Optimum Digital Detector – Alternative Representation
Notations
\( \vec{z} \): A complex column vector in \( \mathbb{C}^k \).
\( \vec{z}^* \): The element-wise complex conjugate of \( \vec{z} \). If \( \vec{z} \) has components \( z_1, z_2, \dots, z_k \), then \( \vec{z}^* \) has components \( z_1^*, z_2^*, \dots, z_k^* \).
\( \vec{z}^T \): The transpose of \( \vec{z} \), converting it from a column vector to a row vector without taking the complex conjugate.
\( \vec{z}^H \): The conjugate transpose (Hermitian transpose) of \( \vec{z} \). This operation involves both transposing and taking the complex conjugate, resulting in a row vector:
$\(
\vec{z}^H = (\vec{z}^T)^* = (\vec{z}^*)^T
\)$
\( \mathbf{M}^{-1} \): The inverse of the covariance matrix \( \mathbf{M} \), which is assumed to be Hermitian (\( \mathbf{M} = \mathbf{M}^H \)) and positive definite.
Comparing \( \vec{z}^T \mathbf{M}^{-1} \vec{z}^* \) and \( -\vec{z}^H \mathbf{M}^{-1} \vec{z} \)
We consider two expressions:
A. \( \vec{z}^T \mathbf{M}^{-1} \vec{z}^* \)
B. \( -\vec{z}^H \mathbf{M}^{-1} \vec{z} \)
Expression A: \( \vec{z}^T \mathbf{M}^{-1} \vec{z}^* \)
Let \( \vec{z} \) be a \( k \times 1 \) complex vector:
\[\begin{split}
\vec{z} = \begin{bmatrix}
z_1 \\
z_2 \\
\vdots \\
z_k
\end{bmatrix}
\end{split}\]
Let \( \mathbf{M}^{-1} \) be a \( k \times k \) Hermitian matrix:
\[\begin{split}
\mathbf{M}^{-1} = \begin{bmatrix}
M_{11}^{-1} & M_{12}^{-1} & \cdots & M_{1k}^{-1} \\
M_{21}^{-1} & M_{22}^{-1} & \cdots & M_{2k}^{-1} \\
\vdots & \vdots & \ddots & \vdots \\
M_{k1}^{-1} & M_{k2}^{-1} & \cdots & M_{kk}^{-1} \\
\end{bmatrix}
\end{split}\]
The complex conjugate of \( \vec{z} \):
\[\begin{split}
\vec{z}^* = \begin{bmatrix}
z_1^* \\
z_2^* \\
\vdots \\
z_k^*
\end{bmatrix}
\end{split}\]
Compute \( \vec{z}^T \mathbf{M}^{-1} \vec{z}^* \)
Perform the matrix multiplication step-by-step.
Multiply \( \vec{z}^T \) with \( \mathbf{M}^{-1} \):
\[
\vec{z}^T \mathbf{M}^{-1} = \begin{bmatrix}
z_1 M_{11}^{-1} + z_2 M_{21}^{-1} + \cdots + z_k M_{k1}^{-1} & z_1 M_{12}^{-1} + z_2 M_{22}^{-1} + \cdots + z_k M_{k2}^{-1} & \cdots & z_1 M_{1k}^{-1} + z_2 M_{2k}^{-1} + \cdots + z_k M_{kk}^{-1}
\end{bmatrix}
\]
Multiply the result with \( \vec{z}^* \):
\[
\vec{z}^T \mathbf{M}^{-1} \vec{z}^* = \sum_{i=1}^{k} \sum_{j=1}^{k} z_i M_{ij}^{-1} z_j^*
\]
This is a scalar value.
Expression B: \( -\vec{z}^H \mathbf{M}^{-1} \vec{z} \)
Recall that:
\[
\vec{z}^H = \begin{bmatrix}
z_1^* & z_2^* & \cdots & z_k^*
\end{bmatrix}
\]
Perform the matrix multiplication step-by-step.
Multiply \( \vec{z}^H \) with \( \mathbf{M}^{-1} \):
\[\begin{split}
\vec{z}^H \mathbf{M}^{-1} = \begin{bmatrix}
z_1^* M_{11}^{-1} + z_2^* M_{12}^{-1} + \cdots + z_k^* M_{1k}^{-1} & z_1^* M_{21}^{-1} + z_2^* M_{22}^{-1} + \cdots + z_k^* M_{2k}^{-1} & \cdots & z_1^* M_{k1}^{-1} + z_2^* M_{k2}^{-1} + \cdots + z_k^* M_{kk}^{-1} \\
\end{bmatrix}
\end{split}\]
Multiply the result with \( \vec{z} \):
\[
\vec{z}^H \mathbf{M}^{-1} \vec{z} = \sum_{i=1}^{k} \sum_{j=1}^{k} z_i^* M_{ij}^{-1} z_j
\]
This is also a scalar value.
Therefore:
\[ \boxed{
\vec{z}^T \mathbf{M}^{-1} \vec{z}^* = \vec{z}^H \mathbf{M}^{-1} \vec{z}
}
\]
Comparing \( \vec{z}^* \vec{z}^T \) and \( \vec{z} \vec{z}^H \)
\( \vec{z}^* \vec{z}^T \):
\[\begin{split}
\vec{z}^* \vec{z}^T =
\begin{bmatrix}
z_1^* \\
z_2^* \\
\vdots \\
z_k^*
\end{bmatrix}
\begin{bmatrix}
z_1 & z_2 & \cdots & z_k
\end{bmatrix}
=
\begin{bmatrix}
z_1^* z_1 & z_1^* z_2 & \cdots & z_1^* z_k \\
z_2^* z_1 & z_2^* z_2 & \cdots & z_2^* z_k \\
\vdots & \vdots & \ddots & \vdots \\
z_k^* z_1 & z_k^* z_2 & \cdots & z_k^* z_k \\
\end{bmatrix}
\end{split}\]
\( \vec{z} \vec{z}^H \):
\[\begin{split}
\vec{z} \vec{z}^H =
\begin{bmatrix}
z_1 \\
z_2 \\
\vdots \\
z_k
\end{bmatrix}
\begin{bmatrix}
z_1^* & z_2^* & \cdots & z_k^*
\end{bmatrix}
=
\begin{bmatrix}
z_1 z_1^* & z_1 z_2^* & \cdots & z_1 z_k^* \\
z_2 z_1^* & z_2 z_2^* & \cdots & z_2 z_k^* \\
\vdots & \vdots & \ddots & \vdots \\
z_k z_1^* & z_k z_2^* & \cdots & z_k z_k^* \\
\end{bmatrix}
\end{split}\]
Therefore
\[
\vec{z}^* \vec{z}^T = \vec{z} \vec{z}^H
\]
Joint Probability Density Function
The noise vector \( \vec{z} \) consists of \( k \) zero-mean complex Gaussian random variables.
The joint PDF for \( \vec{z} \) is given by:
\[
p(\vec{z}) = \frac{1}{\pi^k \det(\mathbf{M})} \exp\left( -\vec{z}^H \mathbf{M}^{-1} \vec{z} \right)
\]
Covariance Matrix \( \mathbf{M} \)
The covariance matrix \( \mathbf{M} \) characterizes the second-order statistical properties of the complex noise vector \( \vec{z} \).
\[
\mathbf{M} = \mathbb{E}\{ \vec{z} \vec{z}^H \}
\]
Probability Density Functions (PDFs) of the Measurement Vector \( \vec{y} \) Under Hypotheses \( H_0 \) and \( H_1 \)
Given the measurement model:
\[
\vec{y} = \vec{u}_i + \vec{z}, \quad i = 0, 1
\]
Under \( H_0 \):
\[
p(\vec{y} | H_0) = \frac{1}{\pi^k \det(\mathbf{M})} \exp\left( -(\vec{y} - \vec{u}_0)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_0) \right)
\]
Under \( H_1 \):
\[
p(\vec{y} | H_1) = \frac{1}{\pi^k \det(\mathbf{M})} \exp\left( -(\vec{y} - \vec{u}_1)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_1) \right)
\]
Likelihood Ratio Test (LRT)
The likelihood ratio \( L(\vec{y}) \) is defined as:
\[
L(\vec{y}) = \frac{p(\vec{y} | H_1)}{p(\vec{y} | H_0)}
\]
Substituting the PDFs:
\[\begin{split}\begin{align*}
L(\vec{y}) &= \frac{ \exp\left( -(\vec{y} - \vec{u}_1)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_1) \right) }{ \exp\left( -(\vec{y} - \vec{u}_0)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_0) \right) } \\
&= \exp\left( -(\vec{y} - \vec{u}_1)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_1) + (\vec{y} - \vec{u}_0)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_0) \right)
\end{align*}
\end{split}\]
Taking the natural logarithm of both sides:
\[
\ln L(\vec{y}) = (\vec{y} - \vec{u}_0)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_0) - (\vec{y} - \vec{u}_1)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_1)
\]
Simplifying the expression:
\[
\ln L(\vec{y}) = 2 \operatorname{Re}\left[ (\vec{u}_1 - \vec{u}_0)^H \mathbf{M}^{-1} \vec{y} \right] - (\vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0)
\]
Simplification of the Log-Likelihood Ratio
Expand each quadratic form:
\[
(\vec{y} - \vec{u}_i)^H \mathbf{M}^{-1} (\vec{y} - \vec{u}_i) = \vec{y}^H \mathbf{M}^{-1} \vec{y} - \vec{y}^H \mathbf{M}^{-1} \vec{u}_i - \vec{u}_i^H \mathbf{M}^{-1} \vec{y} + \vec{u}_i^H \mathbf{M}^{-1} \vec{u}_i
\]
Given that \( \mathbf{M} \) is Hermitian (\( \mathbf{M} = \mathbf{M}^H \)), the following holds:
\[
\vec{u}_i^H \mathbf{M}^{-1} \vec{y} = \left( \vec{y}^H \mathbf{M}^{-1} \vec{u}_i \right)^*
\]
For simplicity in the derivation, we’ll assume that the signals \( \vec{u}_0 \) and \( \vec{u}_1 \) are such that the imaginary parts cancel out when taking the real part of the expression.
Substituting the expanded forms back into \( \ln L(\vec{y}) \):
\[\begin{split}
\begin{aligned}
\ln L(\vec{y}) &= -\left[ \vec{y}^H \mathbf{M}^{-1} \vec{y} - \vec{y}^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_1^H \mathbf{M}^{-1} \vec{y} + \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 \right] \\
&\quad + \left[ \vec{y}^H \mathbf{M}^{-1} \vec{y} - \vec{y}^H \mathbf{M}^{-1} \vec{u}_0 - \vec{u}_0^H \mathbf{M}^{-1} \vec{y} + \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0 \right] \\
&= -\vec{y}^H \mathbf{M}^{-1} \vec{y} + \vec{y}^H \mathbf{M}^{-1} \vec{u}_1 + \vec{u}_1^H \mathbf{M}^{-1} \vec{y} - \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 \\
&\quad + \vec{y}^H \mathbf{M}^{-1} \vec{y} - \vec{y}^H \mathbf{M}^{-1} \vec{u}_0 - \vec{u}_0^H \mathbf{M}^{-1} \vec{y} + \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0 \\
&= (\vec{y}^H \mathbf{M}^{-1} \vec{u}_1 + \vec{u}_1^H \mathbf{M}^{-1} \vec{y}) - (\vec{y}^H \mathbf{M}^{-1} \vec{u}_0 + \vec{u}_0^H \mathbf{M}^{-1} \vec{y}) \\
&\quad - (\vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0)
\end{aligned}
\end{split}\]
Group the terms involving \( \vec{y} \) and those that are independent of \( \vec{y} \):
\[
\ln L(\vec{y}) = \left[ \vec{y}^H \mathbf{M}^{-1} (\vec{u}_1 - \vec{u}_0) + (\vec{u}_1 - \vec{u}_0)^H \mathbf{M}^{-1} \vec{y} \right] - \left[ \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0 \right]
\]
Recognizing that \( \vec{u}_1^H \mathbf{M}^{-1} \vec{y} = (\vec{y}^H \mathbf{M}^{-1} \vec{u}_1)^* \), and assuming that the final decision statistic is real, we can express the first bracket as twice the real part of a complex quantity:
\[
\ln L(\vec{y}) = 2 \operatorname{Re}\left[ \vec{y}^H \mathbf{M}^{-1} (\vec{u}_1 - \vec{u}_0) \right] - \left[ \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0 \right]
\]
Thus, the log-likelihood ratio simplifies to:
\[ \boxed{
\ln L(\vec{y}) = \underbrace{2 \operatorname{Re}\left[ (\vec{u}_1 - \vec{u}_0)^H \mathbf{M}^{-1} \vec{y} \right]}_{\text{Weighted Sum of } \vec{y}} - \underbrace{\left[ (\vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1) - (\vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0) \right]}_{\text{Independent of } \vec{y}}
}
\]
Weighted Sum of \( \vec{y} \):
\[
2 \operatorname{Re}\left[ (\vec{u}_1 - \vec{u}_0)^H \mathbf{M}^{-1} \vec{y} \right]
\]
This term represents a linear combination of the measurement vector \( \vec{y} \), weighted by the difference between the signal vectors under \( H_1 \) and \( H_0 \), scaled by the inverse of the covariance matrix \( \mathbf{M}^{-1} \).
Component Independent of \( \vec{y} \):
\[
(\vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1) - (\vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0)
\]
This term is a constant offset that depends solely on the signal vectors and the noise covariance.
It represents the difference in the energy or signal strength between the two hypotheses, normalized by the noise covariance.
Decision Rule
General Decision Rule Using Likelihood Statistic
The general decision rule using the likelihood ratio \( L(\vec{y}) \) is defined as:
\[
L(\vec{y}) \underset{H_0}{\overset{H_1}{\gtrless}} \eta
\]
Here, \( \eta \) is the threshold that determines the decision boundary between the two hypotheses.
Decision Rule Using Log-Likelihood Statistic
Alternatively, the decision rule can be expressed using the log-likelihood ratio \( \ell(\vec{y}) = \ln L(\vec{y}) \):
\[
\ell(\vec{y}) \underset{H_0}{\overset{H_1}{\gtrless}} \ln \eta \triangleq \eta_L
\]
Here, \( \eta_L = \ln \eta \) is the logarithm of the threshold.
Simplifying the Log-Likelihood Ratio
Define a constant term \( C \):
\[
C = \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0
\]
Since \( C \) is independent of \( \vec{y} \), it can be incorporated into the threshold \( \eta \):
\[
\eta_0 = \frac{\eta_L + C}{2}
\]
Simplified Decision Rule
\[
\operatorname{Re}\left\{ (\vec{u}_1 - \vec{u}_0)^H \mathbf{M}^{-1} \vec{y} \right\} \underset{H_0}{\overset{H_1}{\gtrless}} \eta_0
\]
Weight Vector \( \vec{h} \)
To further simplify the decision rule, we define a weight vector \( \vec{h} \) as follows:
\[
\vec{h} = \mathbf{M}^{-1} (\vec{u}_1 - \vec{u}_0)
\]
Purpose of \( \vec{h} \): Acts as a linear filter that projects the measurement vector \( \vec{y} \) onto the direction defined by the difference in signal vectors \( \vec{u}_1 - \vec{u}_0 \), scaled by the inverse of the noise covariance matrix.
Substituting the definition of \( \vec{h} \) into the log-likelihood ratio:
\[\begin{split}
\begin{aligned}
\ell(\vec{y}) &= 2 \operatorname{Re}\left[ (\vec{u}_1 - \vec{u}_0)^H \mathbf{M}^{-1} \vec{y} \right] - \left( \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0 \right) \\
&= 2 \operatorname{Re}\left[ \vec{h}^H \vec{y} \right] - \left( \vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0 \right)
\end{aligned}
\end{split}\]
By defining a weight vector \( \vec{h} \) and incorporating the constant term into the threshold \( \eta_0 \), the decision rule is simplified to:
\[
\boxed{ \operatorname{Re}\left\{ \vec{h}^H \vec{y} \right\} \underset{H_0}{\overset{H_1}{\gtrless}} \eta_0 }
\]
Where:
\[
\vec{h} = \mathbf{M}^{-1} (\vec{u}_1 - \vec{u}_0)
\]
and
\[
\eta_0 = \frac{\ln \eta + (\vec{u}_1^H \mathbf{M}^{-1} \vec{u}_1 - \vec{u}_0^H \mathbf{M}^{-1} \vec{u}_0)}{2}
\]
