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Abstract

In the DOA estimation of monostatic L-shaped array MIMO radar, Multiple Signal Classification algorithm is efficient. But the peak searching process of Multiple Signal Classification algorithm needs large amount of spectrum calculation. Focusing on the spectrum peak searching process of Multiple Signal Classification, an iterative search approach to reduce the calculation amount is proposed. The first- and second-order derivatives of Multiple Signal Classification spectrum functions are achieved and the calculation amount is analyzed. Two-dimensional Newton iteration methods are applied with multisearching threads and derivation information. The searching approach can greatly reduce the computational complexity of Multiple Signal Classification spectrum peak searching. The total calculation amount of the first and second derivatives is about 15 times of the spectrum function. However, in the two-dimensional searching, especially in the high accuracy processes, the amount of searched points can be reduced by ten hundreds times, and the computation is much lower than the common spectrum peak searching method. The simulation results show that when the search thread number reaches 100, the searching process can effectively achieve the entire spectrum peak and get the correct DOA estimation.

Keywords

MIMO radar Multiple Signal Classification DOA estimation spectrum peak searching Newton iteration

Introduction

The MIMO idea was brought out from communication systems in the 1970s,^1,2 and the original idea of MIMO radar was derived from the wireless communication system. Bliss, Forsythe, Rabideau, Parker, Obey and Coutts et al. proposed the concept of MIMO radar at the beginning of this century. Fishier et al. published similar results in the same period, which clarified the idea of spatial diversity and explained the performance of MIMO radar and its advantages of overcoming RCS (Radar Cross section) flicker.^3,4 At the same time, it quickly set off the upsurge of MIMO radar DOA estimation methods. The literature⁵ uses TDS technology and MUSIC (Multiple Signal Classification) algorithm to estimate aim angle of MIMO radar. The literature^6,7 studies on a variety of algorithms to detect the direction of space targets, including Capon, Amplitude and Phase Estimation (APES), Capon Amplitude and Phase Estimation (CAPES) and Generalized Likelihood Ratio Test (GLRT).

The subspace algorithms perform well in DOA estimation algorithm for MIMO radar at present. These algorithms need to search the whole solution space and calculate the spectrum value of each searched point. When the demanded precision is high, the solution space will become very large and the amount of calculation will be also expanding especially. The amount of computation is not very obvious for line array radar. But for planar array radar, the solution space is two dimensional, the computational complexity of spectrum searching is very large, and the amount of calculation is improved by the square when the demanded accuracy increased. Therefore, it is of great theoretical and engineering significance to reduce the needed searching points and the total calculation amount of the DOA estimation.

Many scholars have proposed a series of methods to improve the efficiency of MUSIC peak searching. The early methods balance the accuracy demand and the amount calculation by modifying the search step size.^1,8–11 However, these methods need to search the entire solution domain; it cannot overcome the inherent shortcomings for large calculation amount. In recent years, there have been a number of spectrum search methods based on intelligent optimization algorithm, such as the genetic algorithm (GA),^12–14 the particle swarm algorithm,^12,16 cuckoo algorithm,¹⁷ niche leapfrog algorithm,^15,18,19 etc. These algorithms do not need to search the whole solution domain, so they can greatly reduce the calculation amount. However, because of the lack of gradient information, it is necessary to calculate the spectrum peak in a wide range and the amount of calculation cannot be further reduced. In the literature,¹⁶ the first- and second- order derivatives of the uniform line array is analyzed, and the MUSIC spectrum search method of the line array is given. This method takes into account both the estimation accuracy and the computational complexity, and makes full use of derivative information to determine the search step, can effectively reduce the amount of calculation while ensuring the accuracy of estimation.

The L-shaped array, cross array and rectangular array are the typical form of the 2D-planar array. The L-shaped array and the cross array need fewer array elements, so they are relatively easy to implement. According to the literature,²⁰ the estimation accuracy of the L-shaped array is 37% higher than cross array with the same number of elements, so it has higher application value. In this paper, a two-dimensional search method for DOA estimation of L-shaped array is presented, which can sharply reduce the calculation amount of peak searching process and maintain high estimation accuracy. The signal model of L-shaped array is discussed firstly, the DOA estimation method based on multiple threads two-dimensional Newton iterative search is proposed and tested by simulating experiments.

The structure of L-shaped array and signal model

Structure of L-shaped array MIMO radar is shown in Figure 1. The transmitting and receiving array is composed of two vertical uniform line arrays. The starting point is the combination point of the transmitting and receiving antenna. Its location is the reference position. The number of transmitting array elements and receiving array elements are M and N, respectively. $d_{T}$ is the spacing of the transmitting element, and $d_{R}$ is the spacing of the receiving element. The direction of the far-field target in space is $(θ, ψ)$ , where $θ$ is azimuth, $ψ$ is the pitch angle and $λ$ is the wavelength of the carrier wave.

Figure 1.

The structure of L-shaped array.

The direction vector of receiving array is

a (θ, ψ) = {[1, e^{j \frac{2 π d_{T}}{λ} \cos (θ) \cos (ψ)} \dots, e^{j \frac{2 π (M - 1) d_{T}}{λ} \cos (θ) \cos (ψ)}]}^{T}

(1)

And the direction vector of transmitting array is

b (θ, ψ) = {[1, e^{j \frac{2 π d_{R}}{λ} \cos (θ) \cos (ψ)} \dots, e^{j \frac{2 π (N - 1) d_{R}}{λ} \cos (θ) \cos (ψ)}]}^{T}

(2)

Let

A (θ, ψ) = a (θ, ψ) b^{T} (θ, ψ)

(3)

The row of $A (θ, ψ)$ indicates that a receiving array element corresponds to all the transmitted signals. $A (θ, ψ)$ has $N \times M$ elements.

The received signal is $S = {[s_{1}, s_{2}, \dots, s_{M}]}^{T}$ within one pulse period.

If there is $P$ aim objects in the space. The total received signal of the pth target in kth pulse periods is expressed as

x (l) = \sum_{p = 1}^{P} β_{p} (l) A (θ_{p}, ψ_{p}) S (t) + w_{l} (t) l = 1, \dots, L

(4)

where

w_{l} (t)

is the vector of Gaussian noise with parameter (0,

σ_{w}^{2}

L

is accumulating times.

β_{p} (l)

is the complex amplitude of the

p

th target in lth pulse period.

MIMO radar treats received signal with matched filter. Output of matched filter is

X (l) = \int_{T} x (l) S^{H} (t) dt = \sum_{p = 1}^{P} β_{p} (l) a (θ_{p}, ψ_{p}) b^{T} (θ_{p}, ψ_{p}) + W_{k} k = 1, \dots, K

(5)

$X (l)$ is $N \times M$ matrix. Vectorize $X (l)$ as,

\hat{X} (l) = v e c [X (l)] = \sum_{p = 1}^{P} C (θ_{p}, ψ_{p}) β (l) + W_{l}

(6)

Angle vector $C (θ_{p}, ψ_{p})$ is

C (θ_{p}, ψ_{p}) = a (θ_{p}, ψ_{p}) \otimes b (θ_{p}, ψ_{p})

(7)

$\hat{X} (l)$ is the $N M \times 1$ matrix. After $L$ times accumulation, the sufficient statistic matrix is:

X = [{\hat{X}}_{1}, {\hat{X}}_{2} \dots, {\hat{X}}_{K}]

(8)

where

X

is the

N M \times L

matrix. Each column of the matrix

X

represents the output of the matched filter for all the received array elements for transmitted signal.

The covariance matrix of $X$ is

R = \frac{1}{L} X X^{H}

(9)

Because of the noise contained in the echo, $R$ is a full rank matrix in general. Do eigen value decomposition with $R$ . The P big eigen value come from object signal, named as signal eigen value. Corresponding eigen vectors are named signal vector. The matrix $U_{S}$ which is composed with signal vector is named signal matrix. Other small eigen value comes from noise, named as noise eigen value. Vector corresponding eigen vectors are named noise vector. And matrix $U_{N}$ which is composed with noise vector is named noise matrix. The space built by noise vector is orthogonal to the angle vector of all aim objects. The spectrum function of MUSIC is expressed as²¹

p_{M U S I C} = \frac{1}{C^{H} (θ, ψ) U_{N} U_{N}^{H} C (θ, ψ)}

(10)

The DOA estimation Newton iterative search approach for L-shaped array MIMO radar

As the gradient information is not used and the search direction is random, the overall calculation is too large when we use the intelligent optimization methods. This section uses the method of multipoint search. The basic idea is scattering multiple starting points (search threads) randomly (or even) when the search starts. Every search thread began iterating search at the same time, climb the nearest peak. If there is a false peak or no peak in the vicinity, then the point can be stopped in a flat area or pseudo peak. It can search for multiple peaks in two-dimensional space when number of searching threads is big enough. As the number of signal sources is known, the joint estimation of the azimuth angle and pitch angle can be obtained from the results.

The objective function and first derivative of the L-shaped array

Due to the form of fraction, direct derivative of MUSIC spectrum formula (10) is complex. So, we transform the formula with logarithm operation as formula (11):

P = - 10 \log [c^{H} (θ, ψ) U_{N} {U_{N}}^{H} c (θ, ψ)] = - 10 \log T

(11)

where

T = c^{H} (θ, ψ) U_{N} {U_{N}}^{H} c (θ, ψ)

(12)

The value of P will be very high near the target angle, and sharply drop far away from the target. P is a single valued real function of the horizontal direction angle $θ$ and the pitch angle $φ$ .

Let

W = U_{N} {U_{N}}^{H}

(13)

Then

T = c^{H} (θ, φ) W c (θ, φ)

(14)

where

W

is H-matrix

The angle expression of the $k M + p$ virtual array is

c (k M + P) = e^{- j ω_{0} (\frac{(k - 1) d_{T}}{c} \sin θ \cos ψ + \frac{(p - 1) d_{R}}{c} \cos θ \cos ψ)} = e^{- j ω_{0} τ_{k M + p}}

(15)

Define the transmitting distance vector as $Γ_{T}$ and the receiving distance vector as $Γ_{R}$ .

Γ_{T} (k M + p) = \frac{(k - 1) d_{T}}{c}

(16)

Γ_{R} (k M + p) = \frac{(p - 1) d_{R}}{c}

(17)

Then

τ_{i} = Γ_{T} (i) \sin θ \cos ψ + Γ_{R} (i) \cos θ \cos ψ

(18)

T = [\sum_{m = 1, n = 1}^{m < M N, n < M N} e^{- j ω_{0} (τ_{m} - τ_{n})} W (m, n)]

(19)

$T$ is the sum of $M N \times M N$ elements. Defining the elements of $T$

T (m, n) = e^{- j ω_{0} (τ_{m} - τ_{n})} W (m, n)

(20)

Derivation of T

d T (m, n) = (- j ω_{0} W (m, n) e^{- j ω_{0} (τ_{m} - τ_{n})} d (τ_{m} - τ_{n})

(21)

Definition matrix $D_{τ}$ with follow element

D_{τ} (m, n) = d (τ_{m} - τ_{n})

(22)

Then:

d T = - j ω_{0} c^{H} (θ, ψ) W • D_{τ} c (θ, ψ)

(23)

Here to analyze the composition of $D_{τ}$ . Derivative of $τ_{i}$ is

d τ_{i} = \frac{d τ_{i}}{d θ} d θ + \frac{d τ_{i}}{d ψ} d ψ = (- Γ_{T} (i) \cos θ \cos ψ + Γ_{R} (i) \sin θ \cos ψ) d θ - (Γ_{T} (i) \sin θ \sin ψ + Γ_{R} (i) \cos θ \sin ψ) d ψ

(24)

The elements of matrices $D_{T}$ and $D_{R}$ are defined as follows:

\begin{array}{l} D_{T} (m, n) = Γ_{T} (m) - Γ_{T} (n) \\ D_{R} (m, n) = Γ_{R} (m) - Γ_{R} (n) \end{array}

(25)

These two matrices are skew symmetric matrices. Name $D_{T}$ and $D_{R}$ as the difference matrix of transmitting and receiving. So,

D_{τ} (m, n) = d (τ_{m} - τ_{n}) = (- D_{T} (m, n) \cos θ \cos ψ + D_{R} (m, n) \sin θ \cos ψ) d θ - (D_{T} (m, n) \sin θ \sin ψ + D_{R} (m, n) \cos θ \sin ψ) d ψ

(26)

Then

\begin{array}{l} \frac{D_{τ}}{d θ} = - D_{X} \cos θ \cos ψ + D_{Y} \sin θ \cos ψ \\ \frac{D_{τ}}{d ψ} = - (D_{X} \sin θ \sin ψ + D_{Y} \cos θ \sin ψ) d ψ \end{array}

(27)

Combining equations (27) and (23), we can get the formula

\begin{array}{l} \frac{d T}{d θ} = j ω_{0} c^{H} (θ, ψ) W • (D_{T} \cos θ \cos ψ - D_{R} \sin θ \cos ψ) c (θ, ψ) \end{array}

(28)

and

\begin{array}{l} \frac{d T}{d ψ} = j ω_{0} c^{H} (θ, ψ) W • (D_{T} \sin θ \sin ψ + D_{R} \cos θ \sin ψ) c (θ, ψ) \end{array}

(29)

Define:

\begin{array}{l} D_{1 θ} = D_{T} \cos θ \cos ψ - D_{R} \sin θ \cos ψ \\ D_{1 φ} = D_{T} \sin θ \sin ψ + D_{R} \cos θ \sin ψ \end{array}

(30)

Then, the formulas (29) and (30) can be simplified as:

\begin{array}{l} \frac{d T}{d θ} = j ω_{0} c^{H} (θ, ψ) W • D_{1 θ} c (θ, ψ) \\ \frac{d T}{d ψ} = j ω_{0} c^{H} (θ, ψ) W • D_{1 φ} c (θ, ψ) \end{array}

(31)

Formula (31) is the first order partial derivative formula of $T$ . Thus, the partial derivative of the objective function P is

\begin{array}{l} \frac{d P}{d θ} = - \frac{10}{T \ln 10} \frac{d T}{d θ} \\ \frac{d P}{d ψ} = - \frac{10}{T \ln 10} \frac{d T}{d φ} \end{array}

(32)

Second order partial derivative of objective function

We have obtained the expression of the first order partial derivative of $T$ . In this section, we analyze the expression of the second order partial derivative $\frac{\partial^{2} T}{\partial θ^{2}}$ , $\frac{\partial^{2} T}{\partial θ \partial ψ}$ , $\frac{\partial^{2} T}{\partial ψ^{2}}$ and $\frac{\partial^{2} T}{\partial ψ \partial θ}$ based on the first order partial derivative.

Calculation of partial derivatives based on $T$ , take partial derivative of formula (20),

\frac{\partial^{2} T (m, n)}{\partial θ^{2}} = - j ω_{0} W (m, n) e^{- j ω_{0} (τ_{m} - τ_{n})} \times [- j ω_{0} ({[\frac{d (τ_{m} - τ_{n})}{d θ}]}^{2} + \frac{\partial^{2} (τ_{m} - τ_{n})}{\partial θ^{2}}]

(33)

\frac{\partial^{2} T (m, n)}{\partial θ \partial ψ} = - j ω_{0} W (m, n) e^{- j ω_{0} (τ_{m} - τ_{n})} × [- j ω_{0} (\frac{d (τ_{m} - τ_{n})}{d θ} \times \frac{d (τ_{m} - τ_{n})}{d ψ} + \frac{\partial^{2} (τ_{m} - τ_{n})}{\partial θ \partial ψ}]

(34)

\frac{\partial^{2} T (m, n)}{\partial ψ^{2}} = - j ω_{0} W (m, n) e^{- j ω_{0} (τ_{m} - τ_{n})} × [- j ω_{0} ({[\frac{d (τ_{m} - τ_{n})}{d ψ}]}^{2} + \frac{\partial^{2} (τ_{m} - τ_{n})}{\partial ψ^{2}}]

(35)

\frac{\partial^{2} T (m, n)}{\partial ψ \partial θ} = - j ω_{0} W (m, n) e^{- j ω_{0} (τ_{m} - τ_{n})} × [- j ω_{0} (\frac{d (τ_{m} - τ_{n})}{d θ} \times \frac{d (τ_{m} - τ_{n})}{d ψ} + \frac{\partial^{2} (τ_{m} - τ_{n})}{\partial ψ \partial θ}]

(36)

These four formulas are the partial derivative of $T (m, n)$ to $θ$ and $ψ$ . The first derivatives $\frac{d (τ_{m} - τ_{n})}{d θ}$ and $\frac{d (τ_{m} - τ_{n})}{d ψ}$ have been obtained in the previous section. The expression of the four partial derivatives of $τ_{i}$ is as follows:

Derivation of formula (22),

\frac{\partial^{2} τ_{i}}{d θ^{2}} = - (Γ_{T} (i) \sin θ \cos ψ + Γ_{R} (i) \cos θ \cos ψ)

(37)

\frac{\partial^{2} τ_{i}}{d ψ^{2}} = - (Γ_{T} (i) \sin θ \cos ψ + Γ_{R} (i) \cos θ \cos ψ)

(38)

\frac{\partial^{2} τ_{i}}{d θ d ψ} = (Γ_{T} (i) \cos θ \sin ψ - Γ_{R} (i) \sin θ \sin ψ)

(39)

\frac{\partial^{2} τ_{i}}{d ψ d θ} = - (Γ_{T} (i) \cos θ \sin ψ - Γ_{R} (i) \sin θ \sin ψ)

(40)

Substituting the formulas (33)–(36), we define:

\begin{array}{l} D_{20} = D_{T} \sin θ \cos ψ + D_{R} \cos θ \cos ψ \\ D_{21} = D_{T} \cos θ \sin ψ - D_{R} \sin θ \sin ψ \end{array}

(41)

$D_{20}$ is the second order derivative delay matrix and $D_{21}$ is the second order partial delay matrix. Finishing (33)–(41), we have:

\frac{\partial^{2} T}{\partial θ^{2}} = - j ω_{0} c^{H} (θ, ψ) W • [- j ω_{0} D_{1 θ} • D_{1 θ} + D_{20}] c (θ, ψ)

(42)

\frac{\partial^{2} T}{\partial ψ^{2}} = - j ω_{0} c^{H} (θ, ψ) W • [- j ω_{0} D_{1 ψ} • D_{1 ψ} + D_{20}] c (θ, ψ)

(43)

\frac{\partial^{2} T}{\partial θ \partial ψ} = - j ω_{0} c^{H} (θ, ψ) W • [- j ω_{0} D_{1 θ} • D_{1 ψ} + D_{21}] c (θ, ψ)

(44)

\frac{\partial^{2} T}{\partial ψ \partial θ} = - j ω_{0} c^{H} (θ, ψ) W • [- j ω_{0} D_{1 θ} • D_{1 ψ} - D_{21}] c (θ, ψ)

(45)

where

•

is dot product operation.

The second order partial derivative of the objective function is:

\frac{\partial^{2} P}{\partial θ^{2}} = - \frac{10}{T^{2} \ln 10} (T \frac{\partial^{2} T}{\partial θ^{2}} - {(\frac{d T}{d θ})}^{2})

(46)

\frac{\partial^{2} P}{\partial ψ^{2}} = - \frac{10}{T^{2} \ln 10} (T \frac{\partial^{2} T}{\partial ψ^{2}} - {(\frac{d T}{d ψ})}^{2})

(47)

\frac{\partial^{2} P}{\partial θ \partial ψ} = - \frac{10}{T^{2} \ln 10} (T \frac{\partial^{2} T}{\partial θ \partial ψ} - \frac{d T}{d θ} \times \frac{d T}{d ψ})

(48)

\frac{\partial^{2} P}{\partial ψ \partial θ} = - \frac{10}{T^{2} \ln 10} (T \frac{\partial^{2} T}{\partial ψ \partial θ} - \frac{d T}{d θ} \times \frac{d T}{d ψ})

(49)

Two-dimensional gradient iteration method

The number of aim object is $P$ which is known before DOA estimation. The goal of spectrum peak searching is to find the highest $P$ peaks. The feature of the peak points is that the first order partial derivatives are 0. The example graph of the spectrum function in the two-dimensional space is shown in Figure 2.

Figure 2.

Two-dimensional search objective function.

Figure 3.

Simulation results of two-dimensional iterative search based on L-shaped array.

There are five peaks corresponding to the five signal sources. The purpose of the searching process is to find the five peaks. For the two-dimensional solution space, two factors must be determined in the search process: direction and step length. Because of noise added to the signal, there are many false peaks in the solution domain. There are many peak searching methods such as GA, particle swarm optimization (PSO) method, Newton iteration method and stochastic parallel gradient descent (SPGD) method. Among those methods, Newton iteration method makes full use of first- and second-order derivates and can give quadratic convergence rate. So, this article uses Newton iteration with first- and second-order derivates to do the peak searching of MUSIC algorithm. Because Newton iteration method can search for only one peak,²² we use multiple thread method with every thread using Newton iteration method.

Mark a point in the solution space $x = {[θ, ψ]}^{T}$ , its first derivative matrix is:

\nabla P = {[\frac{d P}{d θ}, \frac{d P}{d ψ}]}^{T}

(50)

Hesse matrix:

\nabla^{2} P = [\begin{array}{l} \frac{\partial^{2} P}{\partial θ^{2}} & \frac{\partial^{2} P}{\partial θ \partial ψ} \\ \frac{\partial^{2} P}{\partial ψ \partial θ} & \frac{\partial^{2} P}{\partial ψ^{2}} \end{array}]

(51)

Newton search strategy can be determined by

x^{n + 1} = x^{n} + k {(a I + \nabla^{2} P)}^{- 1} \nabla P

(52)

where

{}^{- 1}

means inverse matrix,

k

is the damping coefficient and

a

is the second order acceleration coefficient. To ensure the search accuracy, the damping coefficient values need to be relatively small. So, if the iteration thread is started in the flat region, it cannot reach the peak region within little steps. To reach all the peaks, it is needed to start multiple threads randomly at the same time. When the iteration step length of a searching thread is smaller than controlled accuracy, the thread stops. After all the searching threads end their searching process, the maximum value of the spectrum function is chosen as the estimation result according to the known source number.

The solution space is $θ \in (0, 180)$ , $φ \in (0, 90)$ . As the plane angle and pitch angle are periodic with the maximum value. When the iteration process is out of range, it should be processed as follows:

θ = {\begin{array}{l} θ - 180 θ > 180 \\ θ 0 \leq θ \leq 180 \\ θ + 180 θ <0 \end{array}

(53)

φ = {\begin{array}{l} φ - 90 φ > 90 \\ θ \leq φ \leq 90 \\ φ + 90 φ <0 \end{array}

(54)

The iteration process is described as follows:

Start multiple searching threads from random points in the angle space.

Calculate and record the spectrum function of every thread.

Calculate the first- and second-order derivates, and compose vector and matrix as equations (50) and (51).

Decide iterating direction and step length for every thread as equation (52).

If the step length is small enough (reference to expected accuracy), then end the thread.

If all the threads are end, go to step 7, else go to step 2.

Pick the P big spectrum function as the MUSIC peak, and the position of those peaks show the DOA estimation. (P is the number of aim object and is known before peak searching).

Calculation analysis of MUSIC spectrum and one or second order derivative

In the process of MUSIC spectrum DOA estimation, the common process is to calculate covariance matrix, and then do the singular value decomposition with covariance matrix. The noise subspace $U_{N}$ is obtained by sorting the eigen values. These operations only occur once at the beginning of MUSIC algorithm, so it contributes little to the whole calculation amount. But spectrum function must be calculated with every searched point. So, the most calculated amount is from spectrum function calculation of searched points. In the process of spectrum peak searching, the spectrum function is calculated by formula (10). Since the intermediate term in the denominator of the formula (10) is a constant, it is necessary to calculate the formula:

W = U_{N} U_{N}^{H}

For L-shaped array MIMO radar, $W$ has $M N \times M N$ dimension. In order to facilitate the analysis, let

X = M N

(55)

According to the formula (10), the MUSIC spectrum of each search point needs $X^{2} + X$ times multiplication and plus operation.

The two matrices of $D_{T}$ and $D_{R}$ which are frequently used in the process of the first and second derivatives, are only related to the array structure, and can be obtained before the beginning of the spectrum peak search process.

$D_{1 θ}$ and $D_{1 φ}$ from formula (31) and (32) need to be calculated firstly. According to the definition (30), ignoring some single operations, this process needs $4 X^{2}$ times multiplication and plus operation. Formula (31) contains $4 X^{2} + 2 X$ times multiplication and plus operation. Thus, the first derivative of each search point needs $8 X^{2} + 2 X$ times multiplication and addition operation.

Since $D_{1 θ}$ and $D_{1 φ}$ have been obtained in the first derivative solution, no repeated calculation of the second order derivative formula (42)–(45) is needed. Therefore, we only need to calculate $D_{20}$ and $D_{21}$ . According to formula (41), this process needs $4 X^{2}$ times multiplication and addition operation. In the calculation process of formula (42)–(45), there needs $4 X^{2} + X$ times multiplication and plus operation. Summarize the analysis above, the second order derivative calculation needs $20 X^{2} + 4 X$ times multiplication and plus operation.

The calculation amount needed for one spectrum calculation:

C M_{M U S I C} = 2 X^{2} + X

(56)

The calculation amount needed for first derivative calculation:

C M_{1 D} = 8 X^{2} + 2 X

(57)

The calculation amount needed for second derivative calculation:

C M_{2 D} = 20 X^{2} + 4 X

(58)

When the first and second derivatives information are used to guide the search process of the spectrum peak, compared with the search procedure which only calculates the spectrum value, the ratio of the calculation of each search point is:

C M_{R} = \frac{30 X^{2} + 7 X}{2 X^{2} + X}

(59)

The derivative information is used to guide the searching process of the spectrum function, and the calculated amount of each search point is 15 times of spectrum functions. In other words, only when the calculated points of the spectrum peak search process is less than 1/15 of the direct search, this method is more efficient than direct searching method. Through the rational arrangement of the search process, as soon as possible to discard fault points, it is easy to achieve. Therefore, it is meaningful to use the derivative information to guide the spectrum peak search to reduce calculation.

Simulation experiment

This section is aimed at the DOA estimation for monostatic L-shaped MIMO radar, from receiving signals with model in section “The structure of L-shaped array and signal model”. Estimate DOA based on the gradient search method proposed in section “The DOA estimation Newton iterative search approach for L-shaped array MIMO radar”. Test the various aspects of the performance of the method and the influence of various factors on the performance of the algorithm.

Experimental condition: L-shaped array with 10 transmit antenna and 10 receiver antenna with 1.5 m between antennas. Signal center frequency is 100 MHz. The number of snapshots is 1024. There are five signal sources with DOA at (15,10), (30,25), (45,40), (105,65) and (155,80).

Experimental method: SNR = 0 dB. Randomly scatter 30 initial search points and the control accuracy is 0.0001(−40 dB). There are $180 \times 10000 \times 90 \times 10000 = 1.62 \times 10^{12}$ times of the peak calculation needed to achieve such precision when using ordinary MUSIC spectrum search method.

Estimate DOA using the Newton iteration method described in section “Two-dimensional gradient iteration method”. Record the points in the iteration process. Points of successful searching threads are labeled red. And points of failed threads are labeled dark.

Experimental result: Search all five local extreme points successfully. A total of 14 search points successfully found the local extreme points. Searching result is shown in Fig. 3. Red points dedicate the succeeded points, and the blue points dedicate fault points. The iteration steps are as follows:

[34,26,51,26,125,169,211,59,55,39,39,67,188,26,40, 43,279,42,206,172]

The maximum number of steps is the 279, the minimum number of steps is only 26. The results show that the method is effective for two-dimensional DOA estimation. The total number of iteration steps is 1897, plus the number of iteration steps which is discarded, the total number of iteration steps is 3482. According to the above analysis, the calculation of this method is far less than the ordinary MUSIC peak search method.

Conclusion

To reduce the calculation amount of two-dimensional DOA estimation of L-shaped array monostatic MIMO radar, this paper proposes a Newton iteration method. It analyzes the spectrum function and obtains the first- and second-order derivates. The calculation amount of derivate calculation is analyzed. The calculation amount of first- and second-order derivates calculation is 15 times as the calculation amount of the spectrum function. So, when the number of searched points is less than 1/15 of spectrum function, the calculation amount of the whole peak searching process is lower than direct searching process. Simulation experiment shows that when the searching error is controlled as 10⁻⁴, just less than, 4000 points need to be searched, so the method can sharply reduce the calculation amount of DOA estimation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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The iterative search approach DOA estimation of monostatic L-shaped array MIMO radar

Abstract

Keywords

Introduction

The structure of L-shaped array and signal model

The DOA estimation Newton iterative search approach for L-shaped array MIMO radar

The objective function and first derivative of the L-shaped array

Second order partial derivative of objective function

Two-dimensional gradient iteration method

Calculation analysis of MUSIC spectrum and one or second order derivative

Simulation experiment

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References