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Noise Modeling of RF Seeker for Homing Guidance Applications Anirban Krishna Bhattacharyya1, Shrabani Bhattacharya2, Tanushree Garai1, Siddhartha Mukhopadhyay3

Abstract— The relative kinematic information between the target and the pursuer required for homing guidance computation are provided often by the on board seeker. Due to its operating characteristics and inherent limitations, the measurements from the sensor are corrupted with noise of varied origin and statistics like eclipsing, Glint, RCS fluctuation, Radome distortion, angular noise and Gyro drift. This precludes the use of seeker measurements directly for guidance computations and necessitates an estimator in the guidance loop, such as the standard Kalman estimator. The modeling of statistical properties of the RF seeker noise and its incorporation in the estimator model is discussed in the present paper. Index Terms— Homing Guidance, Radio Frequency Seeker, nonstatonary stochastic process, correlated noise, AR model.

I.

INTRODUCTION

T

he relative kinematic information between the target and the pursuer required for homing guidance computation are provided often by the on-board RF seekers. These sensors produce measurements of relative range-rate, gimbal angles and slight line rates. However, due to their operating characteristics and inherent limitations, the measurements from these sensors are corrupted with noise elements of varied origin and statistics. The major noise components affecting the measurements are due to target RCS variation, glint, RF propagation medium, receiver noise, thermal noise and signal processing etc., some of the contributions being multiplicative and some additive. In addition, due to unavailability of receiver gate during transmission period, the received signal becomes eclipsed in case of high PRF seekers, leading to degraded measurements. Thus the seeker measurements cannot be directly used for guidance computation and an estimator, such as the Kalman filter is essential. Kalman filters are optimal with white noise sequences. Therefore in presence of colored noise a noise model is often included within the filter to retain optimality. It is therefore important to estimate a noise model from simulation or offline experimental data and to include it within the filter 1 Anirban Krishna Bhattacharyya and Tanushree Garai are with Department of Electrical Engineering at IIT, Kharagpur, India. 2 Shrabani Bhattacharya is with Integrated Test Range, Defence Research & Development Organization (DRDO), India. 3 Siddhartha Mukhopadhyay is with Department of Electrical Engineering at IIT, Kharagpur, India as a Professor; (e-mail: [email protected]).

formulation. In this paper, a procedure for estimating such a model is given. Realistic simulation results are presented on the improvement in miss distance demonstrated by Monte Carlo simulation. Statistical analysis of the RF seeker noise characteristics reveals that the noise power is significant and non-stationary, due to variations in the relative distance. Further, the noise sequences are correlated due to the dynamics involved in the sensing, signal processing and bore sight tracking systems within the seeker. It is therefore imperative to characterize this noise. The occurrence of the eclipsing phenomenon is indicated by a flag as an output from the seeker, which is directly used in the estimator for varying the measurement dimension to cater to the absence of LOS rate measurements during the fullyeclipsed duration. In addition, the degraded SNR strength during partial eclipsing period is taken into account as a varying gain by using the SNRdB information as yielded by the seeker. Hence, a noise model has been developed for LOS rate measurements to account for the properties of the rest of the noise components. Results of numerical simulation which establish that incorporation of appropriate noise model of the RF seeker provides significant improvement in estimator performance in terms of reduced estimation error and miss-distance are also provided. II. METHODOLOGY The standard practice in handling non-stationary noise statistics in Kalman estimator is to incorporate varying mean and standard deviation of the noise sequence in the measurement update process. However, in this paper the nonstationarity has been handled by making the noise sequence stationary by normalizing it using varying standard deviation and mean, which are obtained by off-line characterization of the noise properties. Thereafter the stationary but correlated signal has been modeled using an Auto-regressive (AR) system with white noise input. This pre-whitening filter modeling the correlation of the LOS rate measurement noise is used in the estimator formulation by augmenting the basestate elements of interest with the state-elements defining the AR system representing the noise dynamics [1, 2, 3]. The statistical properties in terms of mean, variance, correlation and stationarity of the measurement noise of sight line rates have been studied in order to derive an insight to the

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here. The seeker locks on and starts tracking the target at different range-to-go values for different runs of the simulation as the noise magnitude differs in each run. Further, the effect of glint and other noise sources are predominantly dependent on the range-to-go. So to get the ensembles of the noise samples from the same statistical distribution, the noise values all at the same range-to-go values from each run of the simulator are collected. This is done by closing the guidance loop with kinematic sight line rate feedback, while using this kinematic feedback data as input to the seeker and getting the noisy outputs from it. With this arrangement, since the guidance loop is closed with kinematic feedback, although lock-on occurs at different range-to-go values from run to run, yet once lock-on is achieved, the range-to-go values for each sample in the different runs are the same. The noise samples for different range-to-go values are obtained from different simulation runs and the corresponding seeker outputs are subtracted from the kinematic sight-line-rates to generate the samples of noise. The mean, standard deviation and correlation of the noise is calculated over the ensembles and the data is plotted as shown in the following figures (Fig. 1. to Fig. 7.). The mean and standard deviation are obtained by taking expectation over ensembles, as given below. Mean of noise ( μ η (R TM ) )

=E [ η(R TM ) ] =

1 n ∑ ηi (R TM ) n i=1

(1)

Standard deviation of noise

=σ η (R TM )=

2 1 N ⎡⎣ ηi (R TM )-μ η (R TM ) ⎤⎦ ∑ N i=1

(2)

where N is the number of ensembles. The correlation matrix is computed by taking expectation over an ensemble of the matrix product of the noise vector ( ηi ), formed at different range-to-go values, i.e. Correlation matrix of noise

1 N ∑ (ηi .ηi' ) N i=1

(3)

As can be seen from the plots, the mean of noise is 0, implying that there is no bias. Thus the correlation matrix is same as the covariance matrix. From Fig.1 and Fig.2, it is clear that the noises can be assumed to be zero mean, as is the prerequisite for Kalman Filters. From Fig.3 and Fig.4 the non-stationary nature of the noise is apparent. The noise power reduces with range-to-go consistently, which is further modeled. Fig.5 and Fig.6 below show the autocorrelation matrices of the noise sequences in two channels of measurements. Z Channel 1 0 -1 Noise Mean

A. Numerical procedure for obtaining ensemble statistics The Eclipsing effect is multiplicative and is predictable. So it is treated separately from the rest of the interferences. To get the statistical properties of the noise, an off-line simulation environment is used, where a model of RF seeker generates ensembles of measurements for a typical engagement scenario. As opposed to the standard method for analysing ensembles of stochastic signal, where time is treated as the independent variable, range-to-go (R TM ) is used for the reasons stated

=

-2 -3 -4 -5 -6 -7

0

2

4 6 8 Range to Go (Km.)

10

12

Fig. 1. Mean of noise (rad./sec.) in the Z Channel. It can be seen that the noise is zero mean, except at the very end of the engagement when the RTM value is small. Y Channel 25 20

Noise Mean

nature of noise model required for the process. The mean is studied to see if the noise is biased. The variance and subsequently its Fourier Transform give the noise power spectrum which is further used to formulate the required filter structure. A study of these properties also brings out whether the statistics are stationary or are they varying over the independent variable, i.e. they are non-stationary.

15 10 5 0 -5

0

2

4 6 8 Range to Go (Km.)

10

12

Fig. 2. Mean of noise (rad./sec.) in the Y Channel. It can be seen that the noise is zero mean, except at the very end of the engagement when the RTM value is small.

From the plots (Fig. 4 and Fig. 5) it is clear that the process is non-stationary. It is also seen that the autocorrelation values are significant only when the differences in the delays are small. This implies the possibility of whitening the noise sequences with a small order filter. This has also been carried out below.

3 Z Channel 0.25

Noise Std

0.2

0.15

0.1

0.05

0

0

2

4 6 8 Range to Go (Km.)

10

12

Fig.3. Std of noise (rad./sec.) in the Z Channel. The noise has a decreasing Standard deviation, except at the very end of the engagement when the RTM value is small. This increase is due to the glint noise.

Fig.6. Auto-correlation matrix

RYY (τ 1 ,τ 2 ) of Noise in Y Channel

Y Channel 0.5

Noise Std

0.4 0.3 0.2 0.1 0

0

2

4 6 8 Range to Go (Km.)

10

12

Fig.4. Std of noise (rad./sec.) in the Y Channel. The noise has a decreasing Standard deviation, except at the very end of the engagement when the RTM value is small. This is because of the glint noise.

The cross-correlation between the two channel noise signals is also investigated. It can be seen from the plot (Fig. 7) that it is very low and can be neglected for all practical purposes.

Fig. 7. Cross-correlation matrix

R YZ (τ1 ,τ 2 ) of noise in Y

Channel and Z Channel. The cross correlation is very low.

non-stationary property of the noise has been modeled using the two following model structures: a) Gaussian White Noise with varying standard deviation depending on the range-to-go. b) Gaussian White Noise with varying standard deviation depending on the range-to-go filtered by time invariant autoregressive filter. (a)Gaussian deviation White Noise

White

Noise

with

σ η (R TM )

time-varying

standard

Seeker Noise

Fig. 8. Seeker noise modeled as white noise of varying variance

Fig. 5. Auto-correlation matrix

R zz (τ1 ,τ 2 ) of Noise in Z Channel

B. Approaches for Modeling Non-Stationary Noise Based on the above studies of the noise sequences, attempt is made to estimate noise model for the seeker noise and correspondingly augment the Kalman filter formulation. The

By analyzing the noise ensembles in off-line, varying noise standard deviation has been quantified as a function of rangeto-go (RTM) by fitting a curve to the variation of standard deviation. In real-time the varying measurement variance (R k ) at each time step k is used in Kalman filter by interpolation from the identified model based on the RTM measurement obtained from the seeker. Kalman gain is then given by Kk = Pk|k-1HkT(HkPk|k-1HkT+Rk)-1 and is used in the estimator.

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(b)Gaussian White Noise with time-varying standard deviation filtered by time-invariant Auto-Regressive filter

White Noise

Time invariant Auto-regressive filter

ση (RTM )

Seeker Noise

Fig. 9. Seeker noise modeled as colored noise of varying variance

For this modelling, the nonstationary noise sequence is first normalized with the corresponding nonstationary noise standard deviation function shown in Fig. 3 and Fig. 4 as shown in the equation below. The result is the normalized colored noise sequence ωi (R TM ) . ωi (R TM )=

ηi (R TM ) σ η (R TM )

Fig. 10. Auto-correlation matrix

Rωω (τ 1 ,τ 2 ) of normalized Noise

in Z Channel. The standard deviation has been made constant at 1.

(4)

Then the auto-correlation matrix of ωi (R TM ) is calculated. Due to the normalization operation the variance of the noise now becomes unity, and thus the data can now be considered to have been generated from a stationary noise process. The normalized auto-correlation matrix is shown in Fig. 10 and Fig. 11. The number of significant singular values of the correlation matrix indicates the order of the whitening autoregressive filter [4]. The singular values of the matrix are therefore obtained to determine the orders of the auto-regressive filters to be used to model the normalized noise sequence ωi (R TM ) as, ωi (R TM )=-a 0 ωi (R TM -1)-a1ωi (R TM -2)…+b0 ξ

(5)

Here, ξ is white noise with unity variance. Using the singular values obtained the order of the AR filter is estimated [4]. A fourth order AR filter is constructed. The AR coefficients of the noise model are determined by the extended – Yule Walker equations [4] and using the Data Least Square technique [5]. Subsequently these coefficients of the AR noise model are used to augment the Kalman filter formulation to include the noise states while the system matrix is augmented by the estimated filter coefficients. Augmentation of states [x(k) η y (k) ηy (k-1) ηy (k-2) ηy (k-3) ηy (k-4)

Fig. 11. Auto-correlation matrix

Rωω (τ 1 ,τ 2 ) of normalized Noise

in Y Channel. The standard deviation has been made constant at 1.

ηz (k) ηz (k-1) ηz (k-2) ηz (k-3) ηz (k-4)]T

where the ‘ η y ’s and ‘ ηz ’s depict noise states in the Y and Z channels respectively and x(k) is the kinematic state of the estimator. The state equation for the augmented system is given by

Fig. 12. Singular values of Auto-correlation matrix

Rωω (τ 1 ,τ 2 ) of normalized Noise in Y Channel.

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0 0 ⎤ ⎡ q(k) ⎢ Q(k)= ⎢ 0 R y (k) 0 ⎥⎥ ⎢⎣ 0 0 R z (k) ⎥⎦

(8)

The measurement equation becomes z(k)=H(k)[x(k) ηy (k) ηy (k-1) ηy (k-2) ηy (k-3) ηy (k-4) ηz (k) ηz (k-1) ηz (k-2) ηz (k-3) ηz (k-4)]T

with

H(k)= [ h(k)

σ yη (k) 0 0 0 0 σ zη (k) 0 0 0 0]

Fig. 13. Singular values of Auto-correlation matrix

Rωω (τ 1 ,τ 2 ) of normalized Noise in Z Channel.

⎡ x(k+1) ⎤ ⎡ x(k) ⎤ ⎡ w x (k) ⎤ ⎢ η (k+1) ⎥ ⎢ η (k) ⎥ ⎢ ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎢ ξ y (k) ⎥ ⎢ ηy (k) ⎥ ⎢ ηy (k-1) ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ηy (k-1) ⎥ ⎢ ηy (k-2) ⎥ ⎢ 0 ⎥ ⎢ η (k-2) ⎥ ⎢ η (k-3) ⎥ ⎢ 0 ⎥ ⎢ y ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ ηy (k-3) ⎥ =F(k) ⎢ ηy (k-4) ⎥ + ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ηz (k+1) ⎥ ⎢ ηz (k) ⎥ ⎢ ξ z (k) ⎥ ⎢ η (k) ⎥ ⎢ η (k-1) ⎥ ⎢ 0 ⎥ ⎢ z ⎥ ⎢ z ⎥ ⎢ ⎥ ⎢ ηz (k-1) ⎥ ⎢ ηz (k-2) ⎥ ⎢ 0 ⎥ ⎢ η (k-2) ⎥ ⎢ η (k-3) ⎥ ⎢ 0 ⎥ ⎢ z ⎥ ⎢ z ⎥ ⎢ ⎥ ⎢⎣ ηz (k-3) ⎥⎦ ⎣⎢ ηz (k-4) ⎥⎦ ⎣ 0 ⎦

where σ η (k) and σ η (k) are noise standard deviation for y

(6)

0 0 0 0 0 0 0 0 0 0 ⎤ -a 0y -a1y -a 2y -a 3y -a4y 0 0 0 0 0 ⎥⎥ 1 0 0 0 0 0 0 0 0 0 ⎥ ⎥ 0 1 0 0 0 0 0 0 0 0 ⎥ 0 0 1 0 0 0 0 0 0 0 ⎥ ⎥ 0 0 0 1 0 0 0 0 0 0 ⎥ 0 0 0 0 0 -a 0z -a1z -a 2z -a 3z -a 4z ⎥ ⎥ 0 0 0 0 0 1 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 0 1 0 0 0 ⎥ 0 0 0 0 0 0 0 1 0 0 ⎥ ⎥ 0 0 0 0 0 0 0 0 1 0 ⎦

z

Y and Z channels, h(k) is the measurement matrix for the base state elements. The absence of explicit measurement noise term in the augmented formulation is noteworthy. This formulation is now used in standard Kalman filter framework for prediction and update of base state and noise state and corresponding covariances. III. RESULTS The results obtained on implementing the modification stated above are given below. The outputs of the seeker are the sight-line-rates (SLR). So the performance is studied with respect to the errors in the SLR estimates (ensemble mean and correlation matrix) and final miss-distance. The results are obtained from 50 ensembles in the first case and 100 in the second case.

Here, the augmented state transition matrix is F(k)= ⎡ f(k) ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎣ 0

,

(7)

(a)Gaussian White Noise with time-varying standard deviation From the figures (Fig. 14 to Fig. 17) given below it can be seen that if the noise information is provided to the Kalman filter the estimation improves. Before the SLR starts to build up and the seeker dynamics come into the picture the mean of SLR estimation error is 0, i.e. on the average accurate estimation is taking place. Fig. 18 gives the histogram of missdistance obtained.

f(k) as the state transition matrix for the base state, a and aiz the AR coefficients for Y and Z channels respectively. w x (k) is the process noise matrix for the base state with covariance q(k) . ξ y (k) and ξ z (k) are the white with

y i

noise components in the noise model for Y and Z channels with corresponding covariance as R y (k)=1 and R z (k)=1 respectively. The resulting process noise covariance matrix is Fig. 14. Mean of SLR estimation error.

6 (b)Gaussian White Noise with time-varying standard deviation filtered by time-invariant Auto-Regressive filter The AR filters obtained for the Y and Z channel are given below Y Channel:

ωi (R TM )=0.0584ω(R TM -1)-0.0405ω(R TM -2) -0.0358ω(R TM -3)+0.0007ω(R TM -4)+0.996ξ Z Channel:

ωi (R TM )=0.0146ω(R TM -1)-0.0405ω(R TM -2) -0.0221ω(R TM -3)-0.017ω(R TM -4)+0.998ξ These filters were used to augment the Kalman filter states as described above and the resultant estimator gave the following results (Fig. 19 to Fig. 23).

Fig. 17. Correlation of SLR estimation error. It can be seen that before the SLR starts building up and seeker dynamics come into play the error has been reduced.

Fig. 15. Mean of SLR estimation error. Fig. 18. Miss distance on using the estimator without noise model.

It can be seen from the figures that with the seeker noise modeled the SLR estimates have improved in the zone where the seeker dynamics do not come into play. The miss-distance has also improved appreciably.

Fig. 16. Correlation of SLR estimation error. It can be seen that before the SLR starts building up and seeker dynamics come into play the error has been reduced.

Fig. 19. Mean of SLR estimation error.

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Fig. 23. Miss distance on using estimator with noise model

Fig. 20. Mean of SLR estimation error.

It can be concluded that miss distance values have improved on modeling the noise. IV. CONCLUSION AND FURTHER WORK The paper discusses the requirements and methodology for modeling the correlated seeker noises for appropriate estimation of relative kinematics during the terminal phase of homing guidance. Efficient modeling of the noise leads to improvement in estimation. Further improvements in performance through detailed gray-box sensor modeling are being investigated by the authors at present. Fig. 21. Correlation of SLR estimation error.

REFERENCES [1] [2] [3] [4] [5]

Fig. 22. Correlation of SLR estimation error.

C K Chui, G Chen, Kalman Filtering with Real-time Application, Springer, 3rd Ed. R G Brown, P Y C Hwang, Introduction to Random Signals and Applied Kalman Filtering,. 3rd Ed., John Wiley and Sons, 1997. Y Bar-Shalom, X R Li, T Kirubarajan, Estimation with Applications to Tracking and Navigation, John Wiley and Sons Inc., 2001. James A. Cadzow, Spectral Estimation: An Overdetermined Rational Model Equation Approach, Proceedings Of The IEEE, Vol. 70, No. 9, pp. 907-939, September 1982. Ronald D. DeGroat and Eric M. Dowling, The Data Least Squares Problem and Channel Equalization, IEEE Transactions on Signal Processing, VOL. 41, No. 1, pp. 407-411, January 1993.