Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 182

thesamespace V h , the only difference being that the former is an orthonormal
basis. Thus, given any function y ( t )

∈ V h , we can write

y ( t )=
n

e ( n ) h new ( t − nT ) , (5 . 32)

where e ( m ) are the coe cients of expansion of y ( t )usingthebasis { h new ( t −

nT ) . Thus we have

∞

}

e ( n ) ∞
−∞

=
n

y ( t ) h new ( t nT ) h new ( t

− mT ) dt h new ( t − − mT ) dt

−∞

=
n

e ( n ) δ ( m − n ) (using Eq. (5.29))

so that

∞

y ( t ) h new ( t − mT ) dt = e ( m ) . (5 . 33)

−∞

Defining the function

g ( t )= h new (

− t ) (5 . 34)

we see that

y ( t ) h new ( t

mT ) dt = y ( t ) g ( mT

− − t ) dt,

so that the preceding equation can be rewritten as
∞

y ( t ) g ( mT − t ) dt = e ( m ) . (5 . 35)

−∞

g ∗ ( nT Note that span the same space V h . The process of

extracting the coe cients e ( m )from y ( t ) can therefore be interpreted as filtering
followed by sampling, as shown in Fig. 5.11.
Summarizing, given the output y ( t ) of the channel h ( t ) as in Fig. 5.12(a), it
can be expressed in the form (5.32) using the orthonormal basis h new ( t

{ h ( t − nT ) } and { − t ) }

nT ).
Thecoe cientsofexpansion e ( n ) can be obtained by filtering y ( t )withthe
receiver filter g ( t ) having frequency response H new ( jω ), that is,

−

H ∗ ( jω )
S h ( e jωT ) G ( jω )= , (5 . 36)

as indicated in Fig. 5.12(a) on the right. This is simply the normalized matched
filter, whose sampled output has its noise component whitened. These samples
e ( m ) contain the full information about y ( t ) in view of Eq. (5.32). Using a noble
identity we can redraw the system as in Fig. 5.12(b). Since e ( m ) contains all the
information about y ( t ), the sampled version r ( n ) of the output of the matched
filter H ∗ ( jω ) also contains this information.