A.23  Vector prediction exploring spatial correlation

Instead of exploring correlation over time, this section discusses methods to explore the so-called spatial correlation: an element of the random vector being estimated based on the other elements of this vector. As signal model, the Gaussian block or “packet” ISI channel [CF97], (page 84, Eq. 4.6) is adopted here, which is given by

where $\mathbf{\text{X}}$ is a zero-mean complex random input $m$-vector, $\mathbf{\text{Y}}$ is a zero-mean complex random output $n$-vector, $\mathbf{\text{N}}$ is a complex zero-mean Gaussian noise $n$-vector independent of $\mathbf{\text{X}}$ and $H$ is the complex $n\times m$ channel matrix [CF97]. For these random vectors, the correlation matrices coincide with covariance matrices. ²⁰

The output covariance matrix is given by

where a matrix superscript $\ast$ denotes Hermitian (transpose conjugate).

A characteristic representation of a random $m$-vector $\mathbf{\text{X}}$ is given by the linear combination of the columns of a matriz $F$, whose determinant is equal to one,²¹ weighted by a vector of uncorrelated random variables $\mathbf{\text{V}}$, that is:

Hence, the covariance matrix of $\mathbf{\text{X}}$ is given by:

where $R_{\mathit{vv}}$ is diagonal (the random variables in $\mathbf{\text{V}}$ are uncorrelated).

There are two alternatives of interest for representing a vector in its characteristic form, the modal and the innovations representation. The first is derived from the eigendecomposition of $R_{\mathit{xx}}$. Given the factorization $R_{\mathit{xx}}=U{\Lambda }_x^2{U}^{\ast }=(U{\Lambda }_x){(U{\Lambda }_x)}^{\ast }$, by comparison to Eq. (A.106), $F$ corresponds to the unitary matrix $U$ from the eigendecomposition, while the uncorrelated vector $\mathbf{\text{V}}$ from Eq. (A.105) corresponds to $U^{-1}\mathbf{\text{X}}$. Meanwhile, the latter (innovations representation) is derived from the Cholesky decompostion. In a similar manner, given the factorization $R_{\mathit{yy}}=L{D}_y^2{L}^{\ast }=(L{D}_y){(L{D}_y)}^{\ast }$, $F$ corresponds to the lower triangular matrix $L$, while $\mathbf{\text{V}}$ corresponds to $L^{-1}\mathbf{\text{X}}$.

The important conclusion yielded by these two representations is that a vector $\mathbf{\text{X}}$ whose random variables are correlated can be whitened by a forward section given by $U^{-1}$, the inverse of the unitary matrix from the eigendecomposition of its covariance matrix, or $L^{-1}$, the inverse of the lower triangular matrix from the Cholesky decomposition of its covariance matrix.

The innovations representation is a natural adaptation of linear prediction over time and is obtained with a Cholesky factorization of $R_{\mathit{yy}}$, while the modal representation can be obtained via eigenanalysis or SVD.

The optimum MMSE linear predictor in this scenario is

where the predictor matrix $P$ is given by

with $L$ being obtained from the innovations representation $R_{\mathit{yy}}=L{D}_y^2{L}^{\ast }$ and $I$ being the identity matrix. It was assumed that $R_{\mathit{yy}}$ is nonsingular, otherwise the pseudo inverse can be used.

Because $L$ is lower triangular and monic, its inverse is also lower triangular and monic. The subtraction of $L^{-1}$ from $I$ makes $P$ to be lower triangular with zeros in the main diagonal. This structure imposes a causal relation among the elements of $\mathbf{\text{Y}}$, such that $\widetilde{\mathbf{\text{Y}}}$ can be obtained recursively.

The error vector is

In general, the sum mean-squared prediction error is

where $R_{\mathit{ee}}=𝔼[\mathbf{\text{E}}{\mathbf{\text{E}}}^{\ast }]$ is the autocorrelation matrix of $\mathbf{\text{E}}$. It can be proved (see, e. g., [BLM04]) that when the optimum linear predictor of Eq. (A.108) is adopted, the error power $\mathrm{\text{trace}}\{R_{\mathit{ee}}\}$ achieves its minimum value given by $\mathrm{\text{trace}}\{D_y^2\}$. This avoids the step of estimating $R_{\mathit{ee}}$ to obtain the prediction gain, which is given by

Hence, making an analogy with prediction over time, repeated here for convenience:

the spatial prediction allows to obtain

which is expressed in matrix notation as $\mathbf{\text{E}}=L^{-1}\mathbf{\text{Y}}$ and $\mathbf{\text{Y}}=L\mathbf{\text{E}}$.

Listing A.15 illustrates an example discussed in [BLM04].

1%Example 10-11 from Barry, 2004 (note a typo in matrix R in the book) 
2Ryy=[16 8 4; 8 20 10; 4 10 21] %noise autocorrelation, correlated 
3%Ryy=[10, 8, 2; 8 10 10; 2 10 10]; %another option, higher gain 
4[L D] = ldl_dg(Ryy)%own LDL, do not use chol(A) because it swaps rows 
5Ryy-L*D*L' %compare with Ryy, should be the same 
6P=eye(size(L))-inv(L) %optimum MMSE linear predictor 
7minMSE=trace(D) %minimum MSE is the trace{Ree} = trace{D} 
8sumPowerX=trace(Ryy); %sum of all "users" 
9predictionGain = 10*log10(sumPowerX/minMSE)
  

In Listing A.15, the original predictor matrix is

P =     [0,         0,         0;
    0.5000,         0,         0;
         0,    0.5000,         0]

and the prediction gain is 0.7463 dB. Adopting a new correlation matrix Ryy=[10, 8, 2; 8 10 10; 2 10 10] leads to

P =     [0,         0,         0;
    0.8000,         0,         0;
   -1.6667,    2.3333,         0]

and a prediction gain of 9.2082 dB.

Note that the first element ${\tilde{y}}_1$ of $\widetilde{\mathbf{\text{Y}}}=[y_1,\dots ,y_V]$ in $\widetilde{\mathbf{\text{Y}}}=P\mathbf{\text{Y}}$ is always zero due to the structure of $P$. Then, the second element ${\tilde{y}}_2$ is a scaled version of the first element $y_1$ of $\mathbf{\text{Y}}$, and so on.