Minimum ℓp- norm method for estimating compactness of FM signal in the FRFT domain

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Original research article

Minimum ℓp− norm method for estimating compactness of FM signal in the FRFT domain Yong Guoa,*, Shuo Wangb, Li-Dong Yangb a b

School of Science, Inner Mongolia University of Science and Technology, Baotou 014010, China School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Fractional Fourier transform Compact fractional Fourier domain ℓp− norm Frequency modulated signal

Frequency modulation (FM) signal is widely used in the sonar, radar, laser and newly developed optics cross-research fields, its compactness (or sparsity) plays an important role in the FM signal sampling, reconstruction, compression and denoising. This paper is devoted to studying the compactness of FM signal, a minimum ℓp− norm method (MPNM) is proposed to estimate the compactness of FM signal in the fractional Fourier transform (FRFT) domain. Compared with the existing methods, the compact FRFT domain of FM signal obtained by this method has better sparsity. Furthermore, the proposed method is successfully applied to the denoising of linear FM signal.

1. Introduction As a typical non-stationary signal, frequency modulation (FM) signal is widely used in the traditional areas of sonar, radar and laser [1–3]. In recent years, it has also been applied to some new cross-field with optics, such as microwave photonics crossed by microwave and optics [4–6], new optical measurement technology crossed by optical measurement and signal processing [7–10]. The compactness (or sparsity) of FM signal has always been among the core issues in the FM signal processing, which plays an important role in the sampling, reconstruction, compression and denoising for FM signal. Fractional Fourier transform (FRFT) is a novel tool for analyzing and processing signal in the time-frequency plane, which widely used in the detection, parameter estimation and separation for FM signal [11–15]. FRFT is very suitable to process FM signal owing to the follwoing reasons: 1) FRFT decomposes the signal with a basis formed by the orthonormal linear FM (LFM) functions, this makes it become one of the best tool for FM signal processing; 2) FRFT is a linear transform, and thus is not interfered by cross-terms in the processing of multi-component FM signal. For above advantages, FRFT is further applied to research the compactness of FM signal, many scholars have devoted to exploring the compact FRFT domain for FM signal [17–21]. In [17], the compact FRFT domain is found by employing windowed fractional Fourier transform and minimum second-order FRFT moment. Vijaya et al. found the compact FRFT domain with the criteria of minimizing the percent root mean square difference (PRD) [18]. The value of compact FRFT domain is optimized by repeated computation of DFRFT, IDFRFT and PRD for different FRFT angle α , which is computationally inefficient. Serbes et al. proposed two criteria for finding compact FRFT domains, i.e., maximum time-sharing bandwidth ratio and minimum basic bandwidth [19]. However, the methods in [19] are computationally demanding and may fail to find the compact FRFT domain under certain conditions. A maximum amplitude based coarse-to-fine algorithm (MACF) was adopted to find the compact FRFT domain with maximum amplitude of the correlation signal [1]. It has been shown that MACF is only suitable for

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Corresponding author. E-mail address: [email protected] (Y. Guo).

https://doi.org/10.1016/j.ijleo.2019.163782 Received 27 August 2019; Accepted 13 November 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Please cite this article as: Yong Guo, Shuo Wang and Li-Dong Yang, Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163782

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estimating the chirp types of signals with a single chirp rate [11]. Subsequently, the minimum norm method (MNM) is proposed by Serbes et al., which used ℓ1− norm as a measure of compactness rather than maximum amplitude [21]. Nevertheless, both methods in [20,21] adopted a coarse-to-fine grid search strategy, which are easily affected by the choose of search step [21]. In this paper, the finding of compact FRFT domain is modelled as an optimization problem with the criteria of minimizing ℓp− norm, named as minimum ℓp -norm method (MPNM). The main contributions of this paper include: 1) ℓp− norm is proposed as a measure of compactness. It has been shown that the method using ℓp− norm (0 < p < 1) is superior to that of ℓ1− norm for signal reconstruction and algorithm reliability [22–25]. Moreover, ℓp− norm (0 < p < 1) makes a closer approximation to the “counting norm” ℓ 0− norm compared with ℓ1− norm [25]; 2) whale optimization algorithm (WOA) is used to solve this optimization model, which can effectively improve the shortcoming of the coarse-to-fine grid search strategy. It has been proved that WOA possesses the advantages of flexibility, gradient-free mechanism, and high local optima avoidance, which is very competitive compared to the state-of-art meta-heuristic algorithms as well as conventional methods [26,27]; 3) LFM signal denoising is presented to show the potential application value of the proposed method. For above contributions, the proposed method can obtain better sparsity for FM signal in the FRFT domain. The remainder of the paper is organized as follows: FRFT is introduced in Section 2. Section 3 describes the proposed minimum ℓp− norm method. Section 4 presents the testing results and simulation analyses. An application of the proposed method in the LFM signal denoising is presented in Section 5. Section 6 gives the conclusions. 2. Fractional Fourier transform The FRFT of signal x (t ) with parameter α is defined as [16]

F α (u) = (F αx )(u) =

+∞

∫−∞

x (t ) K α (u, t ) dt

(1)

with the kernel 2

K α (u, t ) =

2

⎧ Aα e j /2(t cot α − 2tucscα + u cot α ), α ≠ kπ δ (t − u), α = 2kπ ⎨ δ (t + u), α = (2k − 1) π ⎩

(2)

where Aα = (1 − j cot α )/2π , α is rotation angle. Some useful properties of FRFT are presented as follows: 1) Linearity: F α [c1 f (t ) + c2 g (t )] = c1 F αf (t ) + c2 F αg (t ) ; 2) Index additivity: F α1+ α2 (u) = F α1 (u)⋅F α2 (u) = F α2 (u)⋅F α1 (u) ; 3) Parseval’s theorem: 〈f (t ), g (t )〉 = 〈F α (u), Gα (u)〉. As a generalization form of Fourier transform (FT), FRFT can be interpreted as the signal rotating counterclockwise at any angle. If FT of the signal indicates that the signal rotates counter-clockwise with π/2 from time axis to frequency axis, then FRFT indicates that the signal rotates counter-clockwise with α from time axis to u axis. For this reason, it provides a continuous mapping of rotation angle α between the time domain and FRFT domain [16,28]. Consequently, how to find the optimal rotation angle α that result in best compact FRFT domain is a key point. 3. Proposed minimum ℓp− norm method The traditional signal representation method in the compact FRFT domain is the minimum ℓ 0− norm. However, because ℓ 0− norm represents the number of non-zero elements in the vector, it will seriously affect the value of ℓ 0 -norm when the signal is contaminated by noise. Since ℓp− norm (0 < p < 1) always yields the same results with ℓ 0− norm in the noiseless case, and has more stronger robustness to noise. Moreover, ℓp− norm (0 < p < 1) makes a closer approximation to the “counting norm” ℓ 0− norm compared with ℓ1− norm. As a result, ℓp− norm is employed to measure the compactness of FM signal in the FRFT domain, i.e.

J (α, p) = || F α (u) ||p

(3)

where J (α, p) denotes the cost function to minimize, || F α (u) ||p denotes the ℓp− norm of the FRFT of signal, ℓp− norm of a vector x = [x (0), …, x (n − 1)] is defined by 1/ p

n

⎞ ⎛ || x ||p = ⎜∑ |x i |p ⎟ ⎠ ⎝ i=1

(4)

The compactness of a signal in FRFT domain becomes better with J (α, p) decreases, so how to find the compact FRFT domain of a signal can be modelled as an optimization problem, i.e.,

αopt = arg min || F α (u) ||p

(5)

α

2

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where the search interval is α ∈ [0, π ]. Since WOA possesses the advantages of flexibility, gradient-free mechanism, and high local optima avoidance [26,27], which motivate our attempts to employ WOA for searching the optimal FRFT angle. In [20,21], both methods adopted a coarse-to-fine grid search strategy, it may fail to find the global minimum within a probability and estimation performance is easily affected by the selection of equidistant step Δαi [21]. For WOA, it utilizes a population of search agents to determine the global optimum for optimization problems, the search process starts with creating a set of random solutions for a given problem. Therefore, WOA is used to solve this optimization problem, and thus make the proposed method has advantages of high local optima avoidance and fast convergence speed. Moreover, compactness is also measured by the number of significant amplitudes (NSA) in addition to the value of ℓp -norm. NSA is a minimum value under the condition of containing 95 % signal energy, thus less NSA means better compactness. For a N point signal f (t ) , its FRFT amplitude spectrum |F αopt (u)| is sorted in descending order firstly, then NSA is calculated by the following formula: M

min {M } s. t .

∑ |(F αopt (u))i |2

= 0.95E (6)

i=1

where M < N , i ∈ {1, 2, …, N }, (F αopt (u))i denotes the i-th largest value of |F αopt (u)|. E represents the energy of signal in the FRFT domain, defined by

E=

∑u

|F a (u)|2

(7)

Owing to the Parseval’s theorem of FRFT, there is no energy loss in the FRFT process, which means the value of E is independent of FRFT angle. The specific algorithm flow of proposed method is given by Step 1: Define the original continuous signal f (t ) , and give corresponding discrete signal with uniform sampling method, denoted as fn . Step 2: Compute the discrete FRFT of fn , and construct the fitness function J (α, p) = || F α (m) ||p . Step 3: Set initial parameters of WOA, obtain the optimal FRFT angle using WOA, denoted as αopt = arg min J (α, p) . α

Step 4: Compute NSA with the optimal FRFT angle αopt .

4. Simulation results All experiments are conducted on a personal computer using MATLAB. In order to show the advantages of proposed method more clearly, it is compared with MACF [20] and MNM [21]. The basis of the comparison is that both methods can operate multicomponent FM signal in the similar way as MPNM. According to the procedures of MACF and MNM, the search parameters of coarseto-fine grid search algorithm are set as [αl, α e] = [π /4, 3π /4], Δαi = π/20 , λ = 0.1, and ε = π/2 × 10-5 . The relevant parameters of WOA algorithm are set as

{N , Nmax , dim , lb, ub} = {100, 5, 1, 0, 2} where N , Nmax , dim , lb, ub represent the search agent number, maximum number of iterations, number of variables, lower and upper bound of variables respectively. The discrete FRFT [28]. proposed by Ozaktas et al. is used to simulate MPNM. In order to find a suitable p value for MPNM, the simulation is conducted with five different p values. The results are further compared with MNM, MACF by ℓp− norm and NSA (see Table 1). From Table 1, it can be found that MPNM has a more stable advantage when p = 0.5. Therefore, the value of p is set to 0.5 in this paper. Table 1 The comparison of αopt and corresponding ℓp− norm, NSA with different p values. p MNM

0.1

0.3

0.5

0.7

0.9

1.8614 1.78× 1026

1.8614 1.96× 108

1.8614 6.40× 104

1.8614 2.54× 103

1.8614 5.00× 102

33 1.8619 1.78× 1026

33 1.8619 1.96× 108

33 1.8619 6.40× 104

33 1.8619 2.54× 103

33 1.8619 4.99× 102

|| ⋅ ||p

33 1.82964 1.65× 1026

33 1.82966 1.81× 108

33 1.82969 6.04× 104

33 1.82978 2.51× 103

33 1.86295 4.98× 102

NSA

21

21

21

21

33

αopt || ⋅ ||p

WACF

NSA αopt

|| ⋅ ||p MPNM

NSA αopt

3

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Fig. 1. J (α ) of a bi-component LFM signal and its optimum FRFT angle.

4.1. LFM signal Firstly, the performance of MPNM is tested by a bi-component LFM signal as follows:

s (t ) = A0 exp [jπ (k 0 t 2 + 2f0 t )] + A1 exp [jπ (k1 t 2 + 2f1 t )]

(8)

where A0 and A1 denote amplitude, k 0 and k1 denote chirp rate, f0 and f1 denote initial frequency. LFM signal is samples with parameters t = n⋅Ts , N = 512 , Ts = 1 N and n = −⌊ (N − 1)/2⌋, …, ⌈ (N − 1)/2⌉. The relevant parameters of LFM signal are assigned by {A0 , k 0, f0 } = {0.2, 0, 0.8} , {A1 , k1, f1 } = {0.3, 0.7, 0} . Firstly, we choose ten values evenly from 0 to π on α , and further J (α, p) of the bi-component LFM are calculated with p = 0.5, the results are shown in the Fig. 1. It can be seen from Fig. 1 that ℓp -norm of the signal has a global minimum value, and the angle corresponding to the minimum point is the optimal FRFT angle αopt . For this bi-component LFM, Fig. 1 also shows that the model we built is correct.

Fig. 2. Bi-component LFM signal representation in the time domain and compact FRFT domain. 4

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Fig. 3. Bi-component LFM signal representation in the compact FRFT domain for three methods.

Secondly, same bi-component LFM signal is used to estimate the compact FRFT domain using MPNM. In order to visualize the compactness of the signal in FRFT domain, the signal representation in the time domain and compact FRFT domain are plotted in the Fig. 2. From Fig. 2, the signal compressed by the MPNM is more compact than the original signal, it proves that the MPNM is effective for estimating compact FRFT domain. Furthermore, the proposed method is compared with MNM and MACF. The results obtained by MNM, MACF and MPNM are shown in the Fig. 3, it can be seen obviously that MPNM give the best compactness for the bi-component LFM signal among three methods. To further compare the performance of MACF, MNM and MPNM, five sets of bi-component LFM signals are selected to simulate. The optimal FRFT angles, corresponding ℓp− norm and NSA obtained by three methods are shown in the Table 2 respectively. From Table 2, the optimal FRFT angles αopt obtained by MPNM are different with MACF and MNM, the corresponding ℓp− norm and NSA are much smaller than that of other two methods. It can be concluded that MPNM can provide more compact FRFT domain than MACF and MNM for bi-component LFM signal. 4.2. QFM signal Moreover, quadratic frequency modulation (QFM) signal is used to evaluate the performance of MPNM, which is generated by

s (t ) = A exp[jπ (σt 3 + kt 2 + 2f0 t )]

(10)

where σ represents the quadratic sweep rate, k denotes the chirp rate, f0 denotes the initial frequency, and A denotes the amplitude. The parameters of the QFM signal are set as follows:

{A, σ , k , f0 } = {1.2, 0.04, 0.4, 0.054} The parameters of QFM signal sampling are consistent with the bi-component LFM signal. Fig. 4 shows the magnitudes spectrum of QFM signal in the compact FRFT domain for three methods. It can be seen from Fig. 4 that MPNM can provides best compact FRFT domain for QFM signal. Furthermore, the optimal FRFT angle, corresponding ℓp -norm and NSA obtained by three methods are listed in the Table 3 respectively. It can be apparently seen that the optimum angle obtained by MPNM is 1.9382, while the optimum angles obtained by MACF and MNM methods are 2.0405 and 2.3562. In addition, the corresponding ℓp− norm and NSA obtained by MPNM are the smallest among three methods, which proves that the compact FRFT domain of QFM signal obtained by MPNM is superior to MACF and MNM. Based on the above simulation results, it can be concluded that the proposed MPNM can successfully give the more compact FRFT domain for both LFM signal and QFM signal. 5. Application LFM signal is inevitably mixed with noise when it is transmitted in the channel, and thus how to effectively eliminate noise is a Table 2 The comparison of the αopt , ℓp− norm and NSA for three methods. {A0 , k 0, f0 } {A1 , k1, f1 }

{0.2,0,0.8} {0.3,0.7,0} {4,0,0.7} {6,0.5,-0.8} {1,0.2,-0.3} {0.6,-0.5,0.2} {5,0.1,-0.3} {8,0.87,0.3} {5.5,0.15,-0.8} {0.5,0.75,-1.2}

ℓp−norm (× 105 )

αopt

NSA

MPNM

MNM

MACF

MPNM

MNM

MACF

MPNM

MNM

MACF

1.9297 1.8560 1.4414 2.0521 1.9923

2.1816 2.0342 1.7682 2.2776 2.2145

1.5708 1.5708 1.1067 2.2872 1.7194

0.311 4.55 1.08 7.85 5.90

0.343 5.25 1.25 8.91 7.72

0.616 8.85 1.81 8.48 12.1

131 107 142 136 101

223 183 260 217 190

324 233 289 239 265

5

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Fig. 4. The magnitudes spectrum of QFM signal in the compact FRFT domain for three methods. Table 3 The comparison of αopt , NSA and ℓp -norms for three methods. Method

αopt

ℓp−norm

NSA

MPNM MNM MACF

1.9382 2.0405 2.3562

7.49 × 104 8.66 × 104 1.22 × 105

97 122 218

key issue in the LFM signal processing. MPNM can concentrates the energy of LFM signal to a minimum interval, which motives us to apply MPNM to LFM signal denoising. In this paper, the application of MPNM on the LFM signal denoising is simulated by single component LFM signal and bi-component LFM signal respectively. 5.1. Single component LFM signal A single component LFM signal is selected with {A0 , k 0, f0 } = {0.2, 0.3, 0.8} (see Fig. 5 (a)), and Gaussian white noise is added to this signal with SNR = 2 dB (see Fig. 5 (b)). Firstly, the noisy LFM signal is compressed into compact FRFT domain using MPNM, most energy of the signal is concentrated in a narrowband, but the noise is still uniformly distributed throughout the bandwidth (see Fig. 5 (c)). Then, the spectrum of signal in the compact FRFT domain is filtered by a bandpass filter. Finally, the denoised signal is obtained by inverse FRFT (see Fig. 5 (d)). A comparison of the Fig. 5(a) (b) and (d) indicates that MPNM can be applied to denoising for single component LFM signal. 5.2. Bi-component LFM signal Gaussian white noise is added to a bi-component LFM signal (see third set of signal in the Table 2) with SNR = 2 dB. The denoising process of the bi-component LFM signal is same as the single component LFM signal, the simulation results are shown in the Fig. 6. It can be concluded from Fig. 6 that MPNM can be applied to denoising for bi-component LFM signal, it further verified that the compact FRFT domain of LFM signal is helpful for signal denoising. 6. Conclusion In this paper, ℓp− norm is raised as an index of signal compactness, and further a minimum ℓp− norm method (MPNM) is proposed to estimate the compact FRFT domain for FM signal. By comparing MPNM with MACF and MNM, the simulation results show that MPNM can give the compact FRFT domain with better sparsity. In addition, MPNM is applied to LFM signal denoising, it indicates that the compact FRFT domain of LFM signal is beneficial for signal denoising. 6

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Fig. 5. (a) original single component LFM signal, (b) noisy signal, (c) amplitude spectrum of the noisy signal in the compact FRFT domain, (d) denoised single component LFM signal.

Fig. 6. (a) original bi-component LFM signal, (b) noisy signal, (c) amplitude spectrum of the noisy signal in the compact FRFT domain, (d) denoised bi-component LFM signal. 7

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