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A Receiver for a Signal of Unknown Phase

Suppose that the signal is known up to an arbitrary phase, $As(t)=A\cos\theta\times s_0(t)+A\sin\theta\times s_1(t)$ where $s_0(t)$ and $s_1(t)$ are known waveforms and $\theta$ is the unknown phase. For simplicity, suppose further that $(s_0\mid s_0)=(s_1\mid s_1)=\sigma^2$ and  $(s_0\mid s_1)=0$, i.e., the waveforms are orthogonal. Again assume that the noise is a stationary (but colored) normal process. Then the likelihood ratio, averaged over a uniform distribution of the unknown phase, is

\begin{displaymath}
\bar{\Lambda} = e^{-A^2\sigma^2/2}I_0(Az)
\end{displaymath} (13.1.293)

where $z^2=x^2+y^2$ with $x=(h\mid s_0)$ and $y=(h\mid s_1)$. Here, $I_0(x)=(2\pi)^{-1}\int_0^{2\pi}e^{x\sin\theta}d\theta$ is the modified Bessel function of the first kind of order zero. Since this function is monotonic, a threshold level can be set on the value of the statistic $z$ rather than $\bar\Lambda$.

When there is no signal present, the statistic $z$ assumes the Rayleigh distribution

\begin{displaymath}
p(z\mid\neg s) = \frac{z}{\sigma^2}\,\exp[-z^2/2\sigma^2]
\end{displaymath} (13.1.294)

and, in the presence of a signal, the statistic assumes a Rice distribution
\begin{displaymath}
p(z\mid s) =
\frac{z}{\sigma^2}\,\exp[-(z^2/\sigma^2+A^2\sigma^2)/2]\,I_0(Az).
\end{displaymath} (13.1.295)

Thus, the false alarm probability for a threshold of $z_\ast$ is
\begin{displaymath}
P(\mbox{false alarm}) = P(z>z_\ast\mid\neg s) = \exp(-z^2_\ast/2\sigma^2)
\end{displaymath} (13.1.296)

and the true detection probability is
\begin{displaymath}
P(\mbox{true detection}) = P(z>z_\ast\mid s) = Q(A\sigma,z_\ast/\sigma).
\end{displaymath} (13.1.297)

The $Q$-function, which is defined by the integral
\begin{displaymath}
Q(\alpha,\beta) = \int_\beta^\infty dx\,
x e^{-(x^2+\alpha^2)/2}I_0(\alpha x),
\end{displaymath} (13.1.298)

has the properties $Q(\alpha,0)=1$, $Q(0,\beta)=e^{-\beta^2/2}$, and the asymptotic forms
\begin{displaymath}
\begin{array}{ll}
Q(\alpha,\beta) \sim 1 - \frac{1}{\alpha...
...}}\,e^{-(\beta-\alpha)^2/2} &
\beta\gg\alpha\gg1.
\end{array}\end{displaymath} (13.1.299)

For any desired value of the false alarm probability, we can calculate the threshold  $z_\ast=\surd[-2\sigma\log P(\mbox{false alarm})]$. Then the probability of true detection can be evaluated for that threshold using the $Q$-function.


next up previous contents
Next: Reception of a Signal Up: The Statistical Theory of Previous: A Receiver for a   Contents
Bruce Allen 2000-11-19