P. A. HUMBLET
2. DETECTION THEORY. The transformation of digital data into a form suitable for
transmission on an analog channel is done in a device called a modulator. The reverse operation is
done by a demodulator. The two devices are usually combined into a single one, called a "modem".
In its most idealized form a modem takes in a group of n binary digits, with values denoted i,
i = 1,2, • • • 2n, and produces in turn one of the waveforms si(t), i = 1,2, • • • 2", that is
transmitted on a channel.
A simple characterization of the channel is as a linear time invariant filter (introducing
deterministic distortion) with impulse response hit), followed by a source of additive noise nit).
For convenience nit) will be modelled as a 0 mean stationary Gaussian process with correlation
We will denote the convolution of si and h by s', and the channel output by rit) irit) =
slit) + nit), for some i). Slif), Slif), Hif) and Kif) will denote corresponding Fourier
transforms. Note that Kif) is real and non negative, as hit) is a correlation function.
The demodulator receives rit), and must produce an estimate of i.
The modern view of this problem is in terms of "single space": the space S of functions
spanned by the slit). Assuming that J\Sl(f)\2/K(f) df is finite for all i, we can make S a
Hilbert space by defining the inner product *(*), yit) — J Xif)Y* if)/Kif) df, where x
and y are in S, and Xif) and Yif) are their Fourier transforms. * denotes complex conjugate.
By using Gram Schmidt's orthonormalization procedure one can find an orthonormal basis for
S and display the waveforms slit) as points in a finite dimensional space of dimension D, typically
smaller than 2^. 5J will denote the jth. coordinate of sl it).
The demodulator processes the received waveform rit). It can find its projection on S iRj
denotes its jth coordinate). Elementary computations reveal that, thanks to the choice of
innerproduct, the component of rit) perpendicular to S is independent of i and irx,r2,...,rD) and
is thus irrelevant to the decision process.
Other elementary calculations show that, conditional on / being transmitted, the rfs are
independent Gaussian random variables with unit variance and mean sj.
At this point the demodulator has all the statistical knowledge required to make a decision,
which need only be based on the rfs. In particular if all the fs are equally likely and minimum
probability of error decoding is desired, one should decide i maximizing the conditional probability