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).