Consider the MSE *D*(*ρ*,*θ*
_{
o
},*θ*) in (9), where *θ* is a random variable (but assumed to remains constant at least for the duration of a single channel codeword), where \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) is given by either (13) or (17). This is the MMSE of an asymptotically optimal HDA-PQ or HDA-TC for a particular (*ρ*,*θ*
_{
o
}). The choice of *ρ* and *θ*
_{
o
} determines how the MMSE varies with the CSNR *θ*. If *θ* is known to the transmitter, *ρ*=0 (purely digital) will achieve the lowest MMSE for any *θ*, since in this case,\(D(0,\theta _{o},\theta)=\sigma _{\epsilon }^{2}(0,\theta _{o})\) can be minimized by choosing *θ*
_{
o
}=*θ*. In this case, both PQ and TC achieve the maximum possible coding gain. If however the receiver CSNR *θ* is not available to the transmitter, a purely digital system must be designed for some *θ*
_{
o
} which will be different to *θ*, resulting in a system that is not robust against CSNR variations. On the one hand, the receiver MSE of such a system remains constant even when *θ*>*θ*
_{
o
} despite the increase in the available channel capacity. On the other hand, the channel code and hence the system fail when *θ*<*θ*
_{
o
}, i.e., system goes into outage. We refer to *θ*
_{
o
} as the outage CSNR of the digital decoder. When the transmitter cannot be adapted to varying *θ*, allocating power to the analog transmission (*ρ*>0) while keeping *θ*
_{
o
} fixed will increase the quantization MSE \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) but will make the overall MSE *D*(*ρ*,*θ*
_{
o
},*θ*) to decrease with *θ*. For fixed *ρ*, increasing *θ*
_{
o
} reduces \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) but will increase the *outage probability* and hence the AMMSE. In order to obtain a robust system which is optimal in some sense over a range of *θ*, we design the transmitter for *ρ* and *θ*
_{
o
} which minimizes the AMMSE *E*{*D*(*ρ*,*θ*
_{
o
},*θ*)} with respect to the distribution of *θ*. Such a design is ideal for a system with a single receiver which experiences slow fading or a broadcast environment with a large number of receivers whose empirical CSNR converges to the fading distribution [26].

The AMMSE of HDA-PQ or HDA-TC is given by

$$ \bar{D}(\rho,\theta_{o})= \left\{ \begin{array}{lc} E \{D(\rho,\theta_{o},\theta)| \theta \geq \theta_{o} \}(1-P_{o})+\sigma_{X}^{2}P_{o} & \rho<1 \\ E\{D_{a}(\theta) \} & \rho=1 \end{array} \right. $$

(18)

where \(D_{a}(\theta)=\frac {\sigma _{X}^{2}}{1+b\theta }\) is the MMSE of the optimal analog system and *P*
_{
o
}=*P*
*r*(*θ*<*θ*
_{
o
}) is the outage probability, and we assume that in the event of an outage, the decoder output is set to \(\hat {x}'_{n}=E\{ x_{n}\}\). It is assumed that the distribution of *θ* is a priori known to the system designer. Our main focus is the Rayleigh fading channel in which the CSNR *θ* is exponentially distributed [27]. The pdf of *θ* is given by

$$ p(\theta)=\frac{1}{\bar{\theta}}exp\left(-\frac{\theta}{\bar{\theta}}\right), $$

(19)

where \(\bar {\theta }= E \{ \theta \}\) is the *mean* CSNR. For the case of Rayleigh fading, from (9), (18), and (19), it follows that

$$\begin{array}{@{}rcl@{}} \bar{D}(\rho,\theta_{o}) & =& \sigma_{\epsilon}^{2}(\rho,\theta_{o}) \int_{\theta_{0}}^{\infty} \frac{exp\left(-\frac{\theta}{\bar{\theta}}\right)}{\bar{\theta}(1+ b\theta \rho)}d\theta + \sigma_{X}^{2}P_{o} \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} &=& \sigma_{\epsilon}^{2}(\rho,\theta_{o}) \frac{exp\left(\frac{1}{b \rho \bar{\theta}} \right)}{b\rho \bar{\theta}}E_{1} \left(\frac{1+b \rho\theta_{o}}{b \rho \bar{\theta}} \right)+ \sigma_{X}^{2}P_{o}, \\ \end{array} $$

(21)

where \(P_{o}=\left (1-exp \left (-\frac {\theta _{o}}{\bar {\theta }} \right)\right)\) and \(E_{1}(x)=\int _{x}^{-\infty }\frac {exp(-t)}{t}dt\) is the *exponential integral* [28]. *E*
_{1}(*x*) is available as a standard function in most numerical software [e.g., expint(x) in Matlab]. The AMMSE depends on the choice of the power allocation *ρ* and the outage probability, or equivalently *θ*
_{
o
}. We define the optimal *robust* HDA system as the one which achieves the minimum AMMSE. The optimal values of *ρ* and *θ*
_{
o
} can be found by solving the problem

$$\begin{array}{@{}rcl@{}} \left(\rho^{*}, \theta_{o}^{*}\right)& = &\arg \min_{\rho,\theta_{o}} \bar{D}(\rho, \theta_{o}) \\ \text{subject to} & & 0 \leq \rho < 1 \\ & & \theta_{0}>0. \end{array} $$

(22)

For fixed *θ*
_{
o
}, \(\bar {D}(\rho, \theta _{o})\) is convex in *ρ*∈(0,1). This can be deduced from (20): \(\sigma _{\epsilon }^{2}(\rho,\theta _{o})\) monotonically increases with *ρ* while the term inside the integral monotonically decreases. This represents the trade-off between the coding gain of PQ or TC due to source memory and the robustness against CSNR variations. There must be a value for *ρ*∈(0,1), which minimizes the AMMSE. Now if *ρ* is fixed, \(\bar {D}(\rho, \theta _{o})\) is quasi-convex in *θ*>0. This is because, as *θ*
_{
o
} is increased (*P*
_{
o
} increases), the first term of the sum in (18) *E*{*D*(*ρ*,*θ*
_{
o
},*θ*)|*θ*≥*θ*
_{
o
}} decreases while the second term \(\sigma _{X}^{2}P_{o}\) increases. A minimum for \(\bar {D}(\rho, \theta _{o})\) occurs for some *θ*
_{
o
}<*∞*. The quasi-convexity follows from the fact that, as *θ*
_{
o
}→*∞*, the system will be always in outage and hence \(\bar {D}(\rho, \theta _{o}) \to \sigma _{X}^{2}\). Figure 2 shows the AMMSEs of HDA-PQ and HDA-TC as a function of *ρ* and *θ*
_{
o
} for the Gauss-Markov process, which we will refer to as the *G*
*M*(*a*) source,

$$ X_{n}={aX}_{n-1}+W_{n}. $$

(23)

Figure 2 illustrates the convexity with respect to *ρ* and quasi-convexity with respect to *θ*
_{
o
}. Below we present an efficient method to determine the optimal solution for (*ρ*,*θ*
_{
o
}) in the case of HDA-PQ. Due to the lack of a closed-form expression for the AMMSE, such a simple procedure cannot be devised for HDA-TQ.

### 4.1 Optimal HDA-PQ

In general, it is difficult to find a closed-form solution to the constrained non-linear minimization problem in (22). In the following, we present a simple *coordinate-descent* (CD) method [15] to solve this problem. In the CD method, \(\bar {D}(\rho, \theta _{o})\) is minimized alternately with respect to *ρ* (for fixed *θ*
_{
o
}) and *θ*
_{
o
} (for fixed *ρ*
_{
o
}), until the solution converges. Unlike the joint minimization problem in (22), these two sub-problems are much easier to solve. Since the solution to each problem is conditionally optimal, the CD algorithm is guaranteed to converge to the minimum of \(\bar {D}\). In actual numerical examples, it was found that this method only required 2–3 iterations to converge. In the following, we present the solutions to two sub-problems solved in each CD iteration.

Before proceeding, it should be noted that in the case of HDA-PQ, an additional constraint is required to ensure that (14) is not violated. This can be stated as

$$ f_{1}(\rho,\theta_{o})<0, $$

(24)

where \(f_{1}(\rho,\theta _{o}) \triangleq c_{1}-\phi (\rho,\theta _{o})\) with *c*
_{1}=*ν*[*h*(1+*A*
^{2})]^{1/b} and *ν*>1 is a sufficiently large constant chosen to ensure that (14) is not violated at low quantization rates. In our experiments, we have used *ν*=2. If the constraint (24) becomes active, the solution is not guaranteed to be optimal. However, note that, as the quantization rate decreases, the HDA-PQ performance approaches that of analog-only transmission. Therefore, when HDA-PQ outperforms purely analog transmission, (24) is unlikely to be active. For example, with both Gauss-Markov sources and speech signals, numerical results presented in Section 6 show that when *ρ* exceeds about 30%, the difference between HDA-PQ and analog systems becomes negligible. It is in this range of *ρ* that (24) becomes active.

#### 4.1.1 Optimal power allocation for fixed outage CSNR

For a fixed *θ*
_{
o
}, optimal power allocation can be found by solving

$$\begin{array}{@{}rcl@{}} \rho^{*} & = &\arg \min_{\rho} \bar{D}(\rho, \theta_{o}) \\ \text{subject to} & & 0 \leq \rho < \rho_{\max}, \end{array} $$

(25)

where *ρ*
_{max}∈(0,1]. In this case, (24) simplifies to

$$ \rho_{\max} < \rho_{1} \triangleq\frac{(1+\theta_{o})c_{1}^{-1}-1}{\theta_{o}}. $$

(26)

and therefore

$$ \rho^{*}=\min \{ \rho',\rho_{1},1 \}, $$

(27)

where *ρ*
^{′} is the solution to \(f_{2}(\rho)\triangleq \partial \bar {D}/\partial \rho =0\). Using (9) and (13), it can be readily shown that *f*
_{2}(*ρ*)=0 is equivalent to

$$\begin{array}{@{}rcl@{}} &&\left[ \frac{b\theta_{o}}{h \sigma_{o}^{2}} \frac{\sigma_{\epsilon}^{2}(\rho,\theta_{o}) \phi^{b}(\rho,\theta_{o})}{1+\rho\theta_{o}} -\frac{(1+b \rho\bar{\theta})}{b\rho^{2}\bar{\theta}} \right]E_{1} \left(\frac{1+b \rho\theta_{o}}{b\rho\bar{\theta}} \right)\\ &&\quad+\frac{exp \left(-\frac{1+b\rho\theta_{o}}{b\rho\bar{\theta}} \right)}{\rho \left(1+b\rho\theta_{o} \right)}=0, \end{array} $$

which can be solved in the interval 0≤*ρ*<*ρ*
_{max} using a single-variable root-finding method.

#### 4.1.2 Optimum outage CSNR for fixed power allocation

For fixed *ρ*, the optimal outage CSNR can be found by solving

$$\begin{array}{@{}rcl@{}} \theta_{o}^{*} & = &\arg \min_{\theta_{o}} \bar{D}(\rho, \theta_{o}) \\ \text{subject to} & & \theta \geq \theta_{o,\min}, \end{array} $$

(28)

where, from (24)

$$ \theta_{o,\min} \geq \theta_{o_{1}} \triangleq \max\left\{ 0,\frac{c_{1}-1}{1-\rho c_{1}}\right \}. $$

(29)

Using (9) and (13), it can be verified that \(\partial \bar {D}/\partial \theta _{o}=0\) is equivalent to

$$\begin{array}{@{}rcl@{}} & &\frac{\sigma_{X}^{2}}{\bar{\theta}}-\sigma^{2}_{\epsilon}(\rho,\theta_{o}) \left[\sigma^{2}_{\epsilon}(\rho,\theta_{o})\frac{(1-\rho)}{h\sigma^{2}_{o}} \frac{(1+\theta_{o})^{b-1}}{(1+\rho \theta_{o})^{b+1}} \right. \\ & & \left.\frac{\exp \left(\frac{(1+b \rho \theta_{o})}{b\rho\bar{\theta}} \right)}{\rho \bar{\theta}} E_{1}\!\left(\!\!\frac{1+b\rho\theta_{o}}{b\rho\bar{\theta}}\!\! \right)\! +\!\frac{1}{\bar{\theta}(1+b \rho \theta_{o})}\!\! \right]\!\,=\,0 \end{array} $$

(30)

for 0<*ρ*<1 and

$$ \sigma_{X}^{2}-\sigma^{2}_{\epsilon}(0, \theta_{o})\frac{ \left(1+b\bar{\theta} \right)}{1+\theta_{o}}=0 $$

(31)

for *ρ*=0. Given *ρ*, optimal *θ*
_{
o
} can be found by locating the root of (30) or (31) in the interval *θ*
_{
o
}∈ [ *θ*
_{
o,min},*θ*
_{
o,max}), where *θ*
_{
o,max} is a suitable value chosen to truncate the pdf *p*(*θ*).