Radar Waveforms - MATLAB Simulations for Radar Systems Design - page 167

The Replica 169

where is the Fourier transform of . Thus, the moduli of and
are identical; however, the phase responses are opposite of each other.

S i () s i () H ()

S i ()

Example:

Compute the maximum instantaneous SNR at the output of a linear filter
whose impulse response is matched to the signal t 2 . x () = exp ( – ⁄ 2 T )

Solution:

The signal energy is

∞
∫

∞
∫

t 2

x () 2

( – ) ⁄ T

E = d = e d = π T Joules

– ∞ – ∞

It follows that the maximum instantaneous SNR is

π T
N 0

SNR = -------------

⁄ 2

where is the input noise power spectrum density. N 0 ⁄ 2

3.8. The Replica

Again, consider a radar system that uses a finite duration energy signal ,
and assume that a matched filter receiver is utilized. The input signal is given
in Eq. (3.76) and is repeated here as Eq. (3.98),

s i ()

x () Cs i = ( t 1 – ) + n i ()

(3.98)

The matched filter output can be expressed by the convolution integral
between the filterÓs impulse response and

y ()

, x ()

∞
∫

y () = x () ht u ( – ) d

(3.99)

– ∞

Substituting Eq. (3.95) into Eq. (3.99) yields

∞
∫

x () s i ∗ τ

y () = ( – t + u ) d = R xs i ( t – τ )

(3.100)

– ∞

where is a cross-correlation between and . Therefore,
the matched filter output can be computed from the cross-correlation between
the radar received signal and a delayed replica of the transmitted waveform. If
the input signal is the same as the transmitted signal, the output of the matched

R xs i ( t – τ ) x () s i ( τ – t )