Radar Waveforms - MATLAB Simulations for Radar Systems Design - page 172

174 MATLAB Simulations for Radar Systems Design

The time delay resolution is

∞
∫

() 2

χ R d

– ∞

∆τ = ----------------------------------- (3.127)

χ 2 ()

Using ParsevalÓs theorem, Eq. (3.127) can be written as

∞
∫

U () 4 d

– ∞

--- ------------------------------------ -- ---

∆τ = 2π

(3.128)

2

∞
∫

U () 2 d

– ∞

The minimum range resolution corresponding to is ∆τ

∆ R ∆τ 2 = ⁄ (3.129)

However, since the signal effective bandwidth is

2

∞
∫

U () 2 d

– ∞

B = -------------------------------------------- (3.130)

∞
∫

U () 4 2π d

– ∞

the range resolution is expressed as a function of the waveformÓs bandwidth as

∆ Rc 2() = ⁄ (3.131)

The comparison between Eqs. (3.116) and (3.126) indicates that the output
of the matched filter and the range ambiguity function have the same envelope
(in this case the Doppler shift is set to zero). This indicates that the matched
filter, in addition to providing the maximum instantaneous SNR at its output,
also preserves the signal range resolution properties.

f d

3.10.2. Doppler Resolution

It was shown in Chapter 1 that the Doppler shift corresponding to the target
radial velocity is

2 vf 0
c

2 v
λ

f d == ------ ---------- (3.132)

where is the target radial velocity, is the wavelength, is the frequency, v λ f 0

and is the speed of light. c