Proceedings of the 9th European Radar Conference
Neural Network based Direction of Arrival Estimation for a MIMO OFDM Radar 1
2
1
Yoke Leen Sit , Marija Agatonovic , Thomas Zwick
Institut f¨ur Hochfrequenztechnik und Elektronik Karlsruhe Institute of Technology Karlsruhe, Germany 1 2 {leen.sit|thomas.zwick}@kit.edu,
[email protected] Abstract—In this paper, the usage of artificial neural networks (ANN) for the estimation of the direction-of-arrival (DOA) in an OFDM-based MIMO configuration radar is explored. For the extension of its range-Doppler estimation functionality, a third dimension of estimation, namely the position of objects in the azimuth plane is considered. Popular subspace-based DOA methods such as MUSIC have been explored, however they required a large processing effort. This added to the latency of the radar processing and thus is deemed to be sub-optimal for real time target localization applications. This paper presents a simulation-based investigation of using ANN for DOA estimation. The results showed that the ANN based algorithm requires less processing time and outperforms the MUSIC algorithm in terms of object separability at the separation angle of less than 5◦ .
I. I NTRODUCTION The OFDM radar and communication system (RadCom) proposed in [1] is extended to a multiple-input-multiple-output (MIMO) configuration for target localization in the azimuth. This third dimension of estimation, besides the range and Doppler estimation, has been implemented using the MUSIC algorithm in [2]. However, the MUSIC processing required a large processing time and effort, thus an alternative direction of arrival (DOA) estimation algorithm was considered. The application of artificial neural networks (ANNs) is considered as an alternative to time-consuming MUSIC algorithm for determining the azimuth angle of arrival. Using ANNs, DOA estimation is treated as a function approximation problem. Being able to map dependence between two data sets, parameters of ANNs are optimized through the learning process in order to have outputs as close as possible to the desired values. Due to their ability to provide responses almost instantaneously ANNs are very convenient as a modeling tool. Compared to standard data fitting/interpolation technique, e.g. polynomial interpolation, ANNs have greater capability of fitting a nonlinear and complex dependency, especially in the cases where increasing the order of the used polynomials does not change fitting accuracy. Owing to the mentioned advantages ANNs have been efficiently applied in a wide range of modeling problems. As it will be shown, the developed ANN models are able to give accurate results providing significant time savings. This paper is organized as follows. First the MIMO antenna array structure, the signal model at the transmitter, channel and receiver are discussed, followed by the radar range processing
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leading to the input covariance matrix for the ANNs. Finally the DOA estimation with ANNs along with the simulation results are then presented. II. MIMO A NTENNA A RRAY The transmitter comprises Nt antennas and the receiver, Nr antennas, which are spaced apart by dr = λ/2, are collocated. The spacing of the transmit antenna elements is set so that dt = Nr dr representing the highest base-line length that can be achieved without producing grating lobes over a full 180◦ azimuth scan. At the receiver, the processing of the received signal involves the transformation of the Nt × Nr antenna elements to a 1 × P equivalent ’pseudo’ receive antenna array where P = Nt Nr . The details of the purpose of this transformation can be found in [3]. In this paper, we used a 4x4 antenna configuration. III. S IGNAL M ODEL A. Transmit Signal The radar signal is an OFDM signal where arbitrary data is modulated onto each orthogonal subcarrier. One OFDM symbol consists of N subcarriers and a series of consecutive M OFDM symbols make up one frame. The time domain signal is thus
x(t) =
M −1 N −1
D (μN + n) exp (j2πfn t) rect
μ=0 n=0
t − μT T
(1) with fn denoting the individual subcarrier frequency, T , the elementary OFDM symbol duration, and D(n), called the ’complex modulation symbol’, is the arbitrary data modulated with phase-shift keying (PSK). Interference between individual subcarriers is avoided based on the condition of orthogonality given by fn = nΔf = Tn where n = 0, ..., N − 1. Interleaving structure To create separate orthogonal channels for each radiating antenna, an interleaving OFDM frame structure is applied whereby selected subcarriers will be allocated to selected transmit antennas. The interleaving structure is depicted in Fig. 1. The reason for adopting such a frame structure is to avoid coupling and interference between subchannels rendering the
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signal-to-interference ratio zero. In this manner, the bandwidth for each transmit antenna is maintained, hence the range resolution will also be conserved. This interleaving structure also helps to avoid directivity of the radiation pattern of the transmit antenna array. The resulting transmit signal at each transmit antenna element is thus
xi (t) =
M −1
N −1
Di (μN + n ˆ ) exp (j2πfnˆ t) ·
μ=0 n ˆ =nNt +T xi −1
t − μT , n = 0, ..., N − 1 (2) T where T xi denotes the index of the transmit antenna element, numbered as 1,2,3... and so forth. Therefore a 2 × 2 system would have an OFDM frame where the first antenna would transmit at the odd subcarriers of the frame and the second antenna, the even subcarriers. rect
IV. R ADAR P ROCESSING Theoretically only one OFDM symbol will be required in the case where Doppler is not considered but since there is an ongoing work to integrate the Doppler component in the DOA estimation, the full mathematical expressions detailing the OFDM signal used to sense range and Doppler have been provided. Based on (5), it can be seen that the distortions due to the channel are fully contained in the received complex modulation symbols Dr (μ, n), which are obtained at the receiver at the output of the OFDM demultiplexer prior to channel equalization and decoding. Thus comparing the transmitted symbol D(μ, n) with the softside received symbol Dr (μ, n) would yield the time-variant frequency domain channel transfer function. This is computed by simply performing an element-wise division Idiv (μ, n) =
Dr (μ, n) D(μ, n)
(6)
In this manner, the acquisition of the range and Doppler profiles will be independent of the payload data. Assuming that the object is stationary, the corresponding channel transfer function is Fig. 1.
Interleaving structure for 4 channels
The cyclic prefix (CP), a part of the tail-end of the individual OFDM symbol prepended at the beginning of every OFDM symbol, has been included but is not mathematically expressed in (1) for simplicity. B. Target & Channel Model In the channel model, an arbitrary number of point scatterers with different radar cross sections (RCS), distance, velocity as well as the angular location to the radar can be defined. For this study, only static scatterers are considered. When the transmit signal of the radar is reflected by an object at the distance of R, the signal experiences a linear phase rotation 2R of exp −j2πnΔf c0 , for all subcarriers. C. Received Signal Depending on the number of transmit antennas Nt , multiple transmit signals experience the linear phase delay with different R due to the slight difference in the antenna array positions. The received signal at one receive antenna is the superimposition of all transmit signals. yi (t) =
Nt
yT xi (t) + W (t)
i = 1, ..., Nt
2R Idiv (n) = exp −j2πnΔf c0
yT xi (t) =
−1 M −1 N
h(k) = IDFT ({Idiv (n)}) =
Dri (μ, n ˆ ) exp (j2πfnˆ t)
(4)
and
2Ri ˆ ) = Di (μ, n ˆ ) exp −j2πfnˆ Dri (μ, n c0
(5)
N −1 1 n Idiv (n) exp j2π k , N n=0 N
k = 0, ..., N − 1 (8)
In principle, the multiple antenna processing techniques can be applied directly to the baseband received signals. In the case of the OFDM radar signals however, the ANN is applied to the processed radar range profiles hi (k) instead to minimize the amount of data to be processed. With the pseudo receive antenna consisting of P elements, the output of all P antenna elements are then arranged in the form of equation (9). A → − covariance matrix is then generated from h (k) which forms the input S for the ANN.
(3)
μ=0 n=0
0 ≤ n ≤ N − 1 (7)
,
The channel impulse response containing the range profile of the object can then be determined by taking an inverse discrete Fourier transform (IDFT) of {Idiv (n)}
→ − h (k) = [h1 (k)
i=1
where W (t) is the noise term and the received signal for one transmit OFDM frame can be expressed as
h2 (k)
...
hP (k)]
(9)
V. P ROCESSING WITH ANN To determine the angular positions of scatterers a standard multilayer perceptron neural network (MLP-NN) can be exploited. The network is built up of a number of elementary processing units, called neurons, which are organized into layers (an input layer, an output layer as well as several hidden layers) (Fig. 2), [4], totaling NL − 1 layers. Every neuron in each layer of the network is connected to all neurons from
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the adjacent layer but no connections are permitted between the neurons belonging to the same layer. Each neuron is characterized by an activation function and its bias, and each connection between two neurons by a weight factor. Input signals propagate gradually from the input layer through the hidden layers up to the output layer. The output of the l-th layer can be written as Yl = F (Wl Yl−1 + Bl )
(10)
where Yl and Yl−1 are outputs of the l-th and (l − 1)-th layer respectively, Wl is a weight matrix between (l − 1)-th and l-th layer and Bl is a bias matrix between the (l − 1)-th and l-th layer.
the training set (generalization capability, which is the most important feature of ANNs). To determine the accuracy of an ANN model, average test error (ATE [%]), worst-case error (WCE [%]), and correlation coefficient between the reference and test data simulated by the ANN are calculated, [4]. The Pearson Product-Moment correlation coefficient is an indicator how the modeled values match the reference ones. It is defined by (pi − p¯i ) (qi − q¯i ) (12) r = 2 2 (pi − p¯i ) (qi − q¯i ) where pi represents the reference value, qi is the ANN computed value, p¯ is the reference sample mean and q¯ is the ANN sample mean. If a correlation coefficient is close to one that means that ANN has an excellent predictive ability whereas a coefficient close to zero indicates poor performance of the network. Since computing of the ANN response can be done practically instantaneously due to performing only basic mathematical operations and calculating elementary functions, neural models are much faster than the computatively intensive DOA algorithms i.e. MUSIC. This ability qualifies them as very suitable to be applied to determine angular positions, as it is shown in this work for the case of two scatteres VI. DOA ESTIMATION WITH ANN
Fig. 2.
Multilayer perceptron neural network (MLP-NN)
Function F is the activation function of each neuron and it is linear for input and output layer and tan-sigmoid for hidden layers F (u) = 1 − e−u / 1 + e−u
(11)
With one or two hidden layers, ANNs can approximate virtually any input-output mapping. A neural network is trained to learn relationship between sets of input-output data that are characteristics of the problem under consideration. The most known training procedure is the backpropagation algorithm and its modifications such as quasy-Newton or Levenberg-Marquardt algorithms. The backpropagation algorithm can be described shortly as follows: after the input vectors are presented to the input neurons, the output vectors are calculated. In the next step, the output vectors are compared with desired values and errors are determined. Error derivatives are then calculated and summed up for each weight and bias until whole training set has been presented to the network. The error derivatives are used to update the weights and biases for neurons in the model. The training process continues until errors are lower than the prescribed values or until the maximum number of epochs (epoch - the whole training set processing) is reached. Once trained, the network is able to provide fast response for different input vectors, even for those not included in
Application of ANNs in the area of DOA estimation is based on the inverse mapping to the one that antenna array performs. That is the mapping from the space of antenna array outputs to the space of azimuth angles of arrival [5], [6]. The input data of ANN is a spatial covariance matrix S of the antenna array outputs, and angular positions of scatterers are ANN responses. The number of neurons in the input layer of the network depends on the dimensionality of S matrix. As S is Nt Nr × Nt Nr square matrix and having in mind that ANNs cannot operate with complex numbers there should be 2(Nt Nr )2 neurons in the input layer of the network. In this work, a dimension-degraded training set is used to develop MLP-NNs. Only the first row of covariance matrix is used to represent signals at the array outputs [5]. Therefore, all inputs are organized into a 2(Nt Nr ) − 1 element vector b that is, before applied to the input layer of the network, normalized with its norm, z = b/b. Consequently, the dimensionality of input vectors is significantly reduced allowing efficient training of MLP-NNs. A number of neural models have been developed and validated independently. The training data set consisted of 859 samples referring to two scatterers moving in the azimuth from -20◦ to 20◦ , in steps of 1◦ . Minimum mutual distance between targets is 1◦ . For each position, the spatial covariance matrix is estimated. Furthermore, MLP-NNs with 31 neurons in the input layer, 1 neuron in the output layer and different number of neurons in two hidden layers have been trained and tested for each scatterer separately. After intensive experimentation it has been found that the neural networks with hidden layers
300
20
of 16 neurons for the first scatterer, and 20 neurons for the second, have demonstrated the best performance. For the chosen ANNs, the Pearson Product-Moment correlation coefficients calculated for the case of the simulation of the DOAs for the training inputs have value of 0.9998. To illustrate accuracy of the proposed model, i.e. to show how much the ANNs have learned the training data and how much are capable to generalize, in the following figures (Fig. 3 and Fig. 4) there are correlation coefficients for two scatterers plotted for training and test data. Blue dots represent the training samples simulated by the ANN models whereas the green dots refer to the 177 test data not included in the training process. A good agreement of the simulated and the reference values for both training and test samples confirms the ability of both ANN models to simulate accurately angular positions for different inputs. Using the ANN models instead of MUSIC algorithm significant time savings can be provided as no spectral search is necessary to determine directions of incoming signals. The other advantage of this approach over MUSIC is its ability to detect targets which are separated less than 5◦ as it is shown in Table I, as compared to a similar study done with MUSIC [2].
ANN response (degrees)
15
ANN response (degrees)
5 0 −5
−10 Training data Test data
Fig. 3.
−10 −5 0 5 10 Reference azimuth angles (degrees)
15
−5 −10 Training data Test data
−20 −20
−15
Fig. 4.
MLP-NN correlation coefficient for the 2nd scatterer
−10 −5 0 5 10 Reference azimuth angles (degrees)
ϕ1 ref -19.75 -16.50 -12.50 -8.50 -7.50 -5.50 -3.75 -2.75 -1.00 -1.50 0.00 1.50 4.75 5.50 8.80 10.25 12.50 15.25 17.50 18.50
10
−15
0
15
20
TABLE I R EFERENCE AND ESTIMATED AZIMUTH POSITIONS AROUND ±20◦
15
−20 −20
5
−15
20
−15
10
ϕ2 ref -18.75 -15.25 -11.25 -9.75 -6.50 4.00 -2.50 -1.75 0.00 0.00 1.50 3.00 5.50 6.50 9.75 11.50 13.50 16.50 18.50 20.00
ϕ1 est -19.7753 -16.3959 -12.4216 -8.3342 -7.3659 -5.3294 -3.6145 -2.6932 -1.1083 -1.4522 0.0017 1.6626 4.6651 5.3147 8.5948 10.3578 12.5774 15.0697 17.5140 18.4313
ϕ2 est -18.6650 -15.1467 -11.2221 -9.5410 -6.2713 -3.8061 -2.8915 -1.5639 -0.1665 -0.1965 1.5650 2.8694 5.3939 6.3561 9.6799 11.5950 13.6257 16.4561 18.4333 19.8891
20
MLP-NN correlation coefficient for the 1st scatterer
VII. C ONCLUSION An ANN-based DOA estimation used with a 4x4 MIMO OFDM radar configuration has been presented. The results when compared to the MUSIC algorithm for the same system setup in [2] has showed that it requires less processing time and effort, as well as providing better azimuth separability. Hence this algorithm has proven to be feasible for use in a real-time target localization system with the proposed MIMO OFDM Radar.
[2] Y. L. Sit, C. Sturm, J. Baier, and T. Zwick, “Direction of Arrival Estimation using the MUSIC algorithm for a MIMO OFDM Radar,” in Radar Conference, 2011 IEEE, May. 2012. [3] K. Schuler, “Intelligente Antennensysteme f¨ur KraftfahrzeugNahbereichs-Radar-Sensorik,” Ph.D. dissertation, Institut f¨ur Hochfrequenztechnik und Elektrotechnik, Universit¨at Karlsruhe (TH), Nov. 2007. [4] Q. J. Zhang and K. C. Gupta, Neural Networks for RF and Microwave Design. Norwood, MA, USA: Artech House Inc., 2000. [5] C. Christodoulou and M. Gerogiopoulos, Applications of Neural Networks in Electromagnetics. Norwood, MA, USA: Artech House Inc., 2000. [6] M. Sarevska, B. Milovanovic, and Z. Stankovic, “Neural Network-based DOA Estimation and Beamforming for Smart Antenna,” in Proc. of Conference ICEST, Jul. 2006, pp. 25–28. [7] S. Caylar, K. Leblebicioglu, and G. Dural, “A New Neural Network Approach to the Target Tracking Problem with Smart Structure,” in Proc. of IEEE AP-S International Symposium and USNC/URCI meeting, Jul. 2006, pp. 1121–1124.
R EFERENCES [1] C. Sturm, T. Zwick, and W. Wiesbeck, “An OFDM System Concept for Joint Radar and Communications Operations,” in Vehicular Technology Conference, 2009. VTC Spring 2009. IEEE 69th, Apr. 2009, pp. 1 –5.
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