多天线高斯信道的容量

y=Hx+n

where H is a r×t matrix and n is zero-mean complex Gaussian noise with independent, equal variance real and imaginary parts. We assume nnT=Ir , that is the noises corrupting the different receivers are independent. The transmitter is constrained in its total power to P ,

{xTx}≤P

因为 xTx=tr(xxT) , and expectation and trace commute,

tr(E{xxT})≤P

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 HH) , chosen according to a probability distribution, and each use of the channel corresponds to an independent realization of HH .

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

   if Q∈Cn×n is non-negative devinite then so is ˆQ∈R2n×2n .

The probability density(with respect to the standard Lebesgue measure on Cn ) of a circularly symmetric complex Gaussian with mean μ and covariance Q is given by

γμ,Q=det(πˆQ)−1/2exp(−(ˆx−ˆμ)TˆQ−1(ˆx−ˆμ))=det(πQ)−1/2exp(−(x−μ)TQ−1(x−μ))

The differential entropy of a complex Gaussian x with covariance Q is given by

H(γQ)=EγQ[−logγQ(x)]=logdet(πQ)+(loge)E[xTQ−1x]=logdet(πQ)+(loge)tr(E[xxT]Q−1)=logdet(πQ)+(loge)tr(I)=logdet(πeQ)

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∈Cr×t . By the SVD theorem y=Hx+n can be written as y=UDVTx+n , where D∈Rr×t .Let ˜y=UTy,˜x=VTx,˜n=UTn .Then, ˜y=D˜x+˜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 λ1/2i,i=1,2,…,min{r,t} , we can write the matrix form of ˜y=D˜x+˜n component-wise, to get ˜yi=λ1/2i˜xi+˜ni,1≤i≤min{r,t} , and the rest of the components of ˜y (if any)are equal to the corresponding components of ˜n . We thus see that ˜yi for i>min{r,t} is independent of the transmitted signal and that ˜xi for i>min{r,t} don't play any role. To maximize the mutual information, we need to choose {˜xi:1≤i≤min{r,t} to be independent, with each ˜xi having independent Gaussian, zero-mean real and imaginary parts. The variances need to be chosen via"water-filling" as

E[Re(˜xi)2]=E[Im(˜xi)2]=12(μ−λ−1i)+

where μ 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

P(μ)=∑i(μ−λ−1i)+,C(μ)=∑i(ln(μλi))+.

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

I(x;y)=H(y)−H(y|x)=H(y)−H(n)

and thus maximizing I(x;y) is equivalent to maximizing H(y) Note that if x satisfies ExxT≤P , so does x−E[x] , so we can restrict our attention to zero-mean x . Furthermore, if x is zero-mean with covariance E[xxT]=Q , then y is zero-mean with covariance E[yyT]=HQHT+Ir , when the x is circular symmetric complex Gaussian, the mutual information is given by:

I(x;y)=logdet(Ir+HQHT)=logdet(Ir+QHTH)

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)≤P and that Q is non-negative definite. Let logdet(I+HQHT)=Ψ(Q,H)