Unimodular Sequence and Receiving Filter Design for Local Ambiguity Function Shaping

Abstract

Ambiguity function (AF), which is used to evaluate the range and Doppler resolutions, plays an important role in radar systems. In this paper, we consider the problems of jointly designing unimodular sequence and receiving filter and also unimodular complementary sequences and corresponding receiving filters with desired AF shapes. The design problems are formulated as the minimization of the weighted integrated sidelobe level (WISL) and the minimization of the complementary integrated sidelobe level (CISL), respectively, under the constraint of the signal-to-noise ratio (SNR) loss. Algorithms based on the alternately iterative minimization method and the majorization-minimization method are developed to tackle the optimization problems. Numerical results are demonstrated to show the superior performance of the proposed algorithms in terms of the achieved objective and running time in comparison with the state-of-the-art algorithms. Meanwhile, the excellent capability of the designed sequences to detect multiple moving targets under the strong clutter is evaluated via simulations. Moreover, the designed sequences are also implemented on an indoor hardware system, and their performance is verified.

Full text

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 60, 2022 5113012
Unimodular Sequence and Receiving Filter Design
for Local Ambiguity Function Shaping
Fulai Wang , Sijia Feng , Jiapeng Yin ,Member, IEEE, Chen Pang, Yongzhen Li , and Xuesong Wang
Abstract— The ambiguity function (AF), which is used to
evaluate the range and Doppler resolutions, plays an important
role in radar systems. In this article, we consider the problems of
jointly designing unimodular sequence and receiving filters, and
also unimodular complementary sequences and corresponding
receiving filters with desired AF shapes. The design problems
are formulated as the minimization of the weighted integrated
sidelobe level (WISL) and the minimization of the complemen-
tary integrated sidelobe level (CISL), respectively, under the
constraint of the signal-to-noise ratio (SNR) loss. Algorithms
based on the alternately iterative minimization method and the
majorization-minimization method are developed to tackle the
optimization problems. Numerical results are demonstrated to
show the superior performance of the proposed algorithms in
terms of the achieved objective and running time in comparison
with the state-of-the-art algorithms. Meanwhile, the excellent
capability of the designed sequences to detect multiple moving
targets under the strong clutter is evaluated via simulations.
Moreover, the designed sequences are also implemented on an
indoor hardware system, and their performance is verified.
Index Terms—Alternate optimization, ambiguity function
(AF), complementary sequences, integrated sidelobe level (ISL),
majorization-minimization, unimodular sequence.
I. INTRODUCTION
AMBIGUITY function (AF) plays an important role in
pulse compression radar systems since it is often used
to evaluate the range and Doppler resolutions and interference
suppression capability [1]–[4]. In addition, the AF also repre-
sents the range-Doppler response of the receiving filter with
respect to different time delays and Doppler-shifted targets [5].
In general, the relationship between the sequence and its AF
is defined by computing the latter from the former. It has been
pointed out in [6] that the inverse problem, that is, constructing
the sequence with the desired AF shape, is considered to be a
trickier issue due to several reasons. One important reason is
that not any AF has a corresponding sequence. For example,
an ideal AF shape should be a 2-D “thumbtack” function with
zero sidelobes on the range-Doppler plane everywhere except
the origin [7], [8]. However, due to the fixed volume invariant
Manuscript received January 14, 2022; revised March 21, 2022; accepted
April 26, 2022. Date of publication April 29, 2022; date of current ver-
sion May 10, 2022. This work was supported by the National Natural
Science Foundation of China under Grant 61921001 and Grant 61971429.
(Corresponding author: Sijia Feng.)
The authors are with the State Key Laboratory of Complex Electromagnetic
Environment Effects on Electronics and Information System, National Univer-
sity of Defense Technology, Changsha 410073, China (e-mail: wflmadman@
outlook.com; fengsijia12@nudt.edu.cn; jiapeng.yin@hotmail.com;
pangchen@nudt.edu.cn; liyongzhen@nudt.edu.cn; wxs1019@vip.sina.com).
Digital Object Identifier 10.1109/TGRS.2022.3171253
property of the AF in the single code field, the “thumbtack”
AF has no corresponding sequences [9], [10].
In order to design a sequence with the desired AF shape,
two main categories of methods are proposed. The first is to
construct a reasonable AF in the single code field with locally
low sidelobe level over certain range-Doppler regions, which
are determined by the prior information provided by auxiliary
systems, so as to realize the detection of targets in the specified
areas [11]. This is of great significance to the cognitive radar
and dual-functional radar-communication systems [6], [12].
The second is to take the advantage of the complementar-
ity between multiple pulses. By constructing complementary
sequences (CSS), the AF can have a low sidelobe level in all
time delays over a certain Doppler extent [13], [14].
Extensive studies about the local AF shaping in the single
code field exist in the literature, and these studies can be
divided into three main categories [15]–[21]. The first category
is to design the transmitted sequence based on the matched
filter scheme. In [22], an efficient gradient descent (EGD)
algorithm based on the fast Fourier transform (FFT) operation
is proposed to suppress the weighted sidelobe energy of the
AF of the unimodular sequence. Subsequently, an accelerated
iterative sequential optimization (AISO) algorithm is proposed
for local AF shaping, and it achieves better performance in
comparison with the algorithm in [9] in terms of the running
time and the objective value.
The second category is to design the receiving filter for
a fixed transmitted sequence, so as to achieve a desired AF
shape [23]–[26]. These methods mainly aim at suppressing the
integrated sidelobe level (ISL) or peak sidelobe level (PSL)
of the AF and seek the optimal receiving filter under the
signal-to-noise ratio (SNR) loss constraint. It has been pointed
out in [27] that the performance of these methods is mainly
determined by the characteristic of the initial transmission
sequence. Generally speaking, the above two categories of
methods only consider the separate design of the transmitter
or receiver sequence and do not make full use of the signal
freedom of joint transceiver processing, resulting in limited
performance.
The third category method, which is to jointly design the
transmitted sequence and receiving filter, attracts more and
more attention since the joint design provides more degrees of
freedom for sidelobes suppression [28]–[31]. In [32], a cyclic
optimization (CO) algorithm is proposed to jointly design a
pair of sequence and receiving filters with the desired AF
shape. However, the CO algorithm cannot control the SNR
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5113012 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 60, 2022
loss caused by using the mismatched filter. Meanwhile, the
CO algorithm is time-consuming and not suitable for long
sequence design.
The reason for studying CSS design is that it is impossible
to construct a unimodular sequence with an impulse-like auto-
correlation function [33], [34]. More specifically, it is noted
that the autocorrelation of a unimodular sequence is always
equal to 1 at the N−1 lag. Even if using the mismatched filter
scheme, it is quite challenging to achieve impulse-like output
under the filter energy constraint in the single code filed [27].
The difficulties encourage researchers to consider the design
of CSS, the autocorrelation of which is the ideal impulse-
like function. However, CSS is extremely Doppler sensitive,
which means that the ideal autocorrelation property would be
destroyed by a slight Doppler shift [35]–[38].
To this end, the AF shaping for CSS attracts signifi-
cant attention in the literature. By rearranging the order
of an expanded version of the existing CSS according to
the Prouhet–Thue–Morse (PTM) sequences, Pezeshki et al.
obtained an AF with low sidelobe level over modest Doppler
shift extent [39]. Later, the PTM method is further developed
and implemented in multiple-input–multiple-output (MIMO)
radar systems [40]. Nevertheless, the aforementioned analyti-
cal methods have a strict restriction on the sequence length and
pulse number of the CSS. In recent work, a numerical opti-
mization algorithm based on the limited-memory Broyden–
Fletcher–Goldfarb–Shanno (L-BFGS) method is proposed to
control the AF shape of CSS [13], [41]. The numerical opti-
mization algorithm has no limitation on the sequence length
and the pulse number. However, the efficiency of the numerical
optimization algorithm and the achieved optimal objective
value can be further exploited.
To solve the abovementioned problems, new efficient algo-
rithms for jointly designing a pair of sequence and receiving
filters and also complementary sequences and receiving filters
with desired AF shapes are presented in this article. Specifi-
cally, in the formulation, the objective function is composed
of the weighted integrated sidelobe level (WISL)/the com-
plementary integrated sidelobe level (CISL) and the penalty
function of the SNR constraint. To make full use of the power
available in the system and avoid trivial solutions, unimodular
constraints on the sequence and energy constraints on the
receiving filter are considered. In the optimization, to han-
dle the resulting nonconvex problem, an alternately iterative
optimization scheme is implemented, and the optimization
problem is converted to two subproblems, which are solved
efficiently by the general majorization-minimization (MM)
method [42], [43]. The major contributions of this article can
be summarized as follows:
1) Problems Formulation: The joint design of unimodular
sequences and receiving filters in single and multiple
codes (it is referred to as the design of CSS and receiving
filters) fields for local AF shaping is considered. The
SNR loss constraint is added to the objective function
in the form of the penalty function. By adjusting the
SNR loss constraint, a tradeoff between the achieved
WISL/CISL and running time can be achieved. It is
noted that, even if the SNR loss constraint is set as 0 dB,
the proposed algorithms still have better performance
than existing methods.
2) Problems Solution: The alternatively iterative scheme is
introduced to transform the joint design problems into
two subproblems. Then, the MM method is developed to
obtain the analytical solutions of the two sequence and
receiving filter design subproblems. The main steps of
proposed algorithms can be implemented by the FFT
operations; thus, they are quite efficient in practice.
Meanwhile, an acceleration scheme based on the squared
iterative method (SQUAREM) is introduced to further
accelerate the proposed algorithms.
3) Experimental Insights and Hardware Validation: Com-
pared with existing methods, which only design the
transmitted sequence or the receiving filter, the proposed
algorithms have better performance in terms of the run-
ning time, the optimal objective value, and the ability to
detect multiple moving targets under the strong clutter.
Meanwhile, the designed sequences are implemented in
an indoor hardware system, and their performance is
validated.
The rest of the article is organized as follows. In Section II,
the joint design problems for suppressing WISL and CISL are
formulated, respectively. In Section III, we derive algorithms
based on the MM algorithm for the above two joint design
problems and then introduce an acceleration scheme. Next,
Section IV presents several numerical results to show the
superior performance of the designed sequences, followed by
the implementation and verification of an indoor hardware
system in Section V. Finally, the conclusions are summarized
in Section VI.
Notations: Throughout the article, boldface upper-case let-
ters denote matrices, boldface lower-case letters represent col-
umn vectors, and standard lower-case letter denote scalars. |·|,
(·)∗,andRe(·)represent the modulus, conjugate, and real parts
of a complex number. Tr(·),(·)T,(·)H,andvec(·)denote the
trace, transpose, complex transpose, and stacking vectorization
of a matrix. A[i,j]denotes the (ith,jth)element of matrix
A,andDiag(a)is a matrix constructed with the vector aas
its principal diagonal. arepresents the 2-norm of the vector
a.Indenotes an n×nidentity matrix. Finally, aand 
represent the smallest integer larger than a(real-valued) and
Hadamard products, respectively.
II. PROBLEM STATEMENT
In this article, we focus on the design of unidomular
sequence and receiving filters, and the design of complemen-
tary sequences and receiving filters with desired AF shapes.
In the following, we first introduce metrics to measure the
performance of the AF. Then, the sequence design prob-
lems are formulated as optimization problems with several
constraints.
A. Design of Unimodular Sequence and Receiving Filter
Consider a complex unimodualr sequence x
x=[x1x2··· xN]T∈CN×1(1)
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WANG et al.: UNIMODULAR SEQUENCE AND RECEIVING FILTER DESIGN 5113012
where
xn=ejφ(n),n=1,...,N(2)
and Nrepresents the length of the sequence. Let h=
[h1h2··· hN]T∈CN×1be the receiving filter; then, the
discrete AF of the sequence xand the receiving filter hcan
be written as [9]
Ax,h(n,f)=hHJnDiag(a(f))x(3)
where
Jn[p,q]=1,q−p=n
0,q−p= n
p,q=1,...,N,n=1−N,...,N−1(4)
and a(f)is the normalized Doppler frequency vector defined
by
a(f)=ej2π1·f
Nej2π2·f
N··· ej2πN·f
NT
f∈[−N/2,N/2).(5)
It is noted that the relationship between the normalized
Doppler frequency fand the nominal Doppler frequency fd
is fd=f/T,whereTis the time width of the waveform.
Then, the WISL of the AF can be defined as
WISL(x,h)=
N−1

n=1−Nf2
f1
w(n,f)hHJnDiag(a(f))x
2df
(6)
where [f1,f2]denotes the minimal frequency extent encom-
passing all regions of interest and w(n,f)is the weighted
coefficient of the range bin nand Doppler shift f.
Compared with the matched filter scheme, more degrees of
freedom can be obtained by jointly designing the transmitted
sequence and receiving filter. In this case, the SNR loss
(SNRL) of the mismatched filter scheme should be considered,
which can be expressed as [17]
SNRL =10log10 x2h2
hHx
2.(7)
It can be observed that SNRL is related to the peak value
of the AF and the energy of the sequence and the receiving
filter. According to (2), the sequence satisfies xHx=N.Ifthe
receiving filter has an energy constraint, i.e., hHh=Nh,the
SNRL can be controlled by introducing a peak cost function
g(x,h)=hHx−amax 
2(8)
where amax is a predefined peak value. To achieve a
desired SNR loss μ,thevalueamax can be computed
by amax =√NhN10−μ/20.
Then, by adopting the Pareto weighting scheme, the
sequence and receiving filter joint design problem for AF
shaping under the SNR loss constraint can be formulated as
min
x,h∈CN×1(x,h)=εWISL(x,h)+(1−ε)g(x,h)
s.t. hHh=Nh
|xn|=1,n=1,...,N(9)
where εis the Pareto weight.
B. Design of Complementary Sequences and
Receiving Filters
Denote Kunimodular sequences and Kreceiving filters
with length Nas
X=[x1x2··· xK]N×K(10)
and
H=[h1h2··· hK]N×K(11)
where
xk=[xk(1)xk(2)··· xk(N)]T,k=1,...,K.(12)
Suppose that a point target resides in a certain range cell within
one coherent processing interval (CPI); then, the AF of the
sequence set Xand the receiving filter set Hcan be expressed
as [40]
˜
AX,Hn,˜
f=
K

k=1
hH
kJnxkejk ˜
f,n=−N+1,...,N−1
(13)
where ˜
f=2πfdTrand Tris the pulse repetition time (PRT).
In this article, to measure the complementarity of Xand Hand
control the shape of the AF, the CISL is considered, which is
defined as
CISL(X,H)=
N−1

n=1−N,n=0˜
f2
˜
f1
K

k=1
hH
kJnxkejk ˜
f
2
d˜
f(14)
where [˜
f1,˜
f2]denotes the Doppler frequency extent of inter-
est. A natural idea to design complementary unimodular
sequence set and receiving filter set is to minimize the CISL
metric under given constraints. Besides, the SNR loss for this
scheme can be expressed as
˜
SNRL =10log10 K
k=1xk2K
k=1hk2
K
k=1hH
kxk
2.(15)
Then, following similar steps from (8) to (9), the complemen-
tary sequences and receiving filters joint design problem for
AF shaping under the SNR loss constraint can be formulated
as:
min
X,H∈CN×K
˜
(X,H)=εCISL(X,H)+(1−ε)˜g(X,H)
s.t.
K

k=1
hH
khk=˜
Nh
|xk(n)|=1,n=1,..., N,k=1,...,K(16)
where
˜g(X,H)=
K

k=1
hH
kxk−˜amax 
2
(17)
and ˜amax is a predefined peak value. To achieve a
desired SNR loss ˜μ,thevalue˜amax can be computed by
˜amax =(˜
NhNK)1/210−˜μ/20.
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5113012 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 60, 2022
III. JOINT DESIGN OF SEQUENCE AND RECEIVING
FILTER VIA MM METHOD
In this section, since optimization problems (9) and (16) are
similar in description, a unified way is expected to solve them
efficiently. Thus, we first perform an equivalent transformation
of problem (16). Define two auxiliary sequences of length
K(2N−1)as follows:
u=xT
10T
N−1··· xT
K0T
N−1T(18)
and
v=hT
10T
N−1··· hT
K0T
N−1T.(19)
Then, the metric CISL in (14) can be rewritten as
CISL(u,v)
=
K(2N−1)−1

n=1−K(2N−1)˜
f2
˜
f1
˜wn,˜
fvHJnDiag˜
a˜
fu
2d˜
f(20)
where
˜
a˜
f=ej2π˜
f1T
N0T
N−1··· ej2πK˜
f1T
N0T
N−1T
(21)
and
˜wn,˜
f=1,1≤|n|≤N−1,˜
f1≤˜
f≤˜
f2
0,else.(22)
Consequently, the optimization problem in (16) can be refor-
mulated as
min
X,H∈CN×K
˜
(X,H)=εCISL(u(X),v(H))
+(1−ε)˜g(u(X),v(H))
s.t. v(H)Hv(H)=˜
Nh
|xk(n)|=1,n=1,...,N,k=1,...,K(23)
where
˜g(u(X),v(H)) =uHv−˜amax
2.(24)
It can be observed that both problems in (9) and (23) are
to optimize the weighted sidelobe energy under unimodular
and energy constraints. To avoid redundancy, in the following,
we take problem (9) as an example for detailed derivation, and
the corresponding steps can also be applied to problem (23)
with slight modification.
A. Design of Unimodular Sequence and Receiving Filter via
MM Method
The joint design problem (9) is nonconvex due to the
unimodular constraint, and it is hard to optimize the sequence
and the receiving filter simultaneously. Thus, the alternately
iterative minimization method is implemented in this article,
which can be summarized by
h(i)=arg min x(i−1),h(25a)
x(i)=arg min x,h(i)(25b)
where h(i)and x(i)are optimal solutions of the objective
function at the ith iteration, respectively. In the following, the
MM algorithm is developed to solve the two suboptimization
problems.
To facilitate the following discussion, we uniformly dis-
cretize the Doppler frequency extent [f1,f2]into Lbins with
grid size f=(( f2−f1)/(L−1)); then, by ignoring the
constant, the objective function (x,h)can be rewritten as
(x,h)=
L

l=1
N−1

n=1−N
w(n,fl)hHJnDiag(a(fl))x
2
+εxHhhHx−2amax RehHx (26)
where fl=f1+(l−1)fand ε=(1−ε)/ε. Then, noticing
that
hHJnDiag(a(fl))x=TrJnPfl(27)
where Pfl=xflhHand xfl=Diag(a(fl))x. The problem (9)
becomes
min
x,h∈CN×1(x,h)=
L

l=1
N−1

n=1−N
w(n,fl)TrJnPfl
2
+εxHhhHx−2amax RehHx
s.t. hHh=Nh
|xn|=1,n=1,...,N.(28)
Since Tr(JnPfl)=vec(PH
fl)Hvec(Jn), the problem in (28) can
be further rewritten as
min
x,h∈CN×1(x,h)=
L

l=1
vecPH
flHQflvecPH
fl
−2amaxεRehHx
s.t. hHh=Nh
|xn|=1,n=1,...,N(29)
where
Qfl=
N−1

n=1−N
w(n,fl)vec(Jn)vec(Jn)H(30)
and
w(n,fl)=w(n,fl)+ε,n=0and fl=0
w(n,fl),else.(31)
It can be observed that the first term in the objective function
(x,h)is a sum of quadratic functions of vec(PH
fl),andthe
matrix Qflis Hermitian. Thus, according to Lemma 1 in [44],
by applying the MM method, the problem in (29) can be
majorized by
min
x,h∈CN×1(x,h)=
L

l=1
2Re
vecPH
flHQfl−γmaxQflI
×vecPH
fl
(i)−2amaxεRehHx
s.t. hHh=Nh
|xn|=1,n=1,...,N(32)
where γmax(Qfl)is the maximum eigenvalue of Qfl, and it is
derived in a close form in [44], that is,
γmaxQfl=max
nw(n,fl)(N−|n|).(33)
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WANG et al.: UNIMODULAR SEQUENCE AND RECEIVING FILTER DESIGN 5113012
For subproblem (25a), which is to optimize the receiving
filter for the fixed sequence, substituting Qflin (30) back
into (32), it can be simplified as
min
h∈CN×1(h)=
L

l=1
2Re
hHRh(i),fl−γmaxQflh(i)xH
flxfl
−2amaxεRexHh
s.t. hHh=Nh(34)
where
Rh(i),fl[m,n]=w(m−n,fl)Ax,h(i)(n−m,fl)∗
m,n=1,...,N.(35)
The majorized problem in (34) is a quadratically constrained
linear program that can be easily solved by the Lagrange
multiplier method, and the optimal solution is given by
h(i+1)
=−




Nh


L
l=1Rh(i),fl−γmaxQflh(i)xH
flxfl−amaxεx


2
·L

l=1Rh(i),fl−γmaxQflh(i)xH
flxfl−amaxεx.(36)
Then, for subproblem (25b), which is to optimize the sequence
for the fixed receiving filter, according to (30) and (32), it can
be simplified as
min
x∈CN×1x=
L

l=1
2Re
xH(Diag(a(fl)))H
·Rx(i),fl−γmaxQflx(i)
flhHh
−2amaxεRehHx
s.t. |xn|=1,n=1,...,N(37)
where
Rx(i),fl[m,n]=w(m−n,fl)Ax(i),h(m−n,fl)
m,n=1,...,N.(38)
It is easy to see that the majorized problem (37) has a closed
form solution, which can be expressed as
x(i+1)
=−ejargL
l=1Diag(a(fl))HRx(i),fl−γmax(Qfl)x(i)
flhHh−amaxεh.(39)
Now, we are ready to summarize the overall algorithm to
jointly design the sequence and receiving filter with a desired
AF shape, and it is given in Algorithm 1. Note that, since
the matrices Rh(i),fland Rx(i),flare Hermitian Toeplitz, the
multiplication terms Rh(i),flxfland Rx(i),flhcan be calculated
by means of the FFT efficiently [33], [45] (readers can refer to
Appendix B of [33] for more details). Thus, the computation
complexity of Algorithm 1 is O(LN log N).
Algorithm 1 Alternately Iterative Algorithm to Jointly Design
the Sequence and Receiving Filter With Desired AF Shape
Input :sequence length N, filter energy Nh, weighted
coefficients {w(n,f)}, the sufferable SNR loss μ
1: Compute amax =√NhN10−μ/20
2: Set i=0, initialize x(0)and h(0)
3: Compute γmaxQflby (33)
4: repeat
5: Compute Ax,h(i)(n,fl)by (3) for a fixed x(i)
6: Update the receiving filter h(i+1)by (36)
7: Compute Ax(i),h(n,fl)by (3) for a fixed h(i+1)
8: Update the sequence x(i+1)by (39)
9: Set i=i+1
10: until convergence
Output :sequence xand receiving filter hwith a desired
AF shape.
B. Design of Complementary Sequences and Receiving
Filters via MM Method
To design complementary sequences and receiving filters
with a desired AF shape, similar steps form (25) to (39) can
be conducted. Applying the alternately iterative minimization
method on problem (23) can be expressed as
v(H)(i)=arg min ˜
u(X)(i−1),v(H)(40a)
u(X)(i)=arg min ˜
u(X),v(H)(i).(40b)
For subproblem (40a), using similar derivations from (26)
to (36), the update of receiving filters can be written as
v(H)(i+1)=−
˜
Nh
pc2(pc)(41)
where
p=
L

l=1Rv(H)(i),˜
fl−γmax˜
Q˜
flv(H)(i)u(X)H
˜
fl
×u(X)˜
fl−˜amax εu(X)(42)
c=1T
N0T
N−1··· 1T
N0T
N−1T(43)
Rv(H)(i),˜
fl[m,n]=˜wm−n,˜
fl
·vH(i)HJn−mDiag˜
a˜
flu(X)∗
m,n=1,...,K(2N−1)(44)
γmax˜
Q˜
fl=max
n˜wn,˜
fl(K(2N−1)−|n|)(45)
and
˜wn,˜
fl=˜wn,˜
fl+ε,n=0and ˜
fl=0
˜wn,˜
fl,else.(46)
For subproblem (40b), using similar derivations from (37)
to (39), the update of complementary sequences can be written
as
u(X)(i+1)=−ejarg(q)c(47)
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5113012 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 60, 2022
where
q=
L

l=1
Diag˜
a˜
flRu(X)(i),˜
fl−γmax˜
Q˜
fl·u(X)(i)
˜
flv(H)H
×v(H)−˜amax εv(H)(48)
and
Ru(X)(i),˜
fl[m,n]=˜wm−n,˜
fl
·v(H)HJm−nDiag˜
a˜
flu(X)(i)∗
m,n=1,...,K(2N−1).(49)
Then, the overall algorithm to jointly design complementary
sequences and receiving filters with a desired AF shape is
given in Algorithm 2. It is also worth noting that the matrices
Rv(H)(i),˜
fland Ru(X)(i),˜
flare Hermitian Toeplitz; the multiplica-
tion terms Rv(H)(i),˜
flu(X)˜
fland Ru(X)(i),˜
flv(H)can be efficiently
calculated by the FFT operation. Thus, the computation com-
plexity of Algorithm 2 is O(LKN log KN).
Algorithm 2 Alternately Iterative Algorithm to Jointly Design
Complementary Sequences and Receiving Filters With Desired
AF Shape
Input :sequence length N, number of sequences K, filter
energy ˜
Nh, weighted coefficients {˜w(n,f)}, the sufferable
SNR loss ˜μ
1: Compute ˜amax =˜
NhNK10−˜μ/20
2: Set i=0, initialize u(X)(0)and v(H)(0)
3: Compute γmax˜
Q˜
flby (45)
4: repeat
5: Compute Rv(H)(i),˜
flby (44) for a fixed u(X)(i)
6: Update the receiving filter set v(H)(i+1)by (41)
7: Compute Ru(X)(i),˜
flby (49) for a fixed v(H)(i+1)
8: Update the sequence set u(X)(i+1)by (47)
9: Set i=i+1
10: until convergence
Output :sequence set Xand receiving filter set Hwith a
desired AF shape.
C. Acceleration Scheme by Applying SQUAREM
In this section, an acceleration scheme that can be used
to accelerate the proposed algorithms is introduced. It is the
squared iterative method, which is usually applied to accelerate
the expectation–maximization (EM) algorithms [46]. Since the
MM algorithm is a generalization of EM, and the updating
procedures in (36), (39), (41), and (47) are one fixed-point
iterations [47], the SQUAREM can also be used to accelerate
the MM algorithm in this article.
Suppose that we have derived the nonlinear fixed-point itera-
tion transformation FMM(·)to minimize the objective function
O(s)based on the MM algorithm, that is,
s(i+1)=FMMs(i).(50)
Then, the accelerated MM algorithm based on the SQUAREM
scheme can be summarized as Algorithm 3. It is noted that
Fig. 1. Evolutions of the NWISL versus the running time (in seconds).
the general SQUAREM scheme may break the nonlinear con-
straint on variables; thus, a projection transformation Ps(·)
that can project wayward points back to the feasible region
is needed. In this article, we have two kinds of constraints
on variables, i.e., the unimodular constraint on the sequence
and the energy constraint on the receiving filter. For the uni-
modular constraint, the projection transformation can simply
be selected as Ps(·)=ejarg(·). For the energy constraint, the
projection transformation can be done by applying the function
Ps(·)=(E/(·)2)1/2(·),whereEis the constraint on the
energy.
Algorithm 3 SQUAREM Acceleration Scheme for MM
Algorithms
1: Set i=0, initialize s(0)
2: repeat
3: s1=FMMs(i)
4: s2=FMM(s1)
5: y=s1−s(i)
6: z=s2−s1−y
7: Compute the step-length by α=−
y/z
8: s=Pss(i)−2αy+α2z
9: while O(s)>Os(i)do
10: α=(α−1)/2
11: s=Pss(i)−2αy+α2z
12: end while
13: s(i+1)=s
14: Set i=i+1
15: until convergence
IV. NUMERICAL RESULTS
To show the performance of the proposed algorithms in
designing sequences and receiving filters with desired AF
shapes, some numerical results are shown for various scenar-
ios. In all simulations, algorithms are initialized with randomly
generated sequences. Meanwhile, it is noted that, since we are
trying to solve nonconvex optimization problems, the selection
of initial sequences does affect the performance of algo-
rithms. To avoid trivial results and find the optimal solutions,
500 Monte Carlo trials (i.e., 500 random initializations)
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WANG et al.: UNIMODULAR SEQUENCE AND RECEIVING FILTER DESIGN 5113012
Fig. 2. AFs of sequences generated by different algorithms at the time of 20 s. (a) EGD algorithm. (b) AISO algorithm. (c) MM-SR algorithm.
Fig. 3. Evolutions of the NWISL versus the running time with different SNR
losses (in seconds).
are conducted for all cases, and the results with the best
performance are kept without additional instructions. For
clarity, we denote the MM algorithms proposed for opti-
mization problems (9) and (23), i.e., Algorithms 1 and 2,
as MM-Sequence Receiving Filter (MM-SR) and MM-
Complementary Sequences Receiving Filters (MM-CSR),
respectively. All results are performed in MATLAB 2016b on
a PC with a 2.6-GHz i7-6700HQ CPU and 8-GB RAM.
A.MinimizationofWISLin(6)
In this section, we compare the performance of the proposed
MM-SR algorithm with the EGD algorithm [22] and the
AISO algorithm [9] in terms of the optimal WISL and the
running time using numerical examples. The MATLAB code
of the AISO algorithm is provided by the original author.
For all cases, the initial sequence and receiving filters x(0)
and h(0)are generated by x(0)={ejφ(0)(n)}N
n=1and h(0)=
(Nh/(hini2))1/2hini ,wherehini ={ejϕ(0)(n)}N
n=1,{φ(0)(n)}N
n=1,
and {ϕ(0)(n)}N
n=1are independent random variables uniformly
distributed in [0,2π]. Subsequently, to quantify of the per-
formance of the algorithm, the normalized WISL (NWISL)
(in dB) at the ith iteration is defined as
NWISL(i)=10log10WISLx(i),h(i)
WISLx(0),h(0).(51)
Suppose that a pair of unimodular sequence and receiving
filter with length N=1024 are required, and the concerning
Fig. 4. Statistical properties of the SNRL and running time with different
sequence lengths of different algorithms.
region of the AF is
w(n,f)=1,|n|∈[1,50],f∈[−3,3]
0,else.(52)
These design parameters are the same as those considered
in [9]. For the proposed algorithm, assume that the energy of
the receiving filter is Nh=N, the Pareto weight is ε=0.1,
and the predefined peak value is amax =N·10−0.5/20,which
means that the SNR loss is set to be μ=0.5dB.Forall
algorithms, we do not stop until the metric NWISL in (51)
goes below −100 dB or after 10000 s.
The evolution curves of the NWISL versus the running time
are shown in Fig. 1. It can be observed that the objective
of the EGD algorithm converges to a local minimum, about
−20 dB in this case. By contrast, both the MM-SR and AISO
algorithms can drive the objective to −100 dB. Compared
with the AISO algorithm, the proposed MM-SR algorithm
has significant advantages in terms of the running time; more
specifically, the MM-SR algorithm takes only 0.66 s, while the
AISO algorithm takes about 80 s. The reason is that the pro-
posed algorithm adopts the mismatched filter scheme, which
has more design degrees of freedom at the expense of the SNR.
Meanwhile, the MM-SR algorithm can be accelerated by the
FFT algorithm, which has high computational efficiency.
Fig. 2 shows the AFs of sequences generated by the three
algorithms within the region of interest at the time of 20 s
(the subgraph in the upper right corner represents the top
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5113012 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 60, 2022
Fig. 5. Range-Doppler plots within the concerning region. (a) AISO sequence. (b) EGD sequence. (c) MM-SR sequence.
view of the AF). Note that the MM-SR algorithm has been
stopped at 0.66 s since the termination condition is achieved.
The results demonstrate that the sidelobe levels of the MM-SR
sequences are suppressed to almost zero (about −120 dB)
within the desired region, while those of the AISO and
EGD sequences are much higher. Specifically, the average
normalized sidelobe levels are about −54.3 dB of the EGD,
−71 dB of the AISO, and −126 dB of the MM-SR algorithm.
Meanwhile, the SNR loss of the sequence and receiving filters
generated by the MM-SR algorithm is almost equal to the
preset value. Then, we change the SNR loss constraint of
the MM-SR algorithm. The corresponding NWISL evolution
curve results are shown in Fig. 3. We can notice that a
slight SNR loss can make the MM-SR algorithm achieve good
performance in a short time, that is, NWISL is less than
−100 dB, and the running time is about 1 s. Meanwhile, even
if the SNR loss constraint is set to μ=0 dB, the running time
of the MM-SR algorithm is only one-third of that of the AISO
algorithm.
Furthermore, the performance of the proposed algorithm is
evaluated with different lengths: N=29,210 ,...,214.The
weighted coefficients are set as
w(n,f)=1,|n|∈[1,0.05N],f∈[−4,4]
0,else.(53)
For the MM-SR algorithm, other parameters are the same
as those in Fig. 1. For all algorithms, we do not stop until
themetricNWISLgoesbelow−60 dB or after 10 000 s.
As mentioned above, 500 Monte Carlo trials are carried out
for each parameter. At this point, the statistical means and
standard deviations of the actual SNR loss and running time
of the MM-SR and AISO (since the AISO algorithm is based
on matched filter scheme, the SNR loss is always equal
to 0 dB) algorithms are shown in Fig. 4. From the figure,
we can see that the proposed MM-SR algorithm is signif-
icantly faster than the AISO algorithm, and this advantage
is more obvious in the long sequence length. It is predicted
that the MM-SR algorithm has the ability to construct quite
long length sequence and receiving filter with the desired
AF shape. Meanwhile, the SNR loss constraint can be well
satisfied.
B. Moving Targets Detection Under Strong Clutters
In this section, a more practical example of using the
designed sequences to detect targets under strong clutters is
presented. Assuming that the clutter is strong, estimating its
delay and Doppler frequency is relatively trivial [23]. Thus,
a simple way to detect targets of interest is to make the AF
of the sequence have low sidelobe levels in the regions where
the target and the clutter may appear.
Suppose that there are three targets located in the range bins
of −90, 10, and 100 with normalized Doppler frequencies
of 3, −1, and −2, respectively. Meanwhile, an isolated and
strong clutter source is located in the range bin 0 with a
normalized Doppler frequency of 1. The powers of the targets
and clutter are 0, 0, 5, and 25 dB, respectively. That is, the
signal-to-clutter ratio (SCR) is 25 dB with respect to the
weakest target. The SNR of the echo from each target is 10 dB.
In this scenario, we set N=1024, and the concerning region
of the AF is
w(n,f)=⎧
⎪
⎨
⎪
⎩
1,|n|∈[1,30],f∈[−5,5]
1,|n|∈[81,110],f∈[−5,5]
0,else.
(54)
Fig. 5(a)–(c) shows the range-Doppler plots of the AISO
sequence, the EGD sequence, and the MM-SR sequence,
respectively. It can be observed that three weak targets are
masked due to the high sidelobes from the clutter in Fig. 5(a).
It means that, in this case, the sidelobe level of the concerning
region of the AF cannot be suppressed effectively by the
AISO algorithm. By contrast, by exploiting the potential of the
local AF optimization design, three targets can be detected in
Fig. 5(b) for the EGD algorithm and in Fig. 5(c) for the MM-
SR algorithm. Meanwhile, as expected, the MM-SR algorithm
can provide lower sidelobe levels within the given range-
Doppler region than the EGD algorithm, which makes targets
easier to be detected.
C. Minimization of CISL in (14)
In this section, we test the performance of the proposed
MM-CSR algorithm and the L-BFGS algorithm [13] in mini-
mizing the metric CISL in (14) and give an example of apply-
ing the above algorithms to design complementary sequences
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WANG et al.: UNIMODULAR SEQUENCE AND RECEIVING FILTER DESIGN 5113012
Fig. 6. Evolutions of the NCISL versus the running time (in seconds).
Fig. 7. AFs of sequences generated by different algorithms at the time of
200 s. (a) L-BFGS algorithm. (b) MM-CSR algorithm.
and receiving filters with a desired AF shape. The initial
complementary sequences X(0)and receiving filters H(0)can
be generated in the same way, as in Section IV-A. To quantify
of the performance of the algorithm, the metric normalized
CISL (NCISL) (in dB) at the ith iteration is defined as
NCISL(i)=10log10CISLX(i),H(i)
CISLX(0),H(0).(55)
In particular, a set of random unimodular sequences of
length N=64 and K=16 is generated as the ini-
tial sequences for both algorithms. The normalized Doppler
Fig. 8. Hardware system.
Fig. 9. Process of the experiment on the hardware system to validate the
performance of designed sequences and receiving filters.
frequency interval of interest is ˜
f∈[−0.2,0.2], which means
that the weighted coefficients are
˜wn,˜
f=1,|n|∈[1,63],˜
f∈[−0.2,0.2]
0,else.(56)
For the MM-CSR algorithm, assume that the energy of the
receiving filter is ˜
Nh=NK, the Pareto weight is ε=0.1,
and the predefined peak value is ˜amax =NK ·10−0.5/20,which
means that the SNR loss is set to be ˜μ=0.5dB.Forall
algorithms, we do not stop until the metric NCISL in (55)
goes below −40 dB or after 10 000 s.
Fig. 6 shows the evolution curves of the NCISL versus the
running time of two algorithms. We can see that the proposed
MM-CSR algorithm drives the metric NCISL to −40 dB at
about 10 s, while the L-BFGS algorithm still has the NCISL
above −30 dB after 500 s and, finally, drives the NCISL
to −40 dB after more than 1900 s. The reason is that the
main steps of the MM-CSR algorithm can be accomplished
by the FFT algorithm. The AFs of complementary sequences
generated by the two algorithms within the region of interest
at the time of 200 s are demonstrated in Fig. 7 (note that
the MM-CSR algorithm has been stopped at 10.72 s since
the termination condition is achieved). The subfigure in the
upper right corner is the zero-Doppler cut of the AF. It can
be observed that the AFs of sequences generated by the
two algorithms have low sidelobe levels in the concerning
region. In the unoptimized region, the sidelobe level of the
AF increases gradually with the increase in the Doppler shift.
More specifically, the PSL of the MM-CSR sequences is
under −60 dB within the desired region, which is about
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5113012 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 60, 2022
Fig. 10. Experimental results of the unimodular sequence and receiving filter. (a) Sampled signal in the time domain. (b) Practical AF within the regionof
interest. (c) Zero-Doppler cut of the AF.
Fig. 11. Experimental results of complementary sequences and receiving filters. (a) Sampled signals in the time domain. (b) Practical AF within the region
of interest. (c) Zero-Doppler cut of the AF.
20 dB lower than that of the L-BFGS sequences. The average
normalized sidelobe levels are about −51.1dBoftheL-BFGS
and −77.2 dB of the MM-CSR algorithm. Meanwhile, we can
notice that the SNR loss of the complementary sequences
and receiving filters optimized by the MM-CSR algorithm is
almost the same as the preset threshold.
V. E XPERIMENTAL RESULTS
To verify the performance of the designed sequences and
receiving filters, some experiments based on a practical hard-
ware system are presented in this section. The hardware system
and the signal processing flowchart of the experiment are
shown in Figs. 8 and 9, respectively. Specifically, the designed
sequence is generated by the vector signal transceiver National
Instruments (NI)-5644R with a 16-bit digital-to-analog con-
verter and passes through the upconverter and the power
amplifier. Then, the amplified signal is attenuated and passes
through the downconverter M9362A and the intermediate fre-
quency (IF) conditioning module M9352A. Finally, the IF sig-
nal is sampled by the vector signal transceiver NI-5644R with
a 16-bit analog-to-digital converter of a 120-MHz sampling
rate for further processing. It is noted that the AFs shown in the
following are obtained by adding different Doppler frequencies
to the sampled signals and doing convolution operations with
the receiving filters.
First, the designed unimodular sequence in Section IV-A
with N=1024 and μ=0.5 dB, i.e., the result shown
in Fig. 2(c), is tested. In this experiment, the carrier frequency
is 10 GHz, the sequence bandwidth is B=20 MHz, and the
pulse duration of the sequence is T=N/B=51.2μs. The
corresponding experimental results are depicted in Fig. 10.
It can be observed that, due to the nonideality of the hardware
system, the practical sidelobe levels of the AF within the
region of interest deteriorate to some extent compared with
the theoretical results. In spite of this, the sidelobe levels of
the AF are all lower than −50 dB within the given region.
Meanwhile, we can see from Fig. 10(c) that the actual SNR
loss is almost the same as the theoretical value.
Next, using the MM-CSR algorithm, we redesign a
set of complementary sequences and receiving filters with
N=128 and K=16. The normalized Doppler frequency
interval of interest is ˜
f∈[−0.3,0.3]. Other parameters are the
same as those in Section IV-C. In the performance verification
based on the hardware system, we set the carrier frequency as
10 GHz, the sequence bandwidth as B=20 MHz, and the
pulse duration of the sequence as T=N/B=6.4μs. The
duty ratio of the transmitted complementary sequences is 0.2.
Fig. 11 shows the sampled signals and the practical AF of
the designed complementary sequences. It can be observed
that the PSL within the region of interest is under −60 dB.
The reason for obtaining the low sidelobe performance on the
actual system may be that the nonideality of the hardware
system can be equivalent to the phase jitter added to the trans-
mitted sequences. The designed Doppler resilient sequences
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WANG et al.: UNIMODULAR SEQUENCE AND RECEIVING FILTER DESIGN 5113012
are tolerant to the phase jitter, resulting in the low sidelobe
level in practice.
VI. CONCLUSION
In this article, the problems of jointly designing a pair
of unimodular sequence and receiving filters and also com-
plementary sequences and receiving filters with desired AF
shapes under the constraint of the SNR loss are addressed.
By using the FFT operations and the SQUAREM scheme,
an alternatively iterative scheme with the framework of the
MM method is proposed to solve the optimization problems
efficiently. Compared with the conventional algorithms, the
proposed algorithms show a significant improvement in terms
of the running time. Numerical results demonstrate that the
WISL/CISL can be markedly reduced at the expense of a
suitable SNR loss in a short time. Moreover, the designed
sequences are implemented on an indoor hardware system,
and their engineering performance is verified.
It has been pointed out in Section IV-B that one potential
application of the designed unimodular and receiving filter
is to detect multiple moving targets under the strong clutter.
When the designed CSS and receiving filters are used to
detect weak targets, the coherence between multiple pulses
is required to guarantee the complementary property of the
CSS and receiving filters. Thus, the CPI cannot be too long,
i.e., the pulse number cannot be too large, so as to avoid the
fluctuation of the scattering characteristics of targets.
Note that the unimodular constraint is considered in this
article. In some applications, a more general constraint, i.e.,
the peak-to-average ratio (PAR) constraint (the unimodular
constraint is a special case of the PAR constraint), is required
[48], [49]. It is predicted that a more relaxed constraint will
result in better correlation performance. On the other hand, the
proposed algorithms can only be used in the local AF shaping
for the system with a single transmitted channel, which limits
their applications in MIMO radar, code-division multiple-
access (CDMA) cellular systems, and other multichannel sys-
tems [50], [51]. Thus, in the future, we will reformulate the
joint design problem with more general modulus constraints
and focus on extending the proposed algorithms to applications
in multichannel systems.
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Fulai Wang received the B.S. degree in com-
munication engineering and the master’s degree
in information and communication engineering
from the National University of Defense Tech-
nology (NUDT), Changsha, China, in 2016 and
2018, respectively, where he is pursuing the Ph.D.
degree with the College of Electronic Science and
Engineering.
His research interests include radar polarimetry,
radar waveform design, and signal processing.
Sijia Feng received the B.S. and M.S. degrees in
information engineering from the National Univer-
sity of Defense Technology, Changsha, China, in
2016 and 2019, respectively, where she is pursuing
the Ph.D. degree with the State Key Laboratory of
Complex Electromagnetic Environment Effects.
Her research interests include synthetic aperture
radar (SAR) image interpretation, feature extraction,
and machine learning.
Jiapeng Yin (Member, IEEE) received the B.Sc.
degree in information engineering from the National
University of Defense Technology (NUDT), Chang-
sha, China, in 2012, and the Ph.D. degree in
atmospheric remote sensing from the Delft Univer-
sity of Technology, Delft, The Netherlands, in 2019.
He is an Assistant Professor with the College
of Electronic Science, NUDT. His research inter-
ests include radar polarimetry, polarimetric weather
radar, radar signal processing, and radar calibration.
Chen Pang received the B.S. degree in infor-
mation engineering and the master’s degree from
the National University of Defense Technology
(NUDT), Changsha, China, in 2009 and 2011,
respectively. In March 2011, he was directly admit-
ted to pursue his Ph.D. degree without having the
M.S. degree, which is a great honor for those top
students.
From November 2012 to November 2014, he was
a Visiting Ph.D. Student with the Delft University of
Technology, Delft, The Netherlands, where he was
with the Geoscience and Remote Sensing Department to extend his research
on polarimetric phased array weather radars. He is an Associate Professor
with the College of Electronic Science and Engineering, NUDT. His research
interests include weather radar signal processing, radar polarimetry, active
array, and electromagnetic simulation.
Yongzhen Li received the B.E. and Ph.D. degrees in
electronic engineering from the National University
of Defense Technology (NUDT), Changsha, China,
in 1999 and 2004, respectively.
He is a Professor with NUDT. His research inter-
ests include radar signal processing, radar polarime-
try, and target recognition.
Xuesong Wang was born in 1972. He received the
B.Sc. and Ph.D. degrees from the College of Elec-
tronic Science and Engineering, National University
of Defense Technology (NUDT), Changsha, China,
in 1994 and 1999, respectively.
He is a Professor with NUDT, where he is also
the Dean of the College of Science. His research
interests concentrate on radar information processing
and target recognition.
Dr. Wang is also a fellow of the Chinese Institute
of Electronics. His Ph.D. dissertation was awarded
as one of the 100 excellent Ph.D. dissertations in China in 2001 (two years
after his graduation).
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