# Detecting complex signal within complex AWGN using Neyman Pearson statistic

I need to construct a Neyman Pearson statistic for detecting a complex signal with known amplitude and unknown phase in additive complex Gaussian noise. My hypotheses are:

\begin{align} \mathcal{H}_0 &: R_i=W_i,& R_i \sim \mathcal{CN} (0,1) \\ \mathcal{H}_1 &: R_i=W_i + A,&R_i \sim \mathcal{CN}(A,1) \end{align}

alternatively,

\begin{align} \mathcal{H}_0 &: R_i=W_i,& W_i \sim \mathcal{CN} (0,1) \\ \mathcal{H}_1 &: R_i -A=W_i,&W_i \sim \mathcal{CN}(0,1) \end{align}

where $$A = A_0e^{j\theta_0}$$ is the signal, and $$i=1,2,...N$$. The signal A occurs over a known region of the data, say $$m_1$$ to $$m_2$$, where $$0 \leq m_1 \lt N, m_1 \lt m_2 \leq N$$ and as such the statistical model is to be applied only over that region.

My first steps are marginalizing the nuisance parameters $$\theta_0$$, and calculating the statistic, assuming $$\theta_0 \sim \mathcal{U}(0,2\pi)$$:

$$\frac{f(\mathbf r|H_1)}{f(\mathbf r|H_0)} = \frac{\displaystyle \prod_{i=1}^{N}\int\limits_0^{2\pi} \frac{1}{2\pi} \frac{1}{\pi} \exp{\left(-\lVert r_i - A_0e^{j\theta_0}\rVert^2\right)}\,\mathrm d\theta_0 }{\displaystyle\prod_{i=1}^N \frac{1}{\pi} \exp{\left(-\lVert r_i\rVert^2\right)}}\text,$$

with $$\mathbf r=(r_1,r_2,\ldots,r_N)$$ being the observation.

Is my construction correct? I can't see how to proceed from here that will allow me to construct a statistic to compare against a threshold. Any suggestions?

• In your formulation of $H_1$, what is circularly-symmetric normal? $A_i$ or $(Y_i-A_i)$? Is it possible that your $A_i=A_0e^{i\theta_0}$ is missing an imaginary unit ($j$) in the exponent? $Y_i$ and $A_i$ are independent, right? – Marcus Müller Jul 1 at 7:37
• Isn't there an integral sign missing in $f(Y|H_0)$? The $\mathrm d\theta_0$ in your integral should be after the exponential, seeing that this is the integrand? – Marcus Müller Jul 1 at 7:42
• Fixed the question. Yes the whole thing is under the integral sign. And I don't see why $f(Y|H_0)$ should have an integral, since it does not have the $\theta_0$ parameter. $Y_i -A_i$ is the circular symmetric variable here. – Razor Jul 1 at 9:16
• While you're right about the integral, this raises a lot of questions. So, let's start small: what is the relationship of the elements in $Y$? I think there's still too much confusion in your notation. For example, your $f(Y|H_1)$ can only be the product of densities of $Y_i-A_i$ if these are independent. But in that case, there's nothing to be done here: in both you $H_0$ and $H_1$ you get the exact same statistic, iid $~\mathcal{CN}(0,1)$, and Neyman-Pearson can't do anything about that. – Marcus Müller Jul 1 at 9:41
• It looks fine, thanks. – Razor Jul 2 at 8:32