Nonlinear FM Waveform Design to Reduction of sidelobe level in Autocorrelation Function

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1 017 5 th Iranian Conerence on Electrical (ICEE) Nonlinear FM Waveorm Design to Reduction o sidelobe level in Autocorrelation Function Roohollah Ghavamirad Department o Electrical K. N. Toosi University o Technology r_ghavamirad@ee.kntu.ac.ir Hossein Babashah Department o Electrical Shari University o Technology ( babashah_hossein@ee.shari.edu) Mohammad Ali Sebt Department o Electrical K. N. Toosi University o Technology sebt@kntu.ac.ir Abstract This paper will design non-linear requency modulation (NLFM) signal or Chebyshev, Kaiser, Taylor, and raised-cosine power spectral densities (PSDs). Then, the variation o peak sidelobe level with regard to mainlobe width or these our dierent window unctions are analyzed. It has been demonstrated that reduction o sidelobe level in NLFM signal can lead to increase in mainlobe width o autocorrelation unction. Furthermore, the results o power spectral density obtained rom the simulation and the desired PSD are compared. Finally, error percentage between simulated PSD and desired PSD or dierent peak sidelobe level are illustrated. The stationary phase concept is the possible source or this error. Keywords: Nonlinear requency modulation; Peak sidelobe level; Mainlobe width; Stationary phase concept. I. INTRODUCTION The principal advantages o pulse compression with NLFM waveorm are high Signal-to-Noise Ratio, easily implementable transmitter due to constant envelope o the transmitted signal, and more reduction o sidelobe level in autocorrelation unction (ACF) with regard to linear requency modulation (LFM) method [1-7]. In this paper, we analyze variation o peak sidelobe level with regard to mainlobe width or our dierent window unctions. In order to reduce sidelobe level o autocorrelation unction, shaping the power spectral density can be used, since autocorrelation unction is the inverse Fourier transorm o the PSD [8], [9]. To achieve this goal, Chebyshev, Kaiser, Taylor, and raised-cosine windows are considered as our desired PSDs. Aterwards, group time delay unction in requency domain is carried out using the stationary phase concept (SPC). This was done by integration o PSD unction and applying boundary conditions [10]. Then group time delay inverse unction is computed yielding requency unction in time domain. The results o autocorrelation unction obtained rom the simulation are presented or all the aorementioned window unctions and comparison between sidelobe level and mainlobe width are made. As the sidelobe level reduces, the mainlobe width increases [11], [1]. This can diminish target detection accuracy. In order to preserve the accuracy, an optimized condition should be considered which is discussed in the paper. Furthermore, the resulting PSD plot and desired PSD plot were compared in order to calculate error percentage. It should be mentioned that the SPC is the possible source or this error due in large to the act that SPC is based on approximated relations. The remainder o the paper is organized as ollows. NLFM signal is designed or our dierent windows in section II. Section III discusses the simulation results; Section IV concludes the paper. II. NLFM WAVEFORM DESIGN A. Method based on Stationary Phase Concept The irst study on using SPC or NLFM was carried out in 1964 by Fowle [8]. To design NLFM signal, a low-pass signal denoted as x(t) is considered. x(t) = a(t)exp (jφ(t)) Where a(t) and φ(t) are amplitude and phase o the signal respectively. The instantaneous requency m at time t m is deined as time derivative o the phase: m = 1 dφ(t m ) π dt We shall write the Fourier transorm o x(t) as X(). Based on SPC, PSD at a requency is large i the rate o requency change at that time is relatively small. Thus, the relation between PSD and requency variation can be expressed as: X( m ) π a (t m ) φ (t m ) According to (3) it is clear that PSD is inversely proportional to second time derivative o phase and it is directly proportional with signal amplitude. Since requency changes with regard to time is linear or LFM, the second time derivative o phase is constant and the PSD depends on signal amplitude. In NLFM approach, signal amplitude is assumed to be constant (a(t) = A, A is constant) and PSD is depended on the value o second order phase derivative. Let us consider X () with an desired PSD such as Z (), where Z () only depends on the second (1) () (3)

2 017 5 th Iranian Conerence on Electrical (ICEE) derivative o phase o x(t) signal. Equation (3) is reproduced in requency domain as: Φ () Z () A π = kz (), k = constant I Z() is deined in the requency range o B (B is the bandwidth), then the irst derivative o phase Φ () is resulted by integral o the second derivative Φ () which can be expressed as: Φ () = Φ (θ)dθ = k Z (θ)dθ The ollowing relation deines group time delay unction T g () as: T g () = 1 π Φ () Substituting o (6) in (5) yields: T g () = k π Z (θ)dθ = k 1 Z (θ)dθ (5) (6) + k Where k 1 and k are the integration constant which are determined using the boundary conditions o group time delay unction. I the signal pulse width is equal to T, then the boundary conditions o group time delay unction written in the orm T g (B ) = T, T g ( ) = T [10]. Moreover, instantaneous requency as a unction o time is given by (t) = T g 1 () Finally, the phase o the designed signal can be obtained rom requency unction using the ollowing equation. t φ(t) = π (θ)dθ T B. Dierent Desired PSDs Analysis 1) Raised Cosine Window NLFM waveorms can be designed using the raised cosine window. For reason o better results, second order raised cosine window is considered in this paper. Thus, PSD relation can be written in the ollowing orm w(n) = k + (1 k)cos ( πn M 1 ), n (10) M 1 Where k is a constant value which can control sidelobe level and M is window length. Assume that = nb (M 1) then, by integration and applying the boundary condition, group time delay unction can be derived as: T g () = T ( k + (1 B 1 + k ) sin ( π π ) B ) (8) (9) (11) (4) (7) The inverse unction o T g () complicated and it does not have closed-orm expression. To illustrate the result, a simulation o inverse unction or T g () was perormed. ) Chebyshev Window The optimal Dolph-Chebyshev window transorm in closedorm is given by: W(m) = cos {Mcos 1 [βcos ( πm )] M cosh[mcosh 1, (β)] m = 0,1,,, M 1 β = cosh [ 1 M cosh 1 (10 α )], α,3,4 (1) (13) Where M and α are window length and parameter or determining the Chebyshev norm o the sidelobes to be 0α db, respectively. Aterwards, zero-phase Dolph-Chebyshev window was calculated using the inverse DFT o W(m) [13]. M 1 w(n) = 1 W(m). exp (jπmn N M ), n M 1 m=0 (14) 3) Kaiser Window The Kaiser window can be computed through approximation o the discrete prolate spheroidal sequence (DPSS) window using Bessel unctions which is written in the ollowing orm: w(n) { I 0 (πα 1 ( n M ) ) n M 1 I 0 (πα) 0 elsewhere (15) In the above equation, I 0 and α represent the zero-order modiied Bessel unction o the irst kind and an arbitrary real number, respectively. In act, α shows the shape o the window in the requency domain and parameter β πα. M is the length o window [13]. The group time delay unction in the Chebyshev and Kaiser windows should be determined numerically. 4) Taylor Window The equation that express Taylor weighting unction is as ollows: n 1 w(n) = 1 + F m m=1 cos ( πmn M 1 ), n M 1 (16) In (16), F m = F(m, n, η) is Taylor coeicients o the mth order. η and n show ratio o mainlobe over sidelobe level and number o sidelobes at equal level, respectively. M also identiies window length [4]. Assume that = nb (M 1), by integration and applying the boundary condition, group time

3 017 5 th Iranian Conerence on Electrical (ICEE) Figure 1. Changes in peak sidelobe level with regard to normalized mainlobe width in autocorrelation unction or our dierent windows. delay unction in Taylor window can be derived as: In this paper n is assumed to be, so group time delay unction is then given by T g () = T ( B + F 1 sin (π π B )) (18) The inverse unction o T g () in the Taylor window should be carried out numerically. III. n 1 T g () = T ( B + 1 π F m m m=1 sin ( πm B )) SIMULATION AND RESULTS Table 1 lists the data that was used in simulation. Comparison between changes in mainlobe width and peak sidelobe level (PSL) are made. Mainlobe width is considered in the order o -3 db then, PSL can be computed by the ollowing equation [14] PSL(dB) = 0 log 10 ( max( R(τ) ) ), z R(0) 1 τ T (17) (19) Where R(τ) is the amplitude o autocorrelation unction o NLFM signal in the time delay o τ second, and z 1 is the irst-null o R(τ). Figure 1 illustrates changes in PSL versus normalized mainlobe o autocorrelation unction o NLFM signal or our dierent windows. Normalized mainlobe width is calculated using mainlobe width o autocorrelation NLFM signal divide by LFM signal. TABLE I. SIMULATION PARAMETERS Parameter Value Unit Pulse width.5 μs Bandwidth 100 MHz Sampling rate 1 GHz The results obtained or raised cosine and irst-order Taylor windows bear a close resemblance due to their similar group time delay unction. At the point where the normalized mainlobe width is equal to 1.364, peak sidelobe level is db or raised cosine and irst-order Taylor indicating better condition with regard to Chebyshev and Kaiser cases. Figure illustrates normalized NLFM and LFM autocorrelation unction amplitude signal or our aorementioned windows. From this igure it can be seen that every window experience maximum sidelobe level reduction. A comparison o the two results reveals that decrease in NLFM sidelobe level is avorably more signiicant than LFM. As mentioned earlier, raised cosine and Taylor autocorrelation unction are similar to one another. The results o PSD obtained rom the simulation and the desired PSD or dierent windows can be compared in Figure 3. The observed dierence between simulated PSD and desired PSD can be explained by the act that SPC is based on approximated relations. To reduce error percentage, error optimization methods can be used. Figure 4 depicts error percentage o the simulated PSD and desired PSD against peak sidelobe level o autocorrelation unction or NLFM signal. It is apparent rom this igure that the reduction o sidelobe level results in lower calculated error percentage.

4 017 5 th Iranian Conerence on Electrical (ICEE) Figure. NLFM and LFM signal autocorrelation unction amplitude or our dierent window unctions. Figure 3. Ampitude o desired PSD and simulated PSD or our dierent window unctions. Figure 4. Error percentage o the simulated PSD and desired PSD against peak sidelobe level o autocorrelation unction or designed NLFM signal.

5 017 5 th Iranian Conerence on Electrical (ICEE) IV. CONCLUSION Based on the results, it can be concluded that reduction o sidelobe level in NLFM signal can lead to increase in mainlobe width o autocorrelation unction, but this increment is not signiicant with regard to sidelobe level. The results obtained also indicate that raised cosine and irst-order Taylor windows perorm better than Chebyshev and Kaiser windows on reduction o sidelobe level. Approximated relations o stationary phase concept result in low error percentage. REFERENCES [1] M. Łuszczyk and A. Łabudzinski, "Sidelobe level reduction or complex radar signals with small base," 01 13th International Radar Symposium, Warsaw, 01, pp [] B. L. Prakash, G. Sajitha, and K. Raja Rajeswari, "Generation o random NLFM signals or radars and Sonars and their ambiguity studies," Indian Journal o Science and Technology, vol. 9, no. 9, Aug [3] Y. K. Chan, M. Y. Chua, and V. C. Koo, "Sidelobes reduction using simple two and tri-stages non linear requency modulation (Nlm)," Progress In Electromagnetics Research, vol. 98, pp. 33 5, 009. [4] Pan Yichun, Peng Shirui, Yang Keeng and Dong Weneng, "Optimization design o NLFM signal and its pulse compression simulation," IEEE International Radar Conerence, 005., 005, pp [5] I. C. Vizitiu, F. Enache and F. Popescu, "Sidelobe reduction in pulsecompression radar using the stationary phase technique: An extended comparative study," 014 International Conerence on Optimization o Electrical and Electronic Equipment (OPTIM), Bran, 014, pp [6] L. R. Varshney and D. Thomas, "Sidelobe reduction or matched ilter range processing," Proceedings o the 003 IEEE Radar Conerence (Cat. No. 03CH37474), 003, pp [7] C. Leśnik, "Nonlinear requency modulated signal design," Acta Physica Polonica A, vol. 116, no. 3, pp , Sep [8] N. Levanon and E. Mozeson, Radar signals. New York, NY, United States: IEEE Computer Society Press, 004. [9] M. Luszczyk, "Numerical Evaluation o Ambiguity Function or Stepped Non-Linear FM Radar Waveorm," 006 International Conerence on Microwaves, Radar & Wireless Communications, Krakow, 006, pp [10] S. Boukea, Y. Jiang and T. Jiang, "Sidelobe reduction with nonlinear requency modulated waveorms," 011 IEEE 7th International Colloquium on Signal Processing and its Applications, Penang, 011, pp [11] S. Alphonse and G. A. Williamson, "Novel radar signal models using nonlinear requency modulation," 014 nd European Signal Processing Conerence (EUSIPCO), Lisbon, 014, pp [1] W. Yue and Y. Zhang, "A novel nonlinear requency modulation waveorm design aimed at side-lobe reduction," 014 IEEE International Conerence on Signal Processing, Communications and Computing (ICSPCC), Guilin, 014, pp [13] K. M. M. Prabhu, Window unctions and their applications in signal processing. Boca Raton, FL, United States: Taylor and Francis, 013. [14] X. Wang, T. Jin, X. Qi and X. Wang, "Research on simultaneous polarimetric measurement based on PN-NLFM signals," IET International Radar Conerence 015, Hangzhou, 015, pp. 1-5.