Aerospace and Electronic Systems Magazine November 2016 Tutorial X - 93

Fasano et al.
Typical values of the PRF range from 3 kHz for Rmax=50 km to 30
kHz for Rmax=5 km.
The selection of radar carrier frequency fc is an important issue,
too. A general rule of thumb suggests that smaller frequencies tend
to suffer less atmospheric absorption, whereas larger frequencies
need smaller antennas to be installed. As a consequence, large aircraft can exploit X band radars (fc=10 GHz) that require antennas
with a diameter of 1 m or more, while small aircraft need to use
smaller Ka band (fc=35 GHz) or W band (fc=94 GHz) radars and
need to increase the level of emitted power.
A first issue that is related to a radar system is range resolution ΔR, i.e., the minimum relative distance with respect to a radar
antenna focal point that permits it to discriminate between two objects that are in the antenna beam. It is worth noting that the resolution can be defined for all terms included in the track state, i.e.,
range, range rate, angles, and angular rates. Since all targets that
are closer than the resolution are imaged as a single target by the
radar, the domain where each term of the track state is defined can
be divided into equal subdomains that have a size equal or larger
than the resolution. These subdomains are called "cells," since they
can contain no more than a single object. During a scan, the radar
emits pulses in order to check if each cell contains an object or not.
The structure that associates the level of power received by the
radar to each cell is called "radar image." If the radar has a high
level of resolution (i.e., fine resolution, small cells with respect to
obstacle size) some object can be extended over neighboring cells.
In this case, a cell association processing step is required. The position of the object can be correlated to the centroid determined by
means of weighted sum of the power measured in each contiguous
cell multiplied by the reference position of each cell.
In order to discriminate two objects that are contiguous in
range, the radar has to receive a combined signal of a pulse backscattered by the two objects that has two different maxima rather
than a single one. For Gaussian-shaped pulses, this event happens
when the Rayleigh condition is verified, i.e., the time difference
between two subsequent backscattered pulses is higher than onehalf of the pulse time duration τ [17]. For τ=0.1 µs pulse duration,
this condition produces a resolution of 15 m.
The uncertainty in the determination of range σR is clearly related to the pulse duration τ. By adopting a frequency modulated
pulse within a bandwidth B, it is possible to carry out processing
dubbed "pulse compression" to reduce pulse duration to 1/B and
generally 1/B < τ. The error in the synchronization of transmitted
and received pulses can be a stated fraction of the pulse duration
[21]. This term can be given by means of (17):

(17)

where S/N is radar signal-to-noise ratio that is discussed in the next
subsection. For bandwidth B = 1 MHz and S/N=3.98, the resulting
theoretical range uncertainty is σR = 53 m. Apart from adopting a
larger bandwidth, in order to reduce the uncertainty, a number of
n subsequent pulses can be transmitted to the same object and the
relevant range can be assumed as the average over the n pulses.
In this case, a reduction of the uncertainty can be achieved in the
order of
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NOVEMBER 2016, Part II of II

Doppler analysis is presently the best option to derivate range rate.
It is based on radar electronics signal processing. If vR is the range
rate and fc is the frequency of the carrier, i.e., the frequency emitted
by the transmitter, the spectrum of the received signal has a shift in
frequency equal to the Doppler frequency fD that is given by (18):

(18)

where λc is the wavelength associated to the carrier. The resulting range rate vR can be estimated by means of a circuit called
frequency-locked loop (FLL) that measures the frequency shift
between the local oscillator and the received signal [17]. As a result of (17), high-frequency carriers allow for better resolution than
low-frequency ones.
Targets that occupy the same range and angle resolution cell
may be resolved in radial velocity by radars that employ coherent
Doppler processing. The velocity resolution ∆V is given by (19)
[21]:

(19).

The uncertainty in the determination of the speed by Doppler processing σV,D depends on the performance of the FLL. In general, it
can be estimated as a function of resolution (19) by means of (20):
(20).
No velocity resolution is defined for simple pulsed radars since,
in this case, the velocity is computed by exploiting more than one
sensor range estimate.
If the range rate is estimated by numerical derivation of the
range history, the uncertainty is by far greater than the one estimated by Doppler processing. In this case, the expression of the
uncertainty σV,N is given by (21) [21]:
(21)
where n is the number of subsequent pulses used to estimate the
range rate.
For a pulse duration, τ =1 µs and S/N=3.98, the resulting theoretical range uncertainty for a Doppler radar with carrier frequency
Ka band radar (fc,Ka=35 GHz) is equal to σV,Ka = 12.3 m/s, whereas
for an X band radar (fc,V=10 GHz) is equal to σV,X = 45 m/s. In order
to achieve a similar performance, non-Doppler radars have to use
more than 100 subsequent pulses.
The angular resolution Δθ of the radar is determined by its half
power beamwidth. For a radar that emits a carrier at wavelength λc
through an antenna that has a characteristic size equal to d, the half
power beamwidth is given by (22):
(22)
If the radar has a parabolic antenna and the angles are measured
in radians, k=1.22 [22]. Small wavelengths, i.e., high frequencies,

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