A New Model for Optimal Waveform Design - Science Direct

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Procedia Engineering

ProcediaProcedia Engineering 00 (2011) Engineering 29 000–000 (2012) 1707 – 1712 www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE)

A New Model for Optimal Waveform Design Bin Wang*, Jinkuan Wang, Xin Song, Yinghua Han Northeastern University at Qinhuangdao, Taishanro. 143, Qinhuangdao 066004, China

Abstract It is an important problem for radar to transmit different waveforms according to different working conditions. In this paper, based on mutual information theory, a new model for optimal waveform design in single-dependent interference is proposed. Through this model, the problem of waveform design can be converted into the problem of optimization. Then we can use convex optimization method to solve the problem of waveform design. Finally, the whole paper is summarized.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology Open access under CC BY-NC-ND license. Keywords: Waveform Design; Single-dependent Interference; Mutual Information; Convex Optimization;

1. Introduction The word “radar” was originally an acronym for “radio detection and ranging”. Today, the technology is so common that the word has become a standard English noun. Early radar development was driven by military necessity, and nowadays the military is still the dominant user and developer of radar technology. Radar now enjoys an increasing range of applications. However, traditional radar systems are lack of adaptivity to the environment. Now the radar working conditions are more and more complex. Modern radar systems should transmit different waveforms according to different environment. So we need to consider the problem of adaptive waveform design. Cognitive radar is a new framework of radar system proposed by Simon Haykin in 2006. It is an advanced form of radar system and may adaptively and intelligently interrogate a propagation channel using all available knowledge [1]. In 2009, Simon Haykin in another paper introduces the realization methods of cognitive radar [2]. In [3], the authors propose a waveform design method that efficiently synthesizes waveforms that provide a trade-off between estimation performance for a Gaussian ensemble

*

* Corresponding author. Tel.: +86-0 15930597253 E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.199

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Bin Wang et al.Yinghua / Procedia 29 (2012) 1707 – 1712 000–000 Bin Wang, Jinkuan Wang, Xin Song, HanEngineering / Procedia Engineering 00 (2011)

of targets and detection performance for a specific target. In [4], Arasaratnam have successfully solved the best approximation to the Bayesian filter in the sense of completely preserving second-order in formation, which is called Cubature Kalman filters. Goodman compares two different waveform design techniques for use with active sensors operating in a target recognition application and proposes the integration of waveform design with a sequential-hypothesis-testing framework that controls when hard decisions may be made with adequate confidence [5]. In [6], the authors present illumination waveforms matched to stochastic targets in the presence of signal-dependent interference. In [7], the authors propose a method to employ waveform agility to improve the detection of low radar-cross section (RCS) targets on the ocean surface that present low signal-to-clutter ratios due to high sea states and low grazing angles. In [8], an algorithm to select and configure linear and nonlinear frequency-modulated waveforms is then proposed. In [9], the authors describe the optimization of an information theoretic criterion for radar waveform design. In [10], the performance of combined constant and swept frequency waveform fusion systems is investigated. In [11], an adaptive, waveform selective probabilistic data association (WSPDA) algorithm for tracking a single target in clutter is presented. In this paper, we propose mutual information model of adaptive waveform design. With this model, we can solve the problem of adaptive waveform design in different radar environment. 2. Theoretical Principle Generally speaking, the design of optimal radar waveform is task-dependent. Different tasks call for different radar waveform. When a target is present, we expect that there will be greater energy in the received signal than when no target is present. So we usually use a threshold test on the energy in the received signal to detect radar targets. A radar system may make measurements of a target in order to determine unknown characteristics of the target. We can also say that a radar system may make measurements of a target in order to decrease the a priori uncertainty about the target. In the analysis of communication channels, information theory has successfully allowed for the information transmission capabilities of a communication channel to be determined [12]. So for cognitive radar, we should consider detection performance and estimation performance simultaneously. We should make a balance between detection performance and estimation performance. Fig. 1 is a figure of basic signal-processing cycle in cognitive radar.

Fig. 1. Basic signal-processing cycle in cognitive radar

3. Mutual Information Model Fig. 2. is signal model of a Gaussian target ensemble in ground clutter. We want to find the mutual information I ( g , y | x ) , that is, the mutual information between the random target impulse response and

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the received radar waveform. Those functions x that maximize I ( y , z | x ) also maximize I ( g , y | x ) . So we maximize

I ( y, z | x ) first.

f x Assume K represents frequency domain sampling point, and k is frequency point. Let k F z correspond to the component of x (t ) with frequency components in k , k correspond to the

z (t ) with frequency components in Fk , and yk correspond to the component of y (t ) F with frequency components in k . So the overall mutual information is

component of

K

I ( y , z | x ) = ∑ I ( yk , z k | x )

(1)

k =1

Fig. 2. Signal model of a Gaussian target ensemble in ground clutter

Fk = [ f k , f k + Δf ] is sufficiently small, so for f ∈ Fk , X ( f ) ≈ X ( f k ) , Z ( f ) ≈ Z ( f k ) , Y ( f ) ≈ Y ( f k ) , D( f ) ≈ D( f k ) . Δf is the bandwidth. Here we have an additive Gaussian noise channel with input Z , a zero mean Gaussian random 2 2 variable with variance σ Z , and additive zero-mean Gaussian noise N with variance σ N , and a zeromean Gaussian random process clutter is a with spectral density Pc ( f ) . Assume the frequency interval

Following we define mutual information. In probability theory and information theory, the mutual information of two random variables express the dependence of them. Mutual information can be defined as from mathematics

I (Y ; Z ) = EY , Z [log

p (Y , Z ) ] p (Y ) p ( Z )

(2)

p (Y , Z ) is joint probability distribution function, p (Y ) and p ( Z ) are marginal probability distribution function of Y and Z , respectively. Intuitively, mutual information contains the total information of Y and Z . Assume H (Y ) represents the marginal entropy of Y , and H (Y | Z ) represents the conditional entropy of Y given Z . So the mutual information can also be expressed as (3) I (Y ; Z ) = H (Y ) − H (Y | Z ) Now Y , being the sum of three zero-mean Gaussian random variables, is itself a zero-mean Gaussian random variables. Since Z , N and D are statistically independent, the variance of Y , is σ Y2 = σ Z2 + σ N2 + σ D2 (4) Following we will solve H (Y ) and H (Y | Z ) respectively. where

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H = (Y ) E[ln p= (Y )]

1 1 πσ Y2 ln 2= ln 2π (σ Z2 + σ N2 + σ D2 ) 2 2

(5)

1 ln 2π (σ N2 + σ D2 ) 2 So the mutual information I (Y ; Z ) is given by the expression = H (Y | Z )

(6)

σ2 1 I (Y ; Z ) = H (Y ) − H (Y | Z ) =ln(1 + 2 Z 2 ) 2 σN +σD Referring again to the signals

(7)

zk , yk , nk and d k with frequency components confined to the interval

= Fk [ f k , f k + Δf ] , we have from the Sampling Theory that each of the signals can be represented by a sequence of samples taken at a uniform sampling rate of 2Δf . Since we assume that the spectra X ( f ) , Z ( f ) , Y ( f ) and D( f ) are smooth and have a constant value (at least approximately) for all f ∈ Fk , the samples of the Gaussian process sampled at a uniform rate 2Δf are statistically independent. The samples zk are independent, identically distributed random variables with aero mean and variables

σ Z2 , we note that the total energy EZ

in

zk is

2

2

EZ = Z ( f k ) ∗ 2Δf = 2Δf X ( f k ) σ G2 ( f k ) (8) Over the time interval T , this energy is evenly spread among 2ΔfT statistically independent samples. Hence, the variance of each sample, σ Z , is 2

2

= σ Z2

2

2Δf X ( f k ) σ G2 ( f k ) EZ = = 2T Δf 2T Δf

X ( f k ) σ G2 ( f k )

(9)

T

nk has total energy EN on the interval T given by EN = ΔfPN ( f k )T (10) And this energy is evenly distributed among the 2T Δf statistically independent, zero-mean samples 2 representing nk . Hence, the variance σ N of each sample is Similarly, the noise process

= σ N2

ΔfPN ( f k )T PN ( f k ) = 2T Δf 2

(11)

Similarly, 2

2

ED = D( f k ) ∗ 2Δf = 2Δf X ( f k ) σ C2 ( f k ) 2

= σ D2

2Δf X ( f k ) σ C2 ( f k ) ED = = 2T Δf 2T Δf

X ( f k ) σ C2 ( f k )

(13)

T

Substituting the results of Eqs. (13), (11) and (9) into (7), we have that for each sample corresponding sample

(12)

2

Z m of zk and

Ym of yk , the mutual information between Z m and Ym is 2

2 X ( f k ) σ G2 ( f k ) 1 σ2 1 ln(1 + 2 Z 2 ) = ln[1 + ] I (Ym ; Z m ) = 2 2 σN +σD 2 PN ( f k )T + 2 X ( f k ) σ C2 ( f k )

(14)

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Now these are interval

2T Δf statistically independent sample values for both zk and yk in the observation

T . Thus, 2

I ( yk , z k | x ) = 2ΔfTI (Ym ; Z m ) = T Δf ln[1 +

2 X ( f k ) σ G2 ( f k ) 2

PN ( f k )T + 2 X ( f k ) σ C2 ( f k )

]

(15)

]

(16)

The over all mutual information is K

2

2 X ( f k ) σ G2 ( f k )

K

I ( y, z | x ) = ∑ I ( y , z | x ) = ∑ T Δf ln[1 +

k k = k 1= k 1

2

PN ( f k )T + 2 X ( f k ) σ C2 ( f k )

Also this is the discrete form of the mutual information between the target response and received signal. Formula (16) is the objective function we should maximize. Considering some actual situations, we should make some constraints. First, in order to confirm certain detection performance, SNR should be greater than certain value. That means

= SNR

Esignal Enoise

≥ SNR0

(17)

Second, transmitted signal should satisfy energy constraint. That means

∫

Tx

0

x 2 (t )dt = Ex

(18)

Third, transmitted signal should satisfy power constraint. That means

∫

f 0 +W f0

2

X ( f ) df ≥ Px

(19)

Finally, other constraints should be satisfied under different radar working conditions. This is the mutual information model of adaptive waveform design. The problem of adaptive waveform design converts to the problem of optimization under some constraints. Then we can use optimization methods to solve the problems of adaptive waveform design. In different environment, the radar can transmit different waveforms. 4. Conclusions Traditional radar systems are lack of adaptivity to the environment and transmit single waveform. However, with the development of modern technology, radar environment are more and more complex. Modern radar systems should transmit different waveforms according to different environment. In this paper, mutual information model of adaptive waveform design is proposed. With this model, different waveforms can be designed adaptively under different radar working conditions. The problem of adaptive waveform design converts to the problem of optimization under some constraints. Then we can use optimization methods to solve the problems of adaptive waveform design. Next we will consider how to use optimization methods and develop new optimization methods to solve the problems of adaptive waveform design. Acknowledgements The authors would like to thank the anonymous reviewers for their insightful comments that helped improve the quality of this paper. This work is supported by the National Natural Science Foundation of China, under Grant no. 60874108, 61004052 and 61104005.

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References [1] Haykin, S.: Cognitive radar: a way of the future. IEEE Signal Processing Magazine, vol. 23, no. 1, pp. 30--40 (2006). [2] Haykin, S.: Cognition is the key to the next generation of radar systems. In: Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop, pp. 463—467. IEEE Press, New York(2009). [3] Haykin, S., Xue, Y.B., Davidson, T.: Optimal waveform design for cognitive radar. In: Asilomar Conference, pp. 1—6. IEEE Press, New York(2008). [4] Arasaratnam, I., Haykin, S.: Cubature Kalman filters. IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 463-467(2009). [5] Goodman, N.A., Venkata, P.R., Neifeld, M.A.: Adaptive waveform design and sequential hypothesis testing for target recognition with active sensors. IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 1, pp. 105--1132(2007). [6] Romero, R.A., Goodman, N.A.: Waveform design in single-dependent interference and application to target recognition with multiple transmissions. IET Radar, Sonar and Navigation, vol. 3, no. 4, pp. 328--340(2009). [7] Sira, S. P., Cochran, D.: Adaptive waveform design for improved detection of low-RCS targets in heavy sea clutter. IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 1, pp. 56--66(2007). [8] Sira, S. P., Papandreou-Suppappola, A., Morrell, D.: Dynamic configuration of time-varying waveforms for agile sensing and tracking in clutter. IEEE Trans. Signal Processing, vol.55, no.7, pp. 3207--3217(2007). [9] Leshem, A., Naparstek, O., Nehorai, A.: Information theoretic adaptive radar waveform design for multiple extended targets. IEEE Journal of Selecte Topics in Signal Processing, vol. 1, no. 1, pp. 42--55(2007). [10] Rago, C., Willett, P., Bar-Shalom, Y.: Detecting-tracking performance with combined waveforms. IEEE Transactions on Aerospace and Electronic Systems, vol.34, no. 2, p. 612--624(1998). [11] Kershaw, D.J., Evans, R.J.: Waveform selective probabilistic data association. IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1180--1188(1997). [12] Bell, M.R.: Information theory and radar waveform. IEEE Transactions on Information Theory, vol. 39, no. 5, pp. 1578-1597(1993).