Classical transceiver techniques - Signal Processing and Optimization for Transceiver Systems - page 181

5.4 Vector space interpretation of matched filtering

We now give an interesting interpretation of matched filtering based on the
language of vector spaces. For this discussion the reader may first want to
review Appendix A on 2 -spaces, L 2 -spaces, etc. We begin by observing that
the noise-free channel output can be expressed as

y ( t )= ∞ s ( n ) h ( t − nT ) , (5 . 25)

n = −∞

and is a linear combination of h ( t ) and shifted versions h ( t nT ) , where the

shifts are in integer multiples of T. So we say that y ( t ) belongs to a space V h of
signals spanned by the basis functions

−

... h ( t + T ) ,h ( t ) ,h ( t − T ) ,h ( t − 2 T ) ,... (5 . 26)

The term “basis” is used here under the assumption that { h ( t − nT ) } are linearly

independent. Assuming that s ( n ) ∈ 2 (space of finite energy sequences), the

space V h is a subspace of L 2 (space of finite energy functions), and we write

V h ⊂ L 2 . Under the condition that

H j ω +2 πk
T

∞

S h ( e jω )= 1
T

2 > 0

(5 . 27)

k = −∞

for all ω, we can construct an orthonormal basis for the space V h , as shown in
Sec. 5.3. 2 For this we define

H ( jω )
S h ( e jωT )

H new ( jω )= (5 . 28)

As we have already shown, h new ( t − nT )isanorthonormalset:

∞

mT ) h new ( t h new ( t − − kT ) dt = cδ ( m − k ) . (5 . 29)

−∞

Observe further that

h new ( t )=
n

c ( n ) h ( t − nT ) , (5 . 30)

where c ( n ) is the inverse transform of 1 / S h ( e jω ) . Thus h new ( t

− kT ) ∈ V h for

all k . Similarly

h ( t )=
n

d ( n ) h new ( t − nT ) , (5 . 31)

where d ( n ) is the inverse transform of S h ( e jω ), which means that h ( t − kT )isin

the space spanned by { h new ( t − kT ) } .Thus { h new ( t − nT ) } and { h ( t − nT ) } span

2 This positivity condition is related to linear independence and bandwidth suciency; see
Lemma 5.3.