多天线高斯信道的容量

\begin{equation} \label{eq:20120322y} \mathbf{y} = H\mathbf{x} + \mathbf{n} \end{equation}

where \(H\) is a \(r\times t\) matrix and \(\mathbf{n}\) is zero-mean complex Gaussian noise with independent, equal variance real and imaginary parts. We assume \(\mathcal{\mathbf{nn^{T}}}= I_r\) , that is the noises corrupting the different receivers are independent. The transmitter is constrained in its total power to \(P\) ,

\begin{equation} \label{eq:20120322varepsilon} \mathcal\{\mathbf{x}^{T} \mathbf{x}\} \le P \end{equation}

因为 \(\mathbf{x}^{T}\mathbf{x} = tr(\mathbf{x}\mathbf{x}^{T})\) , and expectation and trace commute,

\begin{equation} \label{eq:20120322tr} tr(\mathcal{E}\{\mathbf{x}\mathbf{x}^{T}\}) \le P \end{equation}

可以看出有功率限制和噪声分布限制。信道矩阵限制 \(H\) ，分三种情况:

1. \(H\) is deterministic.
2. \(H\) is a random matrix (for which we shall use the notation \(\pmb{H}\)) , chosen according to a probability distribution, and each use of the channel corresponds to an independent realization of \(\pmb{H}\) .

3. \(H\) is a random matrix, but is fixed once it is chosen.

   if \(Q\in \mathbb{C}^{n\times n}\) is non-negative devinite then so is \(\hat{Q}\in \mathbb{R}^{2n\times 2n}\) .

The probability density(with respect to the standard Lebesgue measure on \(\mathbb{C}^{n}\) ) of a circularly symmetric complex Gaussian with mean \(\mu\) and covariance \(Q\) is given by

\begin{eqnarray} \label{eq:20120323gamma} \gamma_{\mu,Q} &=& \det(\pi \hat{Q})^{-1/2} \exp(-(\hat{x} - \hat{\mu})^{T}\hat{Q}^{-1}(\hat{x} -\hat{\mu}) ) \nonumber \\ &=& \det(\pi {Q})^{-1/2} \exp(-( {x} - {\mu})^{T} {Q}^{-1}( {x} - {\mu}) ) \end{eqnarray}

The differential entropy of a complex Gaussian \(\mathbf{x}\) with covariance \(Q\) is given by

\begin{eqnarray} \label{eq:20120323hgammaq} \mathcal{H}(\gamma_Q) &=& \mathcal{E}_{\gamma_Q}[-\log\gamma_Q(\mathbf{x})] \nonumber \\ &=& \log \det(\pi Q) + (\log e)\mathcal{E}[\mathbf{x}^{T}Q^{-1}\mathbf{x}] \nonumber \\ &=& \log \det(\pi Q) + (\log e)tr(\mathcal{E}[\mathbf{x}\mathbf{x}^{T}]Q^{-1}) \nonumber \\ &=& \log \det(\pi Q) + (\log e)tr(I) \nonumber \\ &=& \log \det(\pi eQ) \end{eqnarray}

For us, the importance of the circulary symmetric complex Gaussians is due to the following lemma: circularly symmetric complex Gaussians are entropy maximizers.

We will first derive an expression for the capacity \(C(H,P)\) of this channel, where \(H\in \mathbb{C}^{r\times t}\) . By the SVD theorem \(\mathbf{y} = H\mathbf{x}+ \mathbf{n}\) can be written as \(\mathbf{y}=UDV^{T}\mathbf{x}+\mathbf{n}\) , where \(D\in \mathbb{R}^{r\times t}\) .Let \(\tilde{\mathbf{y}} = U^{T}\mathbf{y}, \tilde{\mathbf{x}} = V^{T}\mathbf{x}, \tilde{\mathbf{n}} = U^{T}\mathbf{n}\) .Then, \(\tilde{\mathbf{y}} = D\tilde{\mathbf{x}} +\tilde{\mathbf{n}}\) . Since \(H\) is of rank at most \(\min\{r,t\}\) , at most \(\min\{r,t\}\) of the singular values of it are non-zero. Denoting these by \(\lambda_i^{1/2},i = 1,2,\dots, \min\{r,t\}\) , we can write the matrix form of \(\tilde{\mathbf{y}} = D\tilde{\mathbf{x}} +\tilde{\mathbf{n}}\) component-wise, to get \(\tilde{y}_i = \lambda_{i}^{1/2}\tilde{x}_{i}+ \tilde{n}_{i}, \quad 1\le i \le \min\{r,t\}\) , and the rest of the components of \(\tilde{y}\) (if any)are equal to the corresponding components of \(\tilde{n}\) . We thus see that \(\tilde{y}_{i}\) for \(i> \min\{r,t\}\) is independent of the transmitted signal and that \(\tilde{x}_i\) for \(i> \min\{r,t\}\) don't play any role. To maximize the mutual information, we need to choose \(\{ \tilde{x}_i:1\le i \le \min\{r,t\}\) to be independent, with each \(\tilde{x}_i\) having independent Gaussian, zero-mean real and imaginary parts. The variances need to be chosen via"water-filling" as

\begin{equation} \label{eq:20120323erexi} \mathcal{E}[\mathfrak{Re}(\tilde{x}_i)^2] = \mathcal{E}[\mathfrak{Im}(\tilde{x}_i)^2] = \frac{1}{2} (\mu -\lambda_i^{-1})^+ \end{equation}

where \(\mu\) is chosen to meet the power constraint. Here \(a^+\) denotes \(\max\{0,a\}\) . The power \(P\) and the maximal mutual information can thus be parameterized as

\begin{equation} \label{eq:20120323pmucmu} P(\mu) =\sum_{i} (\mu - \lambda_{i}^{-1})^+, \qquad C(\mu) =\sum_{i} (\ln (\mu\lambda_i))^+. \end{equation}

\subsubsection{Alternative Derivation of the capacity} The mutual information \(\mathcal{I}(\mathbf{x};\mathbf{y})\) can be written as

\begin{equation} \label{eq:20120323Ixy} \mathcal{I}(\mathbf{x};\mathbf{y}) = \mathcal{H}(\mathbf{y}) - \mathcal{H}(\mathbf{y}| \mathbf{x}) = \mathcal{H}(\mathbf{y}) - \mathcal{H}(\mathbf{n}) \end{equation}

and thus maximizing \(\mathcal{I}(\mathbf{x};\mathbf{y})\) is equivalent to maximizing \(\mathcal{H}(\mathbf{y})\) Note that if \(\mathbf{x}\) satisfies \(\mathcal{E}{\mathbf{x}\mathbf{x}^T} \le P\) , so does \(x-\mathcal{E}[x]\) , so we can restrict our attention to zero-mean \(\mathbf{x}\) . Furthermore, if \(\mathbf{x}\) is zero-mean with covariance \(\mathcal{E}[\mathbf{x}\mathbf{x}^T] = Q\) , then \(\mathbf{y}\) is zero-mean with covariance \(\mathcal{E}[\mathbf{y}\mathbf{y}^T] = HQH^T + I_r\) , when the \(\mathbf{x}\) is circular symmetric complex Gaussian, the mutual information is given by:

\begin{equation} \label{eq:20120323Ixy2} \mathcal{I}(\mathbf{x};\mathbf{y}) = \log\det(I_r + HQH^T)= \log\det(I_r + QH^{T}H) \end{equation}

where the second equality follows from the determinant identity \(\det(I+AB) = \det(I+BA)\) , and it only remains to choose \(Q\) to maximize this quantity subject to the constraints \(tr(Q)\le P\) and that \(Q\) is non-negative definite. Let \(\log\det(I+ HQH^{T})= \Psi(Q,H)\)