4.26 Discrete-time detection: problems

Assume that observations of a sinusoidal signal $s(l)=A\sin (2\pi fl)$ , $l\in \{0, , L-1\}$ , are contaminated by first-order colored noise as decribed in the example .

In space-time decoding systems, a common bit stream is transmitted over several channels simultaneously butusing different signals. $r^{\left(k\right)}$ denotes the signal received from the $k^{\mathrm{th}}$ channel, $k\in \{1, , K\}$ , and the received signal equals $s^{(k,i)}+n^{\left(k\right)}$ . Here $i$ equals 0 or 1, corresponding to the bit being transmitted. Each signalhas length $L$ . $n^{\left(k\right)}$ denotes a Gaussian random vector with statistically independent components having mean zero andvariance ${}_k^2$ (the variance depends on the channel).

The performance for the optimal detector in white Gaussian noise problems depends only on the distancebetween the signals. Let's confirm this result experimentally. Define the signal under one hypothesis to be aunit-amplitude sinusoid having one cycle within the 50-sample observation interval. Observations of this signal arecontaminated by additive white Gaussian noise having variance equal to 1.5. The hypotheses are equally likely.

Physical constraints imposed on signals can change what signal set choices result in the best detectionperformance. Let one of two equally likely discrete-time signals be observed in the presence of white Gaussian noise(variance/sample equals $^2$ ). $${}_0:r(l)=s^{\left(0\right)}(l)+n(l)l\in \{0, , L-1\}$$ $${}_1:r(l)=s^{\left(1\right)}(l)+n(l)l\in \{0, , L-1\}$$ We are free to choose any signals we like, but there are constraints. Average signals power equals $\sum \frac{s(l)^2}{L}$ , and the peak power equals $\max\{l , s(l)^2\}$ .