# Adaptive Waveform Design for Multiple Radar Tasks Based on Constant Modulus Constraint

Authors: Wang Bin; Yang Wenfang; Wang Jinkuan
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Cognitive radar is an intelligent system, and it can adaptively transmit waveforms to the complex environment. The intelligent radar system should be able to provide different trade-offs among a variety of performance objectives. In this paper, we investigate the mutual information (MI) in signal-dependent interference and channel noise. We propose a waveform design method which can efficiently synthesize waveforms and provide a trade-off between estimation performance and detection performance. After obtaining a local optimal waveform, we apply the technique of generating a constant modulus signal with the given Fourier transform magnitude to the waveform. Finally we obtain a waveform that has constant modulus property.


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## Body

## 1. Introduction

Cognitive radar (CR) is a new concept of radar system proposed by Haykin in 2006 [1, 2]. In CR, the radar continuously learns about the environment through experience gained from interactions of the receiver with the environment, the transmitter adjusts its illumination of the environment in an intelligent manner and the whole radar system constitutes a closed-loop dynamic system. Therefore, adaptive waveform design is important to the performance of radar system. Recently, advances in flexible waveform generators and high-speed signal processing hardware have made it possible for transmitted waveforms to vary with the complex environment.Many researches focused on waveform design for different tasks, for example, target detection, estimation, tracking, and recognition. An early attempt to the problem of matched waveform design for detecting a known target in additive Gaussian noise was addressed via the SNR criterion in [3]. From the frequency domain approach, the SNR-based optimal matched waveform for a known target in signal-dependent interference was derived in [4]. Information theory is also an important tool for waveform design. Bell [5] firstly proposed the method of maximizing the mutual information between the received signal and target impulse response to optimize the waveform, and many articles also used mutual information as the optimal criterion for waveform design [6, 7]. Because a more flexible design framework is required, CR should be able to provide different trade-offs among a variety of performance objectives. Haykin et al. [8] proposed a waveform design method that efficiently synthesizes waveforms which provide a trade-off between estimation performance for a Gaussian ensemble of targets and detection performance for a specific target in channel-noise-only environment.In this paper, we will consider a situation when the signal-dependent interference is not negligible, and provide an optimal trade-off between the detection and estimation criteria. Thus we seek to maximize the mutual information between a random target impulse response and the received radar waveform, subjected to a lower bound on theSINR for the target and energy constraints. We assume that the target hypotheses are statistically characterized by known power spectral density (PSD) as in [9]. Therefore, the actual target realization is an unknown sample function generated from the PSD of the true target class.One consideration in forming practical radar waveforms is the constant modulus constraint, which permits efficient use of the front-end power amplifier [10]. With proper manipulation of the waveforms in the temporal domain, it should be possible to design constant modulus waveforms that approximate MI-based waveform spectrum with some loss of optimality. Pillai et al. give us a technique of generating a constant modulus signal with the given Fourier transform magnitude in [11]; thus we can use this method to get a waveform that has constant modulus property.This paper is organized in the following manner. Section2 describes the target model for waveform design in signal-dependent interference. Section 3 explains how to generate the constant modulus waveform from a given Fourier transform magnitude. Section 4 shows the derivation of the mutual information between the random target impulse response and the received radar waveform in signal-dependent interference and waveform design technique for target detection and estimation in signal-dependent interference. Section 5 shows some simulation results. The whole paper is summarized in Section 6.


## 2. Signal Model

The block diagram in Figure1 represents the signal model of a target ensemble in ground clutter being considered. Let x(t) be a finite-energy waveform with duration T. Let g(t) be a zero-mean extended target with energy spectral variance σG2(f). Let Tg be the time duration where most of the target impulse’s energy resides. It is necessary to have Tg>T to capture the target impulse response’s energy. The clutter c(t) is a zero-mean complex Gaussian random process with power spectral density (PSD) σC2(f), and n(t) is the zero-mean receiver noise process with one-sided PSD Pn(f). In addition, n(t) is assumed to be statistically independent of the transmitted waveform x(t), the target impulse response g(t), and the clutter c(t).Figure 1
Signal model of a target ensemble in signal-dependent interference.The waveform received at the receiver is filtered by the ideal lowpass filterB(f), passing only frequencies in the band ω. This is just a statement of the fact that we assume that the transmitted signal has no significant energy outside the frequency interval w=[-W,W]. Since z(t) and d(t) are the response of a linear time-invariant system to the transmitted signal, they do not have significant energy outside the frequency interval w=[-W,W]. Hence we will not consider frequencies outside this interval.Lety(t) be the received signal given by
(1)y(t)=z(t)+d(t)+n(t).z
(
t
) and d(t) are defined by
(2)z(t)=x(t)*g(t),d(t)=x(t)*c(t),
where * denotes the convolution operator.


## 3. Constant Envelope Signals with Given Fourier Transform Magnitude

Pillai et al. give us a technique of generating a constant modulus signal with the given Fourier transform magnitude in [11]. It is summarized as follows.LetCM denote the set of functions {g(t)} that have the prescribed Fourier transform magnitude M(ω) over a prescribed frequency set Ω. The operator PM will assign every arbitrary function x(t) a “nearest neighbor” PMx(t) that belongs to CM such that there exists no other element g∈CM for which ∥x-g∥<∥x-PMx∥ is satisfied.Given an arbitrary functionx(t), its corresponding Fourier transform is X(ω)=|X(ω)|ejΩ(ω) and the magnitude projection of x(t) is defined as
(3)PMx(t)⟷{M(ω)ejΩ(ω),ω∈ΩX(ω),ω∈Ω′.For a constant envelope signalx(t), it can be expressed as
(4)x(t)=Aejθ(t),
where A is a suitable positive constant that can be used to maintain a prescribed energy level for x(t).Interestingly, constant envelope signals also share properties similar to the Fourier transform magnitude situation. Notice that ifCA denotes the set of functions {g(t)} which have constant envelope level A, the operator PA will assign every arbitrary function x(t) a nearest neighbor PAx(t) that belongs to CA such that no other element g∈CA satisfies ∥x-g∥<∥x-PAx∥.Given an arbitrary signalx(t)=a(t)ejθ(t), the projection procedure is
(5)PAx(t)={Aejθ(t),t∈Tx(t),otherwise.The magnitude and amplitude projection are combined according to(6)xk+1=PAPMxk,
where xk is the kth iterative function. After a number of magnitude and amplitude projections, the function xk satisfies the constant modulus property exactly while approximately maintaining the prescribed Fourier transform magnitude.


## 4. Waveform Design Based on Constant Modulus Constraint

We note thatx(t) is a deterministic waveform. It is explicitly denoted in I(y(t);g(t)∣x(t)) because the mutual information is a function of x(t), and we are interested in finding those functions x(t) that maximize I(y(t);g(t)∣x(t)) under constrains on their energy and bandwidth.Bell proposed the derivation of the mutual information in the channel-noise-only case and derived the information-based waveform solution. Here we provide the derivation of the mutual information in the presence of signal-dependent clutter.Here we have a channel (as shown in Figure2) with input Z (a zero-mean Gaussian random variable with variance σZ2), clutter D (a zero-mean Gaussian random variable with variance σD2), and additive zero-mean Gaussian noise N with variance σN2. The mutual information I(Y;Z) between Y and Z is
(7)I(Y;Z)=H(Y)-H(Y∣Z).Figure 2
Channel model in the presence of clutter and additive Gaussian noise.The differential entropiesH(Y) and H(Y∣Z) are
(8)H(Y)=12ln2πσY2=12ln2π(σZ2+σN2+σD2),H(Y∣Z)=12ln2π(σN2+σD2).Thus the mutual information is given by the expression(9)I(Y;Z)=H(Y)-H(Y∣Z)=12ln(1+σZ2σN2+σD2).Consider again the signal model of Figure1. Assume that z^k(t), y^k(t), d^k(t), and n^k(t) are the sample signal in the frequency band Fk=[fk,fk+Δf] and the sampling rate is 2Δf. The samples z^k(t) are independent, identically distributed random variables with zero mean and variance σZ2. Note that the total energy of z^k(t) is
(10)εZ(Fk)=2Δf|X(fk)|2σG2(fk).The factor 2 in the previous formula is due to the fact thatX(fk) is the two-sided spectrum of x(t) and that we are carrying out our calculations using only positive frequencies. In the time interval Ty, the total samples statistically independent are 2ΔfTy. So the variance of each sample is
(11)σZ2=εZ(Fk)2ΔfTy=2Δf|X(fk)|2σG2(fk)2ΔfTy=|X(fk)|2σG2(fk)Ty.The clutter process has the total energy on the intervalTy given by
(12)εD(Fk)=2Δf|X(fk)|2σc2(fk)Ty.The variance of each sample is(13)σD2=εD(Fk)2ΔfTy=2Δf|X(fk)|2σC2(fk)Ty2ΔfTy=|X(fk)|2σC2(fk).The noise process has the total energy on the intervalTy given by
(14)εN(Fk)=ΔfPn(f)Ty.Hence, the varianceσN2 of each sample is
(15)σN2=εN(Fk)2ΔfTy=ΔfPn(f)Ty2ΔfTy=Pn(f)2.The mutual information between each sampleZm of z^k(t) and the corresponding sample Ym of y^k(t) is
(16)I(Ym;Zm)=12ln[1+2|X(fk)|2σG2(fk)Ty{Pn(fk)+2|X(fk)|2σC2(fk)}].Now there are2ΔfTy statistically independent sample values for both z^k(t) and y^k(t) in the observation interval Ty. Thus,
(17)I(y^k(t);z^k(t)∣x(t))=2ΔfTyI(Ym;Zm)=ΔfTyln[1+2|X(fk)|2σG2(fk)Ty{Pn(fk)+2|X(fk)|2σC2(fk)}].If we now consider the frequency intervalω=[0,W], partition it into a large number of disjoint intervals of bandwidth Δf; then let the number of intervals increase as Δf→0, in the limit we obtain an integral for the mutual information I(y(t);z(t)∣x(t)), where we assume the x(t), y(t), and z(t) are confined to the frequency interval ω. Hence the mutual information I(y(t);z(t)∣x(t)) is
(18)I(y(t);z(t)∣x(t))=Ty∫Wln(1+2|X(f)|2σG2(f)Ty{Pn(f)+2|X(f)|2σC2(f)})df,
as
(19)I(y(t);g(t)∣x(t))=I(y(t);z(t)∣x(t)).Thus the mutual information between the random target impulse response and the received radar waveform is(20)I(y(t);g(t)∣x(t))=Ty∫Wln(1+2|X(f)|2σG2(f)Ty{Pn(f)+2|X(f)|2σC2(f)})df.For the MI waveform derivation, we treat the receiver filter as an ideal lowpass filter with approximate time durationTB≤T and TB≤Tg. Therefore TB can be effectively ignored, and the receive filter simply becomes an explicit statement that the radar system is band limited. Therefore, Ty is
(21)Ty=T+Tg.The mutual information between the random target impulse response and the received radar waveform is shown in formula (20). The output SINR is defined to be the ratio of the average power of the signal component to the average power of the noise and interference component [12]. Thus, SINR is expressed as
(22)SINR=∫W2|X(f)|2σG2(f)Pn(f)+2|X(f)|2σC2(f)df.We can assume a lower boundSINR0 on the SINR for the target
(23)∫W2|X(f)|2σG2(f)Pn(f)+2|X(f)|2σC2(f)df≥SINR0.The energy constraint in the bandω=[0,W] is expressed as
(24)∫W|X(f)|2df≤Ex.With these constraints in mind, we can now formulate the arbitrary waveform design problem as the following constrained optimization problem:(25)maxTy∫Wln(1+2|X(f)|2σG2(f)Ty{2|X(f)|2σC2(f)+Pn(f)})dfs.t.∫W2|X(f)|2σG2(f)Pn(f)+2|X(f)|2σC2(f)df≥SINR0s.t.∫W|X(f)|2df-Ex≤0.The previous constrained problem can be formulated as a convex optimization problem by introducing the autocorrelation sequence of the transmitted signal. Then an interior-point method can be used to carry out the optimization task. Although the solution to this formulation is local optimal, this process is complicated. We need to solve the nonlinear constrained maximization problem.After applying the technique of generating a constant modulus signal with the given Fourier transform magnitude to the above obtained waveform, we can get a waveform under multiple constraints.Figure3 represents the closed-loop radar system in signal-dependent interference proposed for target detection and estimation. In this figure, the CR signal processing involved is best described by a block labeled “COGNITIVE RADAR”. CR is an intelligent system. Through sensing the environment, CR transmits the waveform suited to the working conditions. The radar returns and environment factors help to construct the new waveform that achieves a trade-off between the mutual information and the SINR for the target, that is, an optimal trade-off between the detection and estimation criteria. Then it reconstructs a signal with constant envelope property in the time domain according to its Fourier transform magnitude. The signal satisfies the constant modulus property exactly while approximately maintaining the prescribed Fourier transform magnitude. Then the waveform is transmitted to the environment. It forms a feed-back loop, and the cycle goes on and on.Figure 3
Closed-loop radar system.


## 5. Simulation

We consider an arbitrary target spectrum and clutter spectrum shown in Figure4. The total energy is 1. The noise variance is 0.1. The lower bound SINR0 is −8 dB. The number of sample points is 128. Sampling frequency is 2. Modulus value is 0.25.Figure 4
Target spectrum and clutter spectrum.Figure5 is energy spectrum of unconstrained waveform. It shows that the optimized radar waveform only selects the dominant frequency components of the target spectrum. However, it does not distribute energy among different modes of the target. Investigating the reason, there are approximately two: one is the spectrum amplitudes scale in order to compensate for the clutter spectrum, and the other is to balance the detection performance. Hence it provides an optimal trade-off between the detection and estimation criteria.Figure 5
Energy spectrum of unconstrained waveform.Figure6 is energy spectrum of constant modulus constrained waveform. It shows that the constant modulus constraint spreads the waveform energy into additional frequency bands, but the four peak amplitudes are maintained. The energy spectrum of constant modulus constrained waveform in Figure 6 is similar to the energy spectrum of unconstrained waveform in Figure 5. Thus it guarantees the performance of the nonconstant modulus optimized waveform.Figure 6
Energy spectrum of constant modulus constrained waveform.Figure7 shows the time domain representation of the signal in the complex domain with real and imaginary parts of each instant plotted as x-axis and y-axis. The figure shows that after applying the constant modulus constraint, the temporal waveform has constant amplitude. Thus the transmitted waveform has no longer high peak amplitude in time domain and can effectively through DAC and PA of transmitter.Figure 7
Complex constellation of constant modulus constrained waveform.


## 6. Conclusions

In this paper, we investigate the mutual information between the target impulse response and received radar waveform in signal-dependent interference and channel noise. Then we discuss the problem of radar waveform design under multiple constraints. Here we consider a situation when the signal-dependent interference is not negligible. An optimal trade-off between the detection and estimation criteria is provided. After applying the technique of generating a constant modulus signal with the given Fourier transform magnitude to the optimal waveform, a waveform that has constant modulus property is obtained. Simulation results have a significant meaning in the waveform design in cognitive radar. They show that the energy spectrum of constant modulus constrained waveform is similar to the energy spectrum of unconstrained waveform. Hence the performance of the nonconstant modulus optimized waveform is guaranteed. The waveform can also be applied to a CR performing target identification.



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