A Parametric Spectral Moments Estimation Algorithm Based on Fitting Autocorrelation Sequence Xiaoguang Lu*,**, Renbiao Wu**, Juan Qin*
[email protected] * School of Electronic Information Engineering, Tianjin University, Tianjin, P. R. China ** Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin, P. R. China
Abstract The first three spectral moments of weather radar echoes are closely correlative with the types and characteristics of meteorological phenomena. The weather echoes and ground clutter have power spectra with shape following closely to Gaussian function. The first three spectral moments can be estimated using the modelled autocorrelation function. A parametric spectral moment estimator is proposed based on fitting the autocorrelation sequence. And the RELAX is used to deal with the scenarios of two or more mixed Gaussian spectrums. Finally, experimental results with simulated weather radar signals and performance analysis demonstrate that the present estimator is efficient with higher resolution. Keywords: Weather Radar; Parametric Spectral Moments Estimation; Autocorrelation Sequence; RELAX
1 Introduction Meteorological target information from echo signal plays an important role in the radar application system. The pulsed Doppler weather radar should supply the three most important spectrum moment estimates[1,2]. These are 1)the echo power or zero moment of the Doppler spectrum (this is an indicator of liquid water content or precipitation rate in the resolution volume), 2) the mean Doppler velocity or the first moment of the spectrum normalized to the zeroth moment (this is equal to the mean motion of scatters weighted by their cross sections), and 3) the spectrum width, the square root of the second moment about the first of the normalized spectrum, a measure of velocity dispersion (i.e., shear or turbulence) within the resolution volume. Accordingly, the atmospheric information can be retrieved from the estimation of these spectral moments. Some other application areas are ultrasound imaging in medicine[3], wind-profiler radar[4,5], synthetic aperture radar (SAR)[6]. Atmosphere objects and ground can usually be represented as ensembles of randomly located scatters. The total electromagnetic field scattered from an object is the result of wave superposition of elementary contributions from each scatter. These elementary waves have random amplitudes and phases. The radar echoes reflected from the distributed targets are the superposition of all the echoes from the scatters in the
resolution cell. Owing to more than thousands of hydrometeors’ presence in a resolution volume, the weather echo has power spectra with shapes following closely the Gaussian function[7]. The three spectral moments can be used to identify the atmosphere. In addition, ground clutter is a major factor causing the performance degradation of the weather radar. With the theoretical and the experience conclusions, the ground clutter and the weather echo can be distinguishable by their spectrum width[7]. The first three spectral moments are of interest[8], many spectral moments estimation approaches have been tested in literature. Pulse Pair Processing[9] uses the first two elements of autocorrelation sequence, which is simple and efficient. Fast Fourier Transform (FFT) is another common method to estimate the spectral parameters, and also some revised FFT based methods are proposed to improve the estimation precision[10]. However, in case of overlapping echoes (i.e., containing two or more Gaussian-shaped spectra), all these nonparametric methods may fail, which are based on an assumption of the scenario with a single echo. The radar signal with two or more Gaussian echoes can be very common in some scenarios, such as the weather echo involved with the ground clutter, UHF wind profiling radar signal with the clear-air echo, the hydrometeor echo and the ground clutter. Some parametric approaches[11-14] are developed, which employ the Gaussian spectra model. The Gaussian spectral is modeled with the referred three spectral moments. Then these algorithms are proposed based on different estimation criteria. The optimum algorithm is Maximum Likelihood estimator. But it can’t be solved explicitly, which can only be solved by search method with extreme computation load. Reference [15] presented a Stochastic Maximum Likelihood (SML) algorithm, which is optimized with a second-order steepest descent method. Reference [16] describes a so-called weighted pseudosubspace fitting estimator. It exploits the low rank properties of the covariance matrix of the data, and it is optimized by iteration. The iteration-like methods are very sensitive to initialized initialization[16]. Essentially speaking, the aforementioned parametric methods are estimators involved with multiple parameters optimization. It can be solved with circular optimization (i.e., a relaxation methodRELAX[17]). In this paper, the model of the autocorrelation sequence of Gaussian signal is used to implement the spectral moments estimation. And RELAX is used for the scenario with overlapped Gaussian spectra.
The paper is organized as follows. In Sect.2, the underlying signal model is given and the estimation problem is present. In Sect.3, our estimator is deduced and presented. In Sect.4, the performance analysis is provided with computer simulation results. In Sect. 5, conclusions are provided.
Let the autocorrelation sequence of the received radar signal is N
¦D r (E , f ) V >0 i
N
¦S ( f ) V
PS ( f )
T
i 1
§ M 1 · § M 1 · º ª 2S 2 Ei ¨§ M 1 ¸· j 2S fi ¨§ M 1 ¸· j 2S f i ¨ 2S 2 Ei ¨ ¸ ¸ 2 ¹ 2 ¹ © © © 2 ¹ © 2 ¹» «e e " 1 " e e « » ¬ ¼
i
(2)
2 n
r ( Ei , fi )
where V is the power of an additive white Gaussian circular noise, N is the assumed number of the Gaussian echoes contained in the power spectrum. Then (3)
where Pi , f i ,V i are the three unknown spectral moments of the ith Gaussian echo. From (2)and (3), the asymptotic autocorrelation sequence of the time series s (t ) is
¦ P exp 2S V W exp j 2S f W V 2
i
2 2 i
i
G (W )
2 n
(7)
r ¦D i r ( Ei , fi )
2
(8)
i 1
And the minimization is a nonlinear least square problem, which can’t be solved explicitly and the searches for the global minimum of the cost function are computationally time-consuming. Generally, the solution can be obtained by relaxation based method[17]. Next, the processing steps are given.
^
It is assumed that Dˆ i , Eˆi , fˆi
`
N
are given or estimated, and
i 1, i z l
let rl
N
r ¦ Dˆ i r ( Eˆi , fˆi )
(9)
i 1 i zl
i 1
is estimate the parameters set ^Pi ,V i , f i ,V n ` . Built upon on the
N
C1 D i , E i , f i ,V n
(4)
where G (W ) denotes the delta function. The interest problem
T
Deduced from aforementioned definition, the spectral moments estimation is to regard r as a new signal, then fit r. Finally the amplitude, scale and frequency shift can be estimated from the data. In the following, the cost function for fitting r is given
2 n
ª 1 § f f ·2 º Pi i exp « ¨ ¸ » 2SV i «¬ 2 © V i ¹ »¼
2
where ^D i , Ei , f i ,V n ` are unknown parameters.
i 1
r (W )
(6)
" 1 " 0@
2 n
i
2
Several echoes may be present in the weather radar signal: the weather echoes, the ground clutter and the white noise: (1) s (t ) w(t ) c(t ) n(t ) where w(t ) , c(t ) and n(t ) denote the three echoes individually. The spectral model of the signal can be modelled as:
N
i
where M is the sequence length (also lags), and
2 Signal model
Si ( f )
r0 " r( M 1) / 2 ]T
[r ( M 1) / 2 "
r1u M
The minimization of C1 is equivalent to the minimization problem 2
fit of the autocorrelation sequence, a spectral moments estimator is deduced in the following.
N
C2 ^D l , E l , f l `
r ¦
Dˆ r ( Eˆ , fˆ ) i
i
(10)
i
i 1 i zl
3 Estimation based on fitting autocorrelation Minimizing C2 ^D l , El , fl ` with respect to ^D l , El , fl ` yields sequence the estimates as
^Eˆ , fˆ `
We rewrite the autocorrelation sequence r (W )
N
¦ P exp 2S V W exp j 2S f W V 2
i
2 2 i
i
l
G (W )
2 n
Here, we symbolize r (W ) as a finite sample estimation of autocorrelation sequence of the time series s (t ) . Let r (W ) exp(2S 2W 2 ) , then r (W )
N
¦D
i
r ( E iW )ex p j 2S f iW V n2G (W )
(5)
i 1
where D i Pi , Ei V i2 . It can be easily seen from (6), the sequence is the superstition of some Gaussian sequences with different amplitudes, scales and phases (also the frequency shift, it can be known from the properties of FT), and all the Gaussian sequences are rooted from one function r (W ) . So, the estimation of ^Pi ,V i , f i ,V n ` is converted to the estimation
of amplitudes, scale and frequency shift of a Gaussian signal, ^D i , Ei , fi ,V n ` . And the problem is a mixed spectral moments estimation.
El , fl
`
1
1
ª r H Eˆˆˆ, fˆˆˆr E , f º r H E , f r l l l l l l l ¬ ¼
Dˆ l
i 1
^
arg max rl H r E l , f l r H E l , f l r E l , f l r H E l , f l rl (11)
l
(12)
With the simple preparations above, our algorithm can be presented as below. Note that we assume the number of Gaussian echoes is already known or estimated.
^
Step 1: Assume N=1. Dˆ l , Eˆl , fˆl
`
l 1
can be obtained by using
(11) and (12). Step 2: Assume N=2. Compute r2 with (9) by using
^Dˆ , Eˆ , fˆ ` l
l
l
l 1
^
obtained in Step 1. Obtain Dˆ l , Eˆl , fˆl
`
l 2
from r2
by using (11) and (12). Next, compute r1 by using
^Dˆ , Eˆ , fˆ ` l
l
l
l 2
^
, and then redetermine Dˆ l , Eˆl , fˆl
`
l 1
from r1.
Iterate the previous two substeps until “practical convergence” is achieved. Remaining Steps: Continue similarly until N is equal to the desired or estimated number of echoes.
4 Experimental results and analysis
35
In this section, numerical examples are provided to demonstrate the performance of the proposed algorithm. The investigation will be conducted in a context where the theoretical model is rigorously valid and a context with simulated radar signal. Experiments with theoretical model are given to verify the algorithm and implement theoretical performance analysis. Simulated radar signals are used to demonstrate the efficiency.
25
Amplitude/dB
20
120
5 0 The FT of theoretical sequence The FT of firstly fitted one The FT of secondly fitted one
-10
-0.2 -0.1 0 0.1 0.2 Normalizea frequency
0.3
0.4
0.5
Fig. 2 The FT Spectra of fitted sequence and the two estimated sequences 35
30 25
Amplitude/dB
20 15 10 5 0
The FT of theoretical sequence The FT of firstly fitted one The FT of secondly fitted one
-5 -0.5
-0.4 -0.3
-0.2 -0.1 0 0.1 0.2 Normalized frequency
0.3
0.4
0.5
Fig.3 The fitted results with closely overlapped echoes ˄ fˆ1 0.1010,Vˆ1 0.1570; fˆ2 0,Vˆ 2 0.02 ˅ 2.2 Experiments of simulated radar signals
100
Amplitude
10
-15 -0.5 -0.4 -0.3
In this part, the theoretical autocorrelation sequence mixed with white noise is fitted to estimate the spectral moments. It is assumed that two Gaussian echoes are present, i.e. one echo with f1 0.25,V 1 0.15 and SNR=10dB, and another echo with f 2 0,V 2 0.02 and SNR=20dB. The spectral moments estimation results are given in Fig.1 and Fig.2. Fig.1 is the comparison between ideal sequence and the two fitted ones. Fig.2 shows the FT spectrums of them. As shown in the figures, the ideal curve and the fitted are almost entirely overlapped. In the hail of the curve, some differences are present because of the white noise. In the caption of Fig.1, the estimated parameters are also provided. In addition, the proposed algorithm has a good resolution. Fig. 3 shows the fitted results when the two Gaussian echoes are closely overlapped in the frequency domain, when the hydrometeor move in rapid dispersion ( f1 0.1,V 1 0.15; f 2 0,V 2 0.02 ).
80 60 40 20 The theoretical sequence
0
The fitted sequence
-30
-20
-10
0 Lags
10
20
30
40
(a) 2
In this part, the autocorrelation sequence is estimated from the simulated radar echoes. The Gaussian signals are generated by stochastic process simulation[18]. Then the estimated autocorrelation sequence is fitted. The sample length of the simulated signal is 128, and two Gaussian echoes are contained. One has a broader spectrum(SNR=10dB), while the other has a narrow spectrum(SNR=10dB). The length of the estimated autocorrelation sequence is 21. Fig.5 is the comparison between the estimated sequence and the two fitted ones. Fig.6 shows the FT spectrum of the sequences in Fig.5. As shown in Fig.5 and Fig.6, the two Gaussian echoes are correctly estimated. 1 The estimated sequence from signal The fitted sequence
0.9
1.5
0.8 0.7
1
0.6
Amplitude
Amplitude
15
-5
4.1 Experiments of theoretical model
-40
30
0.5
0
23
24
25
26
Lags
(b) Zoom of Fig. (a) Fig.1 The ideal and fitted autocorrelation sequence ( fˆ1 0.25,Vˆ1 0.1570; fˆ2 0,Vˆ 2 0.02 )
0.5 0.4 0.3 0.2 0.1 0 -20
-15
-10
-5
0 Lags
5
10
15
20
Fig.5 The autocorrelation sequences before and after estimation ˄ fˆ1 0.003,Vˆ1 0.019; fˆ2 0.387,Vˆ 2 0.107 ˅
10
5 0
Amplitude/dB
-5 -10 -15 -20 -25 The FT of estimated sequence The FT of firstly fitted one The FT of secondly fitted one
-30 -35 -0.5 -0.4 -0.3
-0.2 -0.1 0 0.1 0.2 Normalized frequency
0.3
0.4
0.5
Fig.6 The FT spectrums of sequences in Fig.5
5 Conclusions A parametric spectral moments estimation approach is proposed in the paper. It is developed by employing the Gaussian characteristics of the spectrum of the weather radar echoes. The spectral moments are estimated from fitting the autocorrelation sequence of the radar signals. A nonlinear least square estimation criterion is established and a relaxation based method is used to solve the minimization problem. The cyclic optimization can be exploited to decrease the computation quantity of search for the parameters of two or more Gaussian echoes. Finally, experimental results verify the excellent estimation performance.
Acknowledgements This work has been supported in part by the National Natural Science Foundation of China under grant 61071194, 60979002 and 61079008, the Fundamental Research Funds for the Central Universities (ZXH2012D006), and the National Key Technology Research and Development Program of China (2011BAH24B12).
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