Is there a situation when maximizing SNR doesn't also maximize probability of detection?

The problem I'm working on is signal detection of a radar signal, but I suspect this problem shows up in many different branches of signal processing.

Background:

From , the factor $I_f$, by which the SIR is improved by an MTI filter (just an FIR filter for non radar people), is given by: $$I_f = \frac{\mathbf{w}^T\mathbf{M}_s\mathbf{w}^*}{\mathbf{w}^T\mathbf{R}_n\mathbf{w}^*},$$ where $\mathbf w \in \mathbb C^N$ is the vector of filter taps, $\mathbf M_s$ is the signal covariance matrix, and $\mathbf R_n$ is the interference covariance matrix. The optimal filter taps are determined by solving the generalized eigenvalue problem: $$\mathbf M_s \mathbf w^* = \gamma \mathbf R_n \mathbf w^*$$ for the eigenvector with the maximum eigenvalue $\gamma_\text{max} = I_f$.

On p.339 of , it states:

When $I_f$ is maximized, the signal-to-interference ratio at the output of the transversal filter is maximized.

This part seems obvious. But then it is followed by the statement that I wasn't sure about:

This results in a maximum $P_d$ when the interference is Gaussian distributed.

Is there a case when an LTI filter that maximizes SIR doesn't also maximize the detection probability $P_d$?

For purposes of this question, make the following assumptions:

• Square law detector, i.e. test statistic is $t(\mathbf v) =|\mathbf w^T \mathbf v|^2$, where $\mathbf v$ is the vector of the last $N$ received pulses
• The probability density of the test statistic for $H_0 =$ "no signal present", i.e. $p(t | H_0)$, is known
• Detection threshold $T$ is chosen to ensure a false alarm probability $P_{fa}$

EDIT:

The application of this is for filtering unwanted surface clutter returns from an airborne radar, so the distribution of $|n_i|$ is likely going to be something like a K-distribution in the case when the clutter dominates the interference term.

References:

 D. C. Schleher, MTI and Pulsed Doppler Radar, with MATLAB, 2nd ed., Artech House, 2010.

• It shouldn't matter what valid distribution your interference is coming from. If your CFAR (your fixed $P_{fa}$) MTI has a good improvement factor and maximizes your SIR, then your $P_d$ will go up. I believe the wording in the book about the interference being Gaussian is to describe the result based on the equation itself. – Envidia Aug 10 '18 at 16:38