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Assume that we have a signal to be detected using the binary testing hypothesis. The received signal is distributed as follows.
\begin{align} \mathcal H_0:& \ x = w\\[2ex] \mathcal H_1:& \ x = s + w \end{align} where $w\sim \mathcal N(\mu_w,C_w)$ and $s\sim \mathcal N(\mu_s,C_s)$

I was trying to apply the NP lemma, but I'm getting stuck somewhere.

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  • $\begingroup$ getting stuck because $C_w \ne C_s$ ? $\endgroup$
    – user28715
    Dec 28, 2017 at 15:06
  • $\begingroup$ It's okay even if you take Cw and Cs as diagonal matrices! $\endgroup$
    – Space crawler
    Dec 28, 2017 at 15:09
  • $\begingroup$ so where stuck? $\endgroup$
    – user28715
    Dec 28, 2017 at 15:13
  • $\begingroup$ In calculating Pd and Pfa $\endgroup$
    – Space crawler
    Dec 28, 2017 at 15:14
  • $\begingroup$ try the scaler case first and then the independent case. The multidimensional integrals can be challenging. There are numerical papers using quadrature rules. There are some tricks in Abramawitz and Stegun. Chervov bounds can also be helpful $\endgroup$
    – user28715
    Dec 28, 2017 at 15:26

1 Answer 1

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You're looking at a Generalized Gaussian Detector$^{\ \text [1]}$. The Neyman-Pearson detector decides $\mathcal H_1$ if: $$ \frac{p(\mathbf x; \mathcal H_1)}{p(\mathbf x; \mathcal H_0)}> \gamma\tag{1} $$ With $\mathbf w$ and $\mathbf x$ independent; you get here: $$\mathbf x\sim \begin{cases} \mathcal N\left(\mu_w, \mathbf C_w\right) &\text{under} \mathcal \quad \mathcal H_0\\[2ex] \mathcal N\left(\mu_s + \mu_w, \mathbf C_s + \mathbf C_w\right) &\text{under} \mathcal \quad\mathcal H_1 \end{cases}$$

The NP inequality in equation $(1)$ gives:

$$ \frac{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_s + C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_s - \mu_w\right)^T\left(\mathbf C_s + \mathbf C_w\right)^{-1}\left(\mathbf x-\mu_s - \mu_w\right)\right]}{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_w\right)^T\mathbf C_w^{-1}\left(\mathbf x-\mu_w\right)\right]}>\gamma $$ Taking the logarithm on both sides gives the needed test statistic $T(\mathbf x)$: $$ \underbrace{\ln\left(\frac{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_s + C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_s - \mu_w\right)^T\left(\mathbf C_s + \mathbf C_w\right)^{-1}\left(\mathbf x-\mu_s - \mu_w\right)\right]}{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_w\right)^T\mathbf C_w^{-1}\left(\mathbf x-\mu_w\right)\right]}\right)}_{T(\mathbf x)}>\ln(\gamma) $$ You can continue from there. Also, have a look at this related question on NP detector.

$^{\ \text [1]}$: Kay, S. Fundamentals of Statistical Signal Processing: Detection Theory.

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  • $\begingroup$ How are $C_s$ and $C_w$ estimated from "real" data? $\endgroup$
    – catch22
    May 26, 2019 at 5:21

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