mohatta2015.pdf

42 MATCHED FILTERING
3. The processing of converting the radio signal to baseband often involves a
chain of filtering operations that, with possibly some additional baseband
filtering, have the effect of matching to the pulse shape.
Such a front end is also convenient in that it allows straightforward, compact formulations
of equalizer filter designs in discrete time.
With partial MF, we match to the pulse shape and then sample with sampling
phase io and sample period T s . This gives a sequence of received samples
which can be modeled as
/
oo
-OO
r{r)v*{T~qT s -t a )dT, (2.57)
v(qT s ) \= TJW S Σ ~ h (l T ° - m T + in)s(m) + ñ(qT a ), (2.58)
m— — oo
where
L-l
~ h (t) = ^g t Rp(.t-T t ). (2.59)
i=o
We can interpret (2.57) as correlating r(r) to a copy of the pulse shape centered at
qT — s.
We will refer to h(t) as the "net" response, which includes the pulse shape at
the transmitter, the medium response, and the initial receiver front-end filter. We
will usually assume that i» = 0, as a nonzero i 0 , can be folded into the medium
path delays. However, we should keep in mind that this implies some form of ideal
synchronization to the path delays or modeling the channel with path delays aligned
to the sampling instances.
When the sample period equals the symbol period (T s = Γ), equalization using
v(qT) is considered a form of symbol-spaced equalization. When the sample period
is less than the symbol period, typically of the form T s = T/Q for integer Q > 1,
we are using a form of fractionally spaced equalization.
2.3.6 Fractionally spaced MF
Suppose we use partial MF to obtain received samples. For a MF receiver, we would
complete the matched filtering process by then matching to the medium response.
Specifically, for symbol mo, we would form decision variable
L-l
*("*(>) = Σ 9¡v{m n T + r e ). (2.60)
Observe that this requires having samples at times m n T + r e ll.
When do we need fractionally spaced MF? First consider a nondispersive channel
(one path). If the receive filter is perfectly matched to the pulse shape and the
receive filter is sampled at the correct time (perfect timing, ί () = τ () ), then fractionally
spaced MF is not needed. The matched filtering is completed by matching