Radar Waveforms - MATLAB Simulations for Radar Systems Design - page 166

168 MATLAB Simulations for Radar Systems Design

Eq. (3.89) tells us that the peak instantaneous SNR occurs when equality is
achieved (i.e., from Eq. (3.88) h s i ∗ ). More precisely, if we assume that =

equality occurs at , and that , then t = t 0 k = 1

h () s i ∗ t 0 = ( – t 1 – u ) (3.90)

and the maximum instantaneous SNR is

∞
∫

2 C 2 ) 2 s i ( t 0 – t 1 – u d

– ∞

SNR t () = ----------------------------------------------------------------

(3.91)

N 0

Eq. (3.91) can be simplified using ParsevalÓs theorem,

∞
∫

2

) 2 E = s i ( t 0 – t 1 – u d (3.92)

– ∞

where denotes the energy of the input signal; consequently we can write the
output peak instantaneous SNR as

E

SNR t () 2 E
N 0

= -------

(3.93)

Thus, we can draw the conclusion that the peak instantaneous SNR depends
only on the signal energy and input noise power, and is independent of the
waveform utilized by the radar.

Finally, we can define the impulse response for the matched filter from Eq.
(3.90). If we desire the peak to occur at

t 0 = t 1

, we get the non-causal matched

filter impulse response,

() s i ∗ h nc = () t (3.94)

Alternatively, the causal impulse response is

() s i ∗ τ h c = ( – t ) (3.95)

where, in this case, the peak occurs at . It follows that the Fourier t 0 = t 1 + τ

transforms of and are given, respectively, by

h nc () h c ()

H nc () S i ∗ () =

(3.96)

– j ωτ

H c () S i ∗ () e

= (3.97)