Aerospace and Electronic Systems Magazine April 2018 - 51

Miralles et al.
Table 1.

FFT Original and Reordered, Index, and Corresponding Angle
Original

1

2

...

8

9

10

...

Reordered

10

11

...

17

1

2

...

Index n

−8

−7

...

−1

0

1

...

−45

−39

...

−5

0

5

...

Angle (°)

sults in a complex spectrum (two sided). These complex points are
sampled at the virtual element positions, which leads to a sampling
frequency in space of
f s,space =

1
≈ 83.3 m −1
dx

(5)

and a frequency spacing for the FFT of
df space =

f s,space
N vir, x

≈ 4.9 m −1.

(6)

The FFT will give Nvir,x bins equally distributed from f = 0 to
f = fs,space. Let α be the angle between the incident wavefront and the
antennas, spanning between −90° and 90°. This means that the first
sample is the component that belongs to a straight incident wave
f s,space
α = 0. The samples above
represent negative angles and
2
have to be shifted according to Table 1. The reordering of the FFT
leads to a representation with α = 0° in the center.
For each frequency sample n of the FFT, the corresponding
angle can be calculated as

α = arcsin ( n ⋅ df space ⋅ λ0 )

(7)

with

N vir, x − 1
N vir, x − 1 
(8)
≤n≤
n ∈ IN −
.
2
2


This leads to azimuth angles from −45° to 45° (Table 1) for the
system.

ramp settings presented in earlier sections, a
3D radar image, which contains information
about the range, azimuth, and elevation, is
calculated. To measure the 3-dB beamwidth of
16
17
the system, a corner cube with a cross-section
of σ0 = 150 m2 is placed in an anechoic cham8
9
ber, at a range R = 23.1 m, with azimuth and
7
8
elevation angles of ϕ = 0° and θ = 0°, respectively. With the FFT processing and FMCW
39
45
ramp settings presented in former sections, a
3D radar image, which contains the information of range, azimuth, and elevation, is calculated.
Figure 7 shows the normalized azimuth (ϕ = 0°) and elevation
(θ = 0°) profiles taken at the range cell of the central target. Under
the assumption that the corner cube is a point target and the distance is big enough to be in the far-field region, the theoretical values of the angular resolution of the MIMO virtual array (calculated
earlier in the Waveform and Timing section) should match with
the measured 3-dB beamwidth shown in Figure 7. The measured
azimuth resolution is Δθ 3dBx = 4.7° and the measured elevation
resolution is Δφ3 dB y = 3.6°, whereas the calculated azimuth resolution is Δθ 3dBx = 4.5° and the calculated elevation resolution is
Δφ3 dB y = 3.5°. As can be seen, the measured and calculated angular
resolutions match well. One thing to consider is that in the estimation along the azimuth direction, the missing element is included,
while in the radar beamforming process, this missing element is
calculated as the average of its neighbors.

RADAR FIELD OF VIEW
The aim of this subsection is to show the dependency between a
target's received power, at a certain distance R, and its angle, in
both azimuth and elevation directions.
A corner cube with a cross-section of σ0 = 150 m2 is placed at
R = 23.1 m, ϕ = 0°, and θ = 0°. Then, the radar is rotated by ϕ =
−60° in the elevation direction, without moving the target. At this
point, a radar capture is started. This process is repeated following
an angular sweep from ϕ = −60°, θ = 0° to ϕ = 60°, θ = 0°, with a

SYSTEM PERFORMANCE VERIFICATION AND
MEASUREMENT RESULTS
This section is divided into two parts: the verification of system
performance and the presentation of measurement results for an
application. The angular variables are θ for azimuth and ϕ for elevation. The coordinate system is displayed in Figure 4.

ANGULAR RESOLUTION
To measure the 3-dB beamwidth of the system, a corner cube with
a cross-section of σ0 = 150 m2 is placed in an anechoic chamber,
at a range R = 23.1 m, with azimuth and elevation angles of ϕ = 0°
and θ = 0°, respectively. With the FFT processing and FMCW
APRIL 2018

Figure 7.

Azimuth and elevation profiles for a central target.

IEEE A&E SYSTEMS MAGAZINE

51



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