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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

function kit).

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