Time-frequency represetation of radar signals using Doppler-Lag block searching Wigner-Ville distribution

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER Time-Frequency Represetation of Radar Signals Using Doppler-Lag Block Searching Wigner-Ville Distribution Muhammad Noor Muhammad HAMDI, Ahmad Zuri SHA’AMERI Department of Electronic and Computer Engineering, School of Electrical Engineering, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia muhdno[email protected], [email protected] DOI: 10.15598/aeee.v16i3.2633 Abstract. Radar signals are time-varying signals where the signal parameters change over time. For these signals, Quadratic Time-Frequency Distribution (QTFD) offers advantages over classical spectrum estimation in terms of frequency and time resolution but it suffers heavily from cross-terms. In generating accurate Time-Frequency Representation (TFR), a kernel function must be able to suppress crossterms while maintaining auto-terms energy especially in a non-cooperative environment where the parameters of the actual signal are unknown. Thus, a new signaldependent QTFD is proposed that adaptively estimates the kernel parameters for a wide class of radar signals. The adaptive procedure, Doppler-Lag Block Searching (DLBS) kernel estimation was developed to serve this purpose. Accurate TFRs produced for all simulated radar signals with Instantaneous Frequency (IF) estimation performance are verified using Monte Carlo simulation meeting the requirements of the CramerRao Lower Bound (CRLB) at SNR > 6 dB. Keywords Adaptive procedure, auto-terms, Cramer-Rao lower bound, cross-terms, kernel function, quadratic time-frequency distribution. 1. Introduction Radar is widely used both in military and non-military applications such as tracking missiles, ships, land vehicles and aircraft, flight control system, ocean surveillance system and geological observations. By definition, Low Probability of Intercept (LPI) radars utilize special emitted waveform that has been specifically designed to avoid detection or interception by noncooperative intercept receiver [1]. This is achieved by integrating additional properties such as ultra-low sidelobe, Advanced Multifunction Radio Frequency Concept (AMRFC), wideband frequency and minimum transmitted energy. The idea of LPI radar is to see and not be seen, meaning it must have the capability to detect targets like any radar while staying invisible to electronic reconnaissance equipment. Intercepting LPI signals is not easy but is not totally impossible. Some of the important properties required in the modern intercept receivers in intercepting LPI signals are channelized receiver, utilization of superheterodyne receiver and sidelobe detection capability [2] and [3]. Signal processing algorithms are the important components of modern intercept receiver that improve the detection and analysis of LPI radar signals. Example of methods used for detecting and analyzing LPI radar signals are adaptive match filtering, parallel filter arrays with higher order statistics, Wigner-Ville Distribution (WVD), quadrature mirror filter bank, and cyclostationary processing [3]. A time-varying signal such as LPI radar signals, the spectral description of which depends on time is best analyzed with Time-Frequency Distribution (TFD). Among TFD classes, Quadratic TFD (QTFD) is appropriately used because it provides high-resolution representation both in time and frequency [4]. Crossterms are introduced in QTFD due to the quadratic nature of the algorithm, which makes it difficult to interpret the true signal characteristics and also exaggerates the effect of noise [5]. Some of the techniques proposed to produce accurate TFR are reduced interference distribution, fractional Fourier transform, Radon-Wigner distribution, adaptive cross WVD and modified B-distribution [6] and [7]. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 318 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER The application of the Fractional Fourier Transform (FrFT) is proven effective in representing and preserving signal components in the Time-Frequency (TF) plane such as Pulse Linear Frequency Modulation (PLFM) [8]. Besides that, the echo of the moving target for airborne Synthetic Aperture Radar (SAR) can also be considered as a LFM signal. Thus, the FrFT can be used to represent and analyze such signals. In the literature, it was mentioned that the FrFT is equivalent to a rotation of the signal either in TF or ambiguity domain. The degree of the TF rotation depends on the fractional power of FrFT [9]. The application of FrFT can be further expanded from this definition for signal parameters estimation and cross-term elimination [10] and [11]. A technique where a combination of a rotated TF plane WVD with a suitable TF filtering is compared with the DLBS-WVD. The rest of the paper is organized as follows. Section 2. presents the signal model and problem definition. The signal characteristics in the time-lag and Doppler-lag domain are discussed in Sec. 3. The relationship between FrFT and TFR is also explained in this section. The simulation result and discussion are presented in Sec. 4. while the field trials results are described in Sec. 5. 2. Signal Model and Problem Definition The four commonly used radar signal types utilized to verify the accuracy of time-frequency representation produced by the Doppler-Lag Block Searching WignerVille Distribution (DLBS-WVD) are: Simple Pulse signal (SP), 4 Costas Coded pulse (CC4) signals, PLFM, and Continuous Wave Linear Frequency Modulation (CW-LFM). The signal parameters are described in Tab. 1. Except for the SP signal, all the other signals can be categorized as LPI radar signal waveforms [1]. The signals are assumed that they have been downconverted from radio frequency to intermediate frequency where they are sampled at the Nyquist rate (sampling frequency, fs= 40 MHz). The use of short pulse repetition period, Tbetween 5 to 20 µs is to simplify the development of the kernel estimation procedure. However, the actual radar signals parameters may vary according to the applications and the detection range [12], [13] and [14]. Typically, Electronic Support (ES) deals with a noncooperative environment where prior knowledge of the true signal characteristics – pulse repetition period, frequency agilities, modulation techniques, pulse width, pulse amplitude – are unknown. Adaptive kernel improves the TFR by estimating the kernel parameters according to the pattern of cross-terms which vary acTab. 1: Signal Parameters. Pulse repetition period (T), pulse width (Tp), lowest frequency (fmin), highest frequency (fmax), bandwidth (BW ). Signal Frequency Parameters Time Parameters Simple Pulse (SP) f= 10 MHz T= 5 µs Tp= 1 µs 4 Costas Coded pulse (CC4) 4 sub-pulse frequencies fb1= 4 MHz fb2= 8 MHz fb3=16 MHz fb4= 12 MHz T= 16 µs Tp= 4 µs Pulse Linear FM (PLFM) fmin = 2 MHz fmax = 17 MHz BW = 15 MHz T= 9 µs Tp= 4 µs Continuous Wave Linear FM (CW-LFM) fmin = 2 MHz fmax = 10 MHz BW = 8 MHz T= 20 µs Tp= 10 µs cording to the signal. The QTFD with the adaptive kernel ensures an accurate TFR for a broad class of signals. 3. Quadratic Time-Frequency Distribution The QTFD produces an energy representation jointly over the time-frequency plane. It is also considered as a related class of filtered WVDs with a specific time–lag kernel function [15]. When expressed with respect to the time-frequency kernel and the WVD, the QTFD is written as [5] ρz(t, f) = γ(t, f)∗ t ∗ fWz(t, f),(1) where γ(t, f)is a time-frequency kernel and Wz(t, f) is the WVD. The WVD can be defined as Wz(t, f) = ∞ Z −∞ Kz(t, τ)e−j2πfτ dτ, (2) where Kz(t, τ)is the bilinear product or Instantaneous Autocorrelation Function (IAF). The bilinear product can be written as Kz(t, τ) = zt+τ 2z∗t−τ 2,(3) where z(t)is the analytical form of the signal. The formulation of the QTFD with the time-lag kernel is given as ρz(t, f) = ∞ Z −∞ G(t, τ)∗ tKz(t, τ)e−j2πfτ dτ, (4) where G(t, τ)is the time-lag kernel. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 319 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER A separable kernel offers independent control of time-smoothing and frequency-smoothing of TFD which can be defined in time-lag function as G(t, τ) = g1(t)g2(τ),(5) where g1(t)is the smoothing function in time and g2(τ) is the smoothing function in lag. By using a separable kernel, the QTFD in Eq. (1) can be described as ρz(t, f) = ∞ Z −∞ g1(t)∗ (t)Kz(t, τ)g2(τ)e−j2πfτ dτ. (6) 3.1. General Signal Characteristics in Time-Lag Domain The IAF defined in Eq. (4) can be viewed as the correlation of signal itself over successive time intervals. Cross-terms are introduced due to the quadratic nature of the IAF which can cause difficulty in the interpretation of the true signal characteristics [16] and [17]. Kernel function as shown in Eq. (4) solved this problem. Various kernel functions gave rise to a range of different TFDs [6] and [15]. The general IAF definition is produced by multiplying the signal with its conjugate as shown in Eq. (7). Each type of input signal, z(t)results in different IAF definition. For example, two pulses of SP signal produce two auto-terms and two cross-terms while two pulses of CW-LFM produce four auto-terms and twelve cross-terms. The IAF obtained by substituting the signal definition in Tab. 1 into Eq. (4) can be expressed in the following form: Kz(t, τ) = hz1t+τ 2+z2t+τ 2−Ti· ·hz∗ 1t−τ 2+z∗ 2t−τ 2−Ti =Kz,11 (t, τ) + Kz,22 (t−T, τ) | {z } auto−terms + Kz,12 t−T 2, τ +T+Kz,21 t−T 2, τ −T | {z } cross−terms , (7) where z1(t)is the first pulse and z2(t)is the second pulse of the signal. The accuracy of TFR depends on the capability of the algorithm to preserve the autoterms energy. Cross-terms can be divided into two categories: intra-pulse and inter-pulses. Intra-pulse cross-terms occur as a result of the correlation of signal between the sub-pulse of the signal while inter-pulse cross-terms are the product of the correlation between different pulses of the signal. Figure 1 shows the time-lag domain of CW-LFM signal and produced by replacing z1(t)and z2(t)in Eq. (7) 1,1 2,2 3,3 4,4 2,1 3,2 1,4 2,41,3 3,42,31,2 4,3 3,1 4,2 4,1 0 Tb 2Tb -Tb -2Tb -3Tb -4Tb 3Tb 4Tb t τ 4Tb 3Tb 2Tb Tb Fig. 1: The time-lag domain two pulses of CW-LFM signal. with two pulses of triangular CW-LFM signal where Tb is the sub-pulse duration. For triangular CW-LFM, the Tbis exactly half of the signal pulse. The dotted diamond shapes represent the intra-pulse cross-terms and the shaded diamond shapes represent the inter-pulse cross-terms. The remaining diamond shapes represent the auto-terms of the signal. Generally, a kernel that is able to suppress the crossterms at |τ|> Tbis acceptable except for CW-LFM signal. Figure 1 shows that there are some cross-terms located between 0≤τ≤Tbwhich could lead to inaccurate TFR of the signal. For complete cross-terms suppression, the applied window or kernel must match perfectly with the auto-terms. To completely separate the auto-terms from the cross-terms in the time-lag domain, alternative cross-terms suppression techniques are proposed in the Doppler-lag domain [18]. 3.2. Radar Signal Characteristics in Ambiguity Domain The AF is related to the IAF by the Fourier transform with respect to time as shown in the following equation: Az(v, τ) = FT t→v[Kz(t, τ)] = ∞ Z −∞ Kz(t, τ)e−j2πvtdt. (8) The complete signal equations for SP, CC4, PLFM and CW-LFM signals in the ambiguity domain can be found in [19]. The AF for the CW-LFM signal is illustrated in Fig. 2. The AF represents two pulses CW-LFM signal where the auto-term are represented by the grey shaded shapes, inter-pulse cross-terms by the blue shaded shapes and intra-pulse cross-terms by the orange shaded shapes. Similar color codes are used in Fig. 3, c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 320 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 0 Tp T -Tp -T -T-Tp T+Tp v τ ` ΔfNTΔfNT T-Tp G2(v ) G2(-v ) g1(-τ ) g1(τ ) (1,1),(2,2), (3,3),(4,4) (1,3),(2,3), (1,4),(2,4) (3,1),(3,2), (4,1),(4,2) (4,3),(2,1) (3,4),(1,2) Fig. 2: The AF for two pulses of CW-LFM. Fig. 4 and Fig. 5. From Fig. 1, the auto-terms are located at τ≤ |Tp|while most of the intra-pulse crossterms are positioned very close to the auto-terms. On the other hand, the inter-pulse cross-terms are located between T−Tp≤τ≤T+Tp. It is worth mentioning here that the energy of the intra-pulse cross-terms for CW-LFM is significantly lower compared to its autoterms. Thus, a small portion of the intra-pulse crossterms in the AF does not cause a major degradation in the TFR. The procedure for suppressing inter-pulse crossterms is much simpler in the Doppler-lag domain compared to the time-lag domain because the location between the inter-pulse cross-terms and auto-terms are well separated. Thus, setting a lag window, g1(τ)at Tpis sufficient for separating the auto-terms from the inter-pulse cross-terms. Figure 3 shows the AF for two pulses of PLFM signal. The position of the auto-terms is at τ≤ |Tp|and the inter-pulse cross-terms are located at T−Tp≤τ≤ T+Tp. The characteristics of PLFM at the AF are similar with the CW-LFM except there are no intrapulse cross-terms for PLFM signal. Due to this, the same cross-terms suppression procedure in CW-LFM signal is applicable for PLFM signal. Figure 4 shows the AF for two pulses of SP signal. Note that the auto-term and inter-pulse cross-terms location are similar with the PLFM and CW-LFM signals which are at τ≤ |Tp|and T−Tp≤τ≤T+Tp respectively. The only difference is the shape of the auto-terms and the cross-terms. Thus, a similar proce0 Tp T -Tp -T -T-Tp T+Tp v τ ∆fNT ` -∆fNT T-Tp G2(v ) G2(-v ) g1(-τ ) g1(τ ) (1,1),(2,2) (2,1) (1,2) Fig. 3: The AF for two pulses of PLFM signal. 0 Tp T -Tp -T -T-Tp T+Tp v τ 1/Tp -1/Tp T-Tpg1(τ ) g1(-τ ) G2(v ) G2(-v ) Fig. 4: The AF for two pulses of SP signal. dure to attenuate the cross-terms of the PLFM signal can be applied to the SP signal. The AF for one pulse of the CC4 signal is shown in Fig. 5 where the auto-terms of the signal are located at τ≤ |Tb|while the intra-pulse cross-terms are scattered on the ambiguity plane at −4Tb≤τ≤4Tb. Setting a lag window at τ≤ |Tb|is enough to extract the autoterms from the cross-terms in the ambiguity domain. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 321 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 0 τ Tb -Tb 2Tb 3Tb 4Tb v -4Tb -3Tb -2Tb ΔfN*T Δf2N*T Δf3N*T -ΔfN*T -Δf2N*T-Δf3N*T G2(v ) G2(-v ) g1(-τ ) g1(τ ) (1,1) (3,4) (2,1) (4,2) (3,1) (4,2) (4,1) (4,3)(1,2) (1,3) (2,4) (1,4) (2,4) Fig. 5: The AF for one pulse of the CC4 signal. 3.3. Estimation of Kernel Parameters Many studies were conducted to estimate the suitable kernel parameters for reducing the interference in the TFR. Adaptive optimal kernel TFR (AOK-TFR) [20] is one of the earliest techniques that incorporates window in the AF and is able to produce accurate TFR even at SNR of 0 dB. However, the capability of the method is limited to multi-component LFM signals. Adaptive Optimal Kernel Smooth-Windowed Wigner-Ville Distribution (AOK-SWWVD) [17] and Adaptive Smoothed Windowed cross WVD (ASWWVD) [15] are also able to produce accurate TFR at a low SNR but these solutions are limited to digitally modulated signals such as FSK and PSK signals. All the methods mentioned above despite proven reliable for signal representation, they lacked the capability to cover a broader class of signals. Thus, the Doppler-Lag Block Searching (DLBS) procedure is introduced in this section to produce an accurate TFR at low SNR while covering a wide class of signals. Before describing the adaptive procedures, it is crucial to first discuss the four ambiguity function quadrants as shown in Fig. 6. In the DLBS approach, it is sufficient to estimate the kernel parameters only in the Q1quadrant due to the symmetrical properties of the ambiguity domain. One of the AF properties exploited is the maximum energy that occurs at the origin of the ambiguity domain. As defined in [21], the AF is highest at the origin in comparison to the other parts of the ambiguity domain according to the following inequality: |Az(v, τ)|≤|Az(0,0)|.(9) Thus, the DLBS initiates the search at the origin and checks for a significant drop in energy in Doppler and lag. This is performed by matching the reference block, 0 v τ Q1 Q2 Q3 Q4 Fig. 6: The quadrant division in the ambiguity domain. Az(0,0) and the analyzed blocks, Az(λ1, λ2)in the AF domain to obtain the kernel parameters. The DLBS algorithm can be expressed as follows: ∆Az=|Az(0,0) −Az(λ1, λ2)|,(10) where ∆Azis the energy difference between Az(0,0) and Az(λ1, λ2)which can be defined as Az(0,0) = ∞ Z −∞ ∞ Z −∞ wa(v, τ)Az(v, τ)dvdτ. Az(λ1, λ2) = ∞ Z −∞ ∞ Z −∞ wa(v−λ1, τ −λ2)Az(v, τ)dvdτ, (11) 0≤λ1<∞,0≤λ2<∞, where wa(v, τ)is the analysis window, λ1is the instant Doppler and λ2is the instant lag. The analysis window can be described as [16] wa(v, τ) = va τa ,(12) 0≤v≤va<1 Tp ,0≤τ≤τa< Tp, where τaand vaare the analysis window size in terms of lag and Doppler respectively, Tpis the signal pulse width. In order for DLBS to accurately estimate the signal pulse width, the analysis window must be set smaller than the expected signal pulse width. All the analyzed blocks that have energy difference, ∆zabove the threshold value, Az,thd are considered as a block with a minimum number of auto-terms and can be excluded from the generation of the TFR. Figure 7 shows the Q1quadrant for PLFM signal and used as an example for DLBS procedure. The DLBS search procedures are as follows: c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 322 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER v τ 1 2 3 4 6 7 8 9 11 12 13 14 16 17 18 19 Tp 1/4Tp 1/2Tp 3/4Tp 5/4Tp 5 10 15 20 21 22 23 24 25 1/Tp 1/0.5Tp Figure 7 Fig. 7: The Q1 of PLFM in DLBS method. •Form the Doppler-lag function, Az(v, τ)for the Q1 quadrant using Eq. (8). •Compute the total energy for block 1 as a reference block, Az(0,0) and set a threshold value, Az,thd. •Compute the energy difference for block 2, Az(1/0.5Tp,1/4Tp). Since the energy difference is below the threshold value continue to the next block on the right. •The energy difference for block 3, Az(1/0.75Tp,1/4Tp)is found to be above the threshold value which means at this particular block the auto-terms energy is low. Hence, the evaluation for row (1/4)Tpis stopped and the evaluation moves to the next row, (1/2)Tp. •The evaluation for new row always starts at one column before the last evaluated column because it will ensure the evaluation start where the autoterms energy is higher. At the row (1/2)Tpthe evaluation starts at block 7. •The evaluation for row (1/2)Tpstarts from block 7 to the right direction until it reaches block 9. Since the energy difference for block 9 is above the threshold value, the search block moves to the next row, (3/4)Tpand start at block 13. •Steps from 3 to 5 are repeated for the other rows until reached row (5/4)Tp. The energy difference for all the blocks in this row are found to be above the threshold value and indicate that the evaluation for PLFM signal is completed. •The kernel parameter is estimated by choosing the location of last evaluated block that has energy difference below the threshold value. For this example, the last evaluated block is block 19 and its location represent the Doppler and lag parameters which is 1/Tpand Tprespectively. The same procedure is applicable for the other signals as their properties in the AF are similar with the PLFM signal. The threshold value of 0.3is chosen because it provides the best Doppler and lag window width for most of the signal [19]. 3.4. Fractional Fourier Transform in Cross-Terms Reduction The FrFT is actually a generalized form of Fourier Transform (FT) with the α-th order of fractional power [22] and is best expressed with the help of transformation kernel. If x(t)is the signal, then the FrFT of x(t) is given as [9] Xα(u) = ∞ Z −∞ x(t)Kα(t, u)dt, (13) where Kα(t, u)is the transformation kernel and can be expressed as Kα(t, u)=r1−jcot α 2πexpjt2+u2 2cot α−tu csc α .(14) Equation (14) is valid if αis not multiple of π. However, if αis a multiple of 2π, the kernel becomes δ(t−u). For (α+π)a multiple of 2π, the kernel becomes δ(t+u). The extension of the FrFT to the TFR of the signal is the rotated version of the WVD of the original signal by the angle θwhich is given as ρxα(t, f) = R−θ{ρx(t, f)},(15) where Rθ{} is the operator which rotates the TF plane clockwise. If the time-varying signal is linearly separable in the time-frequency plane, the cross-terms can be separated with the appropriate rotation angle and suitable TF filtering technique [5]. 3.5. IF Estimate Instantaneous Frequency (IF) estimation can be used to describe the frequency characteristics of the signal. Normally, a good estimator has to be consistent while statistically and computationally efficient [23]. Rao and Taylor proved that WVD peak based IF estimation is optimal for linear FM signals for moderate and high SNR although the estimator performance degrades significantly at low SNR [24]. Peak based IF estimator can be expressed as arg max fρ(t, f),(16) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 323 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER where fi(t)is the IF and ρ(t, f)is the TFR. Peak based IF estimator is able to produce a decent estimation as its capability to localize energy along the IF law [23]. For the purpose of measuring the performance of any unbiased parameter estimator, the Cramer–Rao Lower Bound (CRLB) is frequently used because it can provide the theoretical limit to the variance of the estimator. The most efficient estimator is the one that can achieve the lower bound on the variance [25]. The general formulation of CRLB for IF estimate is [15] var ˆ fl≥24 (2π)2γN (N2−1),(17) where Nis the average window width, and γis the SNR. With the assumption that the actual IF of the signal is known, and the signal is in discrete form, the variance of the IF estimated from measurement can be expressed as [15] var ˆ fi=1 N N−1 X n=0 ˆ fi(n)−fi(n)2,(18) where Nis the total number of samples, ˆ fi(n)is the actual IF and fi(n)is the actual IF. 4. Results and Discussion This section describes the TFRs and IF estimates for SP, CC4, LFM and CW-LFM signals using the DLBSWVD and FrFT followed by the performance of the IF estimator benchmarked with the CRLB. The performance of DLBS-WVD is presented at SNR of 5dB. This value is chosen because SNR above 10 dB is considered as high SNR from previous work on IF estimation [15]. 4.1. TFR and IF Estimate Performance By using the DLBS-WVD, the TFR plot and IF estimate for the signals using the parameters presented in Tab. 1 is produced. The energy of auto-terms components is highest at the origin and decrease positioned away from the origin. Due to this, DLBS-WVD has a difficulty to preserve the entire energy of the signal components especially when the auto-terms are located far from the origin. This problem becomes obvious for PLFM and CW-LFM signals compared to SP and CC4 signals as discussed in [19]. A small portion of the auto-terms components are suppressed for PLFM and CW-LFM signals which produced minor errors in the TFR and IF estimate. (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-Frequency Representation Frequency (Hz) 00.002 0.004 0.006 0.008 0.01 0.012 0 5 10 15 x 106 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 x 10 6 Instantaneous Freqeuncy Time (ms) Frequency (Hz) (a) TFR using DLBS. (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-Frequency Representation Frequency (Hz) 00.002 0.004 0.006 0.008 0.01 0.012 0 5 10 15 x 106 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 x 10 6Instantaneous Frequency Time (ms) Frequency (Hz) (b) IF using DLBS. Fig. 8: TFR plot and IF estimate for two pulses SP signal. Figure 8 shows the TFR plot and IF estimate for a two pulses SP signal. The IF of the signal is accurately estimated at 10 MHz. This is contributed from a clean TFR of the signal due to the successful suppression of the cross-terms in the AF domain. The TFR and IF estimate for a two pulses CC4 signal is shown in Fig. 9. The frequency components of the signal are correctly estimated given that the Costas sequence of the signal is [1 2 4 3]. In this paper, the CC4 signal is used to illustrate the functionality of the DLBS-WVD on the Costas coded class of signals. Figure 10 shows the analysis results for a two pulses PLFM signal. The frequency component of the signal is estimated from 2.27–16.75 MHz while Tand Tpare 13 ms and 4ms respectively. The time components of the signal are accurately estimated but the frequency component of the signal is estimated with 1.6% erc 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 324 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER (b) IF using DLBS (a) TFR using DLBS Time-Frequency Representation F requenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 5 10 15 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 2 4 6 8 10 12 14 16 x 10 6 Instantaneous Freqeuncy Time (ms) Frequency (Hz) (a) TFR using DLBS. (b) IF using DLBS Time-Frequency Representation F requenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 5 10 15 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 2 4 6 8 10 12 14 16 x 10 6 (a) TFR using DLBS Instantaneous Frequency Time (ms) Frequency (Hz) (b) IF using DLBS. Fig. 9: TFR plot and IF estimate for two pulses CC4 signal. ror. This error is due to inevitable suppression of the small part of signal auto-terms in the ambiguity domain. However, the estimation error is too small and can be considered insignificant. The TFR and IF estimate is shown in Fig. 11 for a two pulses CW-LFM signal. The minimum and maximum frequency of the signal is estimated at 2.19 MHz and 5.83 MHz respectively. There is about 4.2% error in the frequency component estimation which is slightly higher than the LFM signal. This is due to the intrapulse cross-terms in the CW-LFM signal which reduces the energy concentration of the signal components and produce errors in the IF estimation. Time (ms) (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-Frequency Representation F requenc y (Hz) 00.005 0.01 0.015 0.02 0.025 0 5 10 15 x 10 6 0 5 10 15 x 10 -3 5 10 15 x 10 6 Instantaneous Freqeuncy Frequency (Hz) (a) TFR using DLBS. Time (ms) (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-Frequency Representation F requenc y (Hz) 00.005 0.01 0.015 0.02 0.025 0 5 10 15 x 10 6 0 5 10 15 x 10 -3 5 10 15 x 10 6Instantaneous Frequency Frequency (Hz) (b) IF using DLBS. Fig. 10: TFR plot and IF estimate for two pulses PLFM signal. 4.2. TFR of WVD Using FrFT Using the same signal as in the previous section, the performance of FrFT in producing an accurate TFR is evaluated. The IF estimate of the signal using FrFT is similar to the result from the previous section given that the cross-terms are successfully removed. It is important to perform a TF rotation in such a way that the fractional frequency spectrum is most compact before applying any TF filtering procedure. The technique used in determining the correct order of FrFT is searching scheme [26]. This method determines the optimal fractional power by evaluating the compactness of the fractional frequency spectrum. The fractional power that provides the most compact fractional spectrum will be used for TF filtering. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 325 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER Time (ms) (b) IF using DLBS (a) TFR using DLBS Time-Frequency Representation F requenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 2 4 6 8 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6x 10 6 Instantaneous Freqeuncy Frequency (Hz) (a) TFR using DLBS. Time (ms) (b) IF using DLBS Time-Frequency Representation F requenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 2 4 6 8 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6x 10 6 (a) TFR using DLBS Instantaneous Frequency Frequency (Hz) (b) IF using DLBS. Fig. 11: TFR plot and IF estimate for two pulses CW-LFM signal. The SP signal does not require any TF rotation since the frequency spectrum is most compact at fractional power zero. The cross-terms that are located between the signal components can be easily removed by applying a TF filtering as shown in Fig. 12(a) with a red dotted line resulting in as shown in Fig. 12(b) a crossterms free TFR of the SP signal. Figure 13 shows a TFR plot for two pulses of CC4 signal which is heavily corrupted by cross-terms. For every pulse of CC4 signal, there are four signal components representing four different frequencies as indicated by red dotted boxes. The localization of crossterms that are very close to the auto-terms result in strong degradation of signal power. There is no suitable rotation angle that can be used to completely remove all the cross-terms. Furthermore, the smearing effect from the cross-terms further reduce the quality of the signal components especially at a frequency of TFR of Original Signal Frequency (Hz) Time (ms) 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (a) (b) Frequency (Hz) Time (ms) TFR After Filteration 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (a) TFR of Original Signal Frequency (Hz) Time (ms) 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (a) (b) Frequency (Hz) Time (ms) TFR After Filteration 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (b) Fig. 12: TFR plot for two pulses SP signal. TFR of Original Signal Frequency (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0 2 4 6 8 10 12 14 16 18 x 10 6 Fig. 13: TFR plot for two pulses CC4 signal. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 326