Parameters Estimation of LFMCW Signals Based on Periodic Fractional Fourier Transform

When using fractional Fourier transform (FrFT) to detect and estimate linear frequency-modulated continuous wave radar signals, two problems will appear: multiple peaks occur in FRFT image and the output SNR at the true parameter values does not increase when the observation time is longer than the signals period. A multiple period LFMCW signals parameters estimation method based on period FRFT (PFRFT) is studied in this paper. The PFRFT formula of multiple period LFMCW signals is given. The relationship among PFRFT output SNR, observation time and sample signals SNR is analyzed. The estimation accuracy formula of PFRFT is derived. At last, numerical simulation shows the effectiveness of the algorithm and PFRFT is superior to FRFT for estimating a multiple periods LFMCW signals.


Introduction
Linear frequency-modulated continuous wave (LFMCW) signal is used in various fields, including radar, sonar and electronic monitoring.Frequency analysis method for the detection of non-stationary signals are commonly used,such as: Wigner-Ville distribution (WVD)、Short-time Fourier transform(STFT), fractional Fourier transform (FrFT) and so on [1].WVD can be effectively used for a single period LFMCW signal parameter estimation, however, in the case of multi-periods signals, cross-term would interfere estimation of signals center frequency and modulation frequency; Smooth Pseudo Wigner-Ville Distribution1(SPWVD) and Time-frequency distribution series(TFDS) can effectively suppress the influence of cross terms while reduce the frequency resolution of the estimated parameters [2].As the FrFT effectively overcomes the two problems above, the algorithm and related derivative algorithm have been widely studied and applied to the field of LFMCW signals detection and parameter estimation.P.R. White and others in-depth researched and analysis of the use of the FrFT performance parameters of chrip signals detection and estimation in the literature [3].Qu Qiang and others studied the chirp signals detection and parameter estimation based on adaptive FrFT in the literature [4].C. Clemente and others use FrFT algorithm to study doppler and CS imaging algorithm.Liu Jiancheng and others detailed analyzed the fractional Fourier domain SNR of LFM signals under Gaussian white noise background in the literature [5].The Researches mentioned above do not give a method to improve the detection and parameter estimation performance of FrFT in the noise environment.
It is presented a method of estimation and detection LFMCW signals parameters based on PFRFT algorithm.When estimation and detection the parameters of a single period LFMCW signal, this algorithm and FRFT have the same advantages, but to estimate the parameters of the multi-periods LFMCW signals, this algorithm is better.

LFMCW Signals Mathematical Model
The LFMCW signals shown in Figure 1, the mathematical expression of the m-th period of the LFMCW signals [7] as follows:

FrFT
There have a variety of definitions of FrFT due to the different application background.Although the physical interpretations of these definitions are not the same, they are equivalent in mathematics.One definition of arbitrary can be derived by other definitions.The FrFT of signal s (t) can be expressed as follows [8] ; Where cot (•) means the cotangent, CSC (•) means the cosecant.
Two different situations due to the LFMCW signals' observation time is discussed as below: (1)When the signals period number M=1, observation time T ≤ Tz, The signal can be expressed as , its FrFT can be expressed as: (2)When the signals period number M>1, observation time T=M × Tz, The signal can be expressed as:


.Based on the Characteristics of FrFT [7] , the FrFT of the signal can be expressed as: By formula (5), it can be seen that the 2 ( , ) . In this case the FM rate estimation is correct.
Due to i=0,1... M-1, u has 5 values, FRFT results will appear M peaks, so the starting frequency can not be accurately estimated.
Using MATLAB to simulate the above situation, the radar echo parameters are shown in the following Signals to noise ratio -5 dB -5dB In figure ( 2

FRFT output SNR
Since the FrFT on the  shaft (0,2 π) symmetry [8] , in order to facilitate analysis, we take .Assuming Gaussian white noise conditions, the observe signals is is Gaussian white noise whose mean is 0 and variance is 2  .Literature [5] proposed that the square of the peak on two-dimensional FrFT domain is signals power, and the square of the peak of the noise as the noise power.The FRFT output SNR of the LFMCW signals can be expressed as [8] : Where [9] : 6) and figure 2, simulated the FRFT output SNR with the parameters in Table 1,the result is shown in Figure 3.The results shows that when the observation time T 0 >T z , the SNR in the peak does not increase, while T 0 <T z , the output SNR increases with the observation time increase.It is considered that FRFT is not optimal parameter estimation algorithm when the numbers of the signals' period M> 1. ,definition of period FrFT is as follows: The physical meaning of the formula (9) , searches the ture period with the multi-period LFMCW signals and do FrFT after fold-weighted with the ture period, then search the peak value in the three-dimensional parameter field and get the coordinates of the peak, at last, calculate the exact value of the signals' parameters.

Figure 4. M=3 LFMCW PFRFT results
To simulate the performance of PFrFT with the parameters in Table 1, the result as shown in Figure 4. in the case of M = 3, there is a peak in the two-dimensional domain consisting of u ,  , PFRFT output of f, g parameter estimates are correct by calculation, and the peak amplitude is 60 which 3 times of the peak amplitude in figure 2,and SNR in the peaks is significantly higher .
is Gaussian white noise whose mean is 0 and variance is 2 0  .Defineds: Where Re(•) is realistic calculation, * is conjugate, ) (t n is Gaussian white noise, so the above equation can be written as: According to formula (13): Comparing the formula ( 17) and formula ( 6

Output SNR Analysis
First-order perturbation method is uesed to analysis the PFrFT output SNR as follews.according to formula ( 11 By the formula (18), due to the introduction of the noise signals, the peak coordinate of    x P moves from the original place ) , , ( With the Taylor formula at 20) can be written as: Therefore, formula (20) ( 21) can be expressed as: The estimation error of start frequency and MF rate can be expressed as: , so according to formula (32) (33), obtains 0 Therefore, the PFrFT multi-periods LFMCW signals parameters estimation is unbiased estimation.As can be seen in formula (36) (37), the estimating mean square error of start frequency and FM rate is decreases with observation time increasing, so, it is useful to improve the signals parameter estimation accuracy by increasing the observation time.It has been shown to be equivalent to the generalized likelihood ratio test (GLRT) and maximum likelihood estimator (MLE) in detection and estimation of LFMCW signals

Conclusion
This paper analyzes the advantages and disadvantages about LFMCW signals parameter estimation by FRFT, and proposes a multi-periods LFMCW signals parameter estimation algorithm basing PFrFT algorithm, the conclusions as follows: (1)When using FRFT to estimate multi-period LFMCW signals parameters, there are a plurality of peaks appearing in converting figure, and the SNR do not increase when the observation time is more than one period.
It is means that FRFT is not optimal multi-period LFMCW signals parameter estimation algorithm.
(2) PFRFT output SNR is (4) It is solved the cross-term problems of WVD, the resolution problems of SPWVD and TFDS, inadequate use of the sample data of FrFT.
Fig. 1.LFMCW Signals ), (a) (b) (c) means the projection on α-axis, μ-axis and two-dimensional map of signals FRFT when M=1.By calculation, the results of simulation and practical results are the same.(d)(e) (f) means the projection on α-axis, μ-axis and two-dimensional map of signals FRFT when M=3. Graph (a) (d) peak coordinates are consistent, that is to say, signals modulation frequency estimation is correct, Fig. (e) (f) have 3 peak values respectively, that will interfere estimation of initial frequency.At the same time, comparing M=1 and M=3, the output signals to noise ratio does not significantly improve while increasing the sample signals observation time.It is considered that FRFT not optimal multi-periods LFMCW signals parameter estimation algorithm.

Figure 3 .
Figure 3. Relationship between FRFT output SNR and observation time

Figure 5
Figure5shows the value curve of the peak when searching the value of period z T ~.Maximum peak is achieved in According to Redefinition of SNR in the literature[5], PFrFT output SNR can be expressed as: of PFRFT could be expressed as: ), PFrFT and FrFT of LFMCW signals have the same output SNR expression.But the value range of 0 T is not the same, 0 T in the formula (6) is not more than z T and 0 T in the formula (17) means sample observation time.It shows that PFrFT and FrFT of the signals have the same output SNR when the sample observation time is less than or equal the period of the signal.But with increasing sample observation time, FrFT output SNR does not increase, while PFrFT output SNR continues increasing, PFrFT takes full advantage of all the information of sample signals.As shown in Figure (6).

Figure 6 .
Figure 6.Compare PFRFT、FRFT output SNR sample SNR is large enough, The PFRFT output SNR at the true parameter values varies approximately linearly with the signal observation time (3) In the noise environment, estimating the multi-period LFMCW signals parameters with PFrFT is unbiased estimates.

B
is the FM bandwidth；j is the imaginary unit；t is time variable, m=1…M,M is the number of the signals' period.The parameter of every LFMCWS signals' period is same.It is used the symbol A, f and g to express the amplitude, starting frequency and frequency modulation rate of the signals in this paper. ),