# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 3 Matrix Applications in Signal Estimation Theory
### Cholesky Decomposition for Whitening the Noise  
The Cholesky decomposition [11] of a Hermitian positive-semidefinite matrix \(\mathbf{A}\) admits the representation:  
\[\mathbf{A}=\mathbf{D}\mathbf{D}^{H},\]  
where \(\mathbf{D}\) is a lower triangular matrix with nonnegative diagonal entries. Every Hermitian positive-definite matrix has a unique Cholesky decomposition.  
Cholesky decomposition is mainly used in whitening the noise for signal estimation and detection applications. Consider a simple vector parameter estimation problem expressed as [12]:  
\[\mathbf{x}=\mathbf{\theta}+\mathbf{\
u}, \tag{10}\]  
where \(\mathbf{x}\in\mathbb{R}^{N}\) stands for the observed data, \(\mathbf{\
u}\in\mathbb{R}^{N}\) denotes the additive noise with Gaussian distribution \(\mathcal{N}(\mathbf{0},\mathbf{\Sigma})\) (\(\mathbf{\Sigma}\) is not diagonal) and \(\mathbf{\theta}\) represents the unknown vector to be estimated. Knowledge of the probability density function (PDF) \(p(\mathbf{x};\mathbf{\theta})\) lies at the basis for several very well-known and used estimators such as the minimum variance unbiased estimator (MVUE) and the maximum likelihood estimator (MLE) [12]. In the present application, \(p(\mathbf{x};\mathbf{\theta})\) stands for a multivariate Gaussian PDF of the form:  
\[p(\mathbf{x};\mathbf{\theta})=(2\pi)^{-\frac{N}{2}}|\mathbf{\Sigma}|^{-\frac{1}{2}}e^ {-\frac{1}{2}(\mathbf{x}-\mathbf{\theta})^{T}\mathbf{\Sigma}^{-1}(\mathbf{x}-\mathbf{ \theta})}, \tag{11}\]  
where \(|\cdot|\) denotes the matrix determinant.  
Interestingly, if we assume the covariance matrix \(\mathbf{\Sigma}\) to be full rank and consider its Cholesky decomposition, i.e., \(\mathbf{\Sigma}=\mathbf{D}\mathbf{D}^{T}\), and multiply with the Cholesky factor \(\mathbf{D}^{-1}\) both sides of (10), we obtain  
\[\begin{split}\mathbf{D}^{-1}\mathbf{x}&=\mathbf{D}^ {-1}\mathbf{\theta}+\mathbf{D}^{-1}\mathbf{\
u},\\ \tilde{\mathbf{x}}&=\tilde{\mathbf{\theta}}+\tilde{\mathbf{ \
u}},\end{split} \tag{12}\]  
where the covariance matrix of \(\tilde{\mathbf{\
u}}\) is now given by \[E(\tilde{\mathbf{\
u}}\tilde{\mathbf{\
u}}^{T})=E(\mathbf{D}^{-1}\mathbf{\
u}\mathbf{\
u}^{T} \mathbf{D}^{-T})=\mathbf{D}^{-1}E(\mathbf{\
u}\mathbf{\
u}^{T})\mathbf{D}^{-T}=\mathbf{D }^{-1}\mathbf{\Sigma}\mathbf{D}^{-T}=\mathbf{D}^{-1}\mathbf{D}\mathbf{D}^{T} \mathbf{D}^{-T}=\mathbf{I}.\]  
This transformation whitens the noise such that the new PDF takes the form:  
\[\begin{split} p(\tilde{\mathbf{\kappa}};\tilde{\mathbf{\theta}})& =(2\pi)^{-\frac{N}{2}}e^{-\frac{1}{2}(\tilde{\mathbf{\kappa}}-\tilde{ \mathbf{\theta}})^{T}(\tilde{\mathbf{\kappa}}-\tilde{\mathbf{\theta}})}\\ &=(2\pi)^{-\frac{N}{2}}e^{-\frac{1}{2}\sum\limits_{i=0}^{N-1}( \tilde{x}_{i}-\tilde{\theta}_{i})^{2}}\\ &=\prod\limits_{i=0}^{N-1}p(\tilde{x}_{i};\tilde{\theta}_{i}). \end{split} \tag{13}\]  
Using representation (13), where the noise is white (not correlated anymore), the estimation task can be performed by processing independently each \(\tilde{\theta}_{i}\) on a one-by-one basis, which, in general, for many applications, leads to a closed form solution. The estimate \(\theta\) can be obtained by \(\mathbf{\theta}=\mathbf{D}\tilde{\mathbf{\theta}}\). In contrast, in (11), we must process the entire vector \(\mathbf{\mathbf{x}}\) due to the dependence caused by the colored noise. Thus, the whitening method significantly simplifies the estimation problem when the noise is colored.