remote sensing
Communication
MIMO FMCW Radar with Doppler-Insensitive
Polyphase Codes
EunHee Kim
Department of Defense System Engineering, Sejong University, 209 Neungdong-ro, Gwangjn-gu,
Seoul 05006, Korea; eunheekim@sejong.ac.kr
Abstract: Co-located MIMO is used to enlarge the antenna aperture virtually and increase the angular
resolution. This paper shows FMCW radar using MIMO VAA. Polyphase codes are designed for
modulating the successive chirps of transmitting signals. The codes are optimized to have low
cross-correlations regardless of the Doppler filter mismatch. Compared with orthogonal codes, the
designed codes show robust performance for Doppler mismatch and lower angle estimation errors.
The entire procedure is explained, and the simulation results are provided.
Keywords: FMCW radar; MIMO VAA; polyphase code; simulated annealing algorithm
1. Introduction
Citation: Kim, E. MIMO FMCW
Radar with Doppler-Insensitive
Polyphase Codes. Remote Sens. 2022,
14, 2595. https://doi.org/10.3390/
rs14112595
Academic Editor: Ashish
Kumar Singh
Received: 21 April 2022
Accepted: 27 May 2022
Published: 28 May 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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Copyright:
© 2022 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Frequency-modulated continuous waveform (FMCW), also known as linear frequency
modulation (LFM), is commonly utilized in the automotive industry. Targets are detected
by the frequency difference between the transmitted and received signals, and the range
resolution is inversely proportional to the bandwidth. Conventional methods use two or
more slopes to resolve the range and velocity but are unsuitable for multiple targets because
they inherently produce ghost targets. Thus, multiple fast chirps are now widely used for
detecting multiple targets, where the distance is determined by the frequency difference
and the velocity (Doppler frequency) is estimated by the phase differences of multiple
consecutive chirps [1]. On the other hand, high-resolution angle estimation capability is
one of the important requirements of automotive radars for collision avoidance or adaptive
cruise control. Because the resolution capability is primarily dependent on the physical
size of the array (i.e., the antenna aperture), it is very difficult for a small-sized automotive
radar to achieve high angular resolution. Several studies on high-resolution angle estimation based on beamforming techniques have been conducted, including subspace-based
algorithms [2–4] like Multiple Signal Classification (MUSIC) and parameter estimation
algorithms using the maximum likelihood (ML) function [5,6]. However, the recent virtual
array antenna (VAA) by the Multiple-Input and Multiple-Output (MIMO) method provides
a more efficient way to achieve high-resolution capability with a large virtual antenna,
which can be combined with conventional methods.
MIMO radar has attracted the attention of many researchers in recent years [7–13], as
it effectively improves radar performance by transmitting multiple signals concurrently.
The MIMO method can be used to increase the antenna aperture in collocated antennas,
which improves the angular resolution [14], or it can be used in a multi-static manner that
is transmitted and received from different sites [15]. Automobile applications typically
use the co-located MIMO method to emulate a much larger aperture than its physical
aperture, which is called VAA. In any case, the essential issue of MIMO is to design the
orthogonal transmit waveforms in at least one domain: the frequency, time, or code domain.
Time division multiplexing (TDM) is the easiest method to achieve orthogonality because
only one waveform is transmitted each time. However, radar signal processing requires
multiple pulses or chirps, and TDM with many transmitters reduces the effective repetition
frequency. Therefore, TDM is inadequate for operations with a high repetition frequency.
Remote Sens. 2022, 14, 2595. https://doi.org/10.3390/rs14112595
https://www.mdpi.com/journal/remotesensing
Remote Sens. 2022, 14, 2595
2 of 12
Remote Sens. 2022, 14, 2595
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frequency. Frequency division multiplexing (FDM) basically means that each transmitter
divides
the division
operating
frequency domain
without overlap.
Particular
FDM with
orthogoFrequency
multiplexing
(FDM) basically
means that
each transmitter
divides
the
nality
in
the
beat
frequency
domain
that
is
compatible
with
FMCW
radar
has
been
operating frequency domain without overlap. Particular FDM with orthogonality introin the
duced
[16–18]. This
approach
used the spectrum
moreradar
efficiently
than
the classical
frebeat frequency
domain
that is compatible
with FMCW
has been
introduced
[16–18].
quency
divisionused
method.
Finally, code
division
multiplexing
uses
orthogonal
code
This approach
the spectrum
more
efficiently
than the(CDM)
classical
frequency
division
modulation
instead
of frequency
modulation. (CDM)
CDM inuses
FMCW
radar used
phase
method. Finally,
code
division multiplexing
orthogonal
codebinary
modulation
shift
keying
(BPSK) ormodulation.
random code
to ensure
eachradar
transmitter
sendsphase
a different
coded
instead
of frequency
CDM
in FMCW
used binary
shift keying
waveform,
provided
the
code
bandwidth
is
small
to
ensure
proper
FMCW
radar
opera(BPSK) or random code to ensure each transmitter sends a different coded waveform,
tion
[19,20].
this bandwidth
paper, we use
the CDM
method
for an
FMCW
radar
using multiple
provided
theIncode
is small
to ensure
proper
FMCW
radar
operation
[19,20].
fast
chirps
by modulating
the orthogonal
polyphase
codes
successive
chirps.
In this
paper,
we use the CDM
method for
an FMCW
radartousing
multiple
fast chirps by
This paper
a design
method
ofto
polyphase
for multi-chirp FMCW ramodulating
the proposes
orthogonal
polyphase
codes
successivecodes
chirps.
dar toThis
improve
angular
resolution
by MIMO
VAA [21].
VAA
beamforming
and angle
paperthe
proposes
a design
method
of polyphase
codes
for multi-chirp
FMCW
radar
estimation
after the range
Doppler
processing.
the codes modulated
to
to improveare
theperformed
angular resolution
by MIMO
VAA
[21]. VAAIfbeamforming
and angle
estimation
are chirps
performed
after the range
Dopplerofprocessing.
If the codes
modulated
to
the
successive
are orthogonal,
the output
the corresponding
Doppler
bin only
the successive
chirps
aresignal.
orthogonal,
the ifoutput
of the
corresponding
Dopplerthe
bincrossonly
contains
a matched
code
However,
the code
is not
perfectly orthogonal,
contains a matched
code
signal.interference
However, and
if thereduces
code isthe
notsignal-to-interference-andperfectly orthogonal, the
correlation
remains as
additive
cross-correlation
as additive
reduces
the
signal-to-interferencenoise
ratio (SINR).remains
Additionally,
if theinterference
center of theand
Doppler
filter
does
not precisely coinand-noise
ratio
(SINR).
Additionally,
of the Doppler
filter does not
precisely
cide
with the
target
velocity,
the same if
is the
true.center
The remaining
cross-correlations
reduce
the
coincide
the target
velocity,
the same
is true. The
remaining
reduce
SINR
andwith
degrade
the angle
estimation
accuracy.
Therefore,
we cross-correlations
designed the polyphase
the SINR
and degrade
angle estimationregardless
accuracy. of
Therefore,
we designed
the polyphase
codes
maintaining
lowthe
cross-correlation
the Doppler
filter mismatch
using
codes
maintaining
low
cross-correlation
regardless
of
the
Doppler
filter
mismatch
using
the simulated annealing (SA) algorithm of the statistical optimization method [22–25].
the
simulated
annealing
(SA)
algorithm
of
the
statistical
optimization
method
[22–25].
Section 2 explains the basic principles of MIMO FMCW, the SA algorithm for code design,
Section
2 explains
the basic
principles
of MIMO
the SAby
algorithm
for code
design,
and
VAA
beamforming.
Section
3 evaluates
theFMCW,
performance
simulation,
and Section
VAA beamforming.
Section
evaluates the performance by simulation, and Section 4
4and
summarizes
and concludes
the3paper.
summarizes and concludes the paper.
2. Basic Principles
2. Basic Principles
2.1.
2.1. MIMO
MIMO FMCW
FMCW
The
source
The source signal
signal for
for all
all transmitters
transmitters is
is the
the multi-chirp
multi-chirp FMCW
FMCW waveform
waveform where
where the
the
chirp
is
repeated
N
times,
as
shown
in
Figure
1.
chirp is repeated N times, as shown in Figure 1.
Figure 1.
1. Multi-chirp
Multi-chirp FMCW
FMCW waveform.
waveform.
Figure
Then, the kth chirp is expressed as
Then, the kth chirp is expressed as
"
2 #
πFBW
𝜋𝐹
𝑇T
S𝑆k (t(𝑡)
) ==S𝑆(𝑡
f c t𝑡+
t − − 𝑘𝑇
− kT , 𝑘𝑇, ≤kT𝑡 ≤
t <+(1)𝑇
k + 1) T
(t −−kT
) ==exp
𝑘𝑇)
expj𝑗 2π
2𝜋𝑓
+
𝑡−
< (𝑘
𝑇T
2 2
(1)
(1)
where
= 0,1,2,
− 1),
where k𝑘 =
0, 1, 2,…. (𝑁
. . (N
− 1)𝑓, f cisisthe
thecenter
centerfrequency,
frequency, 𝐹FBW is
is the
the bandwidth,
bandwidth, 𝑇
T is
is the
the
repetition
repetition period,
period, and
and N
N is
is the
the number
number of
of chirps.
chirps. Each
Each transmitter
transmitter modulates
modulates N
N chirps
chirps by
by
its
its own
own code
code [26].
[26]. We
We designed
designed the
the MIMO
MIMO system
system with
with different
different transmitting
transmitting phase
phase codes
codes
as
2. The
The mth
mth transmitter’s
transmitter’s code
code C𝐂 and
can
expressed
andsignal
signalS 𝑆 (, t)(𝑡)
as shown
shown in
in Figure
Figure 2.
can
bebe
expressed
as
m
m,k
as
h
iT
Cm =
C
C
.
.
.
C
m=
=1,1,…. .𝐿. Lt
(2)
(2)
m,0
𝐂 = 𝐶 , m,1𝐶 ,
… 𝐶
m,( ,(
N −1)) , 𝑚
𝑆 (, t(𝑡)
, 𝑆 (𝑡) = 𝐶 , 𝑆(𝑡 − 𝑘𝑇)
Sm,k
) ==C𝐶
m,k Sk ( t ) = Cm,k S ( t − kT )
(3)
(3)
Remote Sens. 2022, 14, 2595
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where 𝐿Lt is
there is
is one
one target,
target, the
the received
received signal
signal after
after
where
is the
the number
number of
of transmitters.
transmitters. IfIf there
dechirping via mixing the transmit and receive signals is represented by the sum of the
dechirping via mixing the transmit and receive signals is represented by the sum of the
reflective signals as follows:
reflective signals as follows:
Lt
2(2(𝑅
R −−vt𝑣𝑡)
)
∗
Sm,k
−−
τ )𝜏)
Sk 𝑆(t)(𝑡)
++
noise,
τ=
Ri,k𝑅(t)(𝑡)
= =∑ Am,i
, , i 𝑖==1,1,. …
. . ,, 𝐿Lr
∗( t (𝑡
̅
𝑛𝑜𝑖𝑠𝑒,
𝜏
=
𝐴
𝑆
,
,
c𝑐
,
m =1
(4)
(4)
where ** means
means the
the complex
complex conjugate,
conjugate,RRisisthe
thedistance
distanceto
tothe
thetarget,
target,ccisisthe
thelight
lightspeed,
speed,
where
number of
of receivers,
receivers, and
and 𝐴
A̅ m,i is
the
complex
reflective coefficient
coefficient from
from the
the mth
mth
is the
the number
is
the
complex
reflective
𝐿Lr is
,
transmitter
to
the
ith
receiver,
assuming
that
it
does
not
fluctuate
during
the
coherent
transmitter to the ith receiver, assuming that it does not fluctuate during the coherent inintegration
time
NT.
Though
weassumed
assumedadditive
additivenoise,
noise,we
wedo
donot
notdescribe
describeititin
indetail
detail
tegration
time
of of
NT.
Though
we
because
it
has
little
influence
on
this
study.
because it has little influence on this study.
Figure
Figure 2.
2. Overall
Overalldescription
descriptionof
ofthe
the MIMO
MIMO FMCW
FMCW radar.
radar.
After discarding
discarding the
the small,
small, high-order
high-order terms
terms and
and combining
combining the
theconstant
constantphase
phaseterms
terms
After
into A
is is
rewritten
with
thethe
following
equation.
into
𝐴 m,i, ,, the
thereceived
receivedsignal
signalininmulti-chirp
multi-chirpFMCW
FMCW
rewritten
with
following
equaThe detailed
derivation
is in Appendix
A: A:
tion.
The detailed
derivation
is in Appendix
Lt
∗∗
R𝑅i,k,(t(𝑡)
)∼
t−
− kT
𝑘𝑇)
+ f𝑓d t𝑡] +
exp2πj
2𝜋𝑗[𝑓
+noise
𝑛𝑜𝑖𝑠𝑒
[ f b ((𝑡
)+
=≅ ∑ A𝐴m,i, C𝐶m,k
, exp
(5)
(5)
m =1
thebeat
beatfrequency
frequencyby
bythe
thetarget
targetdistance
distanceand
andf d𝑓is the
is the
Doppler
frequency:
where
where 𝑓
f b isisthe
Doppler
frequency:
𝐹 2𝑅
2𝑣
(6)
𝑓 = FBW 2R and 𝑓 = −𝑓 2v .
(6)
fb =
𝑐 and f d = − f c 𝑐 .
𝑇
T
c
c
𝑓 + 𝑓 is estimated via the first range FFT processing of fast-sampling data in one
estimated
range
FFT processing
of fast-sampling
in one
chirp,fand
measuredvia
viathe
thefirst
second
Doppler
FFT of inter-chirp
data with data
a sampling
b + f d𝑓 isis
chirp,
and
f d waveform
is measured
via the second
Doppler
data
a sampling
rate
of T.
The
is usually
designed
for 𝑓 FFT
and of𝑓 inter-chirp
to differ by
twowith
or more
orders
rate
of T. The waveform is usually designed for f b and f d to differ by two or more orders
of
magnitude.
of magnitude.
A single transmit signal is extracted by multiplying its code to the consecutive chirps
single transmit
signalDoppler
is extracted
by multiplying its code to the consecutive chirps
of theAreceived
signal before
processing:
of the received signal before Doppler processing:
Remote Sens. 2022, 14, 2595
∗ R (t)
Cj,k
i,k
4 of 12
Lt
∗ C
= ∑ Am,i Cj,k
m,k exp 2πj [ f b ( t − kT ) + f d t ] + noise
m =1






Lt
∗
= exp 2πj[ f b (t − kT ) + f d t] A j,i N + ∑ Am,i Cj,k Cm,k exp 2πj( f d t) + noise


m =1


(7)
(m6= j)
The second term in Equation (7) is removed after Doppler processing if Cm and C j are
orthogonal and the center of the filter is exactly matched with f d . Otherwise, this term will
not be zero and will remain as interference.
Figure 2 describes the whole structure of the proposed MIMO FMCW radar. Each
transmitting signal Sm,k (t), expressed in Equation (3), is modulated by the different phase
code, and the received signals are downconverted by dechirping, which is represented by
Ri,k (t) in Equation (5). Because the range FFT is a linear transformation performed within
one chirp with the same code, it is executed before decoding to reduce the computational
power. Then, the code decoding for successive chirps and Doppler processing are followed.
VAA beamforming is performed last.
2.2. Design of the Doppler-Insensitive Polyphase Code
The polyphase code in Equation (2) with the unit amplitude is represented by
2π
2π
2π
Cm,k = exp( j ϕm (k)), ϕm (k) ∈ 0,
, 2·
, . . . , ( M − 1) ·
M
M
M
(8)
where M is the number of distinct phases in [0 2π ) [22]. If M = 2, then the code becomes a
biphase code with 1 and −1. The auto-correlation (AC) and the cross-correlation (CC) are
defined by
N
AC =
∗
Cm,k = N,
∑ Cm,k
k =1
N
CC jm =
∗
Cm,k ( j 6= m)
∑ Cj,k
(9)
k =1
and CC is zero if the codes are orthogonal with each other.
As shown in the second term of Equation (7), if the received signal from the moving
targets has Doppler shift, and the center of the Doppler filter does not exactly coincide with
it, the outputs of the correlators are changed by the mismatch ∆ f as follows:
N
N
2πk
2πk
∗
AC(∆ f ) = ∑ exp j
∆ f , CC jm (∆ f ) = ∑ Cj,k Cm,k exp j
∆f
N
N
k =1
k =1
(10)
where ∆ f is a dimensionless number which is the frequency mismatch between the target
Doppler and the center of the Doppler filter divided by the Doppler resolution. The average
signal-to-interference ratio (SIR) at the ith receiver from Equation (7) can be defined by
SIR(∆ f ) = 
A j,i AC(∆ f )
2
2
=
power of signal
.
average power of interference
(11)
1 L

 ∑ t ∑ Lt
Am,i CC jm (∆ f ) 
 L t j =1

m=1
(m 6= j)
A j,i is the complex reflective coefficient from the jth transmitter to the ith receiver. Because the transmitters and the receivers are collocated at the same site, A j,i and Am,i (m 6= j)
Remote Sens. 2022, 14, 2595
5 of 12
are assumed to be the same in amplitude and only be different in phase. The phase
difference is explained in Section 2.3. Given that
Lt
RemoteSens.
Sens.2022,
2022,14,
14,2595
2595
Remote
∑
Am,i CC jm (∆ f ) ≤
m =1
(m6= j)
Lt
∑
Lt
CC jm (∆ f ) ,
∑
5 of 12
Am,i CC jm (∆ f ) = A j.i
(12)
5=
of112
m
(m6= j)
m =1
(m6= j)
Then, the low bound of SIR can be written as follows.
𝐿
𝐿𝑡 𝑡
𝐿
𝐿𝑡 𝑡
𝐿
𝐿𝑡 𝑡
(Δ𝑓)||≤≤ ∑
(Δ𝑓)|==|𝐴
∑ 𝐴𝐴𝑚,𝑖CC
CC𝑗𝑚(Δ𝑓)
∑ |𝐴
|𝐴𝑚,𝑖CC
CC𝑗𝑚(Δ𝑓)|
|𝐴𝑗.𝑖| | ∑
∑ 2|CC𝑗𝑚 (Δ𝑓)|,
|| ∑
[AC
𝑚,𝑖
𝑗𝑚
𝑚,𝑖
𝑗𝑚
𝑗.𝑖( ∆ f )]|CC𝑗𝑚 (Δ𝑓)|,
𝑚=1
𝑚=1
SIR(𝑚=1
∆
f ) ≥ SIRmin (∆ f )𝑚=1
=
(𝑚≠𝑗)
(𝑚≠𝑗)
(𝑚≠𝑗)
(𝑚≠𝑗)

𝑚=1
𝑚=1
(𝑚≠𝑗)
(𝑚≠𝑗)
Then,the
thelow
lowbound
boundof
ofSIR
SIRcan
canbe
bewritten
written
as follows.
 1 as
 L
Then,
 t
 ∑ Lt follows.
jm ( ∆ f )2
j[AC(Δ𝑓)]
=1  ∑22m = 1 CC[AC(Δ𝑓)]
 Lt [AC(Δ𝑓)]
[AC(Δ𝑓)]2
(Δ𝑓)
SIR(Δ𝑓)
≥
SIR
=
=
𝑚𝑖𝑛(Δ𝑓) =
2
SIR(Δ𝑓) ≥ SIR 𝑚𝑖𝑛
(Δ𝑓)
𝐼𝑚𝑎𝑥(Δ𝑓)
(m(Δ𝑓)|
6= j2) = 𝐼𝑚𝑎𝑥
𝑡
11 𝐿𝐿𝒕 𝒕 [∑𝐿𝐿𝑡𝑚=1
|CC𝑗𝑚(Δ𝑓)|
𝑗=1[∑ 𝑚=1 |CC𝑗𝑚
[[𝐿𝑡∑∑𝑗=1
]] ]]
𝐿
(𝑚≠𝑗)
𝑡
(12) [AC(∆ f )]2
(12)
2  =
Imax (∆ f )
(13)
 
 
 
(13)
(13)
(𝑚≠𝑗)
Typically, the goal of orthogonal code design is to make CC zero at ∆ f = 0. One of
Typically,the
thegoal
goalof
oforthogonal
orthogonalcode
codedesign
designisisto
tomake
make CC
CC zero
zeroatat Δ𝑓
Δ𝑓==0.0.One
Oneof
of
Typically,
the
conventional
methods
design orthogonal
codes
is using
the [27].
Hadamard matrix [27].
the
conventional
methods
todesign
designto
orthogonal
codesisisusing
usingthe
theHadamard
Hadamard
matrix
the
conventional
methods
to
orthogonal
codes
matrix
[27].
Figure
3 shows
quadratic
phase orthogonal
with
36 lengths
by the Hadamard
Figure
shows
quadratic
phaseorthogonal
orthogonal
codeswith
withcodes
36lengths
lengths
designed
bythe
thedesigned
HadaFigure
33shows
quadratic
phase
codes
36
designed
by
Hadamard
matrix.
Four
codes
among
the
length
of
36
are
displayed.
Figure
4a
shows
that
the
matrix.
Four
codes
among
the
length
of
36
are
displayed.
Figure
4a
shows
that the demard matrix. Four codes among the length of 36 are displayed. Figure 4a shows that the
denominator 𝐼𝐼𝑚𝑎𝑥(Δ𝑓)
(Δ𝑓)in
inEquation
Equation(13)
(13)isiszero
zeroatat Δ𝑓
Δ𝑓==00 but
butgrows
growswith
with Δ𝑓.
Δ𝑓.As
Asaaresult,
result,
denominator
nominator𝑚𝑎𝑥Imax
∆
f
in
Equation
(13)
is
zero
at
∆
f
=
0
but
grows
with
∆
f
.
As a result,
( )
(Δ𝑓) drops
SIR 𝑚𝑖𝑛(Δ𝑓)
dropsabruptly,
abruptly,as
asshown
shownin
inFigure
Figure4b.
4b.
SIR
𝑚𝑖𝑛
SIR
min ( ∆ f ) drops abruptly, as shown in Figure 4b.
Figure3.3.Example
Exampleofoforthogonal
orthogonalquadratic
quadratic phasecodes
codes oflength
lengthofof36.
36.
Figure
Figure 3. Example
of orthogonalphase
quadraticofphase
codes
(a)
(a)
of length of 36.
(b)
(b)
Figure
Performance
degradation
orthogonal
code
byfilter
filtermismatch.
mismatch.
(a)
ACand
andCC
CCaccording
according
Figure
4.4.Performance
degradation
ofoforthogonal
by
AC
Figure
4. Performance
degradation
ofcode
orthogonal
code by(a)
filter
mismatch.
(a) AC
dopplermismatch
mismatch(b)
(b)Minimum
MinimumSIR.
SIR.
totodoppler
and CC according
to doppler mismatch (b) Minimum SIR.
Therefore,in
inthis
thispaper,
paper,we
wedesign
designthe
thepolyphase
polyphasecodes
codesof
ofwhich
whichthe
thecross-correlacross-correlaTherefore,
tionisislow
low
andinsensitive
insensitive
toDoppler
Doppler
mismatch
Δ𝑓.the
Thecomputational
computational
costfor
for
searching
Therefore,
in this
paper,mismatch
we design
polyphase
codes
ofsearching
which the cross-correlation
tion
and
to
Δ𝑓.
The
cost
the
best
polyphase
codeset
setwith
with
set
sizeof
of 𝐿𝐿mismatch
of.N,
N,
andthe
thedistinct
distinctphase
phase cost for searching
𝑡 , a code length
isbest
low
and insensitive
to
Doppler
∆
f
The
computational
the
polyphase
code
aaset
size
of
and
𝑡 , a code length
𝑡 ×𝑁) , which grows exponennumberof
ofM
Mthrough
throughexhaustive
exhaustivesearch
searchisisto
tothe
theorder
orderof
of 𝑀𝑀(𝐿(𝐿𝑡×𝑁)
number
,
which
grows
exponenthe best polyphase code set with a set size of Lt , a code length of N, and the distinct
tiallywith
with 𝑁𝑁 and
and 𝐿𝐿𝑡. .Thus,
Thus,we
weused
usedthe
thestatistical
statisticaloptimization
optimizationalgorithm
algorithmproposed
proposedin
in
tially
𝑡 of M through
phase
number
exhaustive
search with
is toathe
order iterative
of M( Lt × N ) , which grows
[22]
which
combines
the
simulated
annealing
(SA)
algorithm
traditional
[22] which combines the simulated annealing (SA) algorithm with a traditional iterative
exponentially with N and Lt . Thus, we used the statistical optimization algorithm proposed
in [22] which combines the simulated annealing (SA) algorithm with a traditional iterative
code selection algorithm. The SA algorithm exploits an analogy between the search for a
minimum of a cost function and the physical process by which a material changes state
Remote Sens. 2022, 14, 2595
6 of 12
while minimizing its energy [23]. The major advantage of the SA algorithm is the ability to
avoid trapping in local optima during the search process because it accepts some changes
that also increase the cost function with a probability of
∆E
P = exp −
(14)
Tc
The control parameter is known as the system “temperature”, which slowly decreases
from a large value to a very small one during the annealing process. The SA algorithm can
find the global optimum of a nonlinear multivariable function by carefully controlling the
change rate of the system temperature.
We define the objective function E(C) to be minimized as the peak of the crosscorrelation for ∆ f ∈ [0, 1):
"
#
N
2πk
∗
E(C) = max
max CCjm (∆ f ) = max
max ∑ Cj,k
Cm,k exp j
∆f
(15)
N
∆ f ∈[0, 1) j,m( j6=m)
∆ f ∈[0, 1) j,m( j6=m) k =1
Therefore, the problem is to find C for minimizing the function E(C). The resulting
codes may not be perfectly orthogonal but have very low cross-correlation and are tolerable
to Doppler mismatching. The SA algorithm used in this paper is summarized in Table 1.
Table 1. Process of polyphase codes design using SA algorithm.
Step
Description
0.
For given Lt , N, and M, generate initial C at random
1.
Set the initial temperature T0 and initialize temperature index c = 0
2.
Evaluate the objective function E0 at Tc with current C
Initialize the index i = 0
3.
Choose one of Lt × N phases in matrix C at random
Perturb the selected phase to another value in M − 1 at random
Increase the index i (i = i + 1)
4.
Evaluate the objective function Ei with perturbed phase matrix
Calculate the increment of functions as ∆E = Ei − Ei−1
Calculate the transition probability by ∆E as follows:
(
exp − ∆E
Tc , i f ∆E > 0
P=
1
, otherwise
Update the matrix C according to the probability
6.
Repeat steps 3–5 during i < Imax
7.
If Ei changes from E0 , go to step 8; otherwise go to step 9
Increase c and reduce temperature 1
Go to step 2
If c > cmax or stop condition is satisfied, then quit; otherwise, go to step 8 2
5.
c = c + 1, Tc = αTc−1 (0 < α < 1)
8.
9.
1
1
Maximum number of iterations Imax in step 6 and α in step 8 determine the convergence speed. We use
Imax = 10 × ( NT ML), α = 0.95 in this work. 2 Stop condition in step 9 is that there are no changes in the objective
function for more than three successive temperature cycles in this work.
The phase difference vectors for the transmit, receiving, and virtual arrays are
𝐯
= [1 𝑒
(
𝐯
= 1 𝑒
(
Remote Sens. 2022, 14, 2595
)
)
𝑒
⋯
(
)
𝑒
((
)
⋯
𝑒
((
)
,
)
)
,
(16)
(17)
7 of 12
and
𝐯=𝐯 ⊗𝐯
= [1 𝑒
𝑒
(
)
⋯𝑒
((
)
)
(18)
where
⨂ isVthe
2.3.
MIMO
AAKronecker product and 𝑘 (= 2𝜋 ⁄ 𝜆) is the wave number. Then, v is identical The
to that
of a ULA of
composed
× 𝐿 byelements.
Therefore,
the
complex
coefficient
beampattern
the VAAof
is 𝐿
made
multiplication
of the
transmit
pattern
and
𝐴
in
Equation
(7)
is
related
to
𝐴
as
,
,
receive pattern. Thus, the virtual aperture is dependent on the configuration of the transmit
)
arrays
arrays.
the VAA
𝐴 , =and
𝑣( receive
𝐴 , = 𝑒To[( maximize
𝐴 ,in a uniform linear array (ULA), one ULA,
(19)
)×
either transmitter or receiver, is built with Lr arrays with an interval d, and its physical
In addition,
VAA beamforming
canthe
beother
performed
by Lt arrays. The final aperture
aperture
d × Lr becomes
the interval of
ULA with
size is then
d
×
L
×
L
.
The
spacing
d
is
set
by
the
field
be
𝑌 , of𝐯 view (FOV) and should(20)
𝑌(𝜃) = [𝑌 , 𝑌 , r ⋯ t 𝑌 , 𝑌 , 𝑌 , ⋯ 𝑌
,
less than or equal to half of the wavelength in order to have the whole FOV of ± 90 deg.
where 5Hshows
is the conjugate
and 𝑌with
the
of the
jth code-correlated
DopFigure
the virtualtranspose
array antenna
3 receive
arrays
and Lt = 4 transmit
, is L
r =output
pler
FFT
in
the
ith
receiver
in
Figure
2.
arrays.
Figure5.5.MIMO
MIMOvirtual
virtualarray
arrayexample
examplewhere
whereL𝐿r ==44and
andLr𝐿==3. 3.
Figure
3. Simulation
The phaseResults
difference vectors for the transmit, receiving, and virtual arrays are
i
3.1. Doppler-Insensitive hCode Design (36 Length, 4 Transmitters)
vtx = 1 e j( Lr kd sin θ ) e j(2Lr kd sin θ ) · · · e j(( Lt −1) Lr kd sin θ ) ,
(16)
According to the process in Table 1, we designed four codes of a length of 36 and
compared them with the orthogonal
code in Figure 3. Figure
h
i 6 shows the phase of the
vrx = 1 e j(kd sin θ ) · · · e j(( Lr −1)kd sin θ ) ,
(17)
optimized codes.
and
h
i
v = vtx ⊗ vrx = 1 e jkd sin θ e j(2kd sin θ ) · · · e j(( Lt Lr −1)kd sin θ )
(18)
where ⊗ is the Kronecker product and k (= 2π/λ) is the wave number. Then, v is identical
to that of a ULA composed of Lr × Lt elements. Therefore, the complex coefficient Am,i in
Equation (7) is related to A1,1 as
Am,i = v(m−1)× Lr +i A1,1 = e j[(m−1) Lr +i−1]kd sin θ A1,1
In addition, VAA beamforming can be performed by
Y (θ ) = Y1,1 Y2,1 · · · YLr ,1 Y2,1 Y2,2 · · · YLr −1,Lt
YLr ,Lt v H
(19)
(20)
where H is the conjugate transpose and Yi,j is the output of the jth code-correlated Doppler
FFT in the ith receiver in Figure 2.
3. Simulation Results
3.1. Doppler-Insensitive Code Design (36 Length, 4 Transmitters)
According to the process in Table 1, we designed four codes of a length of 36 and
compared them with the orthogonal code in Figure 3. Figure 6 shows the phase of the
optimized codes.
Figure 7 shows Imax (∆ f ) defined by the denominator in Equation (13). The optimized
codes have a lower value than that of the orthogonal codes if the filter mismatch ∆ f is
greater than 0.075. It also indicates a reasonably consistent value within the entire interval.
The black line represents the mean value of 1000 randomly generated codes. As a result,
SIRmin (∆ f ) of the optimized codes was greater than that of the orthogonal codes or random
codes. It was about 15 dB higher than that of the random codes, as shown in Figure 8.
Remote Sens. 2022, 14, 2595
Remote Sens. 2022, 14, 2595
8 of 12
8 of 12
Figure
6. SA
Optimized
codes
(𝐿 (𝐿
==
4, 𝑀
==
4, 𝑁
==
36).
Figure
6. SA
Optimized
codes
4, 𝑀
4, 𝑁
36).
Figure
7 shows
(Δ𝑓)
defined
byby
thethe
denominator
in inEquation
(13).
The
optiFigure
7 shows𝐼𝑚𝑎𝑥
𝐼𝑚𝑎𝑥
(Δ𝑓)
defined
denominator
Equation
(13).
The
optimized
codes
have
a
lower
value
than
that
of
the
orthogonal
codes
if
the
filter
mismatch
mized codes have a lower value than that of the orthogonal codes if the filter mismatch
Δ𝑓Δ𝑓is is
greater
than
0.075.
It It
also
indicates
a reasonably
consistent
value
within
thethe
entire
greater
than
0.075.
also
indicates
a reasonably
consistent
value
within
entire
interval.
The
black
line
represents
thethe
mean
value
of of
1000
randomly
generated
codes.
AsAs
interval.
The
black
line
represents
mean
value
1000
randomly
generated
codes.
(Δ𝑓)
a result,
𝑆𝐼𝑅
of
the
optimized
codes
was
greater
than
that
of
the
orthogonal
codes
𝑚𝑖𝑛
a result, 𝑆𝐼𝑅𝑚𝑖𝑛 (Δ𝑓) of the optimized codes was greater than that of the orthogonal codes
oror
random
codes.
It It
was
about
1515
dBdB
higher
than
that
ofof
thethe
random
codes,
asas
shown
in in
random
codes.
was
about
higher
than
that
random
codes,
shown
Figure
8.
Figure 8.
Figure 6. SA Optimized codes (L = 4, M = 4, N = 36).
Figure 6. SA Optimized codes (𝐿 = 4, 𝑀 = 4, 𝑁 = 36).
Figure 7 shows 𝐼𝑚𝑎𝑥 (Δ𝑓) defined by the denominator in Equation (13). The optimized codes have a lower value than that of the orthogonal codes if the filter mismatch
Δ𝑓 is greater than 0.075. It also indicates a reasonably consistent value within the entire
interval. The black line represents the mean value of 1000 randomly generated codes. As
a result, 𝑆𝐼𝑅𝑚𝑖𝑛 (Δ𝑓) of the optimized codes was greater than that of the orthogonal codes
or random codes. It was about 15 dB higher than that of the random codes, as shown in
Figure 8.
Figure
7. Comparison
of of
the
interference
power
𝐼𝑚𝑎𝑥
(Δ𝑓).
Figure
7. Comparison
Comparison
of the
the
interference
power
∆ f ).
Figure
7.
interference
power
𝐼I𝑚𝑎𝑥
max ((Δ𝑓).
Figure 7. Comparison of the interference power 𝐼𝑚𝑎𝑥 (Δ𝑓).
(Δ𝑓).
Figure
8. Comparison
of of
𝑆𝐼𝑅
Figure
8. Comparison
Comparison
of
SIR
( ∆ f ).
𝑚𝑖𝑛min
Figure
8.
𝑆𝐼𝑅
𝑚𝑖𝑛 (Δ𝑓).
3.2. Performance according to M
Figure 9 represents SIRmin at ∆ f = 0.5 according to the number of phases M for
different code lengths of 32, 64, and 128.
The performance of the optimized codes improved as the number of phases M increased, but the rate of improvement gradually reduced. Aside from the longer time for
optimization, utilizing more phases was restricted by the hardware configuration. Thus,
there must be trade-offs for implementation. Figure 9 also shows that SIRmin was improved
with
code length
N, similar
(Δ𝑓). to coherent integration.
Figure 8.the
Comparison
of 𝑆𝐼𝑅
𝑚𝑖𝑛
Remote Sens. 2022, 14, 2595
9 of 12
3.2. Performance according to M
Remote Sens. 2022, 14, 2595
Figure 9 represents 𝑆𝐼𝑅𝑚𝑖𝑛 at Δ𝑓 = 0.5 according to the number of phases M for9difof 12
ferent code lengths of 32, 64, and 128.
Figure
Figure9.9.Improvement
Improvementofofthe
theperformance
performanceaccording
accordingtotothe
thenumber
numberofofphases
phasesM.
M.
performance
optimized
improved as the number of phases M in3.3. The
FMCW
Simulationof
andthe
MIMO
Angle codes
Accuracy
creased,
but
the
rate
of
improvement
gradually
reduced.
from
the longer
time for
In this section, we show the simulation results
of theAside
MIMO
FMCW
radar using
the
optimization,
utilizing
more
phases
was
restricted
by
the
hardware
configuration.
Thus,
optimized codes. The simulation was performed by Monte Carlo methods in 1000 runs
for
there
be trade-offs
implementation.
Figure
9 also shows
SIR
was2.imeach must
SNR. The
processingfor
followed
Figure 2, and
the parameters
arethat
listed
in𝑚𝑖𝑛
Table
proved with the code length N, similar to coherent integration.
Table 2. Simulation parameters.
3.3. FMCW Simulation and MIMO Angle Accuracy
In this section,Parameter
we show the simulation results of the MIMOValue
FMCW radar using the
Operatingcodes.
frequency
fc )
77 (GHz)
optimized
The( simulation
was performed
by Monte Carlo methods in 1000 runs
Bandwidth
F
1 (GHz)
(
)
BW processing followed Figure 2,
for each SNR. The
and the parameters are listed in Table 2.
Chirp period ( T )
30 (usec)
Number of transmitters ( Lt )
4
Table 2. Simulation parameters.
Number of receivers ( Lr )
3
1 (N)
64
Number of coherent chirps
Parameter
Value
Number of phases in code ( M)
4
Operating
frequency (𝑓 )
77 (GHz)
Sampling frequency ( Fs 𝑐)
50 (MHz)
Bandwidth
(𝐹𝐵𝑊 )
1 (GHz)
Target velocity
5.57 (m/s
) (∆ f = 0.5)
Initial
target (𝑇)
distance
40 (m) 30 (usec)
Chirp
period
Array interval
0.5 λ 4
Number
of transmitters (𝐿 )
𝑡
1
Remote Sens. 2022, 14, 2595
It is equaloftoreceivers
the code length.
Number
(𝐿𝑟 )
3
Number of coherent chirps 1 (𝑁)
64
Figure 10 shows the root mean square (RMS) error of the estimated angle for a single 10 of 12
Number of phases in code (𝑀)
4
target at a direction of 10 degrees. The angle was calculated in MIMO VAA via conventional
Sampling
frequency
(𝐹
)
𝑠
beamforming, which was
described in Section 2.3. 50 (MHz)
Target velocity
5.57 (m/s) (Δ𝑓 = 0.5)
Initial target distance
40 (m)
Array interval
0.5 𝜆
1
It is equal to the code length.
Figure 10 shows the root mean square (RMS) error of the estimated angle for a single
target at a direction of 10 degrees. The angle was calculated in MIMO VAA via conventional beamforming, which was described in Section 2.3.
Figure
ComparisonofofRMS
RMS
errors
according
to SNR.
Figure 10.
10. Comparison
errors
according
to SNR.
The RMS error is normally dependent on the SNR and, in this case, the SINR. When
the SNR was low, the noise power was dominant over the interference power. Thus, the
RMS error of the VAA was almost the same as 12 ULAs, either using orthogonal codes or
optimized codes. However, as the SNR rose, the interference became dominant. This in-
The RMS error is normally dependent on the SNR and, in this case, the SINR. When
the SNR was low, the noise power was dominant over the interference power. Thus, the
RMS
Remote
Sens. error
2022, 14, of
2595the VAA was almost the same as 12 ULAs, either using orthogonal codes
10 ofor
12
optimized codes. However, as the SNR rose, the interference became dominant. This interference could not be eliminated and remained as angle error because it consisted of the
The RMS error is normally dependent on the SNR and, in this case, the SINR. When
cross-correlations of multiple
transmit signals and was proportional to the signal power.
the SNR was low, the noise power was dominant over the interference power. Thus, the
At Δ𝑓 = 0.5, the remaining
RMS
in almost
Figurethe10same
are as
0.112and
0.2either
for the
and
RMS error
of theerrors
VAA was
ULAs,
usingoptimized
orthogonal codes
or optimized codes.
However, as codes
the SNRoutperformed
rose, the interference
dominant.
This
orthogonal codes, respectively.
The optimized
thebecame
orthogonal
codes.
interference could not be eliminated and remained as angle error because it consisted of the
Of course, the performance of the orthogonal codes was better at Δ𝑓 = 0.01 with only a
cross-correlations of multiple transmit signals and was proportional to the signal power.
slight mismatch and At
approached
12 ULA
optimal
∆ f = 0.5, the the
remaining
RMScase.
errorsThe
in Figure
10 arecodes
0.1 andshowed
0.2 for the similar
optimizedperand
orthogonal
respectively.
The optimized
formance for Δ𝑓, which
is alsocodes,
indicated
in Figures
7 andcodes
8. outperformed the orthogonal codes.
course, the
performance of
orthogonal
was better
at ∆ f is= included
0.01 with only
The interferenceOfpower
is dependent
onthethe
target codes
direction,
which
in
a slight mismatch and approached the 12 ULA case. The optimal codes showed similar
𝐴𝑚,𝑖 in Equations (7)performance
and (19). for
As∆af , result,
remaining
RMS7error
which is the
also indicated
in Figures
and 8. is dependent on the
The interference
target
direction, The
whicherror
is included
in
direction. Figure 11 depicts
the RMS power
errorisindependent
relationontothethe
direction.
of the
A
in
Equations
(7)
and
(19).
As
a
result,
the
remaining
RMS
error
is
dependent
on
the
m,i
optimized codes was smaller
than that of the orthogonal codes, and the angle impact was
direction. Figure 11 depicts the RMS error in relation to the direction. The error of the
less noticeable.
optimized codes was smaller than that of the orthogonal codes, and the angle impact was
less noticeable.
Figure 11. RMS
errortarget
according
to the target
direction
at ∆for
f =high
0.5 forSNR.
high SNR.
Figure 11. RMS error according
to the
direction
at Δ𝑓
= 0.5
4. Conclusions
4. Conclusions
In this paper, we suggest an MIMO FMCW radar system including the design method
of polyphase
multipleFMCW
transmitters,
receive
processing
chains, andthe
VAAdesign
antenna
In this paper, we
suggestcodes
an for
MIMO
radar
system
including
configuration. Polyphase codes with low cross-correlations were optimized by the SA
method of polyphasealgorithm
codes for
multiple transmitters, receive processing chains, and VAA
and modulated to the successive chirps of transmitting signals. Compared
antenna configuration.
codes
low cross-correlations
were
withPolyphase
the orthogonal
codes, with
the proposed
optimum codes showed
robustoptimized
performanceby
for
Doppler
mismatching.
As
a
result,
the
performance
of
VAA
angle
estimation
using
the
the SA algorithm and modulated to the successive chirps of transmitting signals. Comproposed codes outperformed that of the orthogonal codes.
pared with the orthogonal
codes, the proposed optimum codes showed robust
Funding: This research received no external funding.
Data Availability Statement: Not applicable.
Conflicts of Interest: The author declares no conflict of interest.
Remote Sens. 2022, 14, 2595
11 of 12
Appendix A
From Equations (1)–(3), the phase of Sk∗ (t − τ )Sk (t) can be written as follows:
2 2 BW
BW
t − T2 − kT − f c t − 2R−c 2vt + F2T
t − T2 − kT − 2R−c 2vt
f c t + F2T
2 2 BW
= f c 2R−c 2vt + F2T
t − T2 − kT − t − T2 − kT − 2R−c 2vt
h i
2R−2vt
BW
= f c 2R−c 2vt + F2T
2 t − T2 − kT − 2R−c 2vt
c
h
i
2R−2vt
2R−2vt
BW
= f c 2R−c 2vt + F2T
2
t
−
kT
−
T
−
(
)
c
c
h
i h
i
FBW
2R−2vt
2R
v
2R
2vt
BW
= fc
+ 2T
2 1 − c t − T − 2kT − c − F2T
2(t − kT ) − T − 2R−c 2vt
c
c
c
h
i h
i
FBW
2R
2R
2R−2vt
2vt
BW
2
t
−
kT
−
T
−
−
2
t
−
kT
−
T
−
= f c 2R−c 2vt + F2T
(
)
(
)
c
c
2T
c
c
h
i h
i
FBW
2R
BW
= f c 2R−c 2vt + F2T
2(t − kT ) − T − 2R
2(t − kT ) − T − 2R−c 2vt 2vt
c
c −
2T
c
h
i FBW
FBW T
2vt
2R
R
t
−
kT
−
= f c 2R
−
f
t
+
f
t
−
kT
+
f
−
T
−
+
−
−
1 + vc 2vc t2
(
)
d
b
b
c
c
T
c
2
c
T
The third and fourth terms can be ignored because the third is small enough within
one chirp period, and the fourth is small due to the high order. Therefore, Equation (5) can
be obtained from Equation (4) as follows:
∗ (t − τ )S (t)
Am,i Sm,k
k
h
h ∗
−T −
t
−
kT
+
f
−
f
t
+
f
= Am,i Cm,k exp f c 2R
(
)
b
d
b
c
2R
c
ii
∗ exp[ f ( t − kT ) − f t ]
= Am,i Cm,k
b
d
i
h t
−
kT
.
where Am,i = Am,i exp f c 2R
−
f
t
+
f
(
)
d
b
c
References
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