Radar Waveforms - MATLAB Simulations for Radar Systems Design - page 156

c
2 n ∆ f

∆ R = ------------

(3.65)

Range ambiguity associated with a SFW can be determined by examining
the phase term that corresponds to a point scatterer located at range . More R 0

precisely,

2 R 0
c

---------

ψ i () 2π f i =

(3.66)

It follows that

4π f i

------------------------------- R 0
c

( – f i )

4π R 0
c

∆ψ
∆ f

+ 1

------

-------- = = ------------- (3.67)

( f i – f i )

+ 1

or equivalently,

∆ψ
∆ f

c
4π

R 0 = -------- ------ (3.68)

It is clear from Eq. (3.68) that range ambiguity exists for . ∆ψ = ∆ψ 2 n π +

Therefore,

∆ψ 2 n π +

∆ f

c
4π

c
2∆ f

 

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

R 0 = = R 0 + n (3.69)

 

and the unambiguous range window is

c
2∆ f

R u = ---------

(3.70)

Hence, a range profile synthesized using a particular SFW represents the rel-
ative range reflectivity for all scatterers within the unambiguous range win-
dow, with respect to the absolute range that corresponds to the burst time delay.
Additionally, if a specific target extent is larger than , then all scatterers fall-
ing outside the unambiguous range window will fold over and appear in the
synthesized profile. This fold-over problem is identical to the spectral fold-
over that occurs when using a Fast Fourier Transform (FFT) to resolve certain
signal frequency contents. For example, consider an FFT with frequency reso-
lution , and size . In this case, this FFT can resolve
frequency tones between and . When this FFT is used to
resolve the frequency content of a sine-wave tone equal to , fold-over
occurs and a spectral line at the fourth FFT bin (i.e., ) appears. There-
fore, in order to avoid fold-over in the synthesized range profile, the frequency
step

R u

∆ f = 50 Hz NFFT = 64

– 1600 Hz 1600 Hz

1800 Hz
200 Hz

∆ f

must be