Design and Characteristic Analysis of Multicarrier Chaotic Phase Coded Radar Pulse Train Signal

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Introduction
The MCPC signal based on OFDM technique [1] is a new wideband radar signal which has attracted much attention recently.It has a flexible structure, a thumbtack ambiguity function.This signal not only has the advantages of narrowband radar but also can synthesize the wideband signal with orthogonal narrowband signals so as to achieve multichannel separation and quick processing.Starting from OFDM signal, Levanon and Mozeson [2] analyzed the structure, spectrum, ambiguity function, autocorrelation, and power spectrum of monopulse, continuous wave, and pulse train of MCPC signal.Besides, the methods for decreasing the peak-to-mean envelope power ratio (PMEPR) of these signals are elaborated.Studies were performed on the waveform design, pulse compression, target detection, and imaging of MCPC signal in [3].Reference [4] introduced a method of least squares for the allocation of a proper phase to obtain desired ambiguity function.In literature [5], the features of MCPC signal and frequency stepped signal were combined to propose the multicarrier phase coded frequency stepped radar signal, which has a high range resolution yet has the defect of range-Doppler coupling.
In order to extract Doppler information, monopulse needs to be accumulated to obtain the pulse train signal.The pulse train signal reserves the high range resolution of monopulse signal while maintaining the velocity resolution performance of continuous wave signal [6].However, the traditional pulse train signal has a poor performance in measuring remote and high speed targets because of range or velocity ambiguity, and the antijamming performance is also low.MCPC signal has a flexible structure, and different pulse train signals can be obtained using different phase coding methods.Studying the MCPC pulse train signal with complex modulation can improve the detection performance and LPI performance of radar.
Like the noise signal, the chaotic signal exhibits continuous power spectrum, initial-value sensitivity, ergodicity, and aperiodicity, and therefore chaotic signal possesses good range and velocity resolutions, thumbtack ambiguity 2 International Journal of Antennas and Propagation  function, excellent antijamming ability, and electromagnetic compatibility.Thus, chaotic signal avoids the problem of range ambiguity.Moreover, it overcomes many shortcomings of the noise signal, such as the difficulties of generation, replication, and application.These properties of chaotic signal conform to the low probability of intercept (LPI) and antijamming requirements for modern radar.In this study, the multicarrier chaotic phase coded pulse train signal is designed using the chaotic biphase code for phase modulation on MCPC pulse train by different modes.The ambiguity function as well as the autocorrelation performance of the signal is simulated, and the LPI performance is also analyzed.

Structure of MCPC Pulse Train Signal
The complex envelope of the monopulse MCPC signal is given by where  is the subcarrier number and  is the phasemodulated bits.Consider  , =   , , where  , is the th phase element of the th subcarrier.One has () ≡ 1 for 0 ≤  <   and zero elsewhere.The frequency difference between two adjacent subcarriers Δ is set equal to the inverse of the bit duration   yielding orthogonal frequency division multiplexing.
is the complex weight associated with the th subcarrier, and |  | is the frequency weighted amplitude.  is the frequency weighted phase, also known as the initial phase.The structure of the MCPC signal is illustrated schematically in Figure 1.
The schematic diagram shows that the MCPC signal consists of  sequences transmitted simultaneously on  subcarriers.Each sequence contains  phase-modulated bits.
The general expression for the MCPC pulse train complex envelope    () is where  is the period number and   is pulse repetition interval.Consider  ,, =   ,, , where  ,, is the th phase element of the th sequence in th pulse.The rest of the parameters are the same as those of monopulse MCPC signal mentioned above.

Design of Multicarrier Chaotic Phase Coded Radar Pulse Train Signal
Chaos is considered as a phenomenon of random behavior in certainty nonlinear dynamical system [7].Chaotic mapping can generate plenty of signals [8] whose rules are difficult to grasp by the interference side.Considering the balance, correlation, and the flatness of power spectrum in band, this study uses Chebyshev mapping [9] to generate the biphase coded sequence.Chebyshev mapping is defined as where  0 ∈ (−1, 1) and Chebyshev mapping order  = 4.
The chaotic signal generated by chaotic maps is decimal.This decimal chaotic signal can be converted to binary sequence using the threshold comparison approach.Suppose   is the binary code after quantization,   is the original decimal chaotic sequence, and  = lim  → ∞ (1/) ∑  =1   is the mean of the sequence, which is the threshold.The binary code   is obtained by the following expression: Because there are many parameters in MCPC pulse train signal, the phase code of the subcarrier in each pulse can be changed to obtain MCPC pulse train signals with different properties.Based on the differences of the phase codes of each pulse and subcarrier in MCPC pulse train signals, the classification of MCPC pulse train signal is made as shown in Table 1.
Considering the features of the chaotic sequence mentioned above, we use the chaotic sequence to perform phase code modulation on the MCPC pulse train so as to generate chaotic modulation NNS MCPC pulse train signal.Because different initial values can result in chaotic sequences with different properties, all the chaotic sequences mentioned in the following refer to the chaotic biphase sequence with maximum main-to-sidelobe ratio as selected by the principle of autocorrelation maximization [10].We design the following two chaotic modulated NNS MCPC pulse train signals with  periods,  subcarriers, and  bits through different chaotic modulation methods.
Figure 2 shows the structure of Chaos NNS MCPC I pulse train.
(2) Each time a pulse is transmitted, the Chebyshev mapping generates  =  ×  chaotic biphase codes with different initial values.The codes are expressed in set form by { 1 ,  2 ,  3 , . . .  }.Suppose the phase coded set of a single MCPC pulse is  , , where  = 1, 2, . . . and  = 1, 2, . . ..The phase of each subcarrier in the present transmitted pulse is modulated based on Therefore, the second type of chaotic modulation NNS MCPC pulse train is obtained and denoted by Chaos NNS MCPC II.The structure of Chaos NNS MCPC II pulse train is shown as Figure 3.

Derivation of Ambiguity Function
The ambiguity function is an important tool for analyzing radar signals.The ambiguity (, V) of a signal () is given by where  is the time shift variable and V is the frequency shift.By substituting (2) into (7), the ambiguity function of NNS MCPC pulse train is obtained: Let ( −  1 )Δ + V =   ,  − ( − 1)  − ( − 1)  =   , and the integral in ( 8) can be simplified as x 1 x 6 x 7 x 8 x 9 x 2 x 3 x 4 x 5 x 10 x 15 x 20 x

Initial value 1 Initial value 2 Initial value
If   > 0, the integral of (10) becomes if   < 0, the integral of (10) becomes According to (11) and ( 12), the result for the integral in (10) can be expressed as ∫ is the cross-ambiguity function of Chaos NNS MCPC pulse train.

Experimental Results
Since the phase codes in NNS MCPC pulse train signal are different, the periodic side lobe ratio of this signal at integer multiple of   is obviously lower than that obtained for INS MCPC pulse train signal.
Figure 6 shows the ambiguity function of the Chaos NNS MCPC I proposed in this paper with initial values 0.35.

Analysis on Autocorrelation
Performance.There are many parameters in multicarrier chaotic phase code pulse train.Changing the number of subcarriers, bits, and periods will affect the autocorrelation performance of the signal.The simulation result is as follows.
(1) Influence of Subcarrier Number on Autocorrelation.While the numbers of periods and bits are kept unchanged ( = 5,  = 13), the number of subcarriers is increased to three times ( = 21) the original number (the sampling rate is also changed to three times the original value).The corresponding autocorrelation performance of Chaos NNS MCPC I and Chaos NNS MCPC II is shown in Figure 8. ( = 5,  = 7), the numbers of bits is increased to three times ( = 39) the original number.The corresponding autocorrelation performance of Chaos NNS MCPC I and Chaos NNS MCPC II is shown in Figure 9.
(3) Influence of Period Number on Autocorrelation.While the numbers of subcarriers and bits are kept unchanged ( = 7,  = 13), the numbers of periods is increased to three times ( = 15) the original value.The corresponding autocorrelation performance of Chaos NNS MCPC I and Chaos NNS MCPC II is shown in Figure 10.Figures 8 and 9 show that the length of chaotic sequence increases whenever  or  grows.Thus, the randomness increases and the autocorrelation performance improves further.In addition, the autocorrelation of Chaos NNS MCPC II is more improved than that of Chaos NNS MCPC I.This is because, under the condition of a short chaotic sequence, increasing the length more obviously improves the randomness.It can be seen from Figure 10 that the chaotic sequence becomes longer as the number of periods increases, and the autocorrelation performance improves.However, due to the increase of period number, optimal initial values are running out of options, which make the autocorrelation performance of chaotic sequence become worse.Therefore, the improvement of autocorrelation performance of Chaos NNS MCPC II signal is not apparent.We can improve the signal autocorrelation performance under large period number by changing the type of chaotic mapping.

Analysis on LPI Performance.
The intercept factor of radar [11] is written as where  is a parameter related to radar and reconnaissance receiver and can be regarded as a constant. denotes the time-bandwidth product. is an important factor that affects the LPI performance of radar.The larger, the better LPI performance is. = /  and  =   are the MCPC pulse

Figure 2 :
Figure 2: Structure of Chaos NNS MCPC I pulse train.

6 International
Journal of Antennas and Propagation Furthermore, we note that different  has different initial values in Chaos NNS MCPC II pulse train.

5. 1 .
Ambiguity Function.The simulation compares the ambiguity of the INS MCPC and NNS MCPC pulse train signals based on P4 code with that of Chaos NNS MCPC I and Chaos NNS MCPC II proposed in this study.Suppose  = 5,  = 7,  = 13,   =  × 10 −6 s, and the duty ratio is 33%.The phase code of INS MCPC pulse train is the P4 code cyclic shift.Figure 4 shows the ambiguity function of INS MCPC based on P4 code.

Figure 7
shows the Chaos NNS MCPC II proposed in this paper.The initial values are [0.75, −0.52, −0.74, 0.15, 0.63].It can be seen from Figures 6 and 7 that the two signals proposed in this paper possess a thumbtack ambiguity function since randomness is introduced into NNS MCPC pulse train signal in each subcarrier and pulse.Thus, it has high range and velocity resolution.The main lobe near the origin is narrow and the periodic side lobe at integer multiple of   is even lower than that obtained with P4 COCS MCPC pulse train signal.That means the correlation in each period attenuates due to randomness.Although Chaos NNS MCPC I and Chaos NNS MCPC II pulse train have about the same autocorrelation performance, Chaos NNS MCPC II increases the complexity and flexibility of signal design because of different chaotic initial values in each pulse, resulting in higher waveform diversity.

Figure 10 :Figure 11 :
Figure 10: Influence of period number on autocorrelation.

Table 1 :
Classification of MCPC pulse train signal.