# Archimedean Spiral Antenna Calibration Procedures to Increase the Downrange Resolution of a SFCW Radar

Authors: Ioan Nicolaescu; Piet van Genderen
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

This paper deals with the calibration procedures of an Archimedean spiral antenna used for a stepped frequency continuous wave radar (SFCW), which works from 400 MHz to 4845 MHz. Two procedures are investigated, one based on an error-term flow graph for the frequency signal and the second based on a reference metallic plate located at a certain distance from the ground in order to identify the phase dispersion given by the antenna. In the second case, the received signal is passed in time domain by applying an ifft, the multiple reflections are removed and the phase variation due to the time propagation is subtracted. After phase correction, the time domain response as well as the side lobes level is decreased. The antenna system made up of two Archimedean spirals is employed by SFCW radar that operates with a frequency step of 35 MHz.


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## Body

## 1. Introduction

Located at the interface between the
propagation media and the electronic device, the antenna is a very important
element of any electronic equipment used either to pass information from one
place to another or for object detection and tracking (radar systems etc.). In
the last decade, a lot of work has been done to find a proper system for
landmine detection. Among other systems, the ground penetrating radar (GPR) is
very promising because of its advantages over other types of sensors as: no
direct contact with the surface, the possibility to detect both metallic and
nonmetallic objects, and so forth. The GPR works either in time domain (video
pulse radar) or in frequency domain (SFCW radar). In the case of SFCW radar,
the antenna system consists of two antennas, one for transmission and the other
one for reception, which will be moved above the ground at a certain distance
(in this specific case 70 cm) to scan the ground surface. The main parameters
of radar are the range and the resolution. The range of SFCW radar is given by
the frequency step and the resolution by the frequency range. For landmine detection,
one needs both deep penetration and high resolution. Taking into account these
requirements and the international regulations with regards to frequency
allocation, the frequency range for SFCW radar has been chosen from 400 MHz to
4845 MHz. The low frequency will provide deep penetration while high frequency
will give the needed resolution. The antenna system has to comply with these frequency
range, has to have a stationary phase point because the distance is phase
embedded, and should have no extra delays within the antenna because they will
worsen the down range resolution of the system. Moreover, because the system is
a bistatic one (one antenna for transmission and one antenna for reception, 52 cm apart),
the coupling signal and the common footprint have to be as low as possible to provide both dynamic
range and cross-range resolution. Several ultra wideband (UWB) antennas had been
investigated for this application and the best fit was found for two
Archimedean spirals with opposite sense of rotation that have a low level of
the leakage signal and provide circular polarization, which is a very important
advantage in this application. However, the Archimedean spiral has a frequency
dependent delay that will worsen the range resolution. In order to remove the
delay, two procedures are investigated in this paper.


## 2. Antenna System—Time Domain Response

In order to
identify the delay within the antenna systems, the two Archimedean antennas had
been connected to a vector network analyzer (VNA) set to work as an SFCW radar
from 400 MHz to 4845 MHz in 128 steps (the frequency step was 35 MHz). One
antenna will transmit 128 frequencies and the other will receive the reflected
signals.If a metallic
plate is placed at a distanced of the antenna system and the power transmitted
by the radar is equalized for all 128 frequencies, then the transmitted signal
can be written like(1)u(t).=1128∑n=1128Aej2πfnt,where A is a
constant. This signal propagates to the metallic plate and is scattered back to
the antenna system. The received signal can be written as(2)s(t,tin).=1128∑n=1128rnAej2πfn(t−tin), where tin denotes the
delay due to the antenna system and due to propagation towards the metallic
plate and back; rn includes the propagation losses as well as the
reflection losses.The delaytin is frequency
dependent because the delays within the antenna system depend on frequency [1].
This happens due to the principle of operation of spiral antenna, which states
that this antenna has an active part that changes with frequency such as the
ratio between the geometrical dimension and the wavelength stays constant. The
lower frequencies have a larger delay than the higher frequencies [2] because
at lower frequencies, the currents have to propagate a longer way before being
radiated than the currents at higher frequencies. This will lead to an
increased time response as can be seen in Figures 1–3, where three situations
are displayed.Figure 1
The received signals, in time domain, for 4445 MHz frequency span (128
frequencies).Figure 2
The received signals, in time domain, for
3745 MHz frequency span (108 frequencies—first 20
removed).Figure 3
The received signals, in time domain, for 3745 MHz frequency span (108
frequencies—last 20 removed).The first one
corresponds to the case when the ifft transform is applied to a set of 128
frequencies that cover the entire operational band of the radar and the
synthesized time response is about 9 nanoseconds. The second case is for 108
frequencies (first 20 frequencies are removed) and the synthesized time
response is around 5.5 nanoseconds. In the third situation, the last 20
frequencies are removed and the synthesized time response is about 8.5 nanoseconds.The signals for the three situations
are given by(3)s1(tin).=1128∑n=1128rnAe−j2πfntin;for all 128 frequencies, (4)s2(tin).=1108∑n=21128rnAe−j2πfntin;for 108 frequencies (first 20
removed), and(5)s3(tin).=1108∑n=1107rnAe−j2πfntin;for 108 frequencies (last 20 removed), where   0≤tin≤1/Δf.


## 3. Calibration Using an Error-Term Flow Graph

### 3.1. Formulation

The time response of the
antenna system is drastically increased due to different delays associated with different frequencies (Figures1–3). The calibration
procedure should remove this phase dispersion introduced by the antenna system.
Because the radar measures amplitude and phase, in other words, it is a vectorial device, the calibration procedure that is in
place for vector network analyser should be followed. A vector network analyser
is calibrated by using an error-term flow graph for the frequency signal like
in Figure 4.Figure 4
Error-term flow graph for one port vector network analyzer;a and b
are reflected waves.The reflection
coefficientr, which is a function of frequency, can be calculated using the well-known formula [3](6)r=rm−S11S12S21−S22(rm−S11).The three
unknowns (S11, S22, and S12S21)
in (6) can be determined by measuring the reflections from three standards,
which for vector network analyzer are: the open, the short, and the matched
load. If we are able to find the three standards, this procedure can be used
for calibration.In order to
apply this procedure to SFCW radar, let us assume that all systematic linear
errors are included in the two-port error network. The correct reflection
coefficient,r, from an object will be given by
formula (6), where S11 represents
the leakage signal and S22 depends on the radar cross section (RCS)
of the antenna and takes into account the multiple ground bounces. The three
unknowns can be computed if three independent calibration targets are
available. According to [4], these are represented by free space
(“empty room,”
r=0), a metal plate (r=−1), and a wire grid. The error coefficients can be
derived as follows:(7)S11=rmo,S22=rmg−rmo+rg(rmg−rmo)rg(rmg−rmm),S12S21=−(rmg−rmo)(1+S22),   where
rmo,
rmm, and rmg denote measured signals for “empty room,”
metal plate, and metal grid and rg is the exact reflection
coefficient of the metal grid. Having the error coefficients, the reflection
coefficient for any object can be easily computed using (7).The disadvantage
of this procedure lies in the difficulty to find the precise value of the
reflection coefficient for the third standard (wire grid). It can be found
either by measurements or by calculation. V. Mikhnev has proposed a procedure
that replaces the third standard by measuring the reflection coefficient from the
same metallic plate placed in one or several shifted locations. If the
reflectivity of the antenna system from the free space is low(S22≪1),
which is true when the antenna system is high enough above the ground, then the
reflection coefficient of the shifted metal plate can be written as(8)rsp(f)=rm(u+vf+wf2)exp(−j2βl), where rsp
is the reflection coefficient of the metallic plate placed in a shifted
position; l is the offset distance, β is the
phase propagation constant and u, v, and w are parameters which depend on l. If
S22 is neglected then, the real value of the reflection coefficient
can be computed using the formula(9)r=rm−S11S12S21,and the unknown parameters u, v, and w can be determined by
solving the optimization problem(10)∑n|rsp(fn)−ro(fn)rm(fn)−ro(fn)−(u+vfn+wfn2)exp(−j2βnl)|→min.


### 3.2. Experimental Results

Having the
values of the unknown parameters in (8), the exact value of the reflection
coefficient of the shifted metallic plate can be computed. This method provides
better results if the offset distance is around a quarter of a wavelength. The
SFCW radar is an ultra wideband device; it operates from 400 to 4845 MHz so it
is not possible to make measurements at a quarter of wavelengths for all
frequencies. This is why several measurements have been made at different
shifted positions (l  =  97; 99; 102; 105; 109; 112; 115; 118; 122; 125; 127 cm)
and the unknown parameters are computed following a process of minimization of (10).
The values obtained for the three unknowns are u=0.96888;v=0.22×10−10; and w=0.The results are
presented in Figures5 and 6. Comparing these
figures with Figure 1, it can be
seen that the time response decreases and the level of the signal just after
the ground reflection is as low as −40 dB below the
main peak. This outcome is quite important because the mines are supposed to be
in this area.Figure 5
CalibratedA-scan using VNA
method; reference distance-97 cm, separation 105 cm.Figure 6
CalibratedA-scan using VNA method; reference distance-97 cm,
separation 127 cm.


## 3.1. Formulation

The time response of the
antenna system is drastically increased due to different delays associated with different frequencies (Figures1–3). The calibration
procedure should remove this phase dispersion introduced by the antenna system.
Because the radar measures amplitude and phase, in other words, it is a vectorial device, the calibration procedure that is in
place for vector network analyser should be followed. A vector network analyser
is calibrated by using an error-term flow graph for the frequency signal like
in Figure 4.Figure 4
Error-term flow graph for one port vector network analyzer;a and b
are reflected waves.The reflection
coefficientr, which is a function of frequency, can be calculated using the well-known formula [3](6)r=rm−S11S12S21−S22(rm−S11).The three
unknowns (S11, S22, and S12S21)
in (6) can be determined by measuring the reflections from three standards,
which for vector network analyzer are: the open, the short, and the matched
load. If we are able to find the three standards, this procedure can be used
for calibration.In order to
apply this procedure to SFCW radar, let us assume that all systematic linear
errors are included in the two-port error network. The correct reflection
coefficient,r, from an object will be given by
formula (6), where S11 represents
the leakage signal and S22 depends on the radar cross section (RCS)
of the antenna and takes into account the multiple ground bounces. The three
unknowns can be computed if three independent calibration targets are
available. According to [4], these are represented by free space
(“empty room,”
r=0), a metal plate (r=−1), and a wire grid. The error coefficients can be
derived as follows:(7)S11=rmo,S22=rmg−rmo+rg(rmg−rmo)rg(rmg−rmm),S12S21=−(rmg−rmo)(1+S22),   where
rmo,
rmm, and rmg denote measured signals for “empty room,”
metal plate, and metal grid and rg is the exact reflection
coefficient of the metal grid. Having the error coefficients, the reflection
coefficient for any object can be easily computed using (7).The disadvantage
of this procedure lies in the difficulty to find the precise value of the
reflection coefficient for the third standard (wire grid). It can be found
either by measurements or by calculation. V. Mikhnev has proposed a procedure
that replaces the third standard by measuring the reflection coefficient from the
same metallic plate placed in one or several shifted locations. If the
reflectivity of the antenna system from the free space is low(S22≪1),
which is true when the antenna system is high enough above the ground, then the
reflection coefficient of the shifted metal plate can be written as(8)rsp(f)=rm(u+vf+wf2)exp(−j2βl), where rsp
is the reflection coefficient of the metallic plate placed in a shifted
position; l is the offset distance, β is the
phase propagation constant and u, v, and w are parameters which depend on l. If
S22 is neglected then, the real value of the reflection coefficient
can be computed using the formula(9)r=rm−S11S12S21,and the unknown parameters u, v, and w can be determined by
solving the optimization problem(10)∑n|rsp(fn)−ro(fn)rm(fn)−ro(fn)−(u+vfn+wfn2)exp(−j2βnl)|→min.


## 3.2. Experimental Results

Having the
values of the unknown parameters in (8), the exact value of the reflection
coefficient of the shifted metallic plate can be computed. This method provides
better results if the offset distance is around a quarter of a wavelength. The
SFCW radar is an ultra wideband device; it operates from 400 to 4845 MHz so it
is not possible to make measurements at a quarter of wavelengths for all
frequencies. This is why several measurements have been made at different
shifted positions (l  =  97; 99; 102; 105; 109; 112; 115; 118; 122; 125; 127 cm)
and the unknown parameters are computed following a process of minimization of (10).
The values obtained for the three unknowns are u=0.96888;v=0.22×10−10; and w=0.The results are
presented in Figures5 and 6. Comparing these
figures with Figure 1, it can be
seen that the time response decreases and the level of the signal just after
the ground reflection is as low as −40 dB below the
main peak. This outcome is quite important because the mines are supposed to be
in this area.Figure 5
CalibratedA-scan using VNA
method; reference distance-97 cm, separation 105 cm.Figure 6
CalibratedA-scan using VNA method; reference distance-97 cm,
separation 127 cm.


## 4. Calibration Procedure Using a Metallic Plate

The other
procedure, which has been investigated, is the single reference procedure. It
supposes to use a single reference, a metallic plate, located at a certain
distance of the antenna system [5]. In this case, the transmitted signal is
given by (1) and the received signal can be calculated using (2). As was mentioned before, the delays within the transmitting and the receiving antennas
are frequency dependent. In fact, this effect can be seen in
Figure 7 where the
phase variation of the antenna is compared with a linear one. The upper line
represents the phase distortion. As can be seen in Figures 1–3, there are
multiple bounces so, in order to make a proper calibration, the first reflection
should be isolated. Time domain gating can do this. In order to decrease the
side lobes level, a Kaiser window, with β=3.15, is applied.Figure 7
Unwrapped phase variation for 128
frequencies (light red: measured; dark blue: linearized; green: unwrapped
with Matlab function; light red: due to propagation; light blue: phase
error).The
received signal after down conversion at the output ofI and Q mixers will be(11)s(tin).=1128∑n=1128rnknAe−j2πfntin,where kn takes into accountthe
variation of the gain along the processing chain for each frequency (channel).
If the reference plate is located at a reference distance rref then
the two-way propagation delay tref is given by(12)tref=2rrefc,where c is the speed of light.The correction
signal that removes the phase dispersion within the antenna system will be(13)scor(tref,tin).=1128∑n=1128e−j2πfn(tref−tin).
This signal is
displayed in Figure 8, in time domain,
and it is the mirror image of the gated
signal.Figure 8
Phase correction
signal in time domain (red line: absolute values, blue line: real values).The calibrated
signal is obtained by multiplying, for each frequency, the measured signal with
the one given by (13). After an ifft is applied, a calibrated range profile as in
Figures 9 and 10 is got. Matched filter calibration procedure has been employed
for SFCW radar because the level of the side lobes in the vicinity of the
ground reflection is lower than for VNA.Figure 9
A range profile before calibration
(blue line) and after calibration (red line) for a distance of 125 cm; data
measured with VNA.Figure 10
A range profile before
calibration (blue line) and after calibration (red line) for a distance of 127 cm; data measured with VNA.


## 5. System Calibration

The SFCW radar works in
frequency domain. After down conversion, at the output of the quadrature
mixers, for each frequency, a complex signal is obtained. The phase and
magnitude of these signals depend on the parameters of the processing chain and
among these, the balance betweenI  and Q channels plays a crucial role. The
calibration of SFCW radar is made in two stages. In the first stage, the delays
within the transmitter and the receiver are removed by making a direct
connection, with a known length cable, between the two and dividing each A scan
by this reference signal.For
one frequency, the output signals1(fk), after the first
stage, is given by(14)s1(fk).=sm(fk).−soff(fk).sm_ref(fk).−soff(fk).e−j2πfk(lref/v),where
soff(fk)  is the DC offset signal for the kth frequency,
sm_ref(fk) is the measured signal with a direct
connection between the transmitter and the receiver, sm(fk)
is the measured signal, lref is the length of the cable used for
direct connection and v is the propagation velocity through the cable. The
measured data, in time domain, after applying a Hamming window in order to
decrease the side lobes level, are presented in Figure 11.Figure 11
A scan (range profile) after
removing the DC offset and the delays within the transmitter and the receiver.In the second stage, the
single reference procedure is followed. To this end, a metallic plate is placed
at 1.27 m separation of the antenna system. The performances that come out can
be seen in Figures12 and 13 where an uncalibrated and a calibrated A scans are
showed. The first peak of the signal on Figure 13 is the coupling signal, the second is the reflection from the ground and the
next are multiple bounces so, the third peak can be used as a measure of the
RCS of the antenna system.Figure 12
A scan measured with SFCW radar before calibration.Figure 13
A scan measured with
SFCW radar after matched filter calibration procedure is applied.


## 6. Conclusions

(1) The Archimedean spiral antenna is a very good candidate for UWB
application but it has a major drawback because it is a dispersive system,
which means that the delay within the antenna is frequency dependent. As a
result, in any application where the shape of the pulse is important as, for
instance in radar systems, a calibration procedure to remove the delays within
the antenna system is needed.(2)
The
SFCW radar works in frequency domain it means it has a synthesized time domain
pulse that defines the downrange resolution of the system. The procedures
described in the paper should be used to calibrate the system for the delays
within the antenna system.(3) For this application, because the landmines are supposed to be
shallowly buried, the enlargement of the time domain response means that the
reflection from the ground surface will cover the weaker signals, which may
come from landmine. For the same reason, the level of the signal just after the
first reflection from the ground has to be as low as possible.(4)   The three standards procedure that is used for the vector network
analyzers calibration cannot be employed because of the difficulty to find a
suitable standard besides the “empty room” and the metallic plate. Nonetheless,
an alternative of this procedure, which supposes to replace the third standard
with the same metallic plate but placed in a shifted position, has been
analyzed. The side lobe level obtained with this procedure is just−40 dB below the main peak and is not enough
for this application.(5) The matched filter procedure needs a large separation between the
antenna system and the reference plate in order to avoid the overlapping of the
second reflection with the first one. Since the side lobes level near the
ground reflection is decreased to around−50 dB (Figures 9
and 10), in comparison with VNA procedures that provides only   −40 dB
(Figures 5
and 6), this procedure is used for SFCW radar calibration.(6) The Archimedean spirals work with circular polarization, which had been
proven very useful for this application because it exhibits the shape of the
detected object (the reflected signal embeds information about both dimensions
of the object).(7)
The resolution of the SFCW radar is given by the time response of the system.
As can be seen on Figures12 and 13, it is improved from tens of cm to cm,
which is in accordance with theoretical down range resolution given by the
bandwidth of the radar.



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*Source: 378285-2008-04-24.xml*