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or spectral constraints
Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints
Journal Pre-proof
Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints Wen Fan, Junli Liang, Guoyang Yu, Hing Cheung So, Guangshan Lu PII: DOI: Reference:
S0165-1684(19)30501-8 https://doi.org/10.1016/j.sigpro.2019.107450 SIGPRO 107450
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
4 September 2019 28 December 2019 30 December 2019
Please cite this article as: Wen Fan, Junli Liang, Guoyang Yu, Hing Cheung So, Guangshan Lu, Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints, Signal Processing (2019), doi: https://doi.org/10.1016/j.sigpro.2019.107450
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H IGHLIGHTS •
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• •
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Waveform design has rec eived considerable attention be cause it has numerous active sensing applications in sonar, radar, and communications. Although existing solvers can produ ce constant modulus waveforms with low PSL, they lack the flexibility to control specified components of the waveform autocorrelation functions, i.e., local PSL And the se methods do not consider designing waveform with low PSL of autocorrelation in a spectrally crowded environment T he wave form autocorrelation can be expressed by using the FFT and IFFT. C omputationally efficient algo rithms to design a constant modulus waveform with minimum local PSL und er specified spectral and/or autocorrelation requirements. U se PMM to tackle the equality constrained optimization problems, split the complicated problem into small problems, and tac kle them individually.
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Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints Wen Fan, Junli Liang, Guoyang Yu, Hing Cheung So, Guangshan Lu
Abstract—The problem of designing waveforms with specified aperiodic/periodic correlation and spectral properties has received much attention because of its wide applicability in sonar, radar, and communications. In particular, a waveform with low peak sidelobe level (PSL) of autocorrelation can improve the detection performance of a weak target masked by a nearby strong target in range compression radar systems, and that with adaptive spectrum nulls/notches can make the sensing systems not be interfered by other electromagnetic equipment. On the other hand, a constant modulus (CM) waveform can maximize the transmit power efficiency of the system. Whereas, it is difficult to design a CM waveform with minimum local PSL and specified autocorrelation and/or spectral properties due to numerous nonconvex inequality and equality constraints. To develop efficient algorithms for this problem, we first express the waveform autocorrelation in the frequency domain, then use the proximal method of multipliers to tackle the resultant nonconvex and nonsmooth constrained optimization. Numerical examples have shown that the proposed methods can produce the waveforms satisfying these challenging requirements. Index Terms—Waveform design, peak sidelobe level, spectral compatibility, constant modulus.
I. I NTRODUCTION Waveform design has received considerable attention because it has numerous active sensing applications in sonar, radar, and communications [1]–[5]. For example, a probing waveform with minimum peak sidelobe level (PSL) of autocorrelation can improve the detection performance of a weak target masked by a nearby strong target in range compression radar [6], [7]. While a waveform with spectral nulls/notches at frequency bands that other systems have occupied can prevent mutual interference of the services/systems [8], [9]. A typical goal of probing waveform design is to make the integrated sidelobe level (ISL) or the PSL of the waveform autocorrelation as low as possible [10]–[13]. In this paper, we focus on the PSL metric, which PN −nis ∗defined as: −1 PSL = maxn {|rn |}N , where r = n n=1 l=1 xl xl+n [11], [14] denotes the autocorrelation of the aperiodic discrete-time probing waveform x , [x0 · · · xN −1 ]T ∈ CN ×1 , | · |, (·)∗ , and (·)T denote the modulus operation, complex conjugate, and transpos, respectively (for periodic x, its PN −1 waveform ∗ autocorrelation is defined as rn = l=0 xl x(l−n) mod N ). It is seen that PSL is quadratic with respect to x and it W. Fan, J. Liang, G. Yu and G. Lu, are with the School of Electronics and Information, Northwestern Polytechnical University, China. (email:[email protected];[email protected]; [email protected]; [email protected]). H. C. So is with Department of Electrical Engineering, City University of Hong Kong, China (e-mail:[email protected]).
is a nonsmooth function, which makes the problem hard to tackle. Another difficulty is that the nonconvex constant modulus (CM) constraint (i.e., |xn |2 = 1, ∀n) must be imposed on the probing waveform to maximize the output efficiency of the nonlinear radio-frequency power amplifiers and avoid waveform nonlinear distortion [15]–[18]. To address the nonsmooth and nonconvex problem, in [11], under the CM constraint, the authors use the lp -metric with a larger value of p to approximate the PSL, and propose a majorization-minimization (MM) based algorithm. In [19], a coordinate-descent framework is devised for low PSL waveform design. Although these solvers can produce CM waveforms with low PSL, they lack the flexibility to control autocorrelation sidelobes, i.e., local PSL (see Equation (1)). As pointed out in [2], [7], [10], reducing the waveform autocorrelation magnitude at certain lags is important for applications such as wireless communications and synthetic aperture radar imaging, where it is desired to suppress the interference caused by a known clutter discrete or a known multipath [2]. To achieve this, [7] designs a sequence set with accurately controlled correlation properties, and [2] proposes weighted cyclic algorithm-new (WeCAN) for waveform generation with certain specified autocorrelation magnitudes minimized. However, the aforementioned methods do not consider designing waveforms in spectrally crowded environments. Besides correlation properties, to prevent interferences from systems already operating in the corresponding frequency band, it is important to design probing waveform with spectral nulls or notches in those bands [20]–[22]. There are extensive works about the spectrally compatible waveform design. For example, [21] presents a least-squares (LS) fitting approach, called SHAPE algorithm. Based on the weighted LS fitting metric, [23] uses a Lagrange programming neural network to tackle the corresponding nonconvex optimization problem; whereas [3] uses cyclic approach. Other spectral compatible probing waveform design methods are given in [1], [10], [24], [25], and the references therein. However, these methods only focus on minimizing the ISL and cannot suppress the PSL of the designed waveform. In this work, we aim at designing CM waveform with minimum local PSL under specified spectral and/or autocorrelation requirements, in a computationally efficient manner. We first express the waveform autocorrelation via fast Fourier transform (FFT) and inverse FFT (IFFT). Then, the proximal method of multipliers (PMM) [26], [27] is introduced to tackle the resultant nonconvex and nonsmooth optimization problems. The PMM framework is a decomposition algorithm, which consists of minimizing primal proximal regulated aug-
3
mented Lagrangian function, followed by a simple dual update relation. Comparing with the alternating direction method of multipliers [28]–[30], it has more robust convergence guarantees and better practical performance [26], [27], [31]–[33]. The contributions of this work are listed as follows: • In the literature, there are methods [2], [10], [21], [34] that minimize the weighted ISL metric to reduce specified waveform autocorrelation magnitudes and pay no attention to PSL. On the other hand, works such as [11], [35] propose to minimize PSL, but they lack the flexibility to control the local PSL. Unlike them, we minimize the local PSL under the CM and correlation constraints. • Besides correlation properties, we also take the spectral power null/notch into account. As pointed out in [20]– [22], to prevent interferences from systems already operating in the corresponding frequency band or avoid strong emitters, it is important to design probing waveform with spectral nulls or notches in those bands. Although existing solvers, e.g., [1], [3], [20], [21], [25], [34], can produce CM waveforms with desired spectral shape, they only focus on minimizing the ISL and cannot suppress the PSL of the designed waveform. While we propose to minimize the local PSL under the CM, correlation, and spectral constraints. • To address the resultant nonconvex and nonsmooth optimization problems and develop fast and efficient algorithms, we first express the waveform autocorrelation via FFT and IFFT, then use the PMM [26], [27] to split the complicated problem into small problems and tackle them individually.
and we propose a Spectral constrained local PSL (SPSL) optimization problem: min max {|rn |}n∈Θm x nH s.t. fm x ≤ u m , m ∈ Ωs ;
|rn | ≤ δ, ∀n ∈ Θs ; |xn |2 = 1, ∀n, (2) where fm = √1M 1, ej2π(m−1)/M , · · · , ej2π(N −1)(m−1)/M ∈ CN ×1 , m = 1, · · · , M denotes the Fourier basis, um denotes the upper bound of the frequency grid point m, Ωs contains the frequency bands that are occupied by other systems. Remark 1: When Θm = {1, 2, · · · , N − 1}, Θs = ∅, and Ωs = ∅, both (1) and (2) reduce to the minimum PSL design problem, and the interested readers are referred to [11], [35] for detail. It is known that for a sequence, its autocorrelation and power spectral density form a Fourier transform pair [10], which implies that the autocorrelation vector r can be efficiently expressed by using FFT and IFFT, i.e., 2 r = F FH x , (3)
where F = [f0 · · · fM −1 ] ∈ CN ×M is the DFT matrix. We consider N -point DFT (M = N ) of x for periodic T waveform synthesis (r = [r0 , r1 , · · · , rN −1 ] ), and M -point DFT (M = 2N ) of x for aperiodic waveform synthesis T (r = [r0 , r1 , · · · , rN −1 , 0, r1−N , · · · , r−1 ] ) [10]. Thus, with (3), (1) can be rewritten as min max {|rn |}n∈Θm x,r n s.t. r = F |FH x|2 ;
|rn | ≤ δ, ∀n ∈ Θs ; |xn |2 = 1, ∀n.
II. P ROBLEM F ORMULATIONS AND A LGORITHMS In some practical applications, it is important to control the waveform autocorrelation magnitude at certain lags [2], i.e., the waveform local PSL. For example, in wireless communications, the maximum delay of the significant channel tap coefficients is usually known [10], which means that we only need to minimize/control certain autocorrelation of the probing waveform. However, existing works such as [2], [10], [21], [34] only focus on minimizing the weighted ISL metric to reduce waveform autocorrelation magnitudes and pay no attention to local PSL, which may result in larger local PSL. Unlike them, our first problem of interest is to minimize local PSL: min max{|rn |}n∈Θm x
n
s.t. |rn | ≤ δ, ∀n ∈ Θs ; |xn |2 = 1, ∀n,
(1)
where Θm and Θs denote the autocorrelation regions to be minimized and specifically controlled, respectively, and δ denotes the maximum allowable correlation level for the region Θs . Besides specified correlation properties, in order to ensure spectral compatibility of the active sensing systems in a spectrally dense environment, the spectrum of the waveform should have specified nulls/notches in certain frequency bands, e.g., for the bands reserved for communications and satellites [10], [22]. This corresponds to our second problem of interest,
(4)
On the other hand, combining (3) and introducing a new variable y, we rewrite (2) as: min max {|rn |}n∈Θm n s.t. r = F FH x |y| ; |rn | ≤ δ, ∀n ∈ Θs ,
r,x,y
FH x = y; |ym | ≤ um , m ∈ Ωs ; |xn |2 = 1, ∀n,
(5)
where denotes the Hadamard matrix product. The optimization problems (4) and (5) are difficult to solve due to the nonsmooth objective function, nonsmooth and nonconvex highly nonlinear quadratic equality and CM constraints. By taking advantage of the PMM, in the following, two efficient optimization algorithms, called PMM-PSL and PMM-SPSL, are derived to solve (4) and (5), respectively. A. PMM-PSL Algorithm First, we construct the proximal augmented Lagrangian for (4) [26]: 1 kr − r(t) k22 L (x, r, λ) = max{|rn |}n∈Θm + n 2γr
2 ρ + ( r − F(|FH x|2 ) + λ 2 − kλk22 ), (6) 2 where γr , ρ, λ, and r(t) are the proximal parameter, step size (penalty parameter), scaled dual variable, and t-th iteration
4
value of r (t denotes the iteration number), respectively. Then the following steps based on the PMM are applied into (6) to determine x, r, and λ alternately [26]: x
(t+1)
(t)
(t)
2
:= arg min L(x, r , λ ) s.t. |xn | = 1, ∀n; x
λ
:=λ
(t)
+r
(t+1)
H
− F(|F x
(7b) (t+1) 2
(7c)
| ),
We refer this algorithm to as PMM-PSL. The above steps
2 are repeated until a stopping criterion, e.g., r(t) − r(t+1) 2 ≤ ε, is reached. The solutions to the subproblems (7a) and (7b) are derived as follows. (t)
1) Solution to (7a): with given r and λ , determine x(t+1) . Defining ¯r(t) = r(t) + λ(t) , and using FH F = I, we have
2
2
(8) min FH x − z(t) s.t. |xn |2 = 1, ∀n, x 2 where z(t) = Re FH ¯r(t) . It is clear that the objective function of the problem (8) is quartic and the constraint is nonconvex set. Hence it is hard to tackle, generally, NPhard [10]. Nevertheless, as mentioned in [10], we solve the following “almost equivalent” optimization problem instead: (t)
min x
M 2 X H (t) fm x − z˜m s.t. |xn |2 = 1, ∀n,
(t)
H
x
(t)
2
¯ (t) it is easy to verify that its solution is x(t+1) = ej∠(Fy ) .
2) Solution to (7b): With the given x(t+1) and λ(t) , op 2 (t) ¯ = F FH x(t+1) − λ(t) , and timize r(t+1) . Defining λ introducing a new variable η, we rewrite (7b) as the following equivalent form:
2 ζ
min η + r − c(t) r,η 2 2 s.t. |rn | ≤ η, ∀n ∈ Θm ; |rn | ≤ δ, ∀n ∈ Θs , (12) It is
¯ (t)
(t)
γr ρ+1 λ +r (t) = γr ρργ γr , and c r +1 (t+1) (t) seen that r0 = c0 , and ∀n ∈
(13)
. (t+1)
Θs the optimal rn
(14)
Thus, substituting (14) into (12) and defining hn , |rn |, we get the following real-valued optimization problem: 2 ζ X hn − |c(t) min η+ n | hn ,n∈Θm ,η 2 n∈Θm
s.t. 0 ≤ hn ≤ η, ∀n ∈ Θm .
(15)
As (15) is a convex optimization problem, and we solve it by using the Karush-Kuhn-Tucker (KKT) theory [36]. The Lagrangian of (15) is ζ X 2 ((hn − |c(t) l(η, {hn , κn , ϕn }n∈Θm ) =η + n |) 2 n∈Θm
+ κn (hn − η) − ϕn hn ).
(16)
Suppose that {η ? , {h?n , κ?n , ϕ?n }n∈Θm } is an optimal solution to (16). Then under the KKT condition, we have
m=1
where y¯m = z˜m ej∠(fm x ) and the equality holds at x = x(t) . Thus, based on (10), we further simplify the problem (9) as:
2
¯ (t) x(t+1) := arg min x − Fy s.t. |xn |2 = 1, ∀n, (11)
where ζ =
∠r = ∠c(t) .
(9)
√ ˜(t) = z(t) . However, as we see, the objective function where z of (9) is nonsmooth and nonconvex due to the modulus operation. Expanding the objective and using Cauchy function Schwartz inequality and Re yH x ≤ yH x , we get 2 H (t) (t) 2 H (t) H 2 x − 2 fm x z˜m + z˜m fm x − z˜m = fm # " 2 H (t) H 2 x xH fm fm (t) (t) zm Re + zm ≤ fm x − 2˜ H x(t) fm 2 H (t) = fm x + y¯m (10) , (t)
( (t) (t) δej∠cn , if δ ≤ |cn |, = ∀n ∈ Θs . (t) cn , otherwise.
While ∀n ∈ Θm , the optimal phase of r follows that of c(t) , i.e.,
r
(t+1)
rn(t+1)
(7a)
r(t+1) := arg min L(x(t+1) , r, λ(t) ) s.t. |rn | ≤ δ, ∀n ∈ Θs ;
is given by
? ? 0 =ζ(h?n − |c(t) n |) + κn − ϕn , ∀n ∈ Θm ; X 0 =1 − κ?n , ∀n ∈ Θm ;
(17)
0
(20)
0
n∈Θm =κ?n (h?n − η ? ), κ?n ≥ 0, ∀n ∈ =ϕ?n h?n , ϕ?n ≥ 0, ∀n ∈ Θm .
Θm ;
(18) (19)
It is easily revealed that, the dual variables ϕn and κn are corresponding to the mutually exclusive constraints, and either of them must be zero. Then we have the following three cases: •
? if ϕ?n = 0, and κ?n > 0, the third condition n = implies h (t) ? ? ? η and the first condition implies κn = ζ |cn | − η ≥ (t)
•
•
0, i.e., |cn | ≥ η ? ; if ϕ?n > 0, and κ?n = 0, the fourth condition implies (t) h?n = 0 and the first condition implies −ζ|cn | = ϕ?n (t) (this is impossible, since |cn | ≥ 0); if ϕ?n = 0, and κ?n = 0, the first condition implies h?n = (t) |cn |.
Based on the above cases, we have ( (t) η? , if η ? ≤ |cn |, ? hn = (t) |cn |, otherwise,
and
κ?n
=
( (t) (t) ζ |cn | − η ? , if η ? ≤ |cn |, 0,
otherwise.
(21)
(22)
Let [α1 , · · · , αK ] be the descending order sequence of (t) N −1 {0, |cn |}n=1 . Combining (22) and (18), we can determine ? η in a closed-form. Specifically, for each k, we define the (t) index set Ωk = {n||cn | ≥ αk , ∀n}. Then, combining (18)
5
and (22) yields X X ? κ?n = 1 − ζ |c(t) 0 =1 − n | − ηk , n∈Θm
into (25) to determine {x, y, r, λ, µ}: (23)
x(t+1) := arg min L(x, y(t) , r(t) , λ(t) , µ(t) ) x
n∈Ωk
the solution to which is given by P (t) |cn | − 1/ζ ? Pk ηk = n∈Ω . n∈Ωk 1
s.t. |xn |2 = 1, ∀n;
(t+1)
y
y
(24)
If ηk? ≤ αk , we get η ? = ηk? ; otherwise, let k = k + 1, and repeat the above calculations. When η ? is obtained, we de(k+1) (k+1) (k+1) j∠c(t) termine hn using (21), and rn = hn e n , ∀n ∈ Θm . Remark 2: Computing x(t+1) requires two FFTs and one vector multiplication and the complexity is O(2N log(N ) + N ); the update of r(t+1) has a complexity of O(2N log(N ) + 2N ). Thus, in each iteration, the computational complexity of the PMM-PSL algorithm is O(4N log(N ) + 3N ).
Theorem 1: [37] Let {x? , r? , λ? } be a sequence generated by the steps (7a)–(7c), i.e., all the subproblems are solved exactly, with ρ > 0 and γr > 0. Assume limt→∞ kλ(t+1) − λ(t) k2 = 0. Then any limit point {x? , r? , λ? } is a KKT point of (4). Proof From (7c) and limt→∞ kλ(t+1) − λ(t) k2 = 0, we get limt→∞ kr(t+1) − F(|FH x(t+1) |2 )k2 = 0. Since the feasible set of x is bounded and closed, it follows from (12) that the sequence {η (t) , r(t) } is also bounded. Thus there (t) (t) (t) exists a stationary point such that limt→∞?{x? , r? , λ } = ? ? ? ? H ? 2 {x , r , λ }, r = F |F x | , ∇x L (x , r , λ ) = 0, and ∇r L (x? , r? , λ? ) = 0. The proof is complete.
Remark 3: The equality in (10) holds at x = x(t) , which implies that the optimal solution to (11) is also the solution to (9), when x = x? . In addition, since (7a) and (9) are “almost equivalence”, ∇x L (x? , r? , λ? ) = ∇x f8 (x? ) = 0 implies ∇x L (x? , r? , λ? ) ≈ ∇x f9 (x? ) ≈ 0 [10], where f8 (x) and f9 (x) denote the objective functions of (7a) and (9), respectively. Thus, Theorem 1 shows the PMM-PSL sequence (i.e., the sequence obtained in (11), (7b), and (7c)) converges to an approximate KKT point of (4) under mild condition.
r
(t+1)
:= arg min L(x r
(t+1)
µ
:=λ
(t)
(t+1)
+r
(t)
:=µ
H
,y
+F x
(26b)
(t+1)
(t)
(t)
, r, λ , µ ) (26c)
H
− F(|F x
(t+1)
(t)
, y, r , λ , µ )
s.t. |rn | ≤ δ, ∀n ∈ Θs ;
(t+1)
(t)
(t)
s.t. |ym | ≤ um , m ∈ Ωs ;
(t+1)
λ
:= arg min L(x
(26a)
(t+1)
−y
(t+1)
(t+1)
.
| |y
(t+1)
|); (26d)
We name the above algorithm as PMM-SPSL. The solutions to the subproblems (26a)–(26c) are derived as follows. ¯ (t) = 1) Solution to (26a): Defining ¯r(t) = r(t) + λ(t) , y (t) (t) H y − µ , and using F F = I, we have
2 ρ
2 ρ1
(t) H 2 y(t) min
y F x − z(t) +
x − F¯ x 2 2 2 2 s.t. |xn |2 = 1, ∀n, (27) where z(t) = Re FH ¯r(t) . Similar to the majorization step derived in (10), we get M
2 2
X
(t) H
(t) H (t) |ym |fm x + y¯m , (28)
y F x − z(t) ≤ 2
m=1
H (t) (t) zm ej∠(fm x ) .
(t) y¯m
= Thus, (27) is majorized by
2
2 ρ ρ1 (t)
2 H ¯ (t) ¯ (t) min
y FH x − y
F x − y
+ x 2 2 2 2 s.t. |xn |2 = 1, ∀n. (29)
where
Expanding the objective of (29) and rearranging the terms yields:
¯ (t) + ρ22 y ρ21 y(t) y ¯ (t)
(t) H min z F x −
x
z(t) s.t. |xn |2 = 1, ∀n. q 2 where z(t) = ρ21 y(t) + (30) is equivalent to
(30)
ρ2 2
and
a b
, [a1 /b1 , · · · , aN /bN ].
˜(t) k s.t. |xn |2 = 1, ∀n; minkFH x − z x
B. PMM-SPSL Algorithm
First, we construct the following proximal augmented Lagrangian for (5): 1 kr − r(t) k22 L (x, y, r, λ, µ) = max{|rn |}n∈Θm + n 2γr
ρ2
FH x − y + µ 2 − kµk22 + 1 ky − y(t) k22 + 2 2 2γy
2 H ρ1
+ r − F F x |y| + λ 2 − kλk22 . (25) 2
Then the following steps based on the PMM are applied
(26e)
˜(t) = where z obtain:
ρ1 2
(31)
|y(t) | y¯ (t) + ρ22 y¯ (t) . Since FFH = I, we then ρ1 (t) 2 + ρ2 | 2 2 |y
x(t+1) := arg min kx − F˜ z(t) k s.t. |xn |2 = 1, ∀n, x
(32)
(t) and its solution is x(t+1) = ej∠(F˜z ) . ¯ (t) = FH x(t+1) +µ(t) , and 2) Solution to (26b): Defining x ignoring irrelevant items in (26b), we have 2 ρ1
(t)
min
¯r − F FH x(t+1) |y| y 2 2
2 ρ2 1
(t)
(t) 2 ¯ − y + + ky − y k2
x 2 2γy 2 s.t. |ym | ≤ um , m ∈ Ωs . (33)
Using the same strategy derived in (27)–(31), we get
2
y(t+1) =: min y − s(t) s.t. |ym | ≤ um , m ∈ Ωs , (34) =
2
ρ1 |FH x(t+1) | +ρ2 + γ1y
j∠(y(t) )
, and q
(t)
e
¯ (t)
X: 4.025 Y: 9.013
X: 3.419 Y: 8.751
PMM-PSL MM-PSL
0 0
2
4
6
8
10
12
(a) Time (s)
(t)
λ +r where ζ = γr ργ1r+1 , and c(t) = γr ρργ . As (36) has the r +1 similar form as (12), thus it can be solved in a similar way. Remark 4: We analyze the computational complexity of the PMM-SPSL algorithm. Updating x(t+1) needs two FFTs and one vector multiplication and the complexity is O(2N log(N ) + N ); Updating y(t+1) and r(t+1) have a complexity of O(2N log(N ) + 2N ). Thus, in each iteration, the computational complexity of the PMM-SPSL algorithm is O(6N log(N )+5N ). Hence, the computational complexity of the proposed algorithm is linear with respect to the iteration number k. In Table I, we summarize the computational complexities of the proposed methods as well as MM-PSL [11], WeCAN [2], and SCAN [34], where T denotes the total iteration number.
TABLE I: Computational complexities of different methods. Method
Computational Complexity
PMM-PSL PMM-SPSL MM-PSL [11] WeCAN [2] SCAN [34]
O(T (4N log(N ) + 3N )) O(T (6N log(N ) + 5N )) O(T (4N log(N ) + 6N )) O(T ((N 2 + 1) log(N ) + 2N )) O(T (4N log(N ) + 2N ))
Theorem 2: Let {x? , y? , r? , λ? , µ? } be a sequence generated by the steps (26a)–(26e) with ρ1 > 0, ρ2 > 0, γy > 0 and γr > 0. Assume limt→∞ kλ(t+1) − λ(t) k2 = 0 and limt→∞ kµ(t+1) − µ(t) k2 = 0. Then any limit point {x? , y? , r? , λ? , µ? } is a KKT point of (5). Proof The proof can be obtained by using the same procedure discussed in Theorem 1. Remark 5: According to Theorem 2 and using the same procedure discussed in Remark 3, we conclude that the PMMSPSL sequence (32), (34), and (26c)–(26e) converges to a KKT point of (5) under some mild conditions. III. N UMERICAL E XPERIMENTS Numerical examples are conducted to evaluate the proposed methods. For notational simplicity, the normalized frequencies (0 to 1) are used. Unless stated otherwise, the dual variables and auxiliary variables are initialized randomly.
X: -106 Y: -28.85
-20
14
16
18
Random Sequence PMM-PSL MM-PSL WeCAN
0
. The solution to (34) is given by: ( (t) (t) um ej∠sm , if um ≤ |sm |, (t+1) ym = (35) (t) sm , otherwise. ¯ (t) = F( FH x(t+1) 3) Solution to (26c): Defining λ (t+1) y )−λ(t) , and introducing a new variable η, the problem (26c) can be equivalently written as:
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min η + r − c(t) r,η 2 2 s.t. |rn | ≤ η, ∀n ∈ Θm ; |rn | ≤ δ, ∀n ∈ Θs , (36) z
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Fig. 1: (a) Cost function value versus time; (b) Waveform autocorrelation for Θs = ∅; (c) Waveform autocorrelation for Θs 6= ∅. A. PMM-PSL Algorithm We verify the performance of the PMM-PSL algorithm by comparing it with the MM-PSL (accelerated by using the SQUAREM) [11] and WeCAN [2] algorithms. Firstly, we assign N = 128, ρ = 1, γr = 2, Θm = {1, 2, · · · , N − 1}, Θs = ∅, and the maximum iteration number Tm = 20000 is used as the stopping criterion for PMM-PSL. The cost function and the normalized autocorrelation levels, i.e., 20 log10 (|rn |/N ), of the designed aperiodic waveforms are plotted in Figs. 1(a) and (b). We see that both PMM-PSL and MM-PSL converge to almost the same cost function value (the PSL of PMM-PSL is 8.754, whereas MMPSL is 9.01), whereas WeCAN has larger PSL. In the second example, we evaluate the PMM-PSL algorithm in controlling the specific autocorrelation level (i.e., local PSL, Θs 6= ∅). We let Θm = {41, · · · , N − 1}, Θs = {1, · · · , 40}, δ = N 10−3 (i.e., −60 dB), and other parameters remain unchanged. The MM-PSL is excluded from the comparison since it cannot control the waveform autocorrelation magnitude at certain lags. The autocorrelation levels of the designed aperiodic waveforms are plotted in Fig. 1(c). We see that the autocorrelation levels of the PMM-PSL in the region Θs are all lower than −60 dB, and they meet the requirements, i.e., |rn | ≤ δ, ∀n ∈ Θs . In addition, the autocorrelation levels of the PMM-PSL in the region Θm almost have the uniformly local PSL, while those of the WeCAN within Θm and Θs are much larger. B. PMM-SPSL To examine the convergence performance of the PMMSPSL, we assign N = 512, Θm = {1, 2, · · · , N − 1}, 420 Θs = ∅, Ωs = { 400 2N , · · · , 2N }, um = −40 dB (∀m ∈ Ωs ), Θm = {1, 2, · · · , 99, 201, · · · , N − 1}, Θs = {100, · · · , 200}, δ = −45 dB), ρ1 = 0.01, ρ2 = 40000, γx = 20, γy = 20. The maximum iteration number Tm = 20000 is set as the stopping criterion. The auxiliary variables and the scaled dual
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Fig. 2: (a) Objective function versus number of iterations; (b) Autocorrelations of the designed waveforms; (c) Spectra of the designed waveforms. the autocorrelation of the initial sequence and the obtained sequence are shown in Fig. 4. From Figs. 3 and 4, it is seen that the proposed method can synthesize the probing waveform with a minimum local PSL and also achieve the specified spectral and autocorrelation requirements. Whereas SCAN has larger stopband level (−16.52 dB) and PSL (−22.32 dB). Spectrum (dB)
variables (i.e., y(0) , r(0) , λ(0) , µ(0) ) are initialized randomly. We run PMM-SPSL 1000 times and plot the evolution curves of the objective value (i.e., PSL=maxn {|rn |}∀n ) with respect to the number of iterations, autocorrelations and spectra of the synthesized waveforms in Fig.2(a)–(c), respectively. It can be seen that the objective value tends to be stable after 8000 iterations, and the autocorrelations and spectra of the designed waveforms align with the specified values, which imply that the proposed method has satisfactory convergence performance. Then, we compare the proposed method with the SCAN [34]. We use N = 512, Θm = {1, 2, · · · , N − 1}, Θs = ∅, and set the Golomb sequence [10] as the initial sequence, and keep the other algorithm parameters unchanged. The spectrum and the autocorrelation of the sequence obtained by using the PMM-SPSL and SCAN are shown in Fig. 3.
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Fig. 3: Periodic sequence design with spectral constraint. Finally, we test the ability of the PMM-SPSL algorithm to control the specific autocorrelation level (i.e., Θm = {1, 2, · · · , 99, 201, · · · , N − 1}, Θs = {100, · · · , 200}, δ = −50 dB), the random sequence is used as initial sequence, and other parameters remain unchanged. The spectrum and
IV. C ONCLUSION We have devised two methods for designing waveform with low local PSL and spectral compatibility, which are computationally simple and have fast convergence. The key ideas of our algorithms are to utilize the waveform autocorrelation which can be expressed by the FFT and IFFT, use PMM to tackle the equality constrained nonconvex nonsmooth optimization problems and split the complicated problem into small problems and tackle them individually. ACKNOWLEDGEMENT This work was supported in part by the China Scholarship Council, in part by the Huawei-Northwestern Polytechnical
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University Academic Scholarship of China, in part by the Natural Science Foundation of China under Grant 61471295, in part by the Aeronautical Science Foundation of China under Grants 20172053017 and 20172053018, and in part by the Central University Funds under Grants G2016KY0308, G2016KY0002, and 17GH030144. R EFERENCES [1] A. Aubry, A. De Maio, M. Piezzo, and A. Farina, “Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 2, pp. 1138–1152, Apr. 2014. [2] P. Stoica, H. He, and J. Li, “New algorithms for designing unimodular sequences with good correlation properties,” IEEE Trans. Signal Process., vol. 57, no. 4, pp. 1415–1425, Apr. 2009. [3] B. Tang and J. Liang, “Efficient algorithms for synthesizing probing waveforms with desired spectral shapes,” IEEE Trans. Aerosp. Electron. Syst., pp. 1–1, 2018. [4] X. Yu, G. Cui, L. Kong, J. Li, and G. Gui, “Constrained waveform design for colocated MIMO radar with uncertain steering matrices,” IEEE Trans. Aerosp. Electron. Syst., vol. 55, no. 1, pp. 356–370, Feb. 2019. [5] G. Cui, Y. Fu, X. Yu, and J. Li, “Local ambiguity function shaping via unimodular sequence design,” IEEE Signal Process. Lett., vol. 24, no. 7, pp. 977–981, Jul. 2017. [6] G. W. Stimson, Introduction to Airborne Radar, Mendham, NJ, USA: Soc. Photo Optical, 1998. [7] G. Yu, J. Liang, J. Li, and B. Tang, “Sequence set design with accurately controlled correlation properties,” IEEE Trans. Aerosp. Electron. Syst., pp. 1–1, 2018. [8] X. Yu, G. Cui, J. Yang, and L. Kong, “Wideband MIMO radar beampattern shaping with space-frequency nulling,” Signal Process., vol. 160, pp. 80 – 87, 2019. [9] B. Tang, J. Li, and J. Liang, “Alternating direction method of multipliers for radar waveform design in spectrally crowded environments,” Signal Process., vol. 142, pp. 398 – 402, 2018. [10] H. He, J. Li, and P. Stoica, Waveform Design for Active Sensing Systems: A Computational Approach. Cambridge University Press, 2012. [11] J. Song, P. Babu, and D. P. Palomar, “Sequence design to minimize the weighted integrated and peak sidelobe levels,” IEEE Trans. Signal Process., vol. 64, no. 8, pp. 2051–2064, Apr. 2016. [12] F. cong Li, Y. nan Zhao, and X. lin Qiao, “A waveform design method for suppressing range sidelobes in desired intervals,” Signal Process., vol. 96, pp. 203 – 211, 2014. [13] Z. Cheng, B. Liao, Z. He, J. Li, and C. Han, “A nonlinear-ADMM method for designing MIMO radar constant modulus waveform with low correlation sidelobes,” Signal Process., vol. 159, pp. 93 – 103, 2019. [14] G. Cui, X. Yu, M. Piezzo, and L. Kong, “Constant modulus sequence set design with good correlation properties,” Signal Process., vol. 139, pp. 75 – 85, 2017. [15] G. Cui, X. Yu, G. Foglia, Y. Huang, and J. Li, “Quadratic optimization with similarity constraint for unimodular sequence synthesis,” IEEE Trans. Signal Process., vol. 65, no. 18, pp. 4756–4769, Sep. 2017. [16] W. Fan, J. Liang, and J. Li, “Constant modulus MIMO radar waveform design with minimum peak sidelobe transmit beampattern,” IEEE Trans. Signal Process., vol. 66, no. 16, pp. 4207–4222, Aug. 2018. [17] W. Fan, J. Liang, G. Yu, H. C. So, and G. Lu, “MIMO radar waveform design for quasi-equiripple transmit beampattern synthesis via weighted lp -minimization,” IEEE Trans. Signal Process., vol. 67, no. 13, pp. 3397–3411, Jul. 2019. [18] X. Yu, G. Cui, J. Yang, L. Kong, and J. Li, “Wideband MIMO radar waveform design,” IEEE Trans. Signal Process., vol. 67, no. 13, pp. 3487–3501, Jul. 2019. [19] M. A. Kerahroodi, A. Aubry, A. De Maio, M. M. Naghsh, and M. Modarres-Hashemi, “A coordinate-descent framework to design low PSL/ISL sequences,” IEEE Trans. Signal Process., vol. PP, no. 99, pp. 1–1, 2017. [20] J. Liang, H. C. So, J. Li, and A. Farina, “Unimodular sequence design based on alternating direction method of multipliers,” IEEE Trans. Signal Process., vol. 64, no. 20, pp. 5367–5381, Oct. 2016. [21] W. Rowe, P. Stoica, and J. Li, “Spectrally constrained waveform design,” IEEE Signal Process. Mag., vol. 31, no. 3, pp. 157–162, May 2014.
[22] M. S. Greco, F. Gini, P. Stinco, and K. Bell, “Cognitive radars: On the road to reality: Progress thus far and possibilities for the future,” IEEE Signal Process. Mag., vol. 35, no. 4, pp. 112–125, Jul. 2018. [23] J. Liang, H. C. So, C. S. Leung, J. Li, and A. Farina, “Waveform design with unit modulus and spectral shape constraints via lagrange programming neural network,” IEEE Journal of Selected Topics in Signal Processing, vol. 9, no. 8, pp. 1377–1386, Dec. 2015. [24] A. Aubry, V. Carotenuto, A. De. Maio, A. Farina, and L. Pallotta, “Optimization theory-based radar waveform design for spectrally dense environments,” IEEE Aerosp. Electron. Syst. Mag., vol. 31, no. 12, pp. 14–25, Dec. 2016. [25] Y. Jing, J. Liang, D. Zhou, and H. C. So, “Spectrally constrained unimodular sequence design without spectral level mask,” IEEE Signal Process. Lett., vol. 25, no. 7, pp. 1004–1008, Jul. 2018. [26] R. Shefi and M. Teboulle, “Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization,” SIAM J. Optim., vol. 24, no. 1, pp. 269–297, 2014. [27] N. K. Dhingra, S. Z. Khong, and M. R. Jovanovi, “The proximal augmented Lagrangian method for nonsmooth composite optimization,” IEEE Trans. Autom. Control, vol. 64, no. 7, pp. 2861–2868, Jul. 2019. [28] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn., vol. 3, no. 1, pp. 1–122, 2011. [29] C. Chen, B. He, Y. Ye, and X. Yuan, “The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent,” Math. Program., vol. 155, no. 1, pp. 57–79, Jan. 2016. [30] J. Liang, H. So, J. Li, A. Farina, and D. Zhou, “On optimizations with magnitude constraints on frequency or angular responses,” Signal Process., vol. 145, pp. 214 – 224, 2018. [31] M. Sun and J. Liu, “The convergence rate of the proximal alternating direction method of multipliers with indefinite proximal regularization,” J. Inequal. Appl., vol. 2017, no. 1, p. 19, Jan. 2017. [32] Z. Peng, J. Chen, and W. Zhu, “A proximal alternating direction method of multipliers for a minimization problem with nonconvex constraints,” J. Glob. Optim., vol. 62, no. 4, pp. 711–728, Aug. 2015. [33] R. I. Bot and D.-K. Nguyen, “The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates,” 2018. [34] H. He, P. Stoica, and J. Li, “Waveform design with stopband and correlation constraints for cognitive radar,” in in Proc. Int. Workshop Cognit. Inf. Process. (CIP), Elba Island, Italy, Jun. 2010, pp. 344–349. [35] Z. Lin, W. Pu, and Z. Luo, “Minimax design of constant modulus MIMO waveforms for active sensing,” IEEE Signal Process. Lett., vol. 26, no. 10, pp. 1531–1535, Oct. 2019. [36] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [37] Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Problems, vol. 28, no. 11, p. 115010, 2012.
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CREDIT AUTHOR STATEMENT Wen Fan: performed writing-original draft, conceptualization, methodology, software, data curation, and funding acquisition. Junli Liang: methodology, funding acquisition, writingreview, project administration, and supervision. Guoyang Yu: investigation and writing, and validation. Hing Cheung So: performed writing-review, and editing. Guangshan Lu: supervision.
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Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints Wen Fan, Junli Liang, Guoyang Yu, Hing Cheung So, Guangshan Lu PII: DOI: Reference:
S0165-1684(19)30501-8 https://doi.org/10.1016/j.sigpro.2019.107450 SIGPRO 107450
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Signal Processing
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Please cite this article as: Wen Fan, Junli Liang, Guoyang Yu, Hing Cheung So, Guangshan Lu, Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints, Signal Processing (2019), doi: https://doi.org/10.1016/j.sigpro.2019.107450
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H IGHLIGHTS •
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Waveform design has rec eived considerable attention be cause it has numerous active sensing applications in sonar, radar, and communications. Although existing solvers can produ ce constant modulus waveforms with low PSL, they lack the flexibility to control specified components of the waveform autocorrelation functions, i.e., local PSL And the se methods do not consider designing waveform with low PSL of autocorrelation in a spectrally crowded environment T he wave form autocorrelation can be expressed by using the FFT and IFFT. C omputationally efficient algo rithms to design a constant modulus waveform with minimum local PSL und er specified spectral and/or autocorrelation requirements. U se PMM to tackle the equality constrained optimization problems, split the complicated problem into small problems, and tac kle them individually.
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Minimum Local Peak Sidelobe Level Waveform Design with Correlation and/or Spectral Constraints Wen Fan, Junli Liang, Guoyang Yu, Hing Cheung So, Guangshan Lu
Abstract—The problem of designing waveforms with specified aperiodic/periodic correlation and spectral properties has received much attention because of its wide applicability in sonar, radar, and communications. In particular, a waveform with low peak sidelobe level (PSL) of autocorrelation can improve the detection performance of a weak target masked by a nearby strong target in range compression radar systems, and that with adaptive spectrum nulls/notches can make the sensing systems not be interfered by other electromagnetic equipment. On the other hand, a constant modulus (CM) waveform can maximize the transmit power efficiency of the system. Whereas, it is difficult to design a CM waveform with minimum local PSL and specified autocorrelation and/or spectral properties due to numerous nonconvex inequality and equality constraints. To develop efficient algorithms for this problem, we first express the waveform autocorrelation in the frequency domain, then use the proximal method of multipliers to tackle the resultant nonconvex and nonsmooth constrained optimization. Numerical examples have shown that the proposed methods can produce the waveforms satisfying these challenging requirements. Index Terms—Waveform design, peak sidelobe level, spectral compatibility, constant modulus.
I. I NTRODUCTION Waveform design has received considerable attention because it has numerous active sensing applications in sonar, radar, and communications [1]–[5]. For example, a probing waveform with minimum peak sidelobe level (PSL) of autocorrelation can improve the detection performance of a weak target masked by a nearby strong target in range compression radar [6], [7]. While a waveform with spectral nulls/notches at frequency bands that other systems have occupied can prevent mutual interference of the services/systems [8], [9]. A typical goal of probing waveform design is to make the integrated sidelobe level (ISL) or the PSL of the waveform autocorrelation as low as possible [10]–[13]. In this paper, we focus on the PSL metric, which PN −nis ∗defined as: −1 PSL = maxn {|rn |}N , where r = n n=1 l=1 xl xl+n [11], [14] denotes the autocorrelation of the aperiodic discrete-time probing waveform x , [x0 · · · xN −1 ]T ∈ CN ×1 , | · |, (·)∗ , and (·)T denote the modulus operation, complex conjugate, and transpos, respectively (for periodic x, its PN −1 waveform ∗ autocorrelation is defined as rn = l=0 xl x(l−n) mod N ). It is seen that PSL is quadratic with respect to x and it W. Fan, J. Liang, G. Yu and G. Lu, are with the School of Electronics and Information, Northwestern Polytechnical University, China. (email:[email protected];[email protected]; [email protected]; [email protected]). H. C. So is with Department of Electrical Engineering, City University of Hong Kong, China (e-mail:[email protected]).
is a nonsmooth function, which makes the problem hard to tackle. Another difficulty is that the nonconvex constant modulus (CM) constraint (i.e., |xn |2 = 1, ∀n) must be imposed on the probing waveform to maximize the output efficiency of the nonlinear radio-frequency power amplifiers and avoid waveform nonlinear distortion [15]–[18]. To address the nonsmooth and nonconvex problem, in [11], under the CM constraint, the authors use the lp -metric with a larger value of p to approximate the PSL, and propose a majorization-minimization (MM) based algorithm. In [19], a coordinate-descent framework is devised for low PSL waveform design. Although these solvers can produce CM waveforms with low PSL, they lack the flexibility to control autocorrelation sidelobes, i.e., local PSL (see Equation (1)). As pointed out in [2], [7], [10], reducing the waveform autocorrelation magnitude at certain lags is important for applications such as wireless communications and synthetic aperture radar imaging, where it is desired to suppress the interference caused by a known clutter discrete or a known multipath [2]. To achieve this, [7] designs a sequence set with accurately controlled correlation properties, and [2] proposes weighted cyclic algorithm-new (WeCAN) for waveform generation with certain specified autocorrelation magnitudes minimized. However, the aforementioned methods do not consider designing waveforms in spectrally crowded environments. Besides correlation properties, to prevent interferences from systems already operating in the corresponding frequency band, it is important to design probing waveform with spectral nulls or notches in those bands [20]–[22]. There are extensive works about the spectrally compatible waveform design. For example, [21] presents a least-squares (LS) fitting approach, called SHAPE algorithm. Based on the weighted LS fitting metric, [23] uses a Lagrange programming neural network to tackle the corresponding nonconvex optimization problem; whereas [3] uses cyclic approach. Other spectral compatible probing waveform design methods are given in [1], [10], [24], [25], and the references therein. However, these methods only focus on minimizing the ISL and cannot suppress the PSL of the designed waveform. In this work, we aim at designing CM waveform with minimum local PSL under specified spectral and/or autocorrelation requirements, in a computationally efficient manner. We first express the waveform autocorrelation via fast Fourier transform (FFT) and inverse FFT (IFFT). Then, the proximal method of multipliers (PMM) [26], [27] is introduced to tackle the resultant nonconvex and nonsmooth optimization problems. The PMM framework is a decomposition algorithm, which consists of minimizing primal proximal regulated aug-
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mented Lagrangian function, followed by a simple dual update relation. Comparing with the alternating direction method of multipliers [28]–[30], it has more robust convergence guarantees and better practical performance [26], [27], [31]–[33]. The contributions of this work are listed as follows: • In the literature, there are methods [2], [10], [21], [34] that minimize the weighted ISL metric to reduce specified waveform autocorrelation magnitudes and pay no attention to PSL. On the other hand, works such as [11], [35] propose to minimize PSL, but they lack the flexibility to control the local PSL. Unlike them, we minimize the local PSL under the CM and correlation constraints. • Besides correlation properties, we also take the spectral power null/notch into account. As pointed out in [20]– [22], to prevent interferences from systems already operating in the corresponding frequency band or avoid strong emitters, it is important to design probing waveform with spectral nulls or notches in those bands. Although existing solvers, e.g., [1], [3], [20], [21], [25], [34], can produce CM waveforms with desired spectral shape, they only focus on minimizing the ISL and cannot suppress the PSL of the designed waveform. While we propose to minimize the local PSL under the CM, correlation, and spectral constraints. • To address the resultant nonconvex and nonsmooth optimization problems and develop fast and efficient algorithms, we first express the waveform autocorrelation via FFT and IFFT, then use the PMM [26], [27] to split the complicated problem into small problems and tackle them individually.
and we propose a Spectral constrained local PSL (SPSL) optimization problem: min max {|rn |}n∈Θm x nH s.t. fm x ≤ u m , m ∈ Ωs ;
|rn | ≤ δ, ∀n ∈ Θs ; |xn |2 = 1, ∀n, (2) where fm = √1M 1, ej2π(m−1)/M , · · · , ej2π(N −1)(m−1)/M ∈ CN ×1 , m = 1, · · · , M denotes the Fourier basis, um denotes the upper bound of the frequency grid point m, Ωs contains the frequency bands that are occupied by other systems. Remark 1: When Θm = {1, 2, · · · , N − 1}, Θs = ∅, and Ωs = ∅, both (1) and (2) reduce to the minimum PSL design problem, and the interested readers are referred to [11], [35] for detail. It is known that for a sequence, its autocorrelation and power spectral density form a Fourier transform pair [10], which implies that the autocorrelation vector r can be efficiently expressed by using FFT and IFFT, i.e., 2 r = F FH x , (3)
where F = [f0 · · · fM −1 ] ∈ CN ×M is the DFT matrix. We consider N -point DFT (M = N ) of x for periodic T waveform synthesis (r = [r0 , r1 , · · · , rN −1 ] ), and M -point DFT (M = 2N ) of x for aperiodic waveform synthesis T (r = [r0 , r1 , · · · , rN −1 , 0, r1−N , · · · , r−1 ] ) [10]. Thus, with (3), (1) can be rewritten as min max {|rn |}n∈Θm x,r n s.t. r = F |FH x|2 ;
|rn | ≤ δ, ∀n ∈ Θs ; |xn |2 = 1, ∀n.
II. P ROBLEM F ORMULATIONS AND A LGORITHMS In some practical applications, it is important to control the waveform autocorrelation magnitude at certain lags [2], i.e., the waveform local PSL. For example, in wireless communications, the maximum delay of the significant channel tap coefficients is usually known [10], which means that we only need to minimize/control certain autocorrelation of the probing waveform. However, existing works such as [2], [10], [21], [34] only focus on minimizing the weighted ISL metric to reduce waveform autocorrelation magnitudes and pay no attention to local PSL, which may result in larger local PSL. Unlike them, our first problem of interest is to minimize local PSL: min max{|rn |}n∈Θm x
n
s.t. |rn | ≤ δ, ∀n ∈ Θs ; |xn |2 = 1, ∀n,
(1)
where Θm and Θs denote the autocorrelation regions to be minimized and specifically controlled, respectively, and δ denotes the maximum allowable correlation level for the region Θs . Besides specified correlation properties, in order to ensure spectral compatibility of the active sensing systems in a spectrally dense environment, the spectrum of the waveform should have specified nulls/notches in certain frequency bands, e.g., for the bands reserved for communications and satellites [10], [22]. This corresponds to our second problem of interest,
(4)
On the other hand, combining (3) and introducing a new variable y, we rewrite (2) as: min max {|rn |}n∈Θm n s.t. r = F FH x |y| ; |rn | ≤ δ, ∀n ∈ Θs ,
r,x,y
FH x = y; |ym | ≤ um , m ∈ Ωs ; |xn |2 = 1, ∀n,
(5)
where denotes the Hadamard matrix product. The optimization problems (4) and (5) are difficult to solve due to the nonsmooth objective function, nonsmooth and nonconvex highly nonlinear quadratic equality and CM constraints. By taking advantage of the PMM, in the following, two efficient optimization algorithms, called PMM-PSL and PMM-SPSL, are derived to solve (4) and (5), respectively. A. PMM-PSL Algorithm First, we construct the proximal augmented Lagrangian for (4) [26]: 1 kr − r(t) k22 L (x, r, λ) = max{|rn |}n∈Θm + n 2γr
2 ρ + ( r − F(|FH x|2 ) + λ 2 − kλk22 ), (6) 2 where γr , ρ, λ, and r(t) are the proximal parameter, step size (penalty parameter), scaled dual variable, and t-th iteration
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value of r (t denotes the iteration number), respectively. Then the following steps based on the PMM are applied into (6) to determine x, r, and λ alternately [26]: x
(t+1)
(t)
(t)
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:= arg min L(x, r , λ ) s.t. |xn | = 1, ∀n; x
λ
:=λ
(t)
+r
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H
− F(|F x
(7b) (t+1) 2
(7c)
| ),
We refer this algorithm to as PMM-PSL. The above steps
2 are repeated until a stopping criterion, e.g., r(t) − r(t+1) 2 ≤ ε, is reached. The solutions to the subproblems (7a) and (7b) are derived as follows. (t)
1) Solution to (7a): with given r and λ , determine x(t+1) . Defining ¯r(t) = r(t) + λ(t) , and using FH F = I, we have
2
2
(8) min FH x − z(t) s.t. |xn |2 = 1, ∀n, x 2 where z(t) = Re FH ¯r(t) . It is clear that the objective function of the problem (8) is quartic and the constraint is nonconvex set. Hence it is hard to tackle, generally, NPhard [10]. Nevertheless, as mentioned in [10], we solve the following “almost equivalent” optimization problem instead: (t)
min x
M 2 X H (t) fm x − z˜m s.t. |xn |2 = 1, ∀n,
(t)
H
x
(t)
2
¯ (t) it is easy to verify that its solution is x(t+1) = ej∠(Fy ) .
2) Solution to (7b): With the given x(t+1) and λ(t) , op 2 (t) ¯ = F FH x(t+1) − λ(t) , and timize r(t+1) . Defining λ introducing a new variable η, we rewrite (7b) as the following equivalent form:
2 ζ
min η + r − c(t) r,η 2 2 s.t. |rn | ≤ η, ∀n ∈ Θm ; |rn | ≤ δ, ∀n ∈ Θs , (12) It is
¯ (t)
(t)
γr ρ+1 λ +r (t) = γr ρργ γr , and c r +1 (t+1) (t) seen that r0 = c0 , and ∀n ∈
(13)
. (t+1)
Θs the optimal rn
(14)
Thus, substituting (14) into (12) and defining hn , |rn |, we get the following real-valued optimization problem: 2 ζ X hn − |c(t) min η+ n | hn ,n∈Θm ,η 2 n∈Θm
s.t. 0 ≤ hn ≤ η, ∀n ∈ Θm .
(15)
As (15) is a convex optimization problem, and we solve it by using the Karush-Kuhn-Tucker (KKT) theory [36]. The Lagrangian of (15) is ζ X 2 ((hn − |c(t) l(η, {hn , κn , ϕn }n∈Θm ) =η + n |) 2 n∈Θm
+ κn (hn − η) − ϕn hn ).
(16)
Suppose that {η ? , {h?n , κ?n , ϕ?n }n∈Θm } is an optimal solution to (16). Then under the KKT condition, we have
m=1
where y¯m = z˜m ej∠(fm x ) and the equality holds at x = x(t) . Thus, based on (10), we further simplify the problem (9) as:
2
¯ (t) x(t+1) := arg min x − Fy s.t. |xn |2 = 1, ∀n, (11)
where ζ =
∠r = ∠c(t) .
(9)
√ ˜(t) = z(t) . However, as we see, the objective function where z of (9) is nonsmooth and nonconvex due to the modulus operation. Expanding the objective and using Cauchy function Schwartz inequality and Re yH x ≤ yH x , we get 2 H (t) (t) 2 H (t) H 2 x − 2 fm x z˜m + z˜m fm x − z˜m = fm # " 2 H (t) H 2 x xH fm fm (t) (t) zm Re + zm ≤ fm x − 2˜ H x(t) fm 2 H (t) = fm x + y¯m (10) , (t)
( (t) (t) δej∠cn , if δ ≤ |cn |, = ∀n ∈ Θs . (t) cn , otherwise.
While ∀n ∈ Θm , the optimal phase of r follows that of c(t) , i.e.,
r
(t+1)
rn(t+1)
(7a)
r(t+1) := arg min L(x(t+1) , r, λ(t) ) s.t. |rn | ≤ δ, ∀n ∈ Θs ;
is given by
? ? 0 =ζ(h?n − |c(t) n |) + κn − ϕn , ∀n ∈ Θm ; X 0 =1 − κ?n , ∀n ∈ Θm ;
(17)
0
(20)
0
n∈Θm =κ?n (h?n − η ? ), κ?n ≥ 0, ∀n ∈ =ϕ?n h?n , ϕ?n ≥ 0, ∀n ∈ Θm .
Θm ;
(18) (19)
It is easily revealed that, the dual variables ϕn and κn are corresponding to the mutually exclusive constraints, and either of them must be zero. Then we have the following three cases: •
? if ϕ?n = 0, and κ?n > 0, the third condition n = implies h (t) ? ? ? η and the first condition implies κn = ζ |cn | − η ≥ (t)
•
•
0, i.e., |cn | ≥ η ? ; if ϕ?n > 0, and κ?n = 0, the fourth condition implies (t) h?n = 0 and the first condition implies −ζ|cn | = ϕ?n (t) (this is impossible, since |cn | ≥ 0); if ϕ?n = 0, and κ?n = 0, the first condition implies h?n = (t) |cn |.
Based on the above cases, we have ( (t) η? , if η ? ≤ |cn |, ? hn = (t) |cn |, otherwise,
and
κ?n
=
( (t) (t) ζ |cn | − η ? , if η ? ≤ |cn |, 0,
otherwise.
(21)
(22)
Let [α1 , · · · , αK ] be the descending order sequence of (t) N −1 {0, |cn |}n=1 . Combining (22) and (18), we can determine ? η in a closed-form. Specifically, for each k, we define the (t) index set Ωk = {n||cn | ≥ αk , ∀n}. Then, combining (18)
5
and (22) yields X X ? κ?n = 1 − ζ |c(t) 0 =1 − n | − ηk , n∈Θm
into (25) to determine {x, y, r, λ, µ}: (23)
x(t+1) := arg min L(x, y(t) , r(t) , λ(t) , µ(t) ) x
n∈Ωk
the solution to which is given by P (t) |cn | − 1/ζ ? Pk ηk = n∈Ω . n∈Ωk 1
s.t. |xn |2 = 1, ∀n;
(t+1)
y
y
(24)
If ηk? ≤ αk , we get η ? = ηk? ; otherwise, let k = k + 1, and repeat the above calculations. When η ? is obtained, we de(k+1) (k+1) (k+1) j∠c(t) termine hn using (21), and rn = hn e n , ∀n ∈ Θm . Remark 2: Computing x(t+1) requires two FFTs and one vector multiplication and the complexity is O(2N log(N ) + N ); the update of r(t+1) has a complexity of O(2N log(N ) + 2N ). Thus, in each iteration, the computational complexity of the PMM-PSL algorithm is O(4N log(N ) + 3N ).
Theorem 1: [37] Let {x? , r? , λ? } be a sequence generated by the steps (7a)–(7c), i.e., all the subproblems are solved exactly, with ρ > 0 and γr > 0. Assume limt→∞ kλ(t+1) − λ(t) k2 = 0. Then any limit point {x? , r? , λ? } is a KKT point of (4). Proof From (7c) and limt→∞ kλ(t+1) − λ(t) k2 = 0, we get limt→∞ kr(t+1) − F(|FH x(t+1) |2 )k2 = 0. Since the feasible set of x is bounded and closed, it follows from (12) that the sequence {η (t) , r(t) } is also bounded. Thus there (t) (t) (t) exists a stationary point such that limt→∞?{x? , r? , λ } = ? ? ? ? H ? 2 {x , r , λ }, r = F |F x | , ∇x L (x , r , λ ) = 0, and ∇r L (x? , r? , λ? ) = 0. The proof is complete.
Remark 3: The equality in (10) holds at x = x(t) , which implies that the optimal solution to (11) is also the solution to (9), when x = x? . In addition, since (7a) and (9) are “almost equivalence”, ∇x L (x? , r? , λ? ) = ∇x f8 (x? ) = 0 implies ∇x L (x? , r? , λ? ) ≈ ∇x f9 (x? ) ≈ 0 [10], where f8 (x) and f9 (x) denote the objective functions of (7a) and (9), respectively. Thus, Theorem 1 shows the PMM-PSL sequence (i.e., the sequence obtained in (11), (7b), and (7c)) converges to an approximate KKT point of (4) under mild condition.
r
(t+1)
:= arg min L(x r
(t+1)
µ
:=λ
(t)
(t+1)
+r
(t)
:=µ
H
,y
+F x
(26b)
(t+1)
(t)
(t)
, r, λ , µ ) (26c)
H
− F(|F x
(t+1)
(t)
, y, r , λ , µ )
s.t. |rn | ≤ δ, ∀n ∈ Θs ;
(t+1)
(t)
(t)
s.t. |ym | ≤ um , m ∈ Ωs ;
(t+1)
λ
:= arg min L(x
(26a)
(t+1)
−y
(t+1)
(t+1)
.
| |y
(t+1)
|); (26d)
We name the above algorithm as PMM-SPSL. The solutions to the subproblems (26a)–(26c) are derived as follows. ¯ (t) = 1) Solution to (26a): Defining ¯r(t) = r(t) + λ(t) , y (t) (t) H y − µ , and using F F = I, we have
2 ρ
2 ρ1
(t) H 2 y(t) min
y F x − z(t) +
x − F¯ x 2 2 2 2 s.t. |xn |2 = 1, ∀n, (27) where z(t) = Re FH ¯r(t) . Similar to the majorization step derived in (10), we get M
2 2
X
(t) H
(t) H (t) |ym |fm x + y¯m , (28)
y F x − z(t) ≤ 2
m=1
H (t) (t) zm ej∠(fm x ) .
(t) y¯m
= Thus, (27) is majorized by
2
2 ρ ρ1 (t)
2 H ¯ (t) ¯ (t) min
y FH x − y
F x − y
+ x 2 2 2 2 s.t. |xn |2 = 1, ∀n. (29)
where
Expanding the objective of (29) and rearranging the terms yields:
¯ (t) + ρ22 y ρ21 y(t) y ¯ (t)
(t) H min z F x −
x
z(t) s.t. |xn |2 = 1, ∀n. q 2 where z(t) = ρ21 y(t) + (30) is equivalent to
(30)
ρ2 2
and
a b
, [a1 /b1 , · · · , aN /bN ].
˜(t) k s.t. |xn |2 = 1, ∀n; minkFH x − z x
B. PMM-SPSL Algorithm
First, we construct the following proximal augmented Lagrangian for (5): 1 kr − r(t) k22 L (x, y, r, λ, µ) = max{|rn |}n∈Θm + n 2γr
ρ2
FH x − y + µ 2 − kµk22 + 1 ky − y(t) k22 + 2 2 2γy
2 H ρ1
+ r − F F x |y| + λ 2 − kλk22 . (25) 2
Then the following steps based on the PMM are applied
(26e)
˜(t) = where z obtain:
ρ1 2
(31)
|y(t) | y¯ (t) + ρ22 y¯ (t) . Since FFH = I, we then ρ1 (t) 2 + ρ2 | 2 2 |y
x(t+1) := arg min kx − F˜ z(t) k s.t. |xn |2 = 1, ∀n, x
(32)
(t) and its solution is x(t+1) = ej∠(F˜z ) . ¯ (t) = FH x(t+1) +µ(t) , and 2) Solution to (26b): Defining x ignoring irrelevant items in (26b), we have 2 ρ1
(t)
min
¯r − F FH x(t+1) |y| y 2 2
2 ρ2 1
(t)
(t) 2 ¯ − y + + ky − y k2
x 2 2γy 2 s.t. |ym | ≤ um , m ∈ Ωs . (33)
Using the same strategy derived in (27)–(31), we get
2
y(t+1) =: min y − s(t) s.t. |ym | ≤ um , m ∈ Ωs , (34) =
2
ρ1 |FH x(t+1) | +ρ2 + γ1y
j∠(y(t) )
, and q
(t)
e
¯ (t)
X: 4.025 Y: 9.013
X: 3.419 Y: 8.751
PMM-PSL MM-PSL
0 0
2
4
6
8
10
12
(a) Time (s)
(t)
λ +r where ζ = γr ργ1r+1 , and c(t) = γr ρργ . As (36) has the r +1 similar form as (12), thus it can be solved in a similar way. Remark 4: We analyze the computational complexity of the PMM-SPSL algorithm. Updating x(t+1) needs two FFTs and one vector multiplication and the complexity is O(2N log(N ) + N ); Updating y(t+1) and r(t+1) have a complexity of O(2N log(N ) + 2N ). Thus, in each iteration, the computational complexity of the PMM-SPSL algorithm is O(6N log(N )+5N ). Hence, the computational complexity of the proposed algorithm is linear with respect to the iteration number k. In Table I, we summarize the computational complexities of the proposed methods as well as MM-PSL [11], WeCAN [2], and SCAN [34], where T denotes the total iteration number.
TABLE I: Computational complexities of different methods. Method
Computational Complexity
PMM-PSL PMM-SPSL MM-PSL [11] WeCAN [2] SCAN [34]
O(T (4N log(N ) + 3N )) O(T (6N log(N ) + 5N )) O(T (4N log(N ) + 6N )) O(T ((N 2 + 1) log(N ) + 2N )) O(T (4N log(N ) + 2N ))
Theorem 2: Let {x? , y? , r? , λ? , µ? } be a sequence generated by the steps (26a)–(26e) with ρ1 > 0, ρ2 > 0, γy > 0 and γr > 0. Assume limt→∞ kλ(t+1) − λ(t) k2 = 0 and limt→∞ kµ(t+1) − µ(t) k2 = 0. Then any limit point {x? , y? , r? , λ? , µ? } is a KKT point of (5). Proof The proof can be obtained by using the same procedure discussed in Theorem 1. Remark 5: According to Theorem 2 and using the same procedure discussed in Remark 3, we conclude that the PMMSPSL sequence (32), (34), and (26c)–(26e) converges to a KKT point of (5) under some mild conditions. III. N UMERICAL E XPERIMENTS Numerical examples are conducted to evaluate the proposed methods. For notational simplicity, the normalized frequencies (0 to 1) are used. Unless stated otherwise, the dual variables and auxiliary variables are initialized randomly.
X: -106 Y: -28.85
-20
14
16
18
Random Sequence PMM-PSL MM-PSL WeCAN
0
. The solution to (34) is given by: ( (t) (t) um ej∠sm , if um ≤ |sm |, (t+1) ym = (35) (t) sm , otherwise. ¯ (t) = F( FH x(t+1) 3) Solution to (26c): Defining λ (t+1) y )−λ(t) , and introducing a new variable η, the problem (26c) can be equivalently written as:
2 ζ
min η + r − c(t) r,η 2 2 s.t. |rn | ≤ η, ∀n ∈ Θm ; |rn | ≤ δ, ∀n ∈ Θs , (36) z
(t)
50
= rn (dB)
where s
(t)
2
¯ (t) + γ1 y(t) ρ1 |FH x(t+1) | q(t) +ρ2 x y
-40 -100
rn (dB)
y
max {r n } (dB)
6
X: -95 0 Y: -24.69
-50
0
50
(b) n
100 PMM-PSL WeCAN
X: -41 Y: -29.55 X: 17 Y: -60
-50 -100
X: -35 -50 Y: -52.88
0
50
100
(c) n
Fig. 1: (a) Cost function value versus time; (b) Waveform autocorrelation for Θs = ∅; (c) Waveform autocorrelation for Θs 6= ∅. A. PMM-PSL Algorithm We verify the performance of the PMM-PSL algorithm by comparing it with the MM-PSL (accelerated by using the SQUAREM) [11] and WeCAN [2] algorithms. Firstly, we assign N = 128, ρ = 1, γr = 2, Θm = {1, 2, · · · , N − 1}, Θs = ∅, and the maximum iteration number Tm = 20000 is used as the stopping criterion for PMM-PSL. The cost function and the normalized autocorrelation levels, i.e., 20 log10 (|rn |/N ), of the designed aperiodic waveforms are plotted in Figs. 1(a) and (b). We see that both PMM-PSL and MM-PSL converge to almost the same cost function value (the PSL of PMM-PSL is 8.754, whereas MMPSL is 9.01), whereas WeCAN has larger PSL. In the second example, we evaluate the PMM-PSL algorithm in controlling the specific autocorrelation level (i.e., local PSL, Θs 6= ∅). We let Θm = {41, · · · , N − 1}, Θs = {1, · · · , 40}, δ = N 10−3 (i.e., −60 dB), and other parameters remain unchanged. The MM-PSL is excluded from the comparison since it cannot control the waveform autocorrelation magnitude at certain lags. The autocorrelation levels of the designed aperiodic waveforms are plotted in Fig. 1(c). We see that the autocorrelation levels of the PMM-PSL in the region Θs are all lower than −60 dB, and they meet the requirements, i.e., |rn | ≤ δ, ∀n ∈ Θs . In addition, the autocorrelation levels of the PMM-PSL in the region Θm almost have the uniformly local PSL, while those of the WeCAN within Θm and Θs are much larger. B. PMM-SPSL To examine the convergence performance of the PMMSPSL, we assign N = 512, Θm = {1, 2, · · · , N − 1}, 420 Θs = ∅, Ωs = { 400 2N , · · · , 2N }, um = −40 dB (∀m ∈ Ωs ), Θm = {1, 2, · · · , 99, 201, · · · , N − 1}, Θs = {100, · · · , 200}, δ = −45 dB), ρ1 = 0.01, ρ2 = 40000, γx = 20, γy = 20. The maximum iteration number Tm = 20000 is set as the stopping criterion. The auxiliary variables and the scaled dual
7
80
0
30
70 20
Spectrum (dB)
-20
50
rn (dB)
Objective value
-10
60
40
-30
-40
30
0 -10 -20 -30
-50
20
10
-40
10 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-60 -500
2
-400
-300
-200
-100
100
200
300
400
500
0
0.1
0.2
n
104
Number of iterations (a)
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized frequency (c)
(b)
Fig. 2: (a) Objective function versus number of iterations; (b) Autocorrelations of the designed waveforms; (c) Spectra of the designed waveforms. the autocorrelation of the initial sequence and the obtained sequence are shown in Fig. 4. From Figs. 3 and 4, it is seen that the proposed method can synthesize the probing waveform with a minimum local PSL and also achieve the specified spectral and autocorrelation requirements. Whereas SCAN has larger stopband level (−16.52 dB) and PSL (−22.32 dB). Spectrum (dB)
variables (i.e., y(0) , r(0) , λ(0) , µ(0) ) are initialized randomly. We run PMM-SPSL 1000 times and plot the evolution curves of the objective value (i.e., PSL=maxn {|rn |}∀n ) with respect to the number of iterations, autocorrelations and spectra of the synthesized waveforms in Fig.2(a)–(c), respectively. It can be seen that the objective value tends to be stable after 8000 iterations, and the autocorrelations and spectra of the designed waveforms align with the specified values, which imply that the proposed method has satisfactory convergence performance. Then, we compare the proposed method with the SCAN [34]. We use N = 512, Θm = {1, 2, · · · , N − 1}, Θs = ∅, and set the Golomb sequence [10] as the initial sequence, and keep the other algorithm parameters unchanged. The spectrum and the autocorrelation of the sequence obtained by using the PMM-SPSL and SCAN are shown in Fig. 3.
20 0
Random Sequence PMM-SPSL
X: 0.3984 Y: -40.65
-20 -40 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized frequency
SCAN PMM-SPSL Golomb
20 X: 0.4063 Y: -16.52
0
rn (dB)
Spectrum (dB)
0
X: 0.4014 Y: -40.16
-20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
rn (dB)
-20
X: 15 Y: -22.32
-40 -60 -500
SCAN PMM-SPSL Golomb
X: 17 Y: -34.72 -400
-300
-200
-100
0
100
-400
X: -153 -300 -200 Y: -50.51
-100
0
100
200
300
400
500
n
Fig. 4: Periodic sequence design with spectral and specific correlation notch constraints
Normalized frequency 0 X: -497 Y: -33.61
-40 -60 -500
-40
Random Sequence PMM-SPSL
X: -91 Y: -32.59
-20
200
300
400
500
n
Fig. 3: Periodic sequence design with spectral constraint. Finally, we test the ability of the PMM-SPSL algorithm to control the specific autocorrelation level (i.e., Θm = {1, 2, · · · , 99, 201, · · · , N − 1}, Θs = {100, · · · , 200}, δ = −50 dB), the random sequence is used as initial sequence, and other parameters remain unchanged. The spectrum and
IV. C ONCLUSION We have devised two methods for designing waveform with low local PSL and spectral compatibility, which are computationally simple and have fast convergence. The key ideas of our algorithms are to utilize the waveform autocorrelation which can be expressed by the FFT and IFFT, use PMM to tackle the equality constrained nonconvex nonsmooth optimization problems and split the complicated problem into small problems and tackle them individually. ACKNOWLEDGEMENT This work was supported in part by the China Scholarship Council, in part by the Huawei-Northwestern Polytechnical
8
University Academic Scholarship of China, in part by the Natural Science Foundation of China under Grant 61471295, in part by the Aeronautical Science Foundation of China under Grants 20172053017 and 20172053018, and in part by the Central University Funds under Grants G2016KY0308, G2016KY0002, and 17GH030144. R EFERENCES [1] A. Aubry, A. De Maio, M. Piezzo, and A. Farina, “Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 2, pp. 1138–1152, Apr. 2014. [2] P. Stoica, H. He, and J. Li, “New algorithms for designing unimodular sequences with good correlation properties,” IEEE Trans. Signal Process., vol. 57, no. 4, pp. 1415–1425, Apr. 2009. [3] B. Tang and J. Liang, “Efficient algorithms for synthesizing probing waveforms with desired spectral shapes,” IEEE Trans. Aerosp. Electron. Syst., pp. 1–1, 2018. [4] X. Yu, G. Cui, L. Kong, J. Li, and G. Gui, “Constrained waveform design for colocated MIMO radar with uncertain steering matrices,” IEEE Trans. Aerosp. Electron. Syst., vol. 55, no. 1, pp. 356–370, Feb. 2019. [5] G. Cui, Y. Fu, X. Yu, and J. Li, “Local ambiguity function shaping via unimodular sequence design,” IEEE Signal Process. Lett., vol. 24, no. 7, pp. 977–981, Jul. 2017. [6] G. W. Stimson, Introduction to Airborne Radar, Mendham, NJ, USA: Soc. Photo Optical, 1998. [7] G. Yu, J. Liang, J. Li, and B. Tang, “Sequence set design with accurately controlled correlation properties,” IEEE Trans. Aerosp. Electron. Syst., pp. 1–1, 2018. [8] X. Yu, G. Cui, J. Yang, and L. Kong, “Wideband MIMO radar beampattern shaping with space-frequency nulling,” Signal Process., vol. 160, pp. 80 – 87, 2019. [9] B. Tang, J. Li, and J. Liang, “Alternating direction method of multipliers for radar waveform design in spectrally crowded environments,” Signal Process., vol. 142, pp. 398 – 402, 2018. [10] H. He, J. Li, and P. Stoica, Waveform Design for Active Sensing Systems: A Computational Approach. Cambridge University Press, 2012. [11] J. Song, P. Babu, and D. P. Palomar, “Sequence design to minimize the weighted integrated and peak sidelobe levels,” IEEE Trans. Signal Process., vol. 64, no. 8, pp. 2051–2064, Apr. 2016. [12] F. cong Li, Y. nan Zhao, and X. lin Qiao, “A waveform design method for suppressing range sidelobes in desired intervals,” Signal Process., vol. 96, pp. 203 – 211, 2014. [13] Z. Cheng, B. Liao, Z. He, J. Li, and C. Han, “A nonlinear-ADMM method for designing MIMO radar constant modulus waveform with low correlation sidelobes,” Signal Process., vol. 159, pp. 93 – 103, 2019. [14] G. Cui, X. Yu, M. Piezzo, and L. Kong, “Constant modulus sequence set design with good correlation properties,” Signal Process., vol. 139, pp. 75 – 85, 2017. [15] G. Cui, X. Yu, G. Foglia, Y. Huang, and J. Li, “Quadratic optimization with similarity constraint for unimodular sequence synthesis,” IEEE Trans. Signal Process., vol. 65, no. 18, pp. 4756–4769, Sep. 2017. [16] W. Fan, J. Liang, and J. Li, “Constant modulus MIMO radar waveform design with minimum peak sidelobe transmit beampattern,” IEEE Trans. Signal Process., vol. 66, no. 16, pp. 4207–4222, Aug. 2018. [17] W. Fan, J. Liang, G. Yu, H. C. So, and G. Lu, “MIMO radar waveform design for quasi-equiripple transmit beampattern synthesis via weighted lp -minimization,” IEEE Trans. Signal Process., vol. 67, no. 13, pp. 3397–3411, Jul. 2019. [18] X. Yu, G. Cui, J. Yang, L. Kong, and J. Li, “Wideband MIMO radar waveform design,” IEEE Trans. Signal Process., vol. 67, no. 13, pp. 3487–3501, Jul. 2019. [19] M. A. Kerahroodi, A. Aubry, A. De Maio, M. M. Naghsh, and M. Modarres-Hashemi, “A coordinate-descent framework to design low PSL/ISL sequences,” IEEE Trans. Signal Process., vol. PP, no. 99, pp. 1–1, 2017. [20] J. Liang, H. C. So, J. Li, and A. Farina, “Unimodular sequence design based on alternating direction method of multipliers,” IEEE Trans. Signal Process., vol. 64, no. 20, pp. 5367–5381, Oct. 2016. [21] W. Rowe, P. Stoica, and J. Li, “Spectrally constrained waveform design,” IEEE Signal Process. Mag., vol. 31, no. 3, pp. 157–162, May 2014.
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CREDIT AUTHOR STATEMENT Wen Fan: performed writing-original draft, conceptualization, methodology, software, data curation, and funding acquisition. Junli Liang: methodology, funding acquisition, writingreview, project administration, and supervision. Guoyang Yu: investigation and writing, and validation. Hing Cheung So: performed writing-review, and editing. Guangshan Lu: supervision.
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