[2]
is the rotation angle, ie the angle between the fractional Fourier domain and time domain. The rotation angle, is the order of fractional Fourier transform. When, it is the classical Fourier transform. Its rotational axis coordinate system shows in Figure 1. is the fractional Fourier transform operator, is the transformation kernel, is an integer, is the impulse function. The purpose of introducing the conversion axis coordinate system is to get the best signal time-frequency representation. Figure1 Time-frequency plane rotation angle coordinate Figure 1 shows: ， corresponds to the classical Fourier transform of signal. Its rotational relationship is.
DOI: 10.1016/j.sigpro.2008.06.017
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[3]
FrFT is more flexibility compared to the traditional Fourier transform, suitable for processing non-stationary signals. Many properties of FrFT on the theoretical study are of great significance. When, the calculation process can be broken down into the following four steps.
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[1]
The original signal is multiplied with a chirp function.
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[2]
Do the Fourier transform.
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[3]
Multiplied with a chirp function.
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[4]
Multiplied by complex amplitude factor.
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[7]
In the above definition, although can be taken any real number, periods of the fractional Fourier transform is 4. So it is usually consider only in [0, 4]. 2. 2 Principle of STFT The basic idea of short-time Fourier transform is that in the framework of the traditional Fourier transform, the non-stationary signals are viewed as superposition of a series of short-time stationary signal. Short of the time is achieved by adding windows and cover the entire domain by translating parameters. For a given non-stationary signals, , short time Fourier transform of signal is defined as.
DOI: 10.1016/j.sigpro.2019.07.024
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[8]
In general, is the real function and its energy of Fourier transform concentrated in the low frequency. It can be viewed as the impulse response of a low-pass filter. As can be seen from the definition of formula, the signal is divided by adding window functions by translating parameter.
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[9]
Do Fourier transform for the signal on the basis of obtaining signal in interval of original signal which is in near time. With the changes of the translation parameters, time window determined by moves on the timeline, so that the signal gradually go into the state to be analyzed. Since the segment signal shows characteristics of the interval of the original signal, so the Fourier transform of the segment signal reflects the frequency distribution of the original signal at the interval. Its advantage is simple and easy to implement the algorithm, the processing time is short and suitable for time-frequency of all non-stationary signal. 2. 3 Principle of JTFA Since the fractional Fourier transform can be rotated any angle compared to Fourier transform, it can improve the resolution of short-time Fourier transform on the basis of fractional Fourier transform. Joint -time Fourier transform is defined as.
DOI: 10.1016/j.sigpro.2008.06.017
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[10]
calculations can be seen as general short-time Fourier transform of signal. The initial window uses the fractional Fourier transform. If the STFT window is too narrow, the resolution will be decreased in frequency domain; if the window is too wide, the resolution will decrease in time domain. Time domain and frequency domain can't get both high resolution simultaneously. The joint time-Frequency analysis enables the energy more concentrated in time-frequency domain. Selecting the appropriate angle of rotation can get the best time-frequency representation, meanwhile, this method is not affected by cross term interference, solving the problem that the resolution of single short-time Fourier transform is low. 3 Simulation Signal forms of adaptive radar are various, and its waveforms are complex. According to the radar function, the waveform is usually designed on the basis of characteristics of the ambiguity function. Ambiguity function is mainly divided into four categories. The paper selected four kinds of signals according to ambiguity function, namely, single-frequency signal, LFM signal, BPSK signals and coherent pulse-sequence signal. Ambiguity function of these four kinds of signals represents four types of ambiguity function. Simulation of the joint time-frequency analysis is shown in Figure 2. (a)single-frequency signal STFT (b) single-frequency signal JTFA (c)LFM signal STFT (d) LFM signal JTFA (e) BPSK signal STFT (f) BPSK signal JTFA (g) coherent pulse-sequence signal STFT (h) coherent pulse train signal JTFA Figure 2 Comparison of Signal STFT and JTFA Simulation environment: Microsoft Windows XP Professional 2002, CPU: 2. 6GHz, HDD: 150G, Memory: 2G. The simulation tool is MATLABR2009b. Simulation signal: pulse width T=2us, carrier frequency fc=2. 5e6Hz, sampling rate: 300MHz. LFM signal modulation factor k = 1e13, coherent pulse-sequence signal PRI=4us. Figure 2 shows that the joint time-frequency analysis method do have a better result than single short-time Fourier transform in the radar pulse signal analysis process. Energy of the joint time-frequency analysis is more concentrated in certain areas, therefore, the joint time-frequency analysis method can be appropriate to cover defects of STFT. In addition, the order of FrFT will affect the signal analysis results when the window function is for fractional Fourier transform. To get the best effect of time-frequency representation, the frequency axis should rotate a right angle. 4 Conclusion This article applies the joint time-frequency analysis method to adaptive radar signal analysis. Simulation results show that the joint time-frequency analysis method is better than the single short time Fourier transform. The joint time-frequency analysis method can effectively analyze the time-frequency characteristic of radar signals and improve the analysis resolution. This distribution can be more accurately reflects the frequency of the signal changes over time. 5 References.
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