# DOA estimation for wideband LFM signals with a few snapshots

- Deliang Liu
^{1}, - Zhuo Li
^{2, 3}Email author, - Xiwei Guo
^{1}and - Shen Zhao
^{1}

**2017**:28

**DOI: **10.1186/s13638-017-0816-4

© The Author(s). 2017

**Received: **14 November 2016

**Accepted: **30 January 2017

**Published: **7 February 2017

## Abstract

The wideband linear frequency modulation (LFM) signals are widely used in information systems. The conventional direction-of arrival (DOA) estimation algorithms of LFM signals rely on a large number of snapshots, some of which are not reliable in numerous practical applications such as underwater array processing. To solve the above problem, we present a modified sparse iterative covariance (MSPICE)-based estimation method in fractional Fourier transform (FrFT) domain to estimate the DOA of wideband LFM signals. First, we extend the original SPICE algorithm in FrFT domain with a specific transform order for wideband LFM signals. Then, we utilize the energy centrobaric modification method to make the original SPICE more accurate without adding more computational complexity. The simulation results demonstrate the effectiveness of the proposed method.

### Keywords

Direction of arrival Fractional Fourier transform Iterative approach Wideband LFM## 1 Introduction

DOA estimation arises an important problem in a number of fields including sonar, radar, and wireless communication in recent years [1–4]. The classical delay-and-sum (DAS) method suffers from low resolution and high sidelobe levels, while the well-known multiple signal classification (MUSIC) method, which requires a large amount of snapshots, provides super-resolution DOA estimation for narrowband uncorrelated signals [5]. Furthermore, the wideband LFM signal, which is an important class of non-stationary signals, has been widely used in the abovementioned information systems. So many DOA estimation approaches have been proposed for them, such as MUSIC method in FrFT domain [6], Estimating signal parameters via rotational invariance techniques (ESPRIT) in FrFT domain [7] and ambiguity-function-based techniques [8], etc. However, none of these methods is able to provide high-angular resolution depending on very low snapshots which is the case when the environment being sensed by the array is stationary for a short duration of time.

IAA, SLIM, and SPICE algorithms [9] proposed by Stoica are sparsity-based techniques that can estimate the DOA depending on a few snapshots with high-resolution and low sidelobes. But they do not apply to wideband LFM signals, because the steering vector of the received signal model is time-variant. So some extensions to this method have been proposed to deal with the wideband LFM signals, such as spatial resampling (SR) method [10], the accuracy of which is not high due to using an approximate model for wideband LFM signals.

In this paper, a FrFT-MSPICE method is proposed to estimate the DOA for wideband LFM signals using a few snapshots based on the modified SPICE algorithm and FrFT. First, wideband LFM signal was represented as a sine wave with a single frequency in FrFT domain, so that the time-invariant steering vector can be obtained for the original SPICE algorithm. Then, we improve the original SPICE for higher accuracy and utilize it to estimate the DOA values in FrFT domain. The accuracy of the SPICE depends on the number of scanning points in the region, but increasing this number for higher accuracy will also dramatically increase the computational complexity. So, we utilize the energy centrobaric modification method [11] to improve the accuracy of SPICE without adding too much computational cost. Finally, the simulation results confirm the effectiveness of our algorithm.

## 2 Array model

*M*sensors uniformly placed along the

*x*axis. The transmitter is located at the origin point. The distance between two adjacent sensors is

*d. T*

_{1},

*T*

_{2}, …,

*T*

_{K}are

*K*Far-field targets at

*θ*, where

*θ*= [

*θ*

_{1},

*θ*

_{2},…,

*θ*

_{K}]. Here

*K*is usually unknown, so it is considered to be the amount of potential targets (scanning points) in the region, and it is much larger than the amount of actual ones. Only a few signal power estimates of the potential targets will be non-zero, so sparsity-based algorithm can be used in array processing applications.

*a*is the signal amplitude,

*f*

_{0}is the centre frequency, and

*μ*is the chirp rate.

*m*can be expressed as the sum of

*K*delayed versions of

*x*(

*t*), given by

*e*

^{ m }(

*t*) for

*m*= 1, 2, …,

*M*is the additive Gauss white noise at the sensor

*m*,

*ρ*

_{k}is the backscattering coefficient of target

*k*for

*k*= 1,2,…,

*K*. \( {\tau}_k^m \) is the time delay of the

*k*th signal traveling to the sensor

*m*relative to the reference sensor (the first sensor), which can be expressed as

*c*is the wave speed.

*s(t*) = [

*s*

_{1}

*(t*),

*s*

_{2}

*(t*), … ,

*s*

_{ K }

*(t*)]

^{ T }is the waveform vector,

*a*= [

*a*(

*θ*

_{1}),

*a*(

*θ*

_{2}),…,

*a*(

*θ*

_{ K })]

^{ T }is the steering vector with

If the bandwidth is small compared to the carrier frequency, *a*(*θ*
_{
k
}) can be considered as time-invariant (i.e., the term exp(*-j2πμτ*
_{
m
}
*t*) in (5) can be neglected), but for wideband LFM, it can not. So, the steering vector a(*θ*
_{
k
}) of wideband LFM depends on the time *t*, and the SPICE algorithm for narrowband signals cannot be applied directly to wideband signals. To solve this problem, we derive FrFT method.

## 3 The fractional Fourier transform of LFM signal

*x*(t) is represented as [7, 12, 13]

with *p* is the transform order, *F*
^{
p
} is the FrFT operator, *K*
_{
α
}(*t*,*u*) is the kernel function, *α* is the rotation angle, *α* = *pπ*/2.

As a generalization of the standard Fourier transform, the FrFT can be regarded as a counterclockwise rotation of the signal coordinates around the origin in the time-frequency plane, and the rotation angle is *α*. When *α* =2*nπ + π*/2, FrFT is equal to Fourier transform.

*x*(

*t*) in (1) about angle

*α*can be represented as

*α*=

*α*

_{ d }= arctan(

*μs*

_{ c }

^{2}), we can get

*s*

_{ c }is the scale factor to normalize signals. In this paper, we use Ozaktas’s fast sampling-type discrete FrFT method [14] to compute the digital values for FrFT, so, here, \( {s}_c=\sqrt{N}/{f}_s \) with

*N*denotes the number of snapshots,

*f*

_{ s }is the sampling frequency. \( C= a \cos {\alpha}_d\sqrt{1+ j \tan {\alpha}_d} \exp \left[- j\pi {f}_{{}_0}^2 \sin {\alpha}_d \cos {\alpha}_d\right] \), which is a constant. Therefore, after the FrFT, the LFM signal becomes a sine wave with a single frequency

*f*=

*f*

_{0}cos

*α*

_{ d }.

*f*

_{0}= 2.4 MHz,

*μ*

_{0}= 8 × 10

^{13}Hz/s,

*f*

_{ s }= 600 MHz, N = 301. The result is shown in Fig. 2. The LFM signal in time domain (Fig. 2a) is changed into a sine signal (Fig. 2b) in FrFT domain with rotation angle

*α*

_{ d }.

*m*, the received signal in FrFT domain which is reflected by target

*k*can be expressed as

\( {X}^m={\left[{X}_1^m\left({\alpha}_d, u\right),{X}_2^m\left({\alpha}_d, u\right),\dots, {X}_K^m\left({\alpha}_d, u\right)\right]}^T,{A}_k={\left[{A}_k^1\left({\theta}_k\right),{A}_k^2\left({\theta}_k\right),\dots, {A}_k^M\left({\theta}_k\right)\right]}^T \)After FrFT, the steering vector will not depend on *u*, so we can use the SPICE algorithm to estimate the DOA values *θ*
_{
k
}.

## 4 DOA estimation by FrFT-MSPICE

*n*= 1, 2,…,

*N*is the number of snapshots.

*P*be a

*K*×

*K*diagonal matrix, whose diagonal contains the power at each angle on the scanning grid. The initial estimates \( {\widehat{P}}_k^{(0)} \) can be obtained using the SFLS method [15]

*R*of

*X*(

*n*) can be given as

*i*+ 1 iteration, the power at each angle on the scanning grid can be updated as

Then iterate (20)–(21) until \( {\widehat{P}}_k \) will not change obviously.

Finally, we search the positions of the spectral peak of \( {\widehat{P}}_k \) which are the final DOA estimates. The details of SPICE algorithm can be found in [9].

It is obvious that the accuracy of the DOA estimation depends very much on the angular scanning grid size. The higher accuracy we need, the smaller the grid size is, which means *K* is larger. But increasing *K* will dramatically increase the computational complexity of the algorithm. In order to make the DOA estimation more accurate without increasing too much complexity, we utilize the energy centrobaric modification method which is commonly used for modification of the frequency estimation in discrete spectrum.

Actually, because the existence of noise, when SPICE scans angles around the real DOA value, the energy estimation (\( {\widehat{P}}_k \)) will not be zero, which means there is “energy leakage” around the real DOA in the angular spectrum.

*k*=

*k** with the energy

*P*

_{ k*}in the angular spectrum. Then, we search for

*P*

_{ k * '}, which is the largest

*P*

_{ k }around

*k**. Finally, we utilize these two spectral lines to estimate

*θ*. But if

*P*

_{ k * + 1}and

*P*

_{ k * − 1}are nearly equal, we consider

*θ*=

*k**, and it does not need modification.

*P*

_{ k * '}= max(

*P*

_{ k * + 1},

*P*

_{ k * − 1}),

*η*is a specified tolerance.

## 5 Simulation results

We evaluate the performance of the proposed FrFT-MSPICE algorithm and compare it with other alternative methods.

### 5.1 Angular spectrum comparison

We investigate the performance of our algorithm about angular resolution, sidelobe level, and accuracy in the angular spectrum. We also compare it with other two algorithms: one is FrFT-MUSIC algorithm [6] which utilizes MUSIC algorithm combined with FrFT. But in order to apply for active radar, the rotation angle *α* has been changed from -arccot*μ* to arctan*μ*; the other is SR-IAA algorithm [10] which uses spatial resampling method combined with IAA.

The LFM signal with center frequency of 12 MHz and a bandwidth of 6 MHz is used in the simulation. The array contains 32 sensors which are uniformly spaced at half wavelength. The sampling frequency is 60 MHz and the number of snapshots is 2. We consider 3 targets at 70, 72, and 110°. The noise is assumed to be Gaussian random processes with zero mean and the SNR is 20 dB. The scanning grid for the three algorithms is uniform in the range from 1 to 180°, with 1° increment between adjacent grid points, so *K* = 180. The number of iterations is 10. One hundred independent runs are simulated to obtain the angular spectrum. Furthermore, each power value is normalized as 10 log 10(*P*
_{
k
}/*P*
_{max}), where *P*
_{max} is the maximum value of *P*
_{
k
}, *k* = 1,2,…,180.

### 5.2 Accuracy comparison between MSPICE and SPICE

## 6 Conclusions

This paper has presented a FrFT-MSPICE method for the DOA estimation of wideband LFM signal. We extend the SPICE algorithm in FrFT domain so that the DOA of wideband LFM signals can be estimated with a few snapshots. The proposed method has high angular resolution and low sidelobe levels. We also utilize the energy centrobaric modification method in order to increase the accuracy of the SPICE algorithm without imposing too much additional computational burden. The simulation results have demonstrated the effectiveness of the proposed method.

## Declarations

### Acknowledgements

This research was supported the by National Natural Science Foundation of China under Grant nos. 61601494, 61602346, 61501493.

### Competing interests

The authors declare that they have no competing interests.

**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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