Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 47

where β is the bandwidth in hertz. In the case of a simple rectangular pulse at a constant carrier frequency, as used in the WSR-88D
[9], the waveform bandwidth is approximately
1

β≈ ,
τ

(3)

where τ is the pulse duration in seconds.

C. CROSS-RANGE RESOLUTION
The amount of energy transmitted by the radar in a given direction
is determined by the antenna directivity pattern. The directivity
pattern of the WSR-88D is shown in Fig. 1. Maximum energy
is transmitted broadside, i.e., θ = ϕ = 0°, where θ and ϕ are the
azimuth and elevation angles, respectively, relative to the antenna
normal. The two-sided antenna beamwidth is defined as the difference between the half-power points on either side of the pattern
maximum, and it is often called the two-sided half-power (or 3
dB) beamwidth. Together, the half-power beamwidths in azimuth
and elevation define the solid angle of primary illumination of the
antenna beam.
The half-power beamwidth determines the cross-range resolution, i.e., the resolution in the azimuthal and elevation dimensions.
Cross-range resolution is the minimum separation between two objects in either of the two angular dimensions such that the objects
are still resolvable, and it may be approximated for the azimuthal
and elevation dimensions, respectively, as
ΔCRθ = R sin θ3 ≈ Rθ3

Pr =

2 2
PG
λσ
t

( 4π )

R 4 L2 ( R ) Ls

(6)

where Pt is the peak transmit power in watts, G is the unitless gain
of the antenna on transmit and receive, λ is the transmit wavelength
in meters, R is the range of the object in meters, L(R) is the oneway propagation loss over range R, Ls includes all system and processing losses, and σ is the radar cross section (RCS) of the object
in square meters. The RCS of an object corresponds to the reflective strength of a target, defined as the effective area of the target
as seen by the radar, assuming the reflected EM wave is isotropic.
However, because objects do not reflect energy isotropically, the
physical scattering area of an object is not the sole governing factor
in determining RCS. Size, shape, orientation, and composition of
the reflecting object are influencing factors as well.
In the case of weather radar, there are usually many scatterers
within a resolution volume, such as water droplets, ice crystals,
and biological scatterers. Assuming uncorrelated scatter between
objects and neglecting multiple scattering, the radar range equation
may be rewritten as

(4)

and
ΔCRφ = R sin φ3 ≈ Rφ3 ,

(5)

where θ3 is the azimuthal half-power beamwidth and ϕ3 is the elevation half-power beamwidth in radians. The approximations in
(4) and (5) are valid for sufficiently small angles.

D. RADAR RANGE EQUATION
The radar range equation predicts the amount of power received
by the radar from a reflecting object. In this section, we start with
a generic form of the radar range equation [7] and develop the
weather version of the radar range equation. In general, the power
received by a monostatic radar from a point target is
JULY 2017, Part II of II

Figure 1.

Representative directivity pattern for the WSR-88D and worst-case
sidelobe envelope. Typically, the sidelobes of the WSR-88Ds are several
decibels below the envelope [10].

IEEE A&E SYSTEMS MAGAZINE

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2017 Tutorial XI