Radar Waveforms - MATLAB Simulations for Radar Systems Design - page 169

Matched Filter Response to LFM Waveforms 171

2 v
c

f d = ------ f 0 (3.108)

and since , we get γ – 1 = 2 vc ⁄

f d = ( γ – 1 ) f 0

(3.109)

Using the approximation and Eq. (3.109), Eq. (3.107) is rewritten as

γ ≈ 1

() e j 2π f d t 0

( – )

s r ≈ st t 0 ( – ) (3.110)

where

– j 2π f 0 t

st t 0 ( – ) = e s 1 ( t – )

(3.111)

0

is given in Eq. (3.101). The matched filter response is given by the con-
volution integral

s 1 ()

∞
∫

s o () = h () s r ( tu – ) du

(3.112)

– ∞

s ∗

For a non-causal matched filter the impulse response

h ()

is equal to

() t

; it

follows that

∞
∫

s ∗ s o () = –( s r u ( tu – ) d (3.113)

– ∞

Substituting Eq. (3.111) into Eq. (3.113), and performing some algebraic
manipulations, we get

∞
∫

s ∗ () e j 2π f d tu 0

( + – )

s o () = st ( + u t 0 – ) d

(3.114)

– ∞

Finally, making the change of variable yields

t ' = tu +

∞
∫

) e j 2π f d t '

( – t 0 )

s ∗ t ' s o () = ( – t ) st ' ( – t 0 d t ' (3.115)

– ∞

It is customary to set . It follows that t 0 = 0

∞
∫

) e j 2π f d t ' st () s ∗ t ' s o ( tf d ; ) = ( – t d t '

(3.116)

– ∞