the vector \(\mathbf{1}^{T}\) is defined as a vector with all elements equal to one. In our previous work [37], we optimized the transmission rate and code blocklength. In this study, we focus on optimizing the transmission threshold of the fading channel.  
The optimization problem (50) is a non-convex problem with multiple constraints, which makes it a significant challenge to solve. To overcome this, we transform the problem into an unconstrained optimization problem.  
**Theorem 5**.: _The problem (50) is equivalent to_  
\[\min_{\{g_{th}(\zeta,k,l)\}}\frac{\sum\limits_{\zeta\in\mathcal{Q}}\sum\limits_ {k=1}^{K}\sum\limits_{l=0}^{L(\zeta)}\pi^{\mathbf{\mathcal{F}}}(\zeta)p(l|k,\zeta) \bar{P}(g_{th}(\zeta,k,l))}{\sum\limits_{\zeta\in\mathcal{Q}}\pi^{\mathbf{ \mathcal{F}}}(\zeta)\zeta}, \tag{51}\]  
_where we have_  
\[\pi^{\mathbf{\mathcal{F}}}(\zeta)=\frac{\sum_{n=1}^{|\mathcal{Q}|}|A_{\zeta+1,n}|} {|\mathbf{A}|}, \tag{52}\]  
\[\mathbf{A}=\mathbf{H}+\mathbf{1}-\mathbf{I}. \tag{53}\]  
_In this equation, \(|\mathbf{A}_{ij}|\) represents the cofactor of the \(i\)-th row and \(j\)-th column. If \(\pi^{\mathbf{\mathcal{F}}}(\zeta)\) has no solution or any of its elements are less than zero, then the optimization problem has no solution._  
Proof.: By introducing a matrix \(\mathbf{E}\) entirely composed of ones, we have  
\[\mathbf{E}\pi^{\mathbf{\mathcal{F}}}=\mathbf{1}^{T}, \tag{54}\]  
\[(H-\mathbf{I})\pi^{\mathbf{\mathcal{F}}}=\mathbf{0}, \tag{55}\]Subsequently, the following equation is derived:  
\[\boldsymbol{\pi}^{\mathcal{F}}=(\boldsymbol{H}+\boldsymbol{E}-\boldsymbol{I})^{-1 }\boldsymbol{1}^{T}. \tag{56}\]  
We define  
\[\boldsymbol{A}=\boldsymbol{H}+\boldsymbol{E}-\boldsymbol{I}=\begin{bmatrix}a_{ 1,1}&...&a_{1,|\mathcal{Q}|}\\ \vdots&\ddots&\vdots\\ a_{|\mathcal{Q}|,1}&...&a_{|\mathcal{Q}|,|\mathcal{Q}|}\end{bmatrix}. \tag{57}\]  
Then, we have  
\[\boldsymbol{\pi}^{\mathcal{F}}=\boldsymbol{A}^{-1}[1,1,...,1]^{T}=\frac{1}{| \boldsymbol{A}|}\begin{bmatrix}|\boldsymbol{A}_{1,1}|&...&|\boldsymbol{A}_{1,| \mathcal{Q}|}|\\ \vdots&\ddots&\vdots\\ |\boldsymbol{A}_{|\mathcal{Q}|,1}|&...&|\boldsymbol{A}_{|\mathcal{Q}|,| \mathcal{Q}|}|\end{bmatrix}[1,1,...,1]^{T}, \tag{58}\]  
where \(|\boldsymbol{A}_{i,j}|\) represents the cofactor of the \(i\)-th row and \(j\)-th column. Then, we have  
\[\pi^{\mathcal{F}}(\zeta)=\frac{\sum_{n=1}^{|\mathcal{Q}|}|\boldsymbol{A}_{ \zeta+1,n}|}{|\boldsymbol{A}|}. \tag{59}\]  
Up to this point, \(\pi^{\mathcal{F}}(\zeta)\) can be expressed as a function of \(\{g_{th}(\zeta,k,l)\}\). Consequently, the problem can be formulated as an unconstrained optimization problem. Then, the theorem is proved.