The Reliability Function for the Additive White Gaussian Noise Channel at Rates above the Capacity

TL;DR

This paper analyzes the error probability behavior of additive white Gaussian noise channels at rates exceeding capacity, establishing exponential decay and deriving the optimal error exponent using the information spectrum approach.

Contribution

It introduces a new analysis of error probabilities above capacity and derives the optimal exponent function using the information spectrum method.

Findings

Error probability tends to one exponentially for rates above capacity

Derivation of the optimal error exponent function

Demonstration of the usefulness of the information spectrum approach

Abstract

We consider the additive white Gaussian noise channels. We prove that the error probability of decoding tends to one exponentially for rates above the capacity and derive the optimal exponent function. We shall demonstrate that the information spectrum approach is quite useful for investigating this problem.

Equations

$$W={\left\{W(y|x)\right\}}_{(x,y)\in \mathcal{X}\times \mathcal{Y}}.$$

$$Y=X+N,X\perp N,$$

$$W(y|x)=p_N(y-x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{\mathrm{e}}^{-\frac{(y-x{)}^2}{{\sigma }^2}}.$$

$$Y_t=X_t+N_t,X_t\perp N_t,\text{mbox}fort=1,2,\cdots ,n,$$

$$W^n={\left\{W^n(y^n|x^n)\right\}}_{(x^n,y^n)\in {\mathcal{X}}^n\times {\mathcal{Y}}^n}.$$

$$W^n(y^n|x^n)=\prod _{t=1}^n{p}_N(y_t-x_t).$$

$${\mathcal{S}}_{\Gamma }^{(n)}$$

$${\mathrm{P}}_{\mathrm{e}}^{(n)}$$

$${\mathrm{P}}_{\mathrm{e}}^{(n)}={\mathrm{P}}_{\mathrm{e}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W)$$

$${\mathrm{P}}_{\mathrm{c}}^{(n)}={\mathrm{P}}_{\mathrm{c}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W)\stackrel{△}{=}1-{\mathrm{P}}_{\mathrm{e}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W)$$

$${\mathrm{P}}_{\mathrm{c}}^{(n)}={\mathrm{P}}_{\mathrm{c}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W)$$

$${\mathrm{P}}_{\mathrm{e}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W)$$

$$C_{\mathrm{AWGN}}(\Gamma |{\sigma }^2)\stackrel{△}{=}\underset{\epsilon \in (0,1)}{\inf }{C}_{\mathrm{AWGN}}(\epsilon ,\Gamma |{\sigma }^2),$$

$$\displaystyle{\rm P}_{{\rm e,{\color[rgb]{0,0,0}m\normalcolor}}}^{(n)}$$

$$\displaystyle{\rm P}_{{\rm e},{\color[rgb]{0,0,0}\rm m\normalcolor}}^{(n)}(\varphi^{(n)},\psi^{(n)}|W)$$

$${C}_{{\color[rgb]{0,0,0}\rm m\normalcolor},\rm AWGN}(\Gamma|\sigma^{2})=\inf_{\varepsilon\in(0,1)}C_{{\color[rgb]{0,0,0}\rm m\normalcolor},\rm AWGN}(\varepsilon,\Gamma|\sigma^{2})$$

$$C(\Gamma |{\sigma }^2)=\underset{\genfrac{}{}{0ex}{}{p_X\in {\mathcal{G}}_1:}{{\mathrm{E}}_{p_X}{X}^2\le \Gamma }}{\max }I(p_X,W)=\frac{1}{2}\log \left(1+\frac{\Gamma }{{\sigma }^2}\right).$$

$$C_{\mathrm{m},\mathrm{AWGN}}(\Gamma |{\sigma }^2)=C_{\mathrm{AWGN}}(\Gamma |{\sigma }^2)=C(\Gamma |{\sigma }^2).$$

$$\underset{n\to \infty }{\lim }{\mathrm{P}}_{\mathrm{e}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W)=1$$

$$\frac{1}{n}\underset{n\to \infty }{\mathrm{liminf}}{M}_n\ge R.$$

$$C_{\mathrm{m},\mathrm{AWGN}}(\epsilon ,\Gamma |{\sigma }^2)=C_{\mathrm{AWGN}}(\epsilon ,\Gamma |{\sigma }^2)=C(\Gamma |{\sigma }^2).$$

$$G^{(n)}(R,\Gamma |{\sigma }^2)$$

$$\stackrel{△}{=}\underset{\genfrac{}{}{0ex}{}{({\phi }^{(n)},{\psi }^{(n)}):}{\genfrac{}{}{0ex}{}{{\phi }^{(n)}(K_n)\in {\mathcal{S}}_{\Gamma }^{(n)},}{(1/n)\log M_n\ge R}}}{\min }\left(-\frac{1}{n}\right)\log {\mathrm{P}}_{\mathrm{c}}^{(n)}({\phi }^{(n)},{\psi }^{(n)}|W),$$

$$G^{\ast }(R,\Gamma |{\sigma }^2)\stackrel{△}{=}\underset{n\to \infty }{\lim }{G}^{(n)}(R,\Gamma |{\sigma }^2).$$

$$G^{(n+m)}(R,\Gamma |{\sigma }^2)$$

$$G^{\ast }(R,\Gamma |{\sigma }^2)\le R.$$

$${\overline{G}}_{\mathrm{DK}}(R,\Gamma |{\sigma }^2)$$

$$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\min_{\scriptstyle q_{XY}\in\mathcal{G}_{2}:\atop{\scriptstyle{\rm E}_{q_{X}}[X^{2}]\leq\Gamma}}\big{\{}[R-I(q_{X},q_{Y|X})]^{+}+D(q_{Y|X}||W|q_{X})\big{\}},$$

$$I(q_X,q_{Y|X})$$

$$D(q_{Y|X}||W|q_X)$$

$$G^{\ast }(R,\Gamma |{\sigma }^2)\le {\overline{G}}_{\mathrm{DK}}(R,\Gamma |{\sigma }^2).$$

$${\Omega }^{(\mu ,\lambda )}(q_X,Q|{\sigma }^2)$$

Full text

I Introduction

It is well known that discrete memoryless channels(DMCs) have a property that the error probability of decoding goes to one as the block length $n$ of transmitted codes tends to infinity at rates above the channel capacity. This property is called the strong converse property. In this paper we study the strong converse property for additive white Gaussian noise channels(AWGNs). Han [1] proved that we have the strong converse property for AWGNs. In [1], Han introduced a new method to study several coding problems of information theory. This method is called the information spectrum method. Based on the information spectrum method, Han [1] gave a simple proof of the strong property for AWGNs.

In this paper for AWGNs we study an asymptotic behavior for the correct probability of decoding to vanish at rates above the capacity. To our knowledge we have had no work on this subject. In the case of DMCs, Arimoto [2] proved that the correct probability of decoding vanishes exponentially at rates above the capacity and derive an explicit lower bound of this exponent. Dueck and Körner [3] determined the optimal exponent. The equivalence of the bound of Arimoto [2] to the optimal bound of Dueck and Körner [3] was proved by Oohama [4]. Derivations of the optimal exponent using information spectrum method was investigated by Nagaoka [5], Hayashi and Nagaoka [6], and Oohama [7].

In this paper we determine the optimal exponent function on the correct probability of decoding at rates above capacity for AWGNs. To obtain this result we use a lower bound of the exponent function derived by Oohama [7]. This lower bound is not computable since it has an expression of the variational problem. We solve this problem by using its min-max structure, obtaining an explicit formula of this lower bound. This formula coincides with an upper bound obtained by another argument, thereby establishing the optimal exponent. Those arguments are quite simple and elementary.

II

The Capacity of the Additive White Gaussian Noise Channels

Let ${\cal X}$ and ${\cal Y}$ be real lines and let $X$ $\in{\cal X},$ and $Y\in{\cal Y}$ be real valued random variables. The set ${\cal X}$ corresponds to a channel input and the set ${\cal Y}$ corresponds to a channel output. We write a conditional probability density function(p.d.f.) of $Y$ on given $X$ as

[TABLE]

Let $N$ be a Gussian random variable with mean 0 and variance ${\sigma^{2}}$. The additive Gaussian noise channel is defined by

[TABLE]

where $X\perp N$ stands for that $X$ and $N$ are independent. In this case $W=W(y|x)=p_{N}(y-x)$ is a probability density function(p.d.f.) given by

[TABLE]

Let $X^{n}=(X_{1},X_{2},\cdots,X_{n})$ be a random vector taking values in ${\cal X}^{n}$. We write an element of ${\cal X}^{n}$ as x^{n}=x_{1}x_{2}$$\cdots x_{n}. Similar notations are adopted for other random variables. Let $Y^{n}\in{\cal Y}^{n}$ be a random variable obtained as the channel output by connecting $X^{n}$ to the input of channel. The additive white Gaussian noise(AWGN) channel we study in this paper is defined by

[TABLE]

where $N_{t},$ $t=1,2\cdots,n$ are independent Gussian random variables having the same distribution as $N$. We write a conditional probability p.d.f. of $Y^{n}$ on given $X^{n}$ as

[TABLE]

By the definition (1) of the AWGN channel, we have

[TABLE]

The AWGN channel is specified with $\sigma^{2}$. Let $K_{n}$ be uniformly distributed random variables taking values in message sets ${\cal K}_{n}$. The random variable $K_{n}$ is a message sent to the receiver. A sender transforms $K_{n}$ into a transmitted sequence $X^{n}$ using an encoder function and sends it to the receiver. In this paper we assume that the encoder function $\varphi^{(n)}$ is a deterministic encoder. In this case, $\varphi^{(n)}$ is a one-to-one mapping from ${\cal K}_{n}$ into ${\cal X}^{n}$. The decoding function at the receiver is denoted by ${\psi}^{(n)}$. This function is formally defined by ${\psi}^{(n)}:{\cal Y}^{n}\to{\cal K}_{n}.$ Let $c:{\cal X}\to[0,\infty)$ be a cost function. The average cost on output of $\varphi^{(n)}$ must not exceed $\Gamma$. This condition is given by $\varphi^{(n)}({K}_{n})\in{\cal S}_{\Gamma}^{(n)}$, where

[TABLE]

We consider the case where the input constrain $c(X)$ is a power constraint give by $c(x)=x^{2}$. The average error probabilities of decoding at the receiver is defined by

[TABLE]

For $k\in{\cal K}_{n}$, set ${\cal D}(k)\stackrel{{\scriptstyle\triangle}}{{=}}\{y^{n}:\psi^{(n)}(y^{n})=k\}.$ The families of sets $\{{\cal D}(k)\}_{k\in{\cal K}_{n}}$ is called the decoding regions. Using the decoding region, ${\rm P}_{\rm e}^{(n)}$ can be written as

[TABLE]

where $x_{t}(k)=[\varphi^{(n)}(k)]_{t}$, $t=1,2,\cdots,n$ are the $t$-th components of $x^{n}=x^{n}(k)$ $=\varphi^{(n)}(k)$ and $|{\cal K}_{n}|$ is a cardinality of the set ${\cal K}_{n}$. Set

[TABLE]

The quantity ${\rm P}^{(n)}_{\rm c}$ is called the average correct probability of decoding. This quantity has the following form

[TABLE]

For given $\varepsilon$ $\in(0,1)$, $R$ is $\varepsilon$-achievable under $\Gamma$ if for any $\delta>0$, there exist a positive integer $n_{0}=n_{0}(\varepsilon,\delta)$ and a sequence of pairs $\{(\varphi^{(n)},\psi^{(n)}):\varphi^{(n)}({K}_{n})\in{\cal S}_{\Gamma}^{(n)}\}_{n=1}^{\infty}$ such that for any $n\geq n_{0}(\varepsilon,\delta)$,

[TABLE]

The supremum of all $\varepsilon$-achievable $R$ under $\Gamma$ is denoted by ${C}_{\rm AWGN}(\varepsilon,\Gamma|\sigma^{2})$. We set

[TABLE]

which is called the channel capacity. The maximum error probability of decoding is defined by as follows:

[TABLE]

Based on this quantity, we define the maximum capacity as follows. For a given $\varepsilon\in(0,1)$, {\color[rgb]{0,0,0}R\normalcolor} is $\varepsilon$-achievable under $\Gamma$, if for any $\delta>0$, there exist a positive integer $n_{0}=n_{0}(\varepsilon,\delta)$ and a sequence of pairs $\{(\varphi^{(n)},\psi^{(n)}):\varphi^{(n)}({K}_{n})\in{\cal S}_{\Gamma}^{(n)}\}_{n=1}^{\infty}$ such that for any $n\geq n_{0}(\varepsilon,\delta)$,

[TABLE]

The supremum of all $\varepsilon$-achievable rates under $\Gamma$ is denoted by C_{{\color[rgb]{0,0,0}\rm m\normalcolor},{\rm AWGN}}(\varepsilon,\Gamma|\sigma^{2}). We set

[TABLE]

which is called the maximum capacity of the AWGN. Let ${\cal G}_{d}$ be a set of all $d$ dimensional Gaussian distributions. Set

[TABLE]

where $I(p_{X},W)$ stands for a mutual information between $X$ and $Y$ when input distribution of $X$ is $p_{X}$. The following is a well known result.

Theorem 1

For any AWGN, we have

[TABLE]

Han [1] established the strong converse theorem for AWGNs. His result is as follows.

Theorem 2** (Han [1])**

If $R>C(\Gamma|\sigma^{2})$, we have

[TABLE]

for any $\{(\varphi^{(n)},\psi^{(n)}):\varphi^{(n)}({K}_{n})\in{\cal S}_{\Gamma}^{(n)}\}_{n=1}^{\infty}$ satisfying

[TABLE]

The following corollary immediately follows from this theorem.

Corollary 1

For each fixed $\varepsilon$ $\in(0,1)$ and any AWGN specified with $\sigma^{2}$, we have

[TABLE]

To examine an asymptotic behavior of ${\rm P}^{(n)}_{\rm c}(\varphi^{(n)},\psi^{(n)}|W)$ for large $n$ at $R>C(\Gamma|{\sigma^{2}})$, we define the following quantities:

[TABLE]

On the above exponent functions, we have the following property.

Property 1

**

• a)

By definition we have that for each fixed $n\geq 1$, $G^{(n)}(R,\Gamma|{\sigma^{2}})$ is a monotone increasing function of $R\geq 0$ and satisfies $G^{(n)}(R,\Gamma|{\sigma^{2}})\leq R$.

• b)

The sequence $\{G^{(n)}(R,\Gamma|{\sigma^{2}})$ $\}_{n\geq 1}$ of exponent functions satisfies the following subadditivity property:

[TABLE]

from which we have that ${G}^{*}(R,\Gamma|{\sigma^{2}})$ exists and is equal to $\inf_{n\geq 1}G^{(n)}(R,\Gamma|{\sigma^{2}})$.

• c)

For fixed $R>0$, the function ${G}^{*}(R,\Gamma|{\sigma^{2}})$ is a monotone decreasing function of $\Gamma$. For fixed $\Gamma>\Gamma_{0}=\min_{x\in{\cal X}}c(x)$, the function ${G}^{*}(R,\Gamma|{\sigma^{2}})$ a monotone increasing function of $R$ and satisfies

[TABLE]

• d)

The function ${G}^{*}(R,\Gamma|{\sigma^{2}})$ is a convex function of $(R,\Gamma)$.

Proof of this property is found in [7].

III Main Results

In this section we state our main results. We first present a result on an upper bound of $G^{*}(R,\Gamma|\sigma^{2})$. Define

[TABLE]

where $[t]^{+}=\max\{0,t\}$ and

[TABLE]

Then we have the following theorem.

Theorem 3

For any $R>0$,

[TABLE]

Proof of this theorem is given in Appendix -A. We next derive a lower bound of $G^{*}(R,\Gamma|{\sigma^{2}})$. To this end we define several quantities. Define

[TABLE]

According to Oohama [7], we have the following theorem.

Theorem 4** (Oohama [7])**

For any AWGN $W$, we have

[TABLE]

To find an explicit formula of $G(R,\Gamma|{\sigma^{2}})$, set

[TABLE]

It is obvious that

[TABLE]

For $\lambda\in[0,1)$, define

[TABLE]

Furthermore, set

[TABLE]

According to Oohama [7], we have the following lemma.

Lemma 1** (Oohama [7])**

For any $q_{X}$,

[TABLE]

The above minimization is attained by the probability density function $Q$ having the form

[TABLE]

where

[TABLE]

is a constant for normalization.

From Lemma 1, we have the following proposition.

Proposition 1

For any $\mu,\lambda\geq 0$, we have

[TABLE]

In particular, we have

[TABLE]

Let $q_{X,\theta}\in{\cal G}_{1}$ be the p.d.f. of the Gaussian distribution with mean 0 and variance $\theta$. We set

[TABLE]

The following proposition is a key result of this paper.

Proposition 2

For every $\lambda\geq 0$ and for every $\mu\in[0,\frac{1+\lambda}{2{\sigma^{2}}}]$, we have

[TABLE]

The input Gaussian distribution $q_{X,\theta}\in{\cal G}_{1}$ attaining $\overline{G}_{\rm AR}^{(\mu,\frac{\lambda}{1+\lambda})}($ $R,\Gamma|\sigma^{2})$ satisfies the following:

[TABLE]

Furthermore, we have

[TABLE]

Proof of this proposition is given in Section IV. We next drive a relation between $\overline{G}_{\rm AR}(R,\Gamma|{\sigma^{2}})$ and $\overline{G}_{\rm DK}(R,\Gamma|{\sigma^{2}})$. To this end we derive two parametric expressions of $\overline{G}_{\rm DK}(R,$ $\Gamma|{\sigma^{2}})$. For $\mu>0$, we define

[TABLE]

For $\mu,\lambda\geq 0$, we define

[TABLE]

According to Oohama [7], we have the following lemma.

Lemma 2** (Oohama [7])**

For any $R>0$,

[TABLE]

For any $\mu\geq 0$, any $R>0$, we have

[TABLE]

The two equalities (16) and (17) imply that

[TABLE]

We can show the following proposition stating that the two quantities $\overline{G}_{\rm AR}(R,\Gamma|{\sigma^{2}})$ and $\overline{G}_{\rm DK}(R,\Gamma|{\sigma^{2}})$ match.

Proposition 3

For any $\mu,\lambda\geq 0$, we have the following:

[TABLE]

In particular, we have

[TABLE]

Proof of this proposition is given in Section IV. From Theorems 3, 4 and Propositions 2, 3, we immediately obtain the following theorem.

Theorem 5

[TABLE]

We finally solve the maximization problem defining $G($ $R,\Gamma|{\sigma^{2}})$ to derive an explicit formula of this exponent function. We set

[TABLE]

Then we have

[TABLE]

where $h(\cdot)$ is the binary entropy function. Since by (23), we can regard $L^{(\mu,\lambda)}(R,\Gamma|{\sigma^{2}})$ as a quantity with parametor $\rho$ and $\nu$, we denote it by $L^{(\rho,\nu)}(R,\Gamma|{\sigma^{2}})$. Then we have

[TABLE]

Solving the optimization problem (24) defining $G(R,\Gamma|\sigma^{2})$, we obtain the following result.

Theorem 6

Let $\nu_{0}$ be the unique positive solution of

[TABLE]

Then, the exponent function $G(R,\Gamma|\sigma^{2})$ has the following parametric form:

[TABLE]

Proof of this theorem is given in Section IV.

IV Proofs of Results

In this section we prove Propositions 2, 3 and Theorem 6 stated in Section III.

For $\theta\geq 0$, we let $\xi=\xi(\mu,\lambda,\theta)$ be the same quantity as that defined by (10). Let $Q_{\xi+{\sigma^{2}}}\in{\cal G}_{1}$ be a Gaussian distribution with mean 0 and variance $\xi+{\sigma^{2}}$. Applying Lemma 1 to $q_{X,\theta}\in{\cal G}_{1}$, we have the following.

Lemma 3

The function $\Omega^{(\mu,\lambda)}(q_{X,\theta},Q|{\sigma^{2}})$ takes the minimum value at $Q=Q_{\xi+{\sigma^{2}}}$. The value is

[TABLE]

Furthermore, we have

[TABLE]

*Proof: * By an elementary computation we have

[TABLE]

which together with Lemma 1 and (8) in this lemma yields that $\Omega^{(\mu,\lambda)}(q_{X,\theta},Q|{\sigma^{2}})$ takes the minimum value at the p.d.f. $Q=Q(y)$ given by

[TABLE]

From (27), we can see that $\kappa$ must satisfy

[TABLE]

The minimum value is

[TABLE]

Step (a) follows from (9) in Lemma 1. Step (b) follows from (28). Step (c) follows from that by the definition of $\xi$, we have

[TABLE]

Hence we have

[TABLE]

completing the proof. ∎

On a lower bound of $\underline{G}(R,\Gamma|{\sigma^{2}})$, we have the following lemma.

Lemma 4

Suppose that $\mu\in[0,\frac{1+\lambda}{2{\sigma^{2}}}]$. We choose $\eta=\eta(\mu,\lambda)$ so that

[TABLE]

Then for any $q_{X}$, we have

[TABLE]

This implies that

[TABLE]

Proof: By an elementary computation we have

[TABLE]

Step (a) follows from (29). Then we have the following chain of equalities:

[TABLE]

Step (a) follows from (30). Hence we have

[TABLE]

Step (a) follows from that by (29), we have

[TABLE]

Step (b) follows from (29). ∎

Proof of Proposition 2: We assume that $\lambda\geq 0$ and $\mu\in[0,\frac{1+\lambda}{2{\sigma^{2}}}]$. In this case by Lemma 3, we have

[TABLE]

which together with Lemma 4 yields the equality (12). We next prove (13). We set

[TABLE]

Then we have

[TABLE]

By the equality (12) already proved, we have

[TABLE]

Hence it suffices to prove $B\leq A$. When $\lambda\geq 0$ and $\mu\geq\frac{1+\lambda}{2{\sigma^{2}}}$, we have the following chain of inequalities:

[TABLE]

Step (a) follows from that by Lemma 3,

[TABLE]

Step (b) follows from $\lambda\geq 0$, $\mu\geq\frac{1+\lambda}{2{\sigma^{2}}}$. Step (c) follows from the definition of $L^{(\mu,\lambda)}(R,\Gamma|{\sigma^{2}})$. Step (d) follows from (33). Thus we have $B\leq A$, completing the proof. ∎

*Proof of Proposition 3: * For $q_{X}\in{\cal G}_{1}$, we set

[TABLE]

For a given joint Gaussian p.d.f. $q=q_{XY}\in{\cal G}_{2}$, we introduce the conditional p.d.f. $q_{X|Y}$ and the p.d.f. ${}q_{Y}$ by

[TABLE]

The above $q_{X|Y}$ is called a backward channel. Set

[TABLE]

Using $(q_{Y},$ $q_{X|Y})$, we obtain the following:

[TABLE]

where $\hat{q}_{X|Y}$ $=\hat{q}_{X|Y}(x|y)$ is a conditional p.d.f. given by

[TABLE]

and $\hat{q}_{Y}$ $=\left\{\hat{q}_{Y}(y)\right\}_{y\in{\cal Y}}$ is a p.d.f. given by

[TABLE]

Hence, by (36) and the non-negativity of divergence, we obtain

[TABLE]

for any $q_{X}\in{\cal G}_{1}$. Hence we have

[TABLE]

We next prove that for any $\lambda\geq 0$,

[TABLE]

Let $q_{X,\theta}$ be a Gaussian distribution with mean 0 and variance $\theta$. We assume that $q_{X,\theta}$ attains $\overline{G}_{\rm AR}^{(\mu,\frac{\lambda}{1+\lambda})}(R,\Gamma|W)$. Then by Proposition 2, we have

[TABLE]

From (39), we have

[TABLE]

From (40), we have

[TABLE]

Computing $\Lambda(y)$ for this $q_{X,\theta}$, we have

[TABLE]

Step (a) follows from (26). Step (b) follows from (41). Using (42), we have the following:

[TABLE]

Using (42), we also have

[TABLE]

Step (a) follows from the first equality of (40). From (43) and (44), we have

[TABLE]

Now we define the function ${V}=V(y|x)$ by

[TABLE]

By (37) and (38), $V(y|x)$ has the following form:

[TABLE]

Taking integral of both sides of (47) on the variable $y$, we obtain

[TABLE]

Step (a) follows from (45). The above equality implies that $V$ is a conditional density function. Furthermore, note that from (46),

[TABLE]

Then, choosing $q_{Y}=\hat{q}_{Y},q_{X|Y}=\hat{q}_{X|Y}$ in (36), we have, for $\lambda\geq 0$,

[TABLE]

completing the proof. ∎

Proof of Theorem 6: We set

[TABLE]

Then we have

[TABLE]

Computing the gradient of $F$, we obtain

[TABLE]

Let

[TABLE]

be the Hessian matrix of $F$. Computing $B$, we have

[TABLE]

Step (a) follows from $\rho\in[\nu/(1+\nu),1)$. Computing $|B|$, we have

[TABLE]

Step (a) follows from $\rho\in[\nu/(1+\nu),1)$. From (50), (51), and (52), we can see that $F(\rho,\nu)$ is a convex function of $(\rho,\nu)$. Hence from (49), we have that if the pair $(\rho^{*},\nu^{*})$ satisfies $\rho^{*}\in[\nu^{*}/(1+\nu^{*}),1)$ and

[TABLE]

then, it attains the minimum of $F(\rho,\nu)$ under $\rho\in[\nu/(1+\nu),1)$. From (53), we have

[TABLE]

Furthermore, for $\nu^{*}\in[0,\nu_{0})$, we have

[TABLE]

From (54)-(56), we can see that for $\nu^{*}\in[0,\nu_{0})$, the pair $(\rho^{*},\nu^{*})$ certainly attains $G(R,\Gamma|\sigma^{2})=F(\rho^{*},\nu^{*})$. From (54) and (55), we have

[TABLE]

Furthermore, we have

[TABLE]

Step (a) follows from (54) and (55). Thus we obtain the parametric expression of $G(R,\Gamma|\sigma^{2})$ given by (57) and (58). ∎

-A Proof of Theorem 3

We fix $\delta\in[0,1/2)$. We consider a Gaussian channel with input $X$ and output $Y$, having the form

[TABLE]

We consider a Gaussian random pair $(X,Y)$ obtained by letting $X$ be Gaussian random variable with $X\sim{\cal N}(0,\theta)$. We assume that $\theta\leq\Gamma$. Let $q_{XY}$ be a probability density function of $(X,Y)$. For the Gaussian channel specified by $q_{Y|X}$, we can construct an $n$-length block code $(\phi^{(n)},\psi^{(n)})$ with message set ${\cal K}_{n}$ satisfying:

• a)

${\rm P}_{\rm c}^{(n)}(\phi^{(n)},\psi^{(n)}|q_{Y|X})\geq 1-\delta$.

• b)

all codewords $\phi^{(n)}(k),k\in{\cal K}_{n}$ satisfy $||\phi^{(n)}(k)||^{2}\leq n\theta$.

• c)

$\frac{1}{n}\log|\mathcal{K}_{n}|\geq\min\{R,I(q_{X},q_{Y|X})-\delta\}$.

By the condition b), we can obtain the following result.

Lemma 5

For every $k\in{\cal K}_{n}$, we have

[TABLE]

Proof: By a direct computation we have

[TABLE]

Step (a) follows from the condition b). ∎

For $k\in{\cal K}_{n}$, we set

[TABLE]

Furthermore, set

[TABLE]

The quantity ${\rm P}_{\rm c}^{(n)}(\phi^{(n)},\psi^{(n)}|{W})$ has a lower bound given by the following Lemma.

Lemma 6

For any $\delta\in[0,1/2)$, we have

[TABLE]

Here we set $\eta_{n}(\delta)\stackrel{{\scriptstyle\triangle}}{{=}}\frac{1}{n}(1-\delta)^{-1}h(1-\delta)$ and $h(\cdot)$ stands for a binary entropy function.

Proof: We have the following chain of inequalities:

[TABLE]

Step (a) follows from Lemma 5. Steps (b) and (c) follow from the log-sum inequality. Step (d) follows from that

[TABLE]

and $\delta\in(0,1/2].$ From (62), we obtain

[TABLE]

completing the proof. ∎

Proof of Theorem 3: We first consider the case where $R\leq I(q_{X},q_{Y|X})-\delta$. In this case we choose $\varphi^{(n)}=\phi^{(n)}$. Then we have

[TABLE]

Step (a) follows from the condition $R+\delta-I(q_{X},q_{Y|X})\leq 0$. Step (b) follows from that

[TABLE]

We next consider the case where $R>I(q_{X},q_{Y|X})-\delta$. Consider the new message set $\widehat{\mathcal{K}}_{n}$ satisfying $|\widehat{\mathcal{K}}_{n}|={\rm e}^{\lfloor nR\rfloor}$. For new message set $\widehat{\cal K}_{n}$, we define $\varphi^{(n)}(k)$ such that $\varphi^{(n)}(k)=\phi^{(n)}(k)$ if $k\in\mathcal{K}_{n}$. For $k\in\widehat{\mathcal{K}}_{n}-\mathcal{K}_{n}$, we define $\varphi^{(n)}(k)$ arbitrary sequence of ${\cal X}^{n}$ having the type $q_{X}$. We use the same decoder $\psi^{(n)}$ as that of the message set $\mathcal{K}_{n}$. Then we have the following:

[TABLE]

Step (a) follows from (61) in Lemma 6. Step (b) follows from $|\mathcal{K}_{n}|\geq{\rm e}^{n[(I(q_{X},q_{Y|X})-\delta]}.$ Step (c) follows from $[a]\leq[a]^{+}$. Combining (63) and (64), we have

[TABLE]

for any $q_{X}\in{\cal G}_{1}$ with ${\rm E}_{q_{X}}[X^{2}]\leq\Gamma$ and q_{Y|X}\in{\cal G}_{1}($$q_{X}). Hence from (65), we have

[TABLE]

We note that $\eta_{n}(\delta)\to 0$ as $n\to\infty$. Hence by letting $n\to\infty$ in (66), we obtain

[TABLE]

Since $\delta$ can be made arbitrary small, we conclude that $G^{*}(R$ $,\Gamma|\sigma^{2})\leq G_{\rm DK}(R,\Gamma|\sigma^{2}).$ ∎