Efficient Computation of Joint Direction-Of-Arrival and Frequency Estimation Y. He 1 , K. Hueske 2 , E. Coersmeier 3 and J. G¨ otze 2 1 Institute for Integrated Systems, Ruhr University of Bochum,44780 Bochum, Germany 2 Information Processing Lab, Dortmund University of Technology, 44221 Dortmund, Germany 3 Nokia Research Center, 44807 Bochum, Germany Abstract— The efficient computation of joint direction-of-arrival (DOA) and frequency estimation from the data matrix obtained from a sensor array is discussed. High-resolution ESPRIT/MUSIC algorithms are used to compute the estimates. A preprocessing step uses a two-sided DFT (computed using FFT) and applies a threshold to generate a sparse matrix from the given data matrix. The Lanczos method is used to compute the SVD/EVD of the sparse matrix. This results in a reduced computational complexity if the complexity of the preprocessing step is small compared to the reduction of the computational effort obtained by exploiting the sparsity of the matrix. We also compare this procedure with the estimations based on one sensor and one snapshot of the sensor array, respectively. In this case we can build Hankel matrices from the data samples and apply ESPRIT/MUSIC methods to these Hankel matrices and these matrices after the preprocessing step, respectively. This also yields a reduced computational complexity (again using Lanczos’ method) but decreases the accuracy of the estimates. We compare the computational effort and the mean square error (MSE) of the estimates of the different approaches. Keywords—DOA and frequency estimation, MUSIC/ESPRIT, DFT, Threshold, Lanczos method. I. INTRODUCTION Direction-Of-Arrival (DOA) and frequency estimation using the output samples of an array of sensors has gained much attention and various methods were advocated for joint esti- mation of these parameters [1] [2]. On the one hand, the parametric methods, like MUlti- ple SIgnal Classification (MUSIC) [3], [4] and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) [5], [6] estimate the parameters from a singular- value decomposition (SVD) of the N × L data matrix (or an eigenvalue decomposition (EVD) of the respective correlation matrices), which is given by the N samples taken at each of the L sensors. The main advantages of MUSIC/ESPRIT are the high-resolution estimates of the DOAs and frequencies, while the computational effort compared to maximum like- lihood method is significantly reduced. However, due to the required computation of the SVD or EVD, the computational complexity is still high compared to simple non-parametric methods based on the Discrete Fourier Transform (DFT). On the other hand, these simple non-parametric methods using the Fast Fourier Transform (FFT) to compute the DFT can be used for computing a rough estimate of the parameters with a very low computational complexity. Based on the Fourier transformed sensor array output data, both, frequencies and DOAs can be estimated using the FFT [7], [8]. In this work the FFT based methods and the MUSIC/ESPRIT methods are combined: 1) FFTs are used to compute the 2D-DFT of the data matrix, i.e. the columns are transformed in the temporal frequency domain and the rows are transformed into the spatial frequency domain. This is used as a preprocess- ing step, where the DFTs compress the harmonics in time/space. 2) In the resulting matrix the matrix elements with small amplitudes are eliminated (set to zero) by using a threshold. This works like a filter function to alleviate the noise. A threshold that is based on the variance of the noise is a good initial choice [9]. 3) The resulting sparse matrix is used for MUSIC/ESPRIT estimation, where Lanczos methods [10] are used for computing the SVD/EVD. The main advantages of the Lanczos methods [10], [11] are that only matrix-vector products have to be computed and the original matrix is not modified during the iteration steps. Thus the Lanczos procedure is useful for large matrices, especially if they are sparse or if fast routines for computing matrix- vector products are available, such as for Toeplitz or Hankel matrices. It is also possible to estimate the DOAs and frequencies sep- arately. For this approach Hankel matrices are constructed with the samples of only one sensor for frequency estimation and with only one sample of each sensor for direction estimation, respectively. Then we will use the preprocessing procedure described above and apply it to both Hankel matrices. The use of Hankel matrices results in a decreased estimation accuracy but reduces the number of required data samples. The performance of the proposed methods is evaluated using simulations. The results show, that the estimation quality is comparable to existing approaches. Furthermore, we compare the computational complexity of the presented methods to show when their application is worthwhile. In [12] Xu, Cho and Kailath use Lanczos methods for parameter estimation. Applying the Lanczos SVD to Hankel matrices, a fast routine for computing matrix-vector multi- plication is used, which leads to a reduced computational complexity. However, there are no considerations regarding the use of FFT as preprocessing step, threshold, or sparse structures. Furthermore, the method can only be applied to problems where the matrices have Hankel structure.