Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article
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Stochastic Process Theory and Its Applications

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Research Article | Open Access

Volume 2020 |Article ID 3529858 |

Bin Wang, Xiaolei Hao, "Robust Waveform Design Based on Bisection and Maximum Marginal Allocation Methods with the Concept of Information Entropy", Mathematical Problems in Engineering, vol. 2020, Article ID 3529858, 23 pages, 2020.

Robust Waveform Design Based on Bisection and Maximum Marginal Allocation Methods with the Concept of Information Entropy

Guest Editor: Wenguang Yu
Received20 May 2020
Accepted16 Jul 2020
Published06 Aug 2020


Cognitive radar can overcome the shortcomings of traditional radars that are difficult to adapt to complex environments and adaptively adjust the transmitted waveform through closed-loop feedback. The optimization design of the transmitted waveform is a very important issue in the research of cognitive radar. Most of the previous studies on waveform design assume that the prior information of the target spectrum is completely known, but actually the target in the real scene is uncertain. In order to simulate this situation, this paper uses a robust waveform design scheme based on signal-to-interference-plus-noise ratio (SINR) and mutual information (MI). After setting up the signal model, the SINR and MI between target and echo are derived based on the information theory, and robust models for MI and SINR are established. Next, the MI and SINR are maximized by using the maximum marginal allocation (MMA) algorithm and the water-filling method which is improved by bisection algorithm. Simulation results show that, under the most unfavorable conditions, the robust transmitted waveform has better performance than other waveforms in the improvement degree of SINR and MI. By comparing the robust transmitted waveform based on SINR criterion and MI criterion, the influence on the variation trend of SINR and MI is explored, and the range of critical value of Ty is found. The longer the echo observation time is, the better the performance of the SINR-based transmitted waveform over the MI-based transmitted waveform is. For the mutual information between the target and the echo, the performance of the MMA algorithm is better than the improved water-filling algorithm.

1. Introduction

Radar uses radio method to find targets and determine their spatial position. However, with the wide application of electromagnetic spectrum, the working environment of radar is more and more complex. The traditional radar has a single transmitting waveform, which is difficult to adapt to the complex and changeable working environment. Cognitive radar is an intelligent radar system concept proposed in recent years. This system can improve the system performance of the radar through using the feedback structure from the receiver to the transmitter to optimize the transmitted waveform based on the recognition of the target and the scene. The whole system forms a closed-loop structure [1]. In view of the leading role of cognitive radar research in the development direction of radar, experts and scholars in various countries have launched research in related fields.

Adaptive waveform design is the key problem in cognitive radar research, which makes cognitive radar transmit the waveform that adapts to the change of environment. In the past decades, many experts and scholars devoted themselves to researching on transmitted waveform to improve the detection and estimation performance of radar system for extended target. In [2], based on information theory, the author proposes a water-filling algorithm to maximize the mutual information between the received radar waveform and the target. The author studies the use of information theory to design the waveform to measure the resonance phenomenon of the extended radar target. For the deterministic target impulse response and the random target impulse response, the radar waveform design problem with waveform energy and duration constraints is solved. The optimal target detection scheme puts as much energy as possible in the maximum target scattering mode to maximize the mutual information between target and the received radar echo. In [3], the authors propose a minimax robust signal processing scheme when the prior knowledge is inaccurate and discuss robust linear filters for signal estimation and signal detection. Related applications and nonlinear methods for robust signal detection and robust estimation are also studied. In [4], considering the uncertainty of the prior information of the radar target in the actual scene, the authors propose a waveform design method based on mutual information to ensure the parameter estimation performance of complex target models. This algorithm is robust to the uncertainty under the layered game model of radar and jammer and can effectively guarantee the parameter estimation performance. In [5], the authors study the relationship between the transmitted waveform and the multitarget mutual information in the two cases of noise only and clutter included according to the maximum mutual information criterion. Compared with the LFM signal, the waveform designed based on the maximum mutual information criterion can make the radar echo contain more information about multiple targets. In [6], the authors derive the convergence of the iterative water-filling algorithm and propose an algorithm that can guarantee its convergence in the presence of various forms of time-varying errors. Simulation results show that under the condition of strong interference, the traditional iterative water-filling algorithm is divergent, but the algorithm in this paper is still convergent. In [7], the authors focus on the transmitted waveform and filter structure of polarimetric radar. The worst-case signal-to-interference-plus-noise ratio is used as the criterion under both a similarity and an energy constraint on the transmit signal. An iterative optimization method for robust design is proposed. In [8], the authors propose a comprehensive theory of matched illumination waveforms for determining extended targets and random extended targets, use signal-to-noise ratio and mutual information as optimization criteria to design matched waveforms, and extensively discuss the waveform design of random targets and known targets with correlated interference based on SNR and MI. In [9], the authors propose a multitarget detection method and adaptive waveform design algorithm for MIMO cognitive radar, which models multitarget detection as multiple hypothesis testing. An adaptive waveform design algorithm based on information theory is proposed. The semidefinite relaxation technique and semidefinite programming are used to solve nonconvex design problems, which improves the efficiency of multiple hypothesis testing. In [10], the authors propose an adaptive orthogonal frequency division multiplexing radar communication waveform design method to improve the efficiency of limited spectrum resources and study the optimization problem of the conditional mutual information between the random target impulse response and the received signal and data information rate for frequency-selective fading channels. In [11], the author investigates the design of orthogonal frequency division multiplexing multiple-input multiple-output radar waveforms with target uncertainty and improves the space-time adaptive processing detection performance of MIMO-OFDM radar in the most unfavorable case. The author proposes a method based on diagonal loading. By using the DL method, the optimization problem can be reduced to a semidefinite programming problem. In [12], the authors propose a robust waveform technique for multistatic cognitive radars in the context of signal-related clutter and derive a new method that directly assumes uncertainty on the radar cross section and Doppler parameter of the clutters. A specific clutter random optimization method using Taylor series approximation is proposed to determine a robust waveform with specific SINR outage constraints. In [13], the authors study the robust waveform design of multiple-input multiple-output cognitive radar and propose a two-step process. First, the covariance matrix of the detection signal is designed. Then, a waveform is synthesized from the obtained covariance matrix. In [14], the authors investigate the design of angle-robust joint transmit waveforms and receive filters for multiple-input multiple-output (MIMO) radars under signal-dependent interference. The method maximizes the output signal-to-interference-plus-noise ratio (SINR) in the most unfavorable case in unknown target angles. Based on rank-relaxed semidefinite programming (SDP) of nonnegative triangular polynomials, a cyclic optimization algorithm is proposed to solve this problem. In [15], the authors extend the traditional Gaussian target response to arbitrary non-Gaussian target distributions, use cognitive radar multiple hypothesis classification algorithms for non-Gaussian targets, and utilize the sparse spectrum of related narrowband target responses. In previous studies, most of them assume that the target spectrum is known. However, the real target spectrum cannot be accurately captured in practice. Even if some researchers consider the uncertainty of target spectrum and use robust technology, the solution process is very complex.

In this paper, we fully consider the uncertainty of the target in practice. Based on the concept of entropy in information theory, we establish robust waveform design model of MI and SINR. Then, we use water-filling method improved by bisection algorithm and maximum marginal allocation algorithm to maximize MI and SINR. Finally, we obtain robust waveforms with better performance than other waveforms. The whole paper is organized as follows. Section 2 is the signal model. Section 3 is the robust waveform design based on MI. Section 4 is the robust waveform design based on SINR. Section 5 gives the search method of Lagrange multipliers based on bisection algorithm. Section 6 introduces the maximum marginal allocation algorithm. Section 7 shows the simulation results and related analysis. Section 8 concludes the whole paper.

2. Signal Model

Assume that the target model is a stationary random process on time interval , with a value 0 outside of . A stationary random process is a random process whose probability distribution at a fixed time and position is the same as that of all times and positions, which means the statistical characteristics of the random process do not change over time. is a generalized stationary Gaussian random process, whose mathematical expectation and variance are independent of time, and its correlation function is only related to time interval. is the rectangle window function, and the duration of the window function is . Therefore, can be constructed, which is a random process with finite duration. Since is generalized stationary, is locally stationary in .

and represent the transmitted waveform and target, respectively, denotes clutter, and represents noise. Suppose that and are Gaussian random processes with zero mean value, and the power spectral density is and , respectively.

The transmitted waveform and the interference signal are convolved with the target to obtain and , respectively. After the addition of the above two with the noise , the echo can be obtained after the ideal low-pass filter, and the duration of is , as shown in Figure 1.

Since the real radar target signal has a finite duration, is a random process with finite energy, and it is assumed that any sample function of can be integrated. The Fourier transform of the sample function is . From Parseval’s theorem,

The energy spectral density of is

The mean and variance are defined as

In this paper, it is assumed that is 0, so the energy spectral density (ESD) and the energy spectral variance (ESV) are equal. ESV describes the average energy of a random process with finite duration and zero mean value.

For the convolution of known signal with random process , such as , the output ESV is

3. Robust Waveform Design Based on MI

3.1. Derivation of Mutual Information Formula

The mutual information to be researched in this paper is between target and echo when the transmitted waveform is known.

Before solving mutual information, the basic knowledge of information theory needs to be introduced. Suppose is a discrete random variable with a value range of , and for each , the probability of is . The empirical and historical data obtained before the experiment of obtaining samples are called prior information. In order to measure the size of the prior information, the self-information of is defined as

In information theory, the logarithm base 2 is often used, and the unit is Bit. In this paper, for the convenience of calculation, the logarithm base e is used, and the unit is Nat. Therefore, the discrete form and continuous form of information entropy are, respectively, as follows:

The mutual information between and iswhere is the sum of three Gaussian random variables with zero mean value, so it is also a Gaussian random variable with zero mean value. Considering that and are statistically independent, and is also statistically independent, the variance of is as follows:where is a Gaussian random variable, so conforms to a normal distribution. Assume that the mean value is and the variance is , then


Create a functionwith known


From the abovementioned equation, we can obtain

Let us replace with ; it can be calculated by


Because of the nature of information entropy, the constant can be ignored, so the information entropy can be obtained as

Similarly, the conditional entropy between and is

Mutual information can be obtained as

For the signals defined on the frequency interval , such as , , , and , according to the sampling theorem, each signal can be replaced by a series of samples obtained from uniform sampling. Suppose that the sampling frequency of the signal is . When is very small, the spectrum , , , is flat in and can be approximated to a constant value. The Gaussian process samples sampled with uniform sampling rate are also statistically independent.

Sample is an independent, identically distributed random variable, with zero mean value and variance , and the total energy of in the frequency interval is

The number of sample points is , and the energy is uniformly distributed on independent sample points with the same distribution. So, the variance of each sample point is

The total energy of the clutter process on time interval is

The variance of the clutter process at each sample point is

Similarly, the total energy of the noise process on time interval is

Considering that the noise is Gaussian white noise, its power spectral density is a constant in the frequency domain, so

The variance of the noise process at each sample point is

Substituting , , and into formula (19), we getthat is,

Within the observation interval , there are statistically independent sample values with the number of , so the mutual information of and in the case of known isthat is,

Within any frequency interval , it is divided into many disjoint intervals, and the interval bandwidth is . When , the number of intervals is infinite, and the integral expression of mutual information can be obtained as

3.2. Robust Water-Filling Waveform

In previous studies on waveform design, it is assumed that the prior information of the target spectrum is completely known. However, in actual scene, the real target spectrum cannot be captured with complete precision. It is assumed that the target spectrum exists in an uncertain range , which is defined by the known upper and lower bounds, that is,where is the sampling frequency, and for each sampling point, there is an upper bound and a lower bound. The confidence band of the target spectrum can be determined by spectrum estimation, so the upper and lower bounds of the estimated waveform spectrum are reasonable. In this paper, the upper and lower bounds are determined according to the uniform distribution function, which refers to the fact that the range of upper and lower bound is the random number consistent with uniform distribution within . So, the difference value of the spectrum corresponding to each sampling frequency may be different between the upper and lower bounds. The greater the difference between the upper and lower bounds of the uncertainty range is, the greater the uncertainty of the target spectrum is. For each specific target spectrum, there is an optimal transmitted waveform. However, the real target spectrum varies in the range of uncertainty, so in this paper, we adopt the maximin robust waveform design scheme.

For this, we first introduce the concept of robustness. Robustness is a term used in statistics to describe the insensitivity of control systems to perturbation of characteristics or parameters. In general, a robust signal processing scheme may not perform as well under nominal conditions as the optimal scheme under nominal conditions, but its overall performance will be good or acceptable relative to the defined feature categories. To achieve this, we must first specify a metric for the overall performance of a solution, which relates to a class of allowable conditions at the time of input. In many cases, a widely used and effective measure is the worst performance of a solution under certain input conditions. Obviously, if it performs well under most unfavorable condition, we can say that the given scheme is robust. Therefore, we propose the best performance scheme in the worst case, which is maximin robust scheme. The worst-case performance of this solution will be acceptable, which refers to the best performance that can be achieved under the most adverse conditions. So, this is not going to be very much lower than the optimal solution in the nominal case.

In the process of designing the robust transmitted waveform based on MI and SINR, the most unfavorable situation is the lower bound of the uncertainty range of the target waveform spectrum, that is, the lower bound of . Assume the real energy spectral density is , the upper bound is , and the lower bound is :

To solve the formula,that is,

The following function is established by using the Lagrange multiplier method:

After removing the integral sign and constant, it is equivalent to maximizing the function with the energy spectrum of transmitted waveform, which can be expressed by the following equation:

The above-given formula is too complicated, so we adopt the method of symbol substitution to simplify the formula. Let , , , , and ; then, equation (37) becomes

Derive to as follows:

Let , and set the derivative function to zero to find stagnation point, sothat is,


Leaving out the minus sign, we can get



Since the power spectrum density of the transmitted signal is nonnegative, the robust waveform can be expressed aswhere can be obtained from the following energy constraints:

3.3. First-Order Taylor Approximation

Assume that

Taylor’s first-order approximation is the first two terms of Taylor expansion:

Let , , and ; then,

Derive to as follows:

Taylor’s expansion at is


Then,that is,where

4. Robust Waveform Design Based on SINR

It can be seen from the signal model in Figure 1 that

The definition of SINR is the ratio of the useful signal power in the echo to the power of the interference signal plus the noise signal:where


Similarly, from the above derivation of MI robust waveform, it can be seen that, for SINR, the spectrum corresponding to each sampling frequency still has upper and lower bounds:

The problem of optimizing the SINR can be expressed as

The following function is established by using the Lagrange multiplier method:

After removing the integral sign and constant, it is equivalent to maximizing the function with the energy spectrum of the transmitted waveform, which can be expressed by the following equation:

The above formula is too complicated, so we take the method of symbol substitution to simplify the formula. Let , , , , and ; then, the above equation becomes

Derive to as follows:

We let and set the derivative function to zero to find stagnation point, sothat is,


Leaving out the minus sign, we get



Since the power spectrum density of the transmitted signal is nonnegative, the robust waveform can be expressed aswhere

can be obtained from the following energy constraints:

5. Search of Lagrange Multipliers Based on Bisection Algorithm

The abovementioned robust transmitted waveform based on SINR and MI has been obtained aswhere , , and have different values based on different criteria, and based on SINR, based on MI. As for how to find the value of , this paper adopts successive bisection algorithm to search it.

The idea of bisection algorithm is continuous split in half, which is a very classic algorithm. We suppose that is the closed interval over the real number field, and define the interval sequence as follows: , , and . And for any natural number , is equal to or , where is the midpoint of . When using bisection algorithm to search approximation, the data should be arranged in order of size.

Examples are given to illustrate the realization process of bisection algorithm, such as using bisection algorithm to find the zero point of the function. Set the function . Because the function is continuous, and , the function must have zero point in the interval . Now, use bisection algorithm to find the zero point, take the midpoint 0.5 of interval , and get . Therefore, we can narrow down the range of interval. Next, take the midpoint 0.75 of interval , and continue to compare with 0, until the error is within the allowable range; then, we can consider this point as the zero point of the function. Figure 2 shows a flowchart of bisection algorithm.

The principle used to search Lagrange multiplier is similar. Find the value interval of , take the midpoint, substitute in to find , and compare the relationship between and . Make successive approximation by bisection algorithm and assume the allowable error is , until , and it can be considered that meets the condition.

In this paper, the scope of is shown as follows:

Let and . By using the energy constraint, the value of the optimal can be approached step by step.

6. Maximum Marginal Allocation

The amount of a certain resource is limited. If resources are invested in a variety of activities, the problem of how to allocate resources to achieve the optimal total effect will arise. This is resource allocation problem. We can regard the energy allocation of transmitted waveform as a resource allocation problem, which solves the optimization problem of a single constraint. That is, how to allocate the energy of transmitted waveform to achieve the most effective result under the condition of energy constraint.

As for how to allocate the energy of the transmitted waveform in a certain frequency band, we can regard it as a multistage decision process of dynamic programming. In each stage, a decision needs to be made on the allocation of energy. The finite energy is allocated to different to maximize the overall mutual information or signal-to-interference-plus-noise ratio.

The integral form of mutual information is discretized to obtain


The above formula is too complex, and we adopt the method of symbol substitution to simplify the formula. Let

So, MI and SINR can be abbreviated as

Energy constraint is

Under the above energy constraints, we seek the maximization ofwhere

The minimum energy distribution unit is defined as , and the number of energy components is defined as .

So, can be selected in set . The core idea of MMA algorithm is to allocate the energy of in each step and allocate all the energy after step . When is set to be very small, is very large, and the energy allocated in each step will have a tiny impact. We choose the serial number corresponding to the impact that maximizes mutual information to allocate, so that each step is optimal, so as to achieve the purpose of the overall optimal.

In the first step, for any , if , it is optimal for . Let , and assign a unit of energy to it. Since the first step has already allocated a share of the energy, in the second step, for , we compare the edge increase by . So, . The second step is to find the maximum value in the above set and assign a unit of energy to it. Similarly, after step , all the energy is allocated, that is,

The flowchart is shown in Figure 3.

In order to understand the algorithm of MMA, an example is given. For example, the frequency domain is discretized into four components (). Let , which is . To maximize ,

According to the different criteria MI and SINR, can be divided into

Let , so with total energy , you can distribute 0, 1, 2, 3, 4, or 5 units of energy to , or .

The corresponding values of and are shown in Tables 1 and 2, when initial allocation of energy. The underlined value is the maximum marginal growth at this time.



Taking mutual information as an example, the realization process of MMA algorithm is illustrated. When the initial energy is allocated by Table 1, for , is 0.0708, 0.0793, 0.0880, 0.0969, respectively. Find the maximum value 0.0969 and allocate a unit of energy to . When the second energy is allocated by Table 3, the key is the marginal growth when . Find the maximum value 0.0880 and allocate a unit of energy to . Similarly, the next steps are the same. Tables 46 summarize the final energy distribution. In the end, the maximum mutual information is 0.3600 and two parts of energy are allocated to , and one part of energy is allocated to all the other parts, which is