Data preprocessing

API references

doatools.estimation.preprocessing.spatial_smooth(R, l, fb=False)

Applies spatial smoothing to the given covariance matrix.

Spatial smoothing can decorrelate coherent sources.

Parameters:

• R – Input covariance matrix. Both real and complex covariance matrices are supported.
• l (int) – Number of subarrays. Must be an integer that is greater than 0 and less than or equal to the size of R.
• fb (bool) – Set to True to enable forward-backward spatial smoothing. Default value is False and only forward mode spatial smoothing is computed.

Returns:

An (m - l + 1) x (m - l + 1) spatially-smoothed covariance matrix, where m is the size of the input covariance matrix.

References

[1] S. U. Pillai and B. H. Kwon, “Forward/backward spatial smoothing techniques for coherent signal identification,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 1, pp. 8-15, Jan. 1989.

doatools.estimation.preprocessing.l1_svd(y, k)

Performs \(l_1\)-SVD to help reduce the dimensionality.

Consider the following measurement model

\[\mathbf{Y} = \mathbf{A}\mathbf{X} + \mathbf{N}, (1)\]

where \(\mathbf{Y} \in \mathbb{C}^{M \times N}\) is the matrix of snapshots, \(\mathbf{A} \in \mathbb{C}^{M \times K}\) is the steering matrix, \(\mathbf{X} \in \mathbb{C}^{K \times N}\) consists of source signals, and \(\mathbf{N} \in \mathbb{C}^{M \times N}\) consists of additive noise.

When the number of snapshots, \(N\), is large, the resulting problem may be computationally expensive to solve. Note that \(\mathrm{rank}(\mathbf{A}\mathbf{X}) = K <= M\). We can decompose \(\mathbf{Y}\) into two parts: the part corresponding to the signal subspace, and the part corresponding to the noise subspace. We can keep just the part corresponding to the signal subspace to reduce the dimensionality without losing too much information.

To do so, we compute the SVD of \(\mathbf{Y}\) as \(\mathbf{U}\mathbf{S}\mathbf{V}^H\). Let \(\mathbf{T}_K = [\mathbf{I}_K \mathbf{0}]\). Multiplying both sides of (1) by \(\mathbf{V} \mathbf{T}_K\), we obtain

\[\mathbf{Y}_\mathrm{sv} = \mathbf{A}\mathbf{X}_\mathrm{sv} + \mathbf{N}_\mathrm{sv}, (2)\]

where \(\mathbf{Y}_\mathrm{sv} = \mathbf{Y}\mathbf{V}\mathbf{T}_K\), \(\mathbf{X}_\mathrm{sv} = \mathbf{X}\mathbf{V}\mathbf{T}_K\), and \(\mathbf{N}_\mathrm{sv} = \mathbf{N}\mathbf{V}\mathbf{T}_K\). We can then use (2) instead of (1) as our new measurement model to perform DOA estimation.

Parameters:

• y – The snapshot matrix, each column of which represents a single snapshot.
• k – Dimension of the signal subspace, cannot be greater than either the number of snapshots or the number of sensors.

Returns:

The reduced measurement matrix.

References

[1] D. Malioutov, M. Cetin, and A. S. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 3010-3022, Aug. 2005.