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

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.

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 the SINR 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. Section 2 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 Figure 1 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 𝜎²_𝐺(𝑓). Let 𝑇_𝑔 be the time duration where most of the target impulse’s energy resides. It is necessary to have 𝑇_𝑔>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) 𝜎²_𝐶(𝑓), and 𝑛(𝑡) is the zero-mean receiver noise process with one-sided PSD 𝑃_𝑛(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 filter 𝐵(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.

Let 𝑦(𝑡) be the received signal given by 𝑦(𝑡)=𝑧(𝑡)+𝑑(𝑡)+𝑛(𝑡). (1)

𝑧(𝑡) and 𝑑(𝑡) are defined by 𝑧(𝑡)(𝑡)(𝑡),𝑑(𝑡)(𝑡)(𝑡),=𝑥∗𝑔=𝑥∗𝑐 (2) 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.

Let 𝐶_𝑀 denote the set of functions {𝑔(𝑡)} that have the prescribed Fourier transform magnitude 𝑀(𝜔) over a prescribed frequency set Ω. The operator 𝑃_𝑀 will assign every arbitrary function 𝑥(𝑡) a “nearest neighbor” 𝑃_𝑀𝑥(𝑡) that belongs to 𝐶_𝑀 such that there exists no other element 𝑔∈𝐶_𝑀 for which ‖𝑥−𝑔‖<‖𝑥−𝑃_𝑀𝑥‖ is satisfied.

Given an arbitrary function 𝑥(𝑡), its corresponding Fourier transform is 𝑋(𝜔)=|𝑋(𝜔)|𝑒^𝑗Ω(𝜔) and the magnitude projection of 𝑥(𝑡) is defined as 𝑃_𝑀𝑥(𝑡)⟷𝑀(𝜔)𝑒^𝑗Ω(𝜔)𝑋(𝜔),𝜔∈Ω,𝜔∈Ω^′. (3)

For a constant envelope signal 𝑥(𝑡), it can be expressed as 𝑥(𝑡)=𝐴𝑒^{𝑗𝜃(𝑡)}, (4) 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 if 𝐶_𝐴 denotes the set of functions {𝑔(𝑡)} which have constant envelope level 𝐴, the operator 𝑃_𝐴 will assign every arbitrary function 𝑥(𝑡) a nearest neighbor 𝑃_𝐴𝑥(𝑡) that belongs to 𝐶_𝐴 such that no other element 𝑔∈𝐶_𝐴 satisfies ‖𝑥−𝑔‖<‖𝑥−𝑃_𝐴𝑥‖.

Given an arbitrary signal 𝑥(𝑡)=𝑎(𝑡)𝑒^{𝑗𝜃(𝑡)}, the projection procedure is 𝑃_𝐴𝑥(𝑡)=𝐴𝑒^{𝑗𝜃(𝑡)}𝑥(𝑡),𝑡∈𝑇,otherwise. (5)

The magnitude and amplitude projection are combined according to 𝑥_𝑘+1=𝑃_𝐴𝑃_𝑀𝑥_𝑘, (6) where 𝑥_𝑘 is the 𝑘th iterative function. After a number of magnitude and amplitude projections, the function 𝑥_𝑘 satisfies the constant modulus property exactly while approximately maintaining the prescribed Fourier transform magnitude.

4. Waveform Design Based on Constant Modulus Constraint

We note that 𝑥(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 Figure 2) with input 𝑍 (a zero-mean Gaussian random variable with variance 𝜎²_𝑍), clutter 𝐷 (a zero-mean Gaussian random variable with variance 𝜎²_𝐷), and additive zero-mean Gaussian noise 𝑁 with variance 𝜎²_𝑁. The mutual information 𝐼(𝑌;𝑍) between 𝑌 and 𝑍 is 𝐼(𝑌;𝑍)=𝐻(𝑌)−𝐻(𝑌∣𝑍). (7)

Figure 2 

Channel model in the presence of clutter and additive Gaussian noise.

The differential entropies 𝐻(𝑌) and 𝐻(𝑌∣𝑍) are 𝐻(𝑌)=12ln2𝜋𝜎²_𝑌=12ln2𝜋𝜎²_𝑍+𝜎²_𝑁+𝜎²_𝐷,𝐻()=1𝑌∣𝑍2ln2𝜋𝜎²_𝑁+𝜎²_𝐷. (8)

Thus the mutual information is given by the expression 𝐼()(𝑌)()=1𝑌;𝑍=𝐻−𝐻𝑌∣𝑍2𝜎ln1+^2𝑍𝜎²_𝑁+𝜎²_𝐷. (9)

Consider again the signal model of Figure 1. Assume that 𝑧_𝑘(𝑡), 𝑦_𝑘(𝑡), 𝑑_𝑘(𝑡), and 𝑛_𝑘(𝑡) are the sample signal in the frequency band 𝐹_𝑘=[𝑓_𝑘,𝑓_𝑘+Δ𝑓] and the sampling rate is 2Δ𝑓. The samples 𝑧_𝑘(𝑡) are independent, identically distributed random variables with zero mean and variance 𝜎²_𝑍. Note that the total energy of 𝑧_𝑘(𝑡) is 𝜀_𝑍𝐹_𝑘=2Δ𝑓𝑋𝑓_𝑘²𝜎²_𝐺𝑓_𝑘. (10)

The factor 2 in the previous formula is due to the fact that 𝑋(𝑓_𝑘) is the two-sided spectrum of 𝑥(𝑡) and that we are carrying out our calculations using only positive frequencies. In the time interval 𝑇_𝑦, the total samples statistically independent are 2Δ𝑓𝑇_𝑦. So the variance of each sample is 𝜎²_𝑍=𝜀_𝑍𝐹_𝑘2Δ𝑓𝑇_𝑦=2Δ𝑓𝑋𝑓_𝑘²𝜎²_𝐺𝑓_𝑘2Δ𝑓𝑇_𝑦=𝑋𝑓_𝑘²𝜎²_𝐺𝑓_𝑘𝑇_𝑦. (11)

The clutter process has the total energy on the interval 𝑇_𝑦 given by 𝜀_𝐷𝐹_𝑘=2Δ𝑓𝑋𝑓_𝑘²𝜎²_𝑐𝑓_𝑘𝑇_𝑦. (12)

The variance of each sample is 𝜎²_𝐷=𝜀_𝐷𝐹_𝑘2Δ𝑓𝑇_𝑦=2Δ𝑓𝑋𝑓_𝑘²𝜎²_𝐶𝑓_𝑘𝑇_𝑦2Δ𝑓𝑇_𝑦=𝑋𝑓_𝑘²𝜎²_𝐶𝑓_𝑘. (13)

The noise process has the total energy on the interval 𝑇_𝑦 given by 𝜀_𝑁𝐹_𝑘=Δ𝑓𝑃_𝑛𝑓𝑇_𝑦. (14)

Hence, the variance 𝜎²_𝑁 of each sample is 𝜎²_𝑁=𝜀_𝑁𝐹_𝑘2Δ𝑓𝑇_𝑦=Δ𝑓𝑃_𝑛𝑓𝑇_𝑦2Δ𝑓𝑇_𝑦=𝑃_𝑛𝑓2. (15)

The mutual information between each sample 𝑍_𝑚 of 𝑧_𝑘(𝑡) and the corresponding sample 𝑌_𝑚 of 𝑦_𝑘(𝑡) is 𝐼𝑌_𝑚;𝑍_𝑚=122𝑋𝑓ln1+^𝑘2𝜎²_𝐺𝑓_𝑘𝑇_𝑦𝑃_𝑛𝑓_𝑘+2𝑋𝑓_𝑘²𝜎²_𝐶𝑓_𝑘. (16)

Now there are 2Δ𝑓𝑇_𝑦 statistically independent sample values for both 𝑧_𝑘(𝑡) and 𝑦_𝑘(𝑡) in the observation interval 𝑇_𝑦. Thus, 𝐼𝑦_𝑘(𝑡);𝑧_𝑘(𝑡)∣𝑥(𝑡)=2Δ𝑓𝑇_𝑦𝐼𝑌_𝑚;𝑍_𝑚=Δ𝑓𝑇_𝑦2𝑋𝑓ln1+^𝑘2𝜎²_𝐺𝑓_𝑘𝑇_𝑦𝑃_𝑛𝑓_𝑘+2𝑋𝑓_𝑘²𝜎²_𝐶𝑓_𝑘. (17)

If we now consider the frequency interval 𝜔=[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 𝐼𝑦(𝑡);𝑧(𝑡)∣𝑥(𝑡)=𝑇_𝑦𝑊2𝑋𝑓ln1+²𝜎²_𝐺𝑓𝑇_𝑦𝑃_𝑛𝑓+2𝑋𝑓²𝜎²_𝐶𝑓𝑑𝑓, (18) as 𝐼𝑦(𝑡);𝑔(𝑡)∣𝑥(𝑡)=𝐼𝑦(𝑡);𝑧(𝑡)∣𝑥(𝑡). (19)

Thus the mutual information between the random target impulse response and the received radar waveform is 𝐼𝑦(𝑡);𝑔(𝑡)∣𝑥(𝑡)=𝑇_𝑦𝑊2𝑋𝑓ln1+²𝜎²_𝐺𝑓𝑇_𝑦𝑃_𝑛𝑓+2𝑋𝑓²𝜎²_𝐶𝑓𝑑𝑓. (20)

For the MI waveform derivation, we treat the receiver filter as an ideal lowpass filter with approximate time duration 𝑇_𝐵≤𝑇 and 𝑇_𝐵≤𝑇_𝑔. Therefore 𝑇_𝐵 can be effectively ignored, and the receive filter simply becomes an explicit statement that the radar system is band limited. Therefore, 𝑇_𝑦 is 𝑇_𝑦=𝑇+𝑇_𝑔. (21)

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 SINR=_𝑊2𝑋𝑓²𝜎²_𝐺𝑓𝑃_𝑛𝑓+2𝑋𝑓²𝜎²_𝐶𝑓𝑑𝑓. (22)

We can assume a lower bound SINR₀ on the SINR for the target _𝑊2𝑋𝑓²𝜎²_𝐺𝑓𝑃_𝑛𝑓+2𝑋𝑓²𝜎²_𝐶𝑓𝑑𝑓≥SINR₀. (23)

The energy constraint in the band 𝜔=[0,𝑊] is expressed as _𝑊𝑋(𝑓)²𝑑𝑓≤𝐸_𝑥. (24)

With these constraints in mind, we can now formulate the arbitrary waveform design problem as the following constrained optimization problem: max𝑇_𝑦𝑊2𝑋𝑓ln1+²𝜎²_𝐺𝑓𝑇_𝑦2𝑋𝑓²𝜎²_𝐶𝑓+𝑃_𝑛𝑓𝑑𝑓s.t._𝑊2𝑋𝑓²𝜎²_𝐺𝑓𝑃_𝑛𝑓+2𝑋𝑓²𝜎²_𝐶𝑓𝑑𝑓≥SINR_0𝑊𝑋(𝑓)²𝑑𝑓−𝐸_𝑥≤0. (25)

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.

Figure 3 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 Figure 4. 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.

Figure 5 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.

Figure 6 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.

Figure 7 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.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. N110323005), the Natural Science Foundation of Hebei Province (no. F2013501075), and the Doctoral Scientific Research Foundation of Liaoning Province (no. 20131030).