Method for parameter estimation of LFM signal and its application

IET Signal ProcessingVolume 13, Issue 5 p. 538-543 Research ArticleFree Access Method for parameter estimation of LFM signal and its application Yong Guo, Corresponding Author Yong Guo guo_yong@imust.edu.cn School of Science, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia, 014010 People's Republic of ChinaSearch for more papers by this authorLidong Yang, Lidong Yang School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia, 014010 People's Republic of ChinaSearch for more papers by this author Yong Guo, Corresponding Author Yong Guo guo_yong@imust.edu.cn School of Science, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia, 014010 People's Republic of ChinaSearch for more papers by this authorLidong Yang, Lidong Yang School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou, Inner Mongolia, 014010 People's Republic of ChinaSearch for more papers by this author First published: 01 July 2019 https://doi.org/10.1049/iet-spr.2018.5435Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract A linear frequency modulation (LFM) signal is a typical non-stationary signal, and its parameter estimation has always been among the core issues in the field of signal processing. In this study, a novel method for parameter estimation of LFM signal is proposed using the simplified linear canonical transform and evaporation-rate-based water cycle algorithm. According to the results obtained from the theoretical analysis and experimental simulation, the proposed method possesses the advantages of easy implementation, high precision and independence on initial values compared with the concise fractional Fourier transform (CFRFT)-based and fractional Fourier transform-based methods. Moreover, the proposed parameter estimation method is successfully applied to optical measurement with improving measurement accuracy. 1 Introduction As a typical non-stationary signal, linear frequency modulation (LFM) signal is widely used in information systems, such as radar, sonar, and communications. In these systems, how to estimate the parameters of the LFM signal rapidly and exactly is a crucial issue. A large number of methods have been proposed for this purpose, such as maximum likelihood estimation (MLE) [1], chirp-Fourier transform (CFT) [2], discrete polynomial phase transform [3], chirplet transform (CT) [4, 5], fractional Fourier transform (FRFT) [6-10] and various time-frequency distributions [11-14]. These methods have advantages and disadvantages. For example, MLE does not require a cost function or prior knowledge about the parameter, but the computational complexity limits its real-time processing in engineering application. Quadratic time-frequency distributions, such as short time Fourier transform (STFT), Wigner–Ville distribution (WVD) and ambiguity function, present attractive features for LFM signal. In terms of time-frequency localisation, WVD has perfect time-frequency concentration and thus is appropriate for processing mono-component LFM signal. However, due to their quadratic forms, WVD-based methods suffer from the disturbance of cross-terms in processing the multi-component LFM signal. FRFT avoids the cross-term disturbance in the parameter estimation of multi-component LFM signal because it employs a linear frequency modulation basis, but it is difficult to balance the computation and precision. Therefore, a numerical optimisation algorithm is applied to reduce the computational complexity and improve the precision. Qi et al. proposes a two-level search algorithm based on FRFT and a quasi-Newton algorithm to achieve the above objectives [7]. A coarse search is first performed with a large step size; then, an iterative search based on the quasi-Newton method is conducted with the initial values obtained in the first step. However, owing to the sensitivity of quasi-Newton method to initial values, the incorrect results may be obtained if the initial values are not properly set. In this paper, the parameter estimation of LFM signal is modelled as a optimisation problem using the simplified linear canonical transform (SLCT). This novel method is characterised by easy implementation, high precision and independence on initial values because (i) SLCT is a generalisation of Fourier transform, which has the characteristics of simple implementation and lower computational complexity than FRFT. (ii) The evaporation-rate-based water cycle algorithm (ER-WCA) is used to solve this optimisation model, which is characterised by fast convergence, high precision and independence of the initial values. Furthermore, the proposed parameter estimation method is successfully applied to optical measurement [15] and thus improves the measurement accuracy. The remaining sections of this paper are organised as follows: we review some basic concepts about SLCT and ER-WCA in Section 2. In Section 3, a novel method for parameters estimation of LFM signal is proposed using SLCT and ER-WCA. The performance analysis of this method is shown in Section 4. Moreover, the proposed method is applied to optical measurement. Finally, conclusions are stated in Section 6. 2 Simplified linear canonical transform The linear canonical transform (LCT) with real parameter matrix of signal is defined as [16-18] (1)with the kernel (2)where the real parameter matrix satisfying and denotes the LCT of . If the parameter matrix is selected as a special matrix , then LCT reduces to a special transform with parameter , i.e. (3)Since is a special case of LCT, it is named as simplified LCT (SLCT). SLCT is also known as concise FRFT [10] or simplified FRFT [19], it can be calculated by one chirp multiplication and FT operations, i.e. (4)where F denotes as Fourier transform operator. Compared with the digital computation of LCT [20], one chirp multiplication and scaling operations are saved. The discrete algorithm of SLCT is a basic extension from theory to application. It is assumed that the time-width and bandwidth of the signal are T and F, i.e. . According to the sampling theorem, it is reasonable to take the bandwidth F as sampling frequency . Similar to discrete FRFT (DFRFT) [21, 22], the dimensional normalisation is also performed in discrete SLCT. A new coordinate system is defined as , where . Based on the new coordinate system, the signal will have the same interval in the time domain and frequency domain, the sample interval changes to , where . Thus, the discrete time SLCT (DTSLCT) of can be calculated by the following equation: (5)By quantising the variable u in the SLCT domain, discrete SLCT (DSLCT) can be written as (6)Furthermore, inverse DSLCT (IDSLCT) can thus be defined as (7)where fft and ifft are the fast Fourier transform and its inverse, respectively. Based on (6), DSLCT requires one chirp multiplication operation and one FFT operation, the computation of DSLCT is , where [19]. However, DFRFT [21] can be obtained by convolution operation (requires at least two FFTs) and two chirp multiplication operations, the computation of DFRFT is , where [23]. Therefore, the DSLCT is easier to implement and has less computation than DFRFT. 3 Proposed method A mono-component LFM signal [7] can be represented as (8)where A is the amplitude, T is the time width, k is the chirp rate and is the initial frequency and is a rectangular function. The schematic of LFM signal detection based on SLCT can be analysed from the point of WVD in time-frequency plane. The relation between and can be interpreted as (9)It means that time coordinate t is rotated with and stretched with , frequency coordinate is rotated with , the new coordinate system is obtained. The distribution of the finite length LFM signal in the time-frequency plane is characterised by linear type, the slope and centre of the line are and , respectively. When the axis u is orthogonal to the line of LFM signal located, i.e. , the projection on the u axis of the line is a point . Therefore, it can be derived that the frequency spectrum of LFM signal in SLCT domain is energy concentration when , otherwise is energy dispersive. The above schematic diagram can be intuitively shown in Fig. 1. The frequency spectrum characteristic of LFM signal in SLCT domain can also be obtained according to the definitions of SLCT and LFM signal, i.e. (10) (i) when , then (11) From the Parseval principle associated with SLCT, i.e. (12)we can conclude that the amplitude spectrum of reaches its global maximum at when . (ii) when , then where Fig. 1Open in figure viewerPowerPoint Schematic diagram of LFM signal detection using SLCT Thus, the frequency spectrum of LFM signals in SLCT domain can be expressed as where and are the Fresnel integral [24]. The amplitude spectrum of LFM signal in SLCT domain can be summarised as the following three situations: when , , the amplitude spectrum is If , we have according to the property of Fresnel integral, the amplitude spectrum is about when , , the amplitude spectrum is it is half of , where denotes the width of support interval; when , , the amplitude spectrum is it is half of , where denotes the width of support interval. With the increasing of , Fresnel integral fluctuate less within the support interval and thus the amplitude spectrum of LFM signal in SLCT domain becomes flat. While, the amplitude spectrum of LFM signal in SLCT domain decreases rapidly outside the support interval, and the shape of the amplitude spectrum is closer to the rectangle. This indicates that the energy of LFM signal in SLCT domain is mainly concentrated in the support interval, while the energy is very small outside the support interval. Therefore, a novel parameter estimation method for LFM signal is proposed based on the energy concentration of LFM signal in SLCT domain, i.e. (13)where , and and are the estimated chirp rate and initial frequency, respectively. In this paper, ER-WCA is used to solve this optimisation model because ER-WCA converges to the global solution faster and offers more accurate results than other optimisers, such as DE, GA, PSO and ABC [25]. Since the quasi-Newton optimisation algorithm is sensitive to the initial values, a parameter estimation method based on the quasi-Newton optimisation algorithm [7] may not converge to the global maximum if the initial values are not set appropriately. For ER-WCA, the initial populations of the design variables are randomly generated between the upper and lower bounds. Therefore, the proposed method always converges to the global solution with appropriate parameters, it means that the proposed method possesses the advantage of independence on initial values. The method described above can be extended to the case of a multi-component LFM signal when combined with signal separation technology [7]. A multi-component LFM signal can be represented as (14)where , T is the time width, is the amplitude of the m th component, and are the chirp rate and initial frequency of the m th component, respectively, and is zero-mean white Gaussian complex noise. For a multi-component LFM signal, the stronger component may impact the detection and parameter estimation of the weaker component. Therefore, a signal separation technique in the SLCT domain is presented to solve this problem. In this algorithm, each component is detected and estimated sequentially from strong to weak according to the model in (13). The stronger component is filtered from the observed signals in the SLCT domain; thus, the detection and parameter estimation performance for the weaker signal is improved. The procedure for this process is expressed as follows and intuitively shown in Fig. 2 : (i) The strongest component is detected if its DTSLCT amplitude spectrum is larger than a certain threshold, and its parameters are estimated according to (13). The obtained parameters are denoted as . (ii) The original multi-component LFM signal is transformed by the DSLCT in (6). The SLCT frequency spectrum is denoted as: (15)where and are the SLCTs of the LFM signal and noise, respectively. Here, the energy of the strongest signal component is mostly concentrated in a narrowband with centre . Consequently, neither noise nor other components will be concentrated in this narrowband. (iii) A filtering operation is applied to , i.e. (16)where is the transfer function of a band-stop filter with central frequency and properly selected bandwidth. Since most of the energy of the strongest component is concentrated in a narrowband with centre , most of the energy of the strongest component is removed by filtering. (iv) The filtered signal is subjected to IDSLCT with angle . Then, the observed signal is approximately expressed as (17)where is the output noise after filtering in the SLCT domain, which can still be approximated as white Gaussian noise. (v) The above steps are repeated to obtain the parameters of all the components. Fig. 2Open in figure viewerPowerPoint Flowchart of proposed method for multi-component LFM signal 4 Experimental results All experiments are conducted on a personal computer using MATLAB version R2014a. The initialisation parameters of ER-WCA are 4.1 Mono-component In the experiment, the parameters of the first type mono-component are: , and the chirp rate varies from to in intervals of . The parameters of the second type mono-component are: , and the initial frequency varies from to in intervals of . These discrete signals are sampled with parameters and . The intervals of angle used in [7, 10] are and , respectively. In this paper, the proposed method is compared with the FRFT-based method and concise fractional Fourier transform (CFRFT)-based method. The mean square errors (MSEs) between the estimated parameters and the exact parameters are shown in Figs. 3 and 4. Compared with the similar FRFT-based method, the proposed method offers more accurate estimation results for both the chirp rate and the initial frequency. Compared with the CFRFT-based method without the optimisation algorithm, the proposed method greatly improves the accuracy of the estimated chirp rate and the initial frequency, especially the initial frequency. Fig. 3Open in figure viewerPowerPoint MSEs of the estimated chirp rate for LFM signals Fig. 4Open in figure viewerPowerPoint MSEs of the estimated initial frequency for LFM signals 4.2 Multi-component In this section, Monte Carlo simulations are presented to support the theoretical results. In the experiment, the multi-component signal contains two components, the length of the signal is 65, and the parameters of each component are and The sampling interval is 1, and the value of the input signal-to-noise ratio (SNR) varies from −6 to 10 dB in intervals of 1 dB. We run 100 Monte Carlo simulations. The accuracy of parameter estimation is determined by the MSE, which is defined as (18)where is the estimated value of and M is the number of Monte Carlo simulations. Fig. 5Open in figure viewerPowerPoint MSEs of the parameter estimates of the strong component, (a) Initial frequency (b) Chirp rate Fig. 6Open in figure viewerPowerPoint MSEs of the parameter estimates of the weak component, (a) Initial frequency (b) Chirp rate The obtained MSEs of the proposed method are shown in Figs. 5 and 6. The overall trend of the MSEs of the strong component and weak component is to increase with increasing SNR. As shown in Figs. 5 and 6, when the input SNR is higher than −4 dB, the estimated MSEs of the chirp rate and initial frequency are both <, which means the proposed method can accurately estimate the parameters of the LFM signal when the input SNR is >−4 dB. 5 Application As a classic optical signal, Newton's rings is widely used in optical measurement, such as for testing spherical or flatting optical surfaces [26], measuring the liquid refractive [27] or curvature radius index [28, 29] and determining length and displacement [30]. The latest measuring devices based on computers and charge coupled devices (CCDs) have achieved measurement automation and instrumental intelligence (see Fig. 7 a). In the following, the actual Newton's rings are obtained using the following parameters: the curvature radius of an convex lens is , wavelength of the sodium lamp is , and resolutions of the CCD camera are 1600 × 1200 pixels (pixel size of 6.4 μm × 6.4 μm), 1920 × 1080 pixels (pixel size of 3.9 μm × 3.9 μm), and 1280 × 720 pixels (pixel size of 5.9 μm × 5.9 μm). For convenience, these three figures are sheared to resolutions of 720 × 720, 1080 × 1080 and 1200 × 1200 (see Figs. 7b–d). Fig. 7Open in figure viewerPowerPoint The Newton's rings experiment device and actual Newton's rings (a) Newton's rings experiment device, (b) 720 × 720, (c) 1080 × 1080, (d) 1200× 1200 Fig. 8Open in figure viewerPowerPoint Different views of amplitude spectrum for 1200 × 1200 Newton's rings (a) 3D view of the amplitude spectrum of Newton's rings, (b) Projection of the amplitude spectrum of Newton's rings on , (c) 3D view of the amplitude spectrum of Newton's rings after filtering, (d) Projection of the amplitude spectrum of Newton's rings on after filtering Without loss of generality, the mathematical model of the Newton's rings is simplified and demonstrated as a multi-component 2D LFM signal [28] (see (19)), where , and T is the physical dimension of the Newton's rings. represents the mean intensity, and is the amplitude of the sine variation of the fringes. R is the curvature radius of convex lens, is the wavelength of incident light and is the centre coordinates of Newton's rings (19) (20) (21)As shown in (20) and (21), each row and column of Newton's rings are 1D LFM signal. Therefore, R and can be obtained using the estimated chirp rate k and initial frequency , i.e. (22)According to theoretical analysis in Section 3, the SLCT frequency spectrum of each column and row of Newton's rings can be identified obvious impulses with matched angle (i.e. ). Therefore, for this special 1D multi-component LFM signal, the matched angles should satisfy the following conditions: (23)where represent the matched angles. As shown above, and are symmetric to . The mean intensity is useless for measuring the curvature radius, but it has a strong influence on the estimated curvature radius. The signal separation technology can be used to reduce the impact of . The Newton's rings is selected as an example to verify the above results. The SLCT frequency spectrum of Newton's rings is shown in Fig. 8 a. The 3D view of frequency spectrum is projected on for convenient observation (see Fig. 8 b). From the observation of Fig. 8 b, it can be found that one peak can be seen at , the other two peaks are symmetric to . The simulation experiments are consistent to theoretical analysis. The 3D view and projection of the SLCT frequency spectrum of Fig. 7 d after filtering are shown in Fig. 8 c and Fig. 8 d, respectively. These figures indicate that the proposed method effectively decreases the impact of the mean intensity. Table 1. Comparison of measurement results Fig. 7 Parameters FRFT FRFT-LST SLCT (a) radius error (%) 1.27 1.27 0.5966 centre (pixel) (325, 193) (324, 192) (323, 192) (b) radius error (%) 0.9 0.9 0.6634 centre (pixel) (487, 290) (487, 289) (487, 289) (c) radius error (%) 0.34 0.34 0.2961 centre (pixel) (577, 453) (576, 452) (576, 452) The curvature radii for three actual Newton's rings (see Figs. 7a–c) obtained using the proposed method are listed in Table 1. The proposed method is compared with the methods based on FRFT [28] and FRFT-LST [29], the corresponding results are also listed in Table 1. For the radius error, the radius errors obtained by the proposed method are less than those of the FRFT and FRFT-LST methods. For the centre position of Newton's rings, the proposed method improves the accuracy compared with FRFT-based method. 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