IF Extraction of Multicomponent Radar Signals based on Time ...

IF Extraction of Multicomponent Radar Signals based on Time-Frequency Gradient Image Haijian Zhang, Guoan Bi, Lei Yang, Sirajudeen Gulam Razul, and Chong Meng See School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore {zhaijian,egbi,yanglei,esirajudeen,samsonsee}@ntu.edu.sg

I. I NTRODUCTION Most of practical radar signals contain time-varying frequencies [1]. A classical problem is to accurately track the instantaneous frequency (IF) from finite radar samples. Accurate IF estimation facilitates many radar applications, e.g. radar signal recognition [2]. With the ever-increasing number of radar activities, a lot of intercepted radar signals in modern electronic warfare, may contain a wide variety of radar waveforms including not only continuous but also stepped frequency-modulated (FM) pulses, whose spectral contents are randomly distributed in the time-frequency (TF) domain. More radar pulses are available within the intercepted signal during a longer observation time. The IF estimation mechanisms for continuous and stepped FM signals are generally different, thus the coexistence of these two types of signals in TF domain makes the IF estimation very difficult. There has not been reported efforts in the literature on IF estimation of radar signals with mixed continuous and stepped FM pulses. A wide range of IF estimation methods have been extensively investigated in recent decades. Generally, they have been categorized into parametric methods and nonparametric methods. The focus of this paper is on time-frequency distribution (TFD) based nonparametric methods for multicomponent radar signals in low SNR environments. A great amount of work has been reported about the IF estimation of monocomponent signals [3]–[8], whereas a few contributions have been made for multicomponent signals. Some traditional methods use signal spectrum decomposition approaches, by which a multicomponent signal is decomposed into a sum of monocomponent ones. Then the methods for monocomponent

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Abstract—The estimation of multicomponent radar signals with overlapped instantaneous frequencies (IFs) in low SNR environments has been a challenging research topic. This paper proposes an IF estimation algorithm for spectrally overlapped radar signals which contain continuous and stepped IF laws. Firstly, we design a gradient image via the time-frequency distribution, based on which two ratio images are derived for detecting the IF segments of continuous and stepped radar components, respectively. Secondly, a graph to link the detected IF segments is constructed, and a Markov random field model is defined on the constructed graph. The final IF estimation process is transformed into the extraction of the best graph labeling. Numerical results are provided to demonstrate the robustness and efficiency of the proposed algorithm. Index Terms—IF estimation, multicomponent radar signals, time-frequency distribution.

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Fig. 1. BD image of a noisy signal with four overlapped radar components: a monopulse component, a linear FM (LFM) component, a sinusoidal FM (SFM) component, and a Costas component (fs =500 MHz, SNR=20 dB).

signals are applied to each separated component. A new development of the signal separation was based on compressive sensing techniques [9], which can efficiently separate components whose IF laws intersect in TF domain. Another approach was to use pattern recognition techniques, such as Hough transform [10], [11]. However, this transform needs a prior information about the class of signal’s IF law and does not work well for the components with different IF laws. Furthermore, this approach is inefficient if the IF law of a signal cannot be mathematically represented. For low SNR signals, the frequency fluctuation becomes large so that spurious estimates (outliers) far from true IFs frequently occur [12]. In [13], the continuous property was used by the IF detection to reject improbable outliers. This nonparametric method requires the IF passing through as many TF points as possible of the Wigner distribution with strong magnitudes and that the IF variations between two consecutive points are not too abrupt. In [14], a variable bandwidth filter was designed to adaptively estimate IF information of multicomponent signals. Nevertheless, most of existing methods cannot work well for multicomponent signals with overlapped IF laws in low SNR environments. The IF estimation of multicomponent radar signals with continuous and stepped IF laws within a short observation session is a difficult task in radar areas. Currently available techniques are inefficient for dealing with these complex radar signals. Thus, sophisticated IF estimation methods are necessarily investigated for possible solutions to the above situation.

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(a) Mean binary ratio image Rb .

(b) Detected IF segments based on Rb .

(c) The constructed graph based on (b).

eg . (d) Mean gradient ratio image Rg and R

eg . (e) Detected IF segments based on R

(f) The constructed graph based on (e). ideal IFs estimated IFs

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Monopulse component SFM component LFM component Stepped FM component

(g) Estimated IFs of continuous FM pulses from (c).

(h) Estimated IFs of stepped FM pulses from (f).

Fig. 2. Step 1: (a)(b)(d)(e) Step 2: (c)(f)(g)(h) (a) The mean binary ratio image Rb . (b) The detected IF information (binary images) based on Rb . (c) The constructed graph based on detected line segments (solid black lines: thinned line segments from (b); dashed red lines: added segments by locally linking the eg (excluding the region within the dashed red curves). (e) The detected IF information detected line segments. (d) The mean gradient ratio image Rg and R eg . (f) The constructed graph based on the thinned line segments in (e). (g) The extracted IFs of continuous FM components by ASA algorithm. based on R (h) The extracted IFs of stepped FM components and monopulse component (N = 500, w0 = 11, ε0 = 0.01).

This paper proposes an algorithm which transforms the traditional IF estimation problem into the line detection problem from image processing point of view. A line detection algorithm was proposed for road detection from synthetic aperture radar (SAR) images in [15]. The algorithm was applied for detection of power transmission network in [16], [17]. In our study, we develop the algorithm in [15] for IF estimation of noisy radar signals consisting of continuous and stepped FM components. This paper is organized as follows. The proposed IF extraction algorithm is elaborated in Section II. In Section III, the proposed algorithm is evaluated with multicomponent radar data. Finally, conclusion is made in Section IV. II. T HE PROPOSED IF ESTIMATION ALGORITHM Herein the B-distribution (BD) [18] is chosen due to its desired TF resolution and significant cross-term suppression. As shown in Fig. 1, the BD image of a sample radar data within a time interval of 5 µs contains four different radar components intersected in the TF domain. The purpose is to estimate the IF information of each radar pulse. The proposed algorithm is a two-step process.

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Step 1: We design two ratio images (Fig. 2 (a)(d)) based on a TF gradient image, and then sequentially detect the linear IF segments belonging to continuous FM pulses and stepped FM pulses, respectively (Fig. 2 (b)(e))1 . Among the detected IF line segments, some of them are true IF information and others are deceptive resulting from strong noise or cross-terms. Besides, some true IF information is missing. The purpose of this step is to maximize the detection probability of real IF information, and meanwhile reduce the probability of false detections. Step 2: This step obtains the entire IF information by recovering the missing IF line segments and meanwhile omitting deceptive ones. Specifically, a graph is constructed by linking the detected segments (Fig. 2 (c)(f)). Note that the missing IF segments are remedied by the linking operation. A Markov random field (MRF) is defined on the constructed graph by using the geometric information of IF trajectories, and the IF estimation

1 We firstly detect the IFs of continuous FM pulses via Fig. 2 (a)(b)(c)(g), and then delete the TF points (the region within the dashed red lines in Fig. 2 (d)) where the continuous FM pulses pass through. Next, the IF estimation of stepped FM components is implemented via Fig. 2 (d)(e)(f)(h).

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process is modeled as the extraction of the best graph labeling. Based on the fact that the IF trajectories of continuous and stepped FM components have distinct geometric structures, the true IF information of their components are separately extracted (Fig. 2 (g)(h)). The final extraction result is obtained through an energy minimization process realized by the adaptive simulated annealing (ASA) algorithm [19]. A. Detection of IF line segments 1) Computation of gradient image: The computation of the gradient vector of each pixel is based on a local least-squares (LS) fitting method [20]. For each TF point (xi , yj ) in the N × c ij (Fig. 1), its neighborhood N(xi , yj ) N noisy BD image BD within a w0 × w0 square window is   N(xi , yj ) = (xi+r , yj+l ) : r, l ∈ [−m : m] , (1) where w0 = 2m + 1. The TF points in this neighborhood are fitted by a local LS plane LSij (x, y) = βb0ij + βb1ij (x − xi ) + βb2ij (y − yj ),

(2)

 m P  c i+r,j+l ,  BD (x, y) ∈ N(xi , yj ), βb0ij = w12   0  r,l=−m   m P Pm c i+r,j+l , (xi+r − xi ) l=−m BD βb1ij = w01Ax  r=−m   m  P Pm ij  1 c i+r,j+l ,  (yj+l − yj ) r=−m BD βb2 = w0 Ay

(3)

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Pm where Ax = r=−m (xi+r − xi )2 and Ay = l=−m (yj+l − ~ij of the LS plane at the pixel yj )2 . The gradient vector V (xi , yj ) is formed by the LS coefficients of the fitted LS plane ~ij = (βbij , βbij ), V 1 2

{i, j} = 1, . . . , N.

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~ ? by adjustNext, we generate a modified gradient image V ~ ing the gradient direction of Vij in (4) ( st rd bij bij ~ ~ij? = (β2 , −β1 ), if Vij in the 1 or 3 quadrant, V ij ij ~ij in the 2nd or 4th quadrant, (−βb2 , βb1 ), if V {i, j} = 1, . . . , N.

(5)

2) Computation of two mean ratio images: A mean gradient ~ and V ~ ? in (4) and (5) ratio image is computed according to V g Rij = βb0ij

~ ? k2 kV ij ~ij k2 } max{ε0 , k V

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~ij and V ~ ? denote the mean vectors of gradient vectors where V ij ~ and V ~ ? , and ε0 is a within a window in the images V small threshold to avoid the ratio computation in (6) being an infinitive value. The normalized Rg for the sample data is shown in Fig. 2 (d). Note that the TF points along real IF curves are considerably intensified, while the TF points stemmed from noise or cross-terms are well attenuated. The IF estimation becomes difficult when the spectral contents of continuous and stepped radar pulses are intersected

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in the TF domain. We propose to sequentially deal with continuous FM components and stepped FM components (including monopulse). To achieve this purpose, we design two binary images B and B ? , which are generated based on ~ and V ~ ? according to the following criterion V ( ~ij in the 1st or 2nd quadrant, +1, if V Bij = ~ij in the 3rd or 4th quadrant, −1, if V ( ~ ? in the 1st quadrant, +1, if V ij ? Bij = ~ ? in the 2nd quadrant, −1, if V ij {i, j} = 1, . . . , N.

(7)

The mean binary ratio based on B and B ? is calculated by b Rij = βb0ij

? |2 |Bij

max{ε0 , |Bij |2 }

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(8)

? are the mean values of binary values where Bij and Bij within a window in the images B and B ? . The normalized Rb on the sample data is shown in Fig. 2 (a). It is seen that the IF information of the stepped FM component and the monopulse component is omitted. The IF information of the SFM component at the positions with zero slope is also erased. In contrast, the TF points at the positions with non-zero slope have significant mean binary ratios. 3) Line segment detection: Figs. 2 (b) and (e) show the detected IF information by using a predefined threshold on Rb eg in Fig. 2 (d), respectively. The width in Fig. 2 (a) and on R of detected IF segments is normally wider than one pixel. The thinning operation in morphological processing can be applied to extract the pixels at the axes of IFs. The thinned IF line segments can be observed in Figs. 2 (c) and (f).

B. Real IF extraction based on MRF model 1) Graph construction: A graph is formed by linking the detected line segments according to the following conditions: • The linking is made only between two endpoints belonging to different line segments; • The two linked line segments are closely located; Figs. 2 (c) and (f) illustrate the constructed graphs by linking the detected line segments. Let us assume the detected line segments and the linked line segments (total Ns segments) as the segment candidates for IF extraction, and all the true IF information is contained in these segment candidates. Note that a coarse detection of line segments in the first step is sufficient since the missing IF information due to the coarse detection is included in the linked line segments. 2) Definition of MRF model: Based on the constructed graph, an MRF model which associates the structural knowledge of IFs and TF image intensity to segment candidates is defined. We define a binary variable Hn to associate the nth segment Sn in the graph, where Hn = 1 indicates that Sn belongs to the true IFs and Hn = 0 means Sn is a spurious IF segment. The problem of IF extraction is transformed into the problem of finding the best graph labeling. The most probable

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b is achieved configuration H, given the observation process D, by maximizing a posterior distribution b = arg max p(H|D) b ∝ p(D|H)p(H), b H H∈H

(9)

b where p(D|H) and p(H) are the conditional distribution and the prior distribution to be estimated, respectively. (a). Conditional distribution b is dependent on the BD intensity The observation field D b image. Supposing that Dn only depends on Hn , we obtain b p(D|H) =

Ns Y

 b n |Hn ) ∝ exp p(D

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(10) b n |Hn ) is the potential where the Vn (D as  b n |Hn = 0) = −|D0 − D b n |, Vn (D     b n |Hn = 0) = |D0 − D b n |, Vn (D b b n |,  Vn (Dn |Hn = 1) = −|D0 − D    b n |Hn = 1) = |D0 − D b n |, Vn (D

function of Sn , defined if if if if

bn :D bn :D b : Dn bn :D

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(11)

where D0 is a threshold value to be evaluated by empirical statistic. Because a true IF segment labeled by 1 should have a b n , this segment will be rewarded. Otherwise, strong value of D b n is less than D0 . In the this segment should be penalized if D b n and similar way, a spurious IF segment with weak (strong) D labeled by 0 is rewarded (penalized). (b). Prior distribution It is known that an MRF model with a random field H has the same distribution as a Gibbs probability law, so we can establish simple potentials by a probability distribution associating with prior geometric knowledge of IFs   X p(H) = exp − U (H) = exp − Vc (H) , (12) c∈C

where C denotes the clique set in the graph. The continuous FM components and the stepped components have different geometric structures. Generally, the IFs of continuous FM components have the following geometric properties: 1) The IF trajectories are long without breakpoints; 2) The IF trajectories have low curvatures without abrupt changes; 3) The IF trajectory of an individual component is a single path without any branch. In contrast to continuous FM components, the IF trajectories of stepped FM components (monopulse) are step-wise (horizontal) with zero-slope for each IF segment. We define the prior distribution by associating the above prior geometric information of IF curves to the potential function Vc (H) in (12). The underlying real IFs can be therefore extracted from the constructed graphs in Fig. 2 (c)(f) by considering the specific structural knowledge of IFs.

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3) Simulated annealing: From (9), (10), and (12), the maximum a posteriori probability estimate equals to the minimization of the following potential energy b = U (H|D)

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(13)

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The ASA algorithm is used to perform the energy minimization. The extracted IF curves are presented in Figs. 2 (g) and (h). Note that the IF of each component in Fig. 1 is successfully separated and accurately estimated. Simulated annealing algorithm simulates the physical annealing procedure in which a physical substance is melted and then iteratively cooled down in search of a minimum energy configuration [21]. Defining a random realization F in a configuration space F has the acceptance probability at the tth iteration  1/Tt , (14) p(Tt ) (F ) = p(F ) where Tt denotes the temperature parameter, and p(F ) is the computed posterior probability at the current iteration. When Tt converges to infinity, p(Tt ) (F ) is a uniform distribution. When Tt = 1, p(Tt ) (F ) = p(F ) and as Tt converges to 0, p(Tt ) (F ) concentrates on the peaks of p(F ). The temperature Tt decreases in accordance with a carefully chosen schedule Tt = αTt−1 ,

(15)

where α is a value between 0 and 1. Choosing a big α increases the reliability of the algorithm to reach the global minimum but it corresponds to a slower cooling rate. We set α = 0.95 for our simulation. The initial temperature T0 is chosen high enough (herein T0 = 500) so that essentially all configuration changes are acceptable. This process iterates until Tt is approaching to zero, i.e., when the system is frozen near the minimum energy. The ASA algorithm is developed to statistically find the best global fit of a nonlinear constrained non-convex cost-function over a high-dimensional space. This algorithm permits an annealing schedule for temperature Tt decreasing exponentially at each annealing-time. The introduction of re-annealing also permits adaptation to changing sensitivities in the multi-dimensional parameter-space. III. N UMERICAL RESULTS Let us consider a more difficult case where the received radar signal has many different types of components within the observation duration (10 µs). Fig. 3 (a) presents a clean radar signal with five component types: one SFM component, four parallel LFM components, three parallel hyperbolic FM (HFM) components, one monopulse (MP) component, and two Costas coding components with different symbol durations. The sampling frequency is 500 MHz. Different components have different values of time-support during the observation time, and the spectral contents of some components are superposed. The noisy signal at SNR=-2 dB is presented in Fig. 3 (b). The ratio image Rb is firstly computed. The detected IF information is thinned and displayed in solid white lines

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(f) The IF estimation results vs. ideal IFs.

The proposed algorithm on a radar signal with 11 components. (solid white lines: detected IF line segments, dashed lines: added line segments).

in Fig. 3 (c). The graph is created by locally connecting the detected lines, and the added linking is indicated by dashed red lines in Fig. 3 (c). The optimization result of the ASA algorithm based upon the graph in Fig. 3 (c) is given in Fig. 3 eg in Fig. 3 (d) is generated (e). Subsequently, the ratio image R g based on R by removing the points along the estimated IFs of continuous FM components. The ASA algorithm results for the stepped FM components and the monopulse component are then combined into Fig. 3 (e) to construct the overall estimation. The estimated IFs of all the components in contrast to ideal IFs are given in Fig. 3 (f). We note that the estimated IF curves of different signal components well fit the ideal ones. The normalized mean absolute errors (NMAEs) of IF estimation for each radar component are provided in Table I for different SNR values. Note that desirable performance is achieved even for a low SNR value. Assuming a consistent IF estimator is applied for this radar data, it is expected that the NMAEs increase as the level of SNR decreases.

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However, the proposed algorithm is able to restore the strong noisy-corrupted IF information by linking the detected IF line segments, which explains why the values of NMAEs with low SNR values in Table I are comparable to those with high SNR values. For lower SNR values (