- Research
- Open Access

# Cognitive radar ambiguity function optimization for unimodular sequence

- Jindong Zhang
^{1}, - Xiaoyan Qiu
^{1}Email author, - Changli Shi
^{1}and - Yue Wu
^{1}

**2016**:31

https://doi.org/10.1186/s13634-016-0325-3

© Zhang et al. 2016

**Received:**22 May 2015**Accepted:**17 February 2016**Published:**8 March 2016

## Abstract

An important characteristic of a cognitive radar is the capability to adjust its transmitted waveform to adapt to the radar environment. The adaptation of the transmit waveform requires an effective framework to synthesize waveforms sharing a desired ambiguity function (AF). With the volume-invariant property of AF, the integrated sidelobe level (ISL) can only be minimized in a certain area on the time delay and Doppler frequency shift plane. In this paper, we propose a new algorithm for unimodular sequence to minimize the ISL of an AF in a certain area based on the phase-only conjugate gradient and phase-only Newton’s method. For improving detection performance of a moving target detecting (MTD) radar system, slow-time ambiguity function (STAF) is defined, and the proposed algorithm is presented to optimize the range-Doppler response. We also devise a cognitive approach for a MTD radar by adaptively altering its sidelobe distribution of STAF. At the simulation stage, the performance of the proposed algorithm is assessed to show their capability to properly shape the AF and STAF of the transmitted waveform.

## Keywords

- Cognitive radar
- Unimodular sequence synthesis
- Optimization algorithm
- Ambiguity function
- Slow-time ambiguity function

## 1 Introduction

In radar systems, unimodular (i.e., constant modulus) sequences are usually exploited and optimized for transmission. The integrated sidelobe level (ISL) of the autocorrelation function (ACF) is often used to express the goodness of the correlation properties of a given sequence. A transmitted sequence with low ISL value reduces the risk that the echo signal of the weak target of interest is drawn in the sidelobes of the strong one or clutter interference [6]. Additionally, the unimodular sequence has low peak-to-average power ratio (PAR) which is especially desired for the transmitter [7]. A lot of literature has been focused on the topic of unimodular sequence synthesis with good properties (in particular, the ACF with low ISL values) and the many references included. These unimodular synthesis methods can be summarized into two types. The first is to use some famous sequences, such as the Golomb sequence [8], Frank sequence [9], and a pseudo random sequence, which have been proved with low sidelobes and applied in the radar systems successfully. The second is to synthesize the sequence with minimized ISL metric by the optimization algorithms [10–12]. Because the problem of reducing the ISL metric may have multiple local minima, the exhaustive search algorithm has been proposed in [12]. The computational burden of this kind of algorithm increases significantly as the sequence length increases. Some optimization algorithms have been designed as the local minimization algorithms to overcome this default [13–17]. Most of these algorithms can obtain fast convergence in descent gradient and provide quick solutions. It is worthwhile to mention that the cyclic algorithms proposed in [13] can design unimodular sequences that have virtually zero autocorrelation sidelobes in a specified lag interval and long sequences.

where *τ* and *f*
_{
d
} denote the time delay and Doppler frequency shift, respectively, and *s*(*t*) is the radar waveform. It describes the matched filter response to the target signature. The shape of AF indicates the range and Doppler resolutions of the radar system. It also demonstrates the matched filter output with respect to the interference produced by unwanted returns. It should come with no surprise that extensive research on AF synthesis exists in the literature [18–23]. Despite so much effort on this problem, few methods can synthesize the desired AF successfully. In [22], the cross ambiguity function was considered instead of AF. A pair of the waveform and receiving filter was developed simultaneously. Aubry et al. [23] deal with the design of phase-coded pulse train, which approximately maximizes the detection performance. A similarity constraint between the ambiguity functions of the devised waveform and the pulse train encoded with the prefixed sequence is required. De Maio et al. [5] also discuss the design problem of phase-coded pulse train. The average value of the STAF of the transmitted signal over some range-Doppler bins is minimized with prior information.

is equal to the energy of *s*(*t*). The volume-invariant property of AF prevents the synthesis of an ideal AF that has a high narrow peak in the origin and zero sidelobes everywhere else. In this paper, we mainly focus on the synthesis of an AF that has a clear area close to the origin or minimized ISL in a certain area on the time delay and Doppler frequency shift plane.

Additionally, it is known that a moving target detecting (MTD) radar system is designed to observe the target in range-Doppler bins [6]. Its detection performance is considerably affected by the range-Doppler response of the waveform used to illuminate the operation environment. Considering that a MTD radar transmits a burst of pulses in slow time, the STAF is defined to evaluate the range-Doppler response.

- (1)
For optimizing the shape of an AF, the optimization algorithm is proposed based on the phase-only conjugate gradient (POCG) and phase-only Newton’s method (PONM), which have been successfully applied in optimizing the phased array radar beam pattern.

- (2)
We extend the PCA and present an algorithm for optimizing the shape of STAF.

- (3)
A cognitive approach for a MTD radar system is also provided in this work. The radar system can adaptively alternate its sidelobe distribution of STAF according to the interested area and clutter distribution on the time delay and Doppler frequency shift plane. This scheme is especially attractive for detecting a target with a small radar cross section (RCS) in a heavy clutter scenario.

The rest of this work is organized as follows. Section 2 discusses the formulation of the ambiguity function synthesis problem of unimodular sequence, and the optimization method based on POCG and PONM are proposed. Section 3 defines the STAF and extends the optimization algorithm for optimizing the shape of STAF of a MTD radar system. A cognitive workflow is also given. Several numerical examples are presented in Section 4. Finally, concluding remarks and directions for future research are presented in Section 5.

## 2 Ambiguity function synthesis

### 2.1 Problem formulation

*N*denotes the number of subpulses,

*s*

_{ k }is the sequence code of the

*k*th subpulse, and

*p*

_{ k }(

*t*) is the pulse-shaping function. The typical form of

*p*

_{ k }(

*t*) is the rectangular pulse and can be expressed as

*t*

_{ p }is the time duration of subpulses and

*s*(

*t*) can be given by

denotes the cross ambiguity function (CAF) of the pulse-shaping functions *p*
_{
k
}(*t*) and *p*
_{
l
}(*t*).

*χ*

_{ s }(

*τ*,

*f*

_{ d }) can be rewritten as

*N*-space, (·)

^{ T }and (·)

^{ H }indicate transpose and conjugate transpose of a vector or matrix, respectively, and

is the subpulse CAF matrix, which is fixed once the pulse-shaping function and the number of subpulses *N* are given. Therefore, the shape of the AF *χ*
_{
s
}(*τ*,*f*
_{
d
}) is directly determined by the sequence codes \(\{s_{k}\}_{k=1}^{N}\) or the phase variables \(\{\phi _{k}\}_{k=1}^{N}\) of \(\{s_{k}\}_{k=1}^{N}\).

*τ*−

*f*

_{ d }plane, is discretized into grids with sufficient precision. The spacing of the grids is

*t*

_{ p }in the time-delay axis and 1/(

*Mt*

_{ p }) in the Doppler frequency shift axis. By substituing

*τ*=

*nt*

_{ p }and

*f*

_{ d }=

*m*/(

*Mt*

_{ p }) in Eq. (7), we have

where **U**
_{
n,m
}=**R**(*nt*
_{
p
},*m*/(*Mt*
_{
p
})).

*τ*−

*f*

_{ d }plane. Considering that the shape of AF can be controlled by the shape of DAF, we exploit the ISL metric of DAF, which is described as

*I*

_{ Ω }is the subset of the range and Doppler bins (

*nt*

_{ p },

*m*/

*Mt*

_{ p }) on

*τ*−

*f*

_{ d }plane. Additionally, the synthesized sequence should have constant modulus, i.e.,

where *ϕ*
_{
k
} is the phase of the *k*th sequence code *s*
_{
k
}. Therefore, we can think of synthesizing the unimodular sequence **s** as minimizing the ISL metric in Eq. (12) over the unimodular sequence set.

The objective function in Eq. (14) is a quartic form, which is relatively difficult to tackle. With the conclusions in [24], the objective function is also a non-convex function. Moreover, the constraint set is a non-convex set. Hence, this problem is a non-convex optimization problem. The paper [25] has suggested that maximum block improvement (MBI) algorithms are capable of providing some good-quality solutions to this kind of problem in polynomial time. A simplified and more practical method relies on the exploitation of a simpler criterion (in particular, a quadratic function) to replace the quartic function [26].

In general, constrained optimization problems such as this one can be difficult to deal with because we must simultaneously perform the optimization and satisfy the constraint. It is worthwhile to point out that unconstrained gradient-based algorithms can be generalized to the constant modular constraint case. Therefore, the constrained optimization problem can be transformed to be unconstrained. With the derivatives of the objective function with respect to the phases, a local optimum can be obtained by gradient-based algorithms, such as the conjugate-gradient method and Newton’s method. However, a local minima can also be found in the gradient equation by successive iterations if the Hessian matrix is (semi) positive definite. Furthermore, the application of the iterative algorithm is computational efficient and easy to realize.

Based on the above considerations, accounting for the complicated form of the objective function, we can obtain the local optimum in the first-order and second-order derivatives instead. Although the Hessian matrix is not (semi) positive definite, we can exploit the diagonal loading technique to make it so.

### 2.2 Optimization analysis

As already highlighted, a highly multi-modal optimization objective inevitably appears in Eq. (14). It is hard for us to obtain the global optimum by the analytical expression or the optimization method. In this section, we expect to find the local optimum for the problem in Eq. (14) and propose a computationally efficient approach.

where \(\mathbf {\phi } = [\phi _{1}, \phi _{2}, \ldots, \phi _{N}]^{T} \in \mathbb {R}^{N}\), \(\mathbb {R}^{N} \) denotes real *N*-space and Re(·) and Im(·) represent the real and imaginary part of a complex number, respectively (see the derivation in Appendix B).

Namely, the Hessian matrix of the ISL metric is required to be (semi) positive definite. With the positive definiteness of \(\frac {\partial ^{2} \text {ISL} }{ \partial \mathbf {\phi } \partial \mathbf {\phi }^{T}}\), the stable points \(\{ \widetilde {\mathbf {s}} \}\) form the set of the local minimum.

**U**

^{ ‡ }. Note that this matrix can be (semi) positive definite using the diagonal loading technique, which implies

where **I** is an identity matrix, *λ* is a constant coefficient, which should satisfy *λ*+*δ*
_{
min
}(**U**
^{
‡
})/*N*
^{2}≥0, and *δ*
_{
min
}(**U**
^{
‡
}) denotes the smallest singular value of the Hessian matrix **U**
^{
‡
}.

**U**

_{0,0}=

**I**. Hence, the corresponding optimization problem in (14) can be transformed to

where *v* is a real number.

### 2.3 Optimization method

*N*-torus, i.e., for a phase-only vector

**s**, direction

**h**

_{ c }=

*∂*ρ/

*∂*

**ϕ**, and step size

*t*

**s**

_{ i }and

**s**

_{ i+1}be the sequence at the

*i*th and (

*i*+1)th iteration. The detailed steps incorporating the phase-only conjugate gradient method and the phase-only Newton’s method are given as follows:

- 1.
Select \(\phi _{0} \in \mathbb {R}_{N}\), compute

**g**_{0}=**h**_{0}=*∂ρ*(**s**_{0})/*∂***ϕ**, and set*i*=0. - 2.For
*i*=0,1,…,*N*_{ c }, compute*t*_{ i }such thatfor all$$ \rho\left(e^{jt_{i} \text{Diag}\left(\mathbf{h}_{i}\right)} \mathbf{s}_{i}\right) > \rho\left(e^{jt \text{Diag}\left(\mathbf{h}_{i}\right)} \mathbf{s}_{i}\right) $$*t*>0 (line optimization). - 3.
Set \(\mathbf {s}_{i+1} = e^{jt_{i} \text {Diag}\left (\mathbf {h}_{i}\right)} \mathbf {s}_{i}\).

- 4.Set$$ \mathbf{g}_{i+1} = \frac{\partial \rho\left(\mathbf{s}_{i+1}\right)}{\partial \mathbf{\phi}} $$$$ \mathbf{h}_{i+1} = \mathbf{g}_{i+1} + \gamma_{i} \mathbf{h}_{i} $$$$ \gamma_{i} = \frac{\left(\mathbf{g}_{i+1}- \mathbf{g}_{i}\right)^{T} \mathbf{g}_{i+1}}{\| \mathbf{g}_{i} \|^{2}}. $$
- 5.
Set

*i*=*i*+1; if*i*<*N*_{ c }, go to Step 2; or else, go to Step 6. - 6.Compute$$ \mathbf{U}^{\ddag}(\mathbf{s}_{i}) = \frac{\partial^{2} \mathrm{\rho(\mathbf{s}_{i})}}{\partial\mathbf{\phi} \partial\mathbf{\phi}^{T}} $$$$ \textbf{h}_{i} = - \mathbf{U}^{\ddag(-1)}(\mathbf{s}_{i}) \mathbf{g}_{i}. $$
- 7.
Set

*i*=*i*+1; go to Step 6 until \(\| \rho (\mathbf {s}_{i+1}) -\rho (\mathbf {s}_{i}) \|_{2}^{2} < \varepsilon \), where*ε*is a predefined parameter.

The algorithm of POCG requires on the order of 8*ℓN*
^{2}+*ℓN* real floating point operations (flops) to form the gradient vector, where *ℓ* is the number of samples available. Per iteration, it requires 8*ℓN*
^{2}+*ℓN* flops to compute the gradient, and 2*N* flops to compute the updated search direction.

The algorithm of PONM requires on the order of 8*ℓN*
^{2}+*N*
^{2} flops to form the Hessian matrix, 2*N*
^{3}/3+*N*
^{2}/4+2*N* flops to perform matrix inversion, and 4*N*(*N*−1) to perform the production of a matrix and a vector.

### 2.4 Selection of parameter *λ*

In optimization algorithms of POCG and PONM, the local/gobal optimum is obtained by successive iterations. It should be pointed out that the Hessian matrix **U**
^{
‡
} varies with the synthesized sequence at the optimization process, and the parameter *λ*
_{
i
} should change with the smallest singular value of **U**
^{
‡
}(**s**
_{
i
}) to guarantee the positive definiteness of the Hessian matrix.

*λ*, and this will make

*λ*a constant value. The second is to calculate the eigenvalues and eigenvector of

**U**

^{ ‡ }(

**s**

_{ i }) at every iteration. Note that the matrix inversion of

**U**

^{ ‡ }(

**s**

_{ i }) is also required at every iteration, and \(\mathbf {U}^{\ddag }(\mathbf {s}_{i}) = \sum _{l=1}^{w_{i}} \delta _{l} \mathbf {v}_{i}^{l} \mathbf {v}_{i}^{l H}\), where

*w*

_{ i }=rank(

**U**

^{ ‡ }(

**s**

_{ i })). The matrix inversion of

**U**

^{ ‡ }(

**s**

_{ i }) after diagonal loading by

*λ*

_{ i }can be given by

where *λ*
_{
i
}+*δ*
_{min}(**U**
^{
‡
}(**s**
_{
i
}))>0.

## 3 Slow-time ambiguity function synthesis in cognitive MTD radar

Motivated by higher performance requirements, the radar system now can exploit different environmental information, such as geographic information database, meteorological data, previous scans and some electromagnetic reflectivity, and spectral clutter models [27]. In this paper, we consider a cognitive MTD radar system which can observe the range and Doppler bins where clutter or interference is foreseen. This radar can then transmit a burst of waveforms whose STAF generates low sidelobe values in those bins.

### 3.1 STAF optimization

*P*slow-time pulses. The transmitted pulses can be written as

where *T*
_{
r
} is the pulse repetition interval, and *T*
_{
r
}≫*Nt*
_{
p
}.

*𝜗*(

*τ*,

*f*

_{ d }) as

*𝜗*(

*τ*,

*f*

_{ d }) can also be written as

Hence, the STAF *𝜗*(*τ*,*f*
_{
d
}) can be regarded as the product of the Doppler weighted function and the AF *χ*
_{
s
}(*τ*,*f*
_{
d
}).

*τ*=

*nt*

_{ p }and

*f*

_{ d }=

*m*/(

*Mt*

_{ p }), the discretized form of

*𝜗*(

*τ*,

*f*

_{ d }) is given by

*𝜗*(

*n*,

*m*) with minimized ISL in the range-Doppler bins where the clutter exists. The ISL metric for STAF can be expressed as

where *I*
_{
C
} is the subset of the range and Doppler bins, whose sidelobes are desired to be suppressed as much as possible at the output of the MTD processor.

The proposed optimization algorithm in Section 2.3 can also be used to solve this problem.

### 3.2 Workflow of a cognitive MTD radar

## 4 Numerical examples

In order to verify the effectiveness of the proposed algorithms, we will present several numerical examples, including the AF synthesis, STAF synthesis, and detection performance of a cognitive MTD radar. In the following examples, we all assume that the unimodular sequence has *N*=100 subpulses with rectangular pulse-shaping. The time duration of each subpulse is *t*
_{
p
} and that of the total waveform is *T*=100*t*
_{
p
}. The pulse repetition interval is *T*
_{
r
}=10*T*, and the number of pulses in a CPI is *P*=64. In AF and STAF, the time delay axis *τ* is normalized by *T* and the Doppler frequency axis *f* is normalized by 1/*T*. The convergence of the proposed algorithm will be tested by using randomly generated sequences in the initialization. In the iteration process, the parameter *ε* is set to be 10^{−3}.

### 4.1 AF synthesis

Suppose that *Ω*={(*τ*,*f*
_{
d
})||*τ*|<0.2,|*f*
_{
d
}|<0.01,*τf*
_{
d
}≠0} is the interested area, which is near the origin but excludes the origin on *τ*−*f*
_{
d
} plane. With randomly generated sequence in the initialization, PCA is applied to minimize the ISL metric of the AF of the synthesized sequence.

*τ*−

*f*

_{ d }plane. The desired low sidelobes in the interested area of AF is obviously obtained in Fig. 4 b. Therefore, the synthesized sequence has a good capability of separating and detecting closely spaced targets.

Figure 4
c,d gives the zero-Doppler range profile cut and zero-delay Doppler profile cut of the AF in Fig. 4
b. The sidelobes in the interested area is suppressed to about −40 dB in the time delay axis with |*τ*|<0.2. Due to the fact that the synthesized sequence has constant modulus, the zero-delay Doppler profile cut is a sinc function.

### 4.2 STAF synthesis

*Ω*

_{1}is minimized and the averaged sidelobe of the obtained sequence is suppressed to about −50 dB in Fig. 4 b.

The desired and synthesized STAFs of the seconde type are plotted in Fig. 5
c,d. The ISL of STAF in *Ω*
_{2} is reduced and the averaged sidelobe of the obtained sequence is suppressed to about −70 dB in Fig. 5
d.

### 4.3 STAF synthesis in a cognitive MTD radar system

*τ*−

*f*

_{ d }plane is plotted and a strong clutter block lies in

For ease of simulation, the clutter in every range-Doppler bin can be treated as a stationary scattering point. Hence, the whole clutter return is the superposition of all the returns from every range-Doppler scattering point.

and consider the underlying scintillation on RCS based on different Swerling models for the moving target. The optimized shape of STAF is plotted in Fig. 6 b, in which a low sidelobe is presented in the target and heavy clutter area.

In Eqs. (46) and (47), *σ* is the value of RCS, and *σ*
_{average} is the mean value of RCS.

where *C*(*τ*,*f*
_{
d
}) is the clutter distribution. In this definition, the average scattering power of the Swerling target model is compared with the average power of all the clutter scattering points.

In Fig. 6 c,d, considering the radar scene in Fig. 6 a, the detection probability versus SCR is given for the Swerling I and III target models, and the detection probability of the optimized Frank and Golomb sequences are compared. As expected, the optimized sequence outperforms Frank and Golomb sequences, showing the performance of higher detection probability and suppressing the interference of the clutter returns from the output of MTD processing. Furthermore, as SCR increases, the detection probability is raised accordingly for both the Swerling I and III models. These two figures highlight the capability of the proposed algorithm to suitably shape the STAF of the transmitted waveform.

## 5 Conclusions

An algorithm was proposed to synthesize a unimodular sequence by minimizing the sidelobe values of AF in certain areas on the time delay and Doppler frequency shift plane. This algorithm can be convergent theoretically and practically and has been shown to be useful for ISL minimization of AF and STAF. The algorithm for synthesizing the unimodular sequence with the desired AF and STAF was built in this work.

A cognitive approach to devise waveforms for a MTD radar system was also put forward in this work. With this approach, the MTD radar system can adaptively optimize the STAF of its transmit waveform by minimizing the ISL metric of the interested area and clutter area on the time delay and Doppler frequency shift plane. The numerical example shows that better detection performance can be achieved by our proposed approach.

We note further that computational efficiency of Newton’s method was limited by matrix inversion. This algorithm is better for the sequence with a length no longer than 10^{4}. Therefore, in the future work, we will try to find a better approach and a computation-saving method.

## 6 Appendix

### 6.1 Subpulse cross ambiguity function

*k*th and

*l*th subpulse CAF expressions as follows:

*t*overlap with each other only when

*τ*=(

*k*−

*l*)

*t*

_{ p }+

*τ*

^{′}, with |

*τ*

^{′}|≤

*t*

_{ p }. The integral in Eq. (7) can be calculated in two cases.

###
**Case**
**1**.

*t*

_{ p }≤

*τ*

^{′}<0

*f*

_{ d }

*τ*

^{′}≪1, the above equation can be simplified to

###
**Case**
**2**.

*τ*

^{′}≤

*t*

_{ p }

### 6.2 Derivatives of ISL

*γ*

_{ n,m }(

**s**)=

**s**

^{ H }

**U**

_{ n,m }

**s**and 1≤

*k*

_{0}≤

*N*, we have

*ϕ*can be given by

### 6.3 Proof of Eq. (20)

**U**

_{ n,m }, we have

**1**=[1,1,…,1]

^{ T }. The second item in Eq. (20) can also be expressed as

### 6.4 Equality proof

**s**

^{∗}⊙

**U**

_{ n,m }

**s**, and given by

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under grant 61201367, the Natural Science Foundation of Jiangsu Province under grant BK2012382, the Aeronautical Science Foundation of China under grant 20142052019, the Fundamental Research Funds for Central Universities under grant NS2016042, and the Cooperative Innovation Foundation Project in Jiangsu Province under grant BY2014003-5.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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