Analysis and Detection of S-Shaped NLFM Signal Based on
Transcription
Analysis and Detection of S-Shaped NLFM Signal Based on
Journal of Communications Vol. 10, No. 12, December 2015 Analysis and Detection of S-Shaped NLFM Signal Based on Instantaneous Frequency Jun Song, Yue Gao, and Demin Gao Department of information science & technology, Nanjing Forestry University, Nanjing 210037, China Email: [email protected] Abstract—The characteristics typical S-shaped Nonlinear Frequency Modulation (S-NLFM) signal are analyzed, and a detection method for this type of signal is also proposed. First of all, the instantaneous frequency of the S-NLFM signal based on sinusoid modulation and tangent modulation is obtained by phase unwrapping respectively. Then, two characteristics are obtained from instantaneous frequency. Consequently, a recognition method is proposed based on the two characteristics obtained in the previous step. At last, the detection method for the S-NLFM signal is implemented. Simulation results indicate that the proposed method is robust and the detection rate is more than 90% under an SNR condition of 0 dB or higher regardless of the modulation parameters. Index Terms—Instantaneous frequency, signal detection, Sshaped NLFM, phase unwrap I. INTRODUCTION Pulse compression is used in many radar and active sonar systems to achieve long-range performance and fine range resolution simultaneously. Long-range detection implies fine noise reduction, requiring long time transmissions. To achieve fine range resolution, the band width of the pulse must be enlarged accordingly. These conflicting requirements can be realized by modulating an S-shaped function (eg. sinusoid or tangent function) with a Linear Frequency Modulation (LFM) [1], thus constructing the typical S-shaped Nonlinear Frequency Modulation signal (S-NLFM). In recent decades, the S-NLFM signal processing has attracted increasing attention in the scope of radar and communication engineering [2]-[11]. In this work, character analysis and detection of the SNLFM signal are implemented. As all known, LFM is a typical pulse compression method, and the side lobes are usually high when the LFM signal is processed by a matched filter. As a result, a weighted filter should be utilized to obtain proper side lobe suppression. However, a weighted filter may result in a decrease in the Signal-toNoise Ratio (SNR). A weighted filter is unnecessary for side lobe suppression when the NLFM signal is processed [1]. For the S-shaped NLFM signal, the chirp rate Manuscript received June 8, 2015; revised December 7, 2015. This work was supported by the Jiangsu Higher Education Natural Science Found under Grant No.13KJB220003. Corresponding author email: [email protected]. doi:10.12720/jcm.10.12.976-982 ©2015 Journal of Communications 976 increases or decreases at the end of a pulse, which is contrary to what occurs in the middle of a pulse. The characteristics above result in timely side lobe suppression [2]. Therefore, the S-NLFM signal is widely utilized in all types of wireless applications [3], [4]. Studies on the wave design and performance analysis of S-NLFM signal in the past several years have attracted proper attentions [2], [5]-[7]. Recently, several studies have focused on the character analysis and detection for this type of signal [9]-[15]. In [6] and [14], researchers proposed a double-characters detection method for NLFM based on fractional Fourier transform (FRFT), and the experiments validated its robust performance. ZHANG et al. [7] analyzed intra-pulse feature by means of fractal dimensions and recognized 10 typical radar signals including NLFM, LFM, BPSK and FSK etc. K. J. You et al. [9] employed Gini’s coefficient and maximum likelihood classifier to measure the frequency inequality and then to divide signal modulation types into four classes. The algorithm of [9] exhibited excellent performance. WANG J. [10] proposed a time-frequency tilling based detector for detection of NLFM and Polynomial Phase Signal (PPS). The method of [10] exploited adaptive wavelet transform and Radon-Wigner Transform (RWT) to get proper detection performance. Most of previous studies exploited time-frequency analysis, such as wavelet, RWT and FRFT, which required heavy computation. In addition, the characters of S-NLFM were not analyzed in previous works. In this paper, two characteristics of typical S-NLFM signal are analyzed based on instantaneous frequency, and then a new detection method is proposed. Unlike existing works on S-NLFM signal detection, the contribution of our work has two aspects. One is to extract two characteristics of the signal’s instantaneous frequency by phase unwrap. The other is to design a signal detection method based on the character analysis above and segment filter. The computation of the proposed method is mainly concentrated on the phase unwrap and segment filter which result in easy implementation in engineering. In addition, the analysis and simulations show that our method for S-NLFM signal detection is robust and is independent of modulation style. The remainder of this paper is organized as follows. The S-NLFM signal model and construction are described in the next section, specifically the construction of the signal phase of sinusoid based and tangent based S- Journal of Communications Vol. 10, No. 12, December 2015 7 B 2 K (m) 2 mt s(t ) A exp j t BT cos( ) (5) m T m 1 T NLFM. Then two characteristics of S-NLFM signal’s instantaneous frequency are analyzed in section III. Consequently, the detection method and flow are provided in Section IV. The simulation and numerical results are presented in Section V. And the conclusion is drawn in the final section. In summary, the sinusoid based NLFM signal can be classified into two types, one is with no weighting coefficients (as in (2)), and the other is accompanied by a set of weightings (as in (4)). The frequency curves of the two types S-NLFM are plotted in Fig. 1. II. SIGNAL CONSTRUCTION OF S-NLFM 7 The frequency functions of S-NLFM signals can be divided typically into two types, one is sinusoid based and the other is tangent based. Firstly, we discuss the sinusoid based S-NLFM signal, and the frequency functions can be expressed as: (1) where B is the bandwidth and T is the duration of the signal pulse. Then the signal can be given as: B 2 2 t t BT cos( ) s1 (t ) A exp j T T s (t ) A exp j B t 2 BT cos( 2 t ) 2 T T (3) 3.74 (t ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 x 10 (6) B(1 ) T t2 BT ln(1 tan(2 t T ) 2 ) 4 tan( ) (7) The wave form of a tangent based S-NLFM signal can be expressed as follows: s(t ) A exp t 0 (4) B(1 ) 2 BT ln(1 tan(2 t T ) 2 ) (8) j t T 4 tan( ) and the frequency curves with different α and γ are shown in Fig. 2. Both the sinusoid based and tangent based S-NLFM signals mentioned above are all typical ones in Thus, the module of the sinusoid based NLFM signal is provided by ©2015 Journal of Communications 3.76 where B is the bandwidth, α is the balance factor, and γ is the weight of tangent function ( 2 ). Similarly, the phase function of the tangent based SNLFM signal can be implemented by integration. That is, (t ) 2 f d T 3.78 2 1 t f (t ) B tan(2 t T ) tan( ) T is the sinusoid weighting coefficients, T is the duration of the pulse, and B is the bandwidth. The phase function of (1) is, K ( m) 2 mt cos( ) m T m 1 3.8 In engineering, another type of S-NLFM signal is based on a tangent function instead of a sinusoid one. Accordingly, the frequency modulation-time function is based on tangent function while mixed with linear one. In addition, the engineering realizability should be taken into consideration just like the implementation of sinusoid based ones. Collins and Atkins [16] presented an extended form of tangent based S-NLFM, which employs a set of weightings on tangent function and linear modulation rate to construct the frequency function. The expression is as follows, 0.0082, 0.0055, 0.004 7 3.82 Fig. 1. The frequency curves of sinusoid based S-NLFM. (2) K (m) 0.1145, 0.0396, 0.0202, 0.0118, t 2 BT 3.84 Time (s) where 3.86 3.7 7 B with weighting coefficients signal-1:without weightings signal-2:without weightings 3.72 where A is the amplitude. The sinusoid based NLFM signal has many advantages such as low probability interception and fine range resolution, but every coins has two sides, it is more sensitive to Doppler effect than LFM signal. In an effort to reduce the side lobe, a weighting window function in frequency domain often must be employed while processing the S-NLFM. For the Taylor weighting method provides an approach to Dolph-Chebyshev function, it is recommended in engineering [1]. According to [1], the frequency function of sinusoid based S-NLFM with Taylor −40 dB pulse compression response can be expressed as follows: 2 mt t f (t ) B K (m)sin T T m 1 x 10 3.88 Instantaneous Frquency (Hz) 2 t t f1 (t ) B T sin T ,0 t T f (t ) B t sin 2 t ,0 t T 2 T T 3.9 977 Journal of Communications Vol. 10, No. 12, December 2015 engineering, and they have similar frequency modulation curves and relevant performance of low probability interception. symmetrical (as Fig. 3). The linear part of the modulation rate can be constructed as follows: B (11) t T The difference in modulation rate between nonlinear and linear frequency modulation can be derived as flinear (t ) 7 Instantaneous Frquency (Hz) 3.95 x 10 1 = -0.5,1 = 1.2 2 = 0.6, 2 = 1.4 3.9 7 f (t ) f (t ) flinear (t ) B K (m)sin 3.85 m 1 The curve of 3.8 f (t ) is odd symmetrical. Consequently, the two sides/areas of curve f (t ) demarcated by central 3.75 line t T / 2 are equal. That is, 3.7 SL SR 3.65 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) x 10 T /2 f (t )dt 0 T T /2 (13) f (t )dt 7 x 10 3.9 LFM Signal NLFM with weighting coefficients NLFM signal-1:without weightings NLFM signal-2:without weightings 3.88 Instantaneous Frquency (Hz) III. CHARACTERISTICS OF S-NFLM SIGNALS The sinusoid-based S-NLFM signal is taken as an example in this section. The frequency modulation function can be derived by differential as follows: df (t ) 1 7 2 m 2 mt (t ) B[ K (m) cos( )] dt T m 1 T T (9) 7 B 2 mt [1 K (m)2 m cos( )] T T m 1 3.86 3.84 3.82 3.8 3.78 3.76 3.74 3.72 The NLFM signal will become an LFM one if all weighting coefficients K (m) are equal to zero and the 3.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) frequency modulation function is a constant B / T . However, if K (m) 0 , the frequency modulation function is a variable of time. That is, various modulation rates can be calculated at three sites of the frequency modulation function, namely, initial, end, and middle time. That is, 1 -4 x 10 Fig. 3. The instantaneous frequency of sinusoid based S-NLFM and LFM signal. 7 Instantaneous Frquency (Hz) 3.95 (10) It can be found out that (0) (T ) (T 2) . These results of (10) indicate that the modulation rate at the initial moment is equal to that of the end moment but is different from that of the middle moment. The difference in modulation rate is an important characteristic of the SNLFM signal. Therefore, it can be utilized as a basis for detection. Given the sinusoid function is an odd one, the front and back halves of the modulation rate curve are also odd ©2015 Journal of Communications As shown in Fig. 3, the conclusion of (13) is clearly validated. 1 -4 Fig. 2. Frequency curves with different α and γ. 7 B (0) [1 K (m)2 m]; T m 1 7 B (T ) [1 K (m)2 m]; T m 1 7 B (T 2) [1 K (m)2 m cos( m)]. T m 1 2 mt (12) T x 10 LFM Signal tangent based S-NLFM signal-1# tangent based S-NLFM signal-2# 3.9 3.85 3.8 3.75 3.7 3.65 0 0.1 0.2 0.3 0.4 0.5 Time (s) 0.6 0.7 0.8 0.9 1 -4 x 10 Fig. 4. The instantaneous frequency of tangent based S-NLFM and LFM signal. As the characteristics of tangent based S-NLFM are similar to those of sinusoid-based ones (as shown in Fig. 4), we will not discuss them repeatedly. 978 Journal of Communications Vol. 10, No. 12, December 2015 According to the analysis above, the characteristics for S-NLFM detection are summarized as follows. 1) The modulation rate of initial moment t 0 is equal to that of the end moment t T . The two moments are different from middle moment t T / 2 . 2) The instantaneous frequency curve is odd symmetrical in the middle moment if the linear part is eliminated. In engineer application, the calculation of instantaneous frequency and modulation rate may be disturbed by noise. As a result, deviation from the theoretical value is observed. Thus, the rules for S-NLFM detection based on the two characteristics above should be modified as follows. 1) The modulation rate of initial moment t 0 is approximately equal to that of end moment t T ; 2) The instantaneous frequency curve is almost odd symmetrical in the middle moment if the linear part is eliminated. In order to weaken the disturbance of noise, the SNLFM signal should be pre-filtered to improve the SNR level. The segment filter algorithm [17] can be implemented to process the S-NLFM signal because there is no sudden break or turn in the instantaneous frequency of this type of signal. Step 2: The considered signal is processed by segment filtering to improve the SNR level. The signal’s phase is unwrapped to obtain instantaneous frequency f (t ) . Bandwidth B is estimated with (15). Step 3: The linear frequency modulation part is constructed as flinear (t ) f (t ) f (t ) flinear (t ) should be calculated. Then, the two sides/areas of curve f (t ) , which is demarcated by central line t T / 2 , are computed. The two areas are denoted as S L and S R . If S L and S R satisfy the statements SL SR SL SR 0.9 SL / SR 1.1 we can regard the statements above as one characteristic for S-NLFM detection. Step 4: The three sections of the instantaneous frequency curve f (t ) are cut out. The three sections are located at the initial, middle, and end moments of f (t ) . Each section is a tenth of the total length of f (t ) . The three sections are then subjected to regression fitting through a linear module. The coefficients of one-degree terms are the correct modulation rates at the three moments above, which are denoted as (0) , (T / 2) , and (T ) . If the following statements are satisfied, 0.5 (T ) / (0) 1.5 and (14) (T / 2) Bandwidth estimation is then considered. By setting tS 0 and tE T , the initial and terminal values of instantaneous frequency are calculated. Then, the estimate of bandwidth B can be obtained from the difference of the values above, that is, t 0 f (t ) t T mean (T ), (0) 2 or (T / 2) mean (T ), (0) 0.5 they can be regarded as another characteristic for SNLFM detection. Step 5: If the two characteristics mentioned in steps 3 and 4 are both fulfilled, the signal can be regarded as Sshaped NLFM. Otherwise, it is another type of signal. (15) The characteristics for S-NLFM signal detection were studied, and the construction of the linear frequency modulation part was also discussed. In conclusion, the SNLFM signal detection flow can be summarized as follows. Step 1: By using the signal arrival time estimation algorithm in [18] and phase unwrapping, pulse width can be estimated as (14). ©2015 Journal of Communications 0.1 and In the foregoing subsections, we discussed two characteristics of S-NLFM signal for detection. According to the analysis above, the linear frequency modulation part of S-NLFM instantaneous frequency should be constructed to allow for further processing. Bandwidth B and pulse width T are essentially required to build the linear one as shown in (11). First, the estimation of pulse width T is considered. The initial time (denoted as t S ) of pulse and end time (denoted as t E ) can be estimated with the signal arrival time estimation algorithm proposed in [18]. Pulse width is estimated by B f (t ) t T The difference, IV. DETECTION OF S-NLFM SIGNAL T tE tS B V. SIMULATIONS AND PERFORMANCE ANALYSIS Computer simulations are conducted to validate the proposed detection algorithms in this section. Experiment 1: In the simulations, sinusoid and tangent based S-NLFM signals are selected with pulse width T = 100 µs and bandwidth B = 2 MHz. The sampling frequency is 200 MHz. The initial frequency is assumed 979 Journal of Communications Vol. 10, No. 12, December 2015 to be f 0 37 MHz, and the coefficients K (m) are similar to those in (3). The coefficients of tangent-based SNLFM are 0.5 and 1.4 . Noise is added as white complex Gaussian noise with zero mean. At each SNR level, 1000 Monte Carlo simulations are performed to obtain the detection results. To improve the detection performance, the signal is pre-processed by a segment filter in almost all the experiments. The original signal without filter processing is also simulated for comparison. The value of input SNR varies from −6 dB to 16 dB at increments of 1 dB. The detection simulation results are shown in Fig. 5. As shown in Fig. 5, the detection performance of sinusoid-based S-NLFM is almost equivalent to that of the tangent-based one. However, the segment filter influences the detection performance significantly. The SNR level should be higher than 9 dB to achieve detection probability that is higher than 90% for the original signal. However, for filtered signal, only 0 dB SNR level or higher is required to achieve the same detection performance. weighting coefficients. Various ones with different parameters are chosen to compare the detection performance. One is with pulse width T= 100 µs and bandwidth B = 2 MHz, and the second is with pulse width T = 150 µs and bandwidth B = 2.5 MHz, the third is with pulse width T = 180 µs and bandwidth B = 3 MHz. In addition, their initial frequencies are different from each other, and are random from 30 MHz to 40 MHz. The detection performance with different SNR levels is plotted in Fig. 6. Experiment 3: In these simulations, two sinusoid-based ones are introduced, one is accompanied by weighting coefficients as in (3) and the other is on the contrary, that is to say there are no weighting coefficients for sinusoid function in the second signal. The input SNR ranges also from −6 dB to 16 dB. The numerical results are shown in Fig. 7. 100 90 80 Detection times 70 100 90 Detection times 80 60 50 40 70 30 60 20 50 10 40 0 sinusoid-based one with weightings sinusoid-based one without weightings -5 0 10 15 Fig. 7. Detection results comparison of two kinds of sinusoid based SNLFM signals. sinusoid-based S-NLFM(Orignal) tangent-based S-NLFM(Orignal) sinusoid-based S-NLFM(Filtered) tangent-based S-NLFM(Filtered) 20 10 0 -5 0 5 10 Experiment 4: Three kinds of tangent-based S-NLFM signals with different α and γ are taken into consideration. The first signal is with α1=−0.5, γ1=1.2, the second is with α2=0.6, γ2=1.4, and the third is with α3=0.7, γ2=1.1. The simulation results are proposed in Fig. 8. 15 SNR (dB) Fig. 5. Detection results of original and filtered S-NLFM in Monte Carlo simulations. 100 100 90 90 80 80 70 70 Detection times Detection times 5 SNR (dB) 30 60 50 40 sinusoid-based S-NLFM 1# sinusoid-based S-NLFM 2# sinusoid-based S-NLFM 3# 30 20 50 40 30 1 = -0.5,1 = 1.2 2 = 0.6, 2 = 1.4 2 = 0.7, 2 = 1.1 20 10 10 0 60 0 -5 0 5 10 15 -5 0 5 10 15 SNR (dB) SNR (dB) Fig. 6. Detection results of three different sinusoid based S-NLFM signals without weightings. Fig. 8. Detection results comparison of three tangent based S-NLFM signals. Experiment 2: In the following experiments, we consider sinusoid-based S-NLFM signals without Experiment 5: In an effort to compare the detection performance of different kinds of S-NLFM signals ©2015 Journal of Communications 980 Journal of Communications Vol. 10, No. 12, December 2015 including sinusoid based and tangent based ones, we perform a serial of experiments at fixed 3dB and 0dB SNR levels. There are totally five signals as mentioned in Experiment 2-4. And the comparison results are listed in Table I and Table II. ACKNOWLEDGMENT The authors would like to thank the support by the Jiangsu Higher Education Natural Science Found under Grant No.13KJB220003. REFERENCES TABLE I: DETECTION RESULTS FOR DIFFERENT SIGNALS AT 3DB SNR Signals sinusoid based without weightings sinusoid based with weightings [1] tangent based sig.2 98 [2] TABLE II: DETECTION RESULTS FOR DIFFERENT SIGNALS AT 0DB SNR [3] Detection times Signals Detection times sig. 1 99 sig. 2 100 sinusoid based without weightings sig. 1 93 sig. 2 95 99 sinusoid based with weightings 94 sig.1 99 tangent based sig.1 94 sig.2 93 [4] [5] Fig. 6 displays that the detection performance of the proposed method is hardly affected by the modulation parameter of sinusoid based S-NLFM signals. And if the SNR level is higher than 0dB, the detection probability is above 90% despite of the modulation parameter. According to Fig. 7, we can find out that two kinds of sinusoid based S-NLFM signals, one is with weighting coefficients and the other is without weightings, are conducted, and our algorithm proposes relevant detection performance for these two kinds signals. That is to say the proposed algorithm maintains a steady performance regardless of the sinusoid weightings. What’s more, the parameters of tangent based S-NLFM signals have little impact on the detection results as indicated in Fig.8. Table I shows that at the same 3dB SNR level, the proposed algorithm performs almost 98%~99% detection performance for three kinds of signals including tangent based S-NLFM signal, sinusoid based one without weightings and sinusoid based one with weightings. Table II shows the detection performance is about 93%~94% at a fixed 0dB SNR level regardless of the modulation style. Accordingly, we can draw a conclusion that modulation style can not influence the robust of our algorithm almost. 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Wang, “Improved denoising algorithm for narrow-band signal and its application,” Journal of Vibration and Shock, vol. 32, pp. 59-63, August 2013. [18] G. B. Hu, Y. Liu, and Z. M. Deng, “Arrival time estimation of signals based on Haar wavelets transform,” System Engineering and Electronics, vol. 31, pp. 1615-1619, July 2009. professor. His research interests include spectral estimation, array signal processing, and information theory. De-min Gao was born in Shandong Province in 1980. He received his B.S and M.S. degree in computer application technology from Jingdezhen Ceramic Institute, Jingdezhen, in 2005 and 2008. During 2011-2012, he pursued his study as a joint PhD student and joined the research lab of Kwan-Wu Chin in School of Electrical Engineering, University of Wollongong, Australia. His research fields contain routing protocols for delay tolerant in wireless sensor networks. Jun Song was born in Jiangsu Province, China, in 1979. He received the B.S. degree and M.S. degree from the China University of Mining and Technology (CUMT), Xuzhou, in 2002 and in 2005 respectively. He received the Ph.D. degree from the Nanjing University of Aeronautics and Astronautics, Nanjing, in 2014, all in electrical engineering. He is currently with the Department of information science and technology, in Nanjing Forestry University as an associate ©2015 Journal of Communications Yue Gao was born in 1995. She is a B.S. degree candidate in electronic engineering in Nanjing Forestry University. Her current research areas include adaptive signal processing and filter design. 982