next up previous contents
Next: A Receiver for a Up: The Statistical Theory of Previous: Maximum Likelihood Receiver   Contents

A Receiver for a Known Signal

Consider the case in which the signal has an exactly known form, $s(t)$. In general, signals can occur with a variety of amplitudes; we let $s(t)$ be the known signal with some fiducial normalization and we write the actual signal (if present) as $As(t)$ where $A$ is the amplitude of the signal. Further, we assume that the noise samples are drawn from a stationary Normal distribution, though there may be correlations amongst the noise events (colored noise). The noise correlations may be expressed in terms of the one-sided noise power spectrum, $\frac{1}{2}S_h(\vert f\vert)\delta(f-f')=\langle\tilde{n}(f)\tilde{n}^\ast(f')\rangle$, where $\tilde{n}(f)$ is the Fourier transform of the noise $n(t)$, and $\ast$ denotes complex conjugation. The probability of obtaining an instant of noise, $n(t)$, is  $p(n)\propto\exp[-\frac{1}{2}(n\mid n)]$, where the inner product $(\cdot\mid\cdot)$ is given by

\begin{displaymath}
(a\mid b) = \int_{-\infty}^\infty df\,
\frac{\tilde{a}^\as...
...lde{b}(f)+\tilde{a}(f)\tilde{b}^\ast(f)}
{S_h(\vert f\vert)}.
\end{displaymath} (13.1.289)

The likelihood ratio is the ratio of the probabilities $P(h\mid s)=p(h-As)$ (since $h(t)=As(t)+n(t)$ given the signal is present) and  $P(h\mid\neg s)=p(h)$:
\begin{displaymath}
\Lambda = e^{Ax-A^2\sigma^2/2}
\end{displaymath} (13.1.290)

where $x=(h\mid s)$ and $\sigma^2=(s\mid s)$. Since the likelihood ratio is a monotonically increasing function of $x$, and since the output $h$ appears only in the construction of $x$, we can set a threshold on the value of $x$ rather than on $\Lambda$.

It is straightforward to compute the false alarm and true detection probabilities for any choice of threshold $x_\ast$. Under the assumption that there is no signal present, the false alarm probability is the probability that $\vert x\vert>x_\ast$:

\begin{displaymath}
P(\mbox{false alarm}) = P(\vert x\vert>x_\ast\mid\neg s) =
{\mathrm{erfc}}[x_\ast/\surd(2\sigma^2)].
\end{displaymath} (13.1.291)

Notice that we have used the absolute value of $x$ since the actual signal may have either positive or negative amplitude. Similarly, the true detection probability (the converse of the false dismissal probability) is the probability that $\vert x\vert>x_\ast$ when a signal is present:
$\displaystyle P(\mbox{true detection})$ $\textstyle =$ $\displaystyle P(\vert x\vert>x_\ast\mid s)$  
  $\textstyle =$ $\displaystyle {\textstyle\frac{1}{2}}\,
{\mathrm{erfc}}[(x_\ast-A\sigma^2)/\sur...
...{\textstyle\frac{1}{2}}\,
{\mathrm{erfc}}[(x_\ast+A\sigma^2)/\surd(2\sigma^2)].$ (13.1.292)

Here, the complementary error function is defined by ${\mathrm{erfc}}(x)=(2/\surd\pi)\int_x^\infty e^{-t^2}dt$. Using these equations, the threshold can be set for any desired false alarm probability, and then the probability of a true detection can be computed.


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