6
P. A. HUMBLET
The sequence of N binary digits is usually parsed in groups of n (with values 1,2, 2"). If
the kth group has value mik) i then the waveform slit**kT) is transmitted (this is just the
waveform sl it) delayed by kT). The received waveform is the sum of the waveforms
corresponding to each group plus the noise thus it has the form
r
(
r
) -
^smik)it-kT)+nit)
.
To generate the most likely sequence the demodulator must find the values m(0), m(l), ,
mi ) minimizing (1). This operation can be rewritten as maximizing
N-n
S irit)Jm(kHt-kT) - ±Mm(k)it~kT)\\2- 2 sm(k,)it-k'T),smik)it-kT)» (2)
k-0
2
k'k
N
The only quantities depending on rit) in this formula are the 2 n innerproducts rit)t
n
slit—kT). For k fixed the demodulator can either evaluate them directly, or first compute the
components of rit) in the space spanned by the slit—kT).
Maximizing (2) would be easy if slit—£T), sHt) were 0 for all ij and £ ^ 0, i.e. if
there were no "intersymbol interference". In that case each mik) can be decoded independently of
the others.
As observed by Nyquist, this condition is met if, for all ij,
£ S'(f + h&\f + UlK{f + •£)
/«_«, 1 1 1
is equal to afj for some atj and for / .
Waveforms are usually designed to (approximately) satisfy this condition, as the task of the
demodulator is then simplified. However it becomes increasingly hard to meet when the passband
of the channel is close to —-, as is the case in high performance modems, or when the channel
response Hif) is not known a priori.
If slit—klT), sjit) is negligible for |fc| L then an elegant algorithm is available to
maximize (2). It is a form of dynamic programming known to communication engineers as
Viterbi's algorithm.
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