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of Rxx. Because stable SVD algorithms are available, several methods rely on decomposing the data matrix instead of diagonalizing the spatial covariance matrix to obtain the basis for the signal subspace. In practice, the N − r smallest eigenvalues will not be precisely σ 2; rather, they will all have small values compared to the signal eigenvalues. This is because the matrix Rxx is not known perfectly, but must be estimated from the data [12]. A common estimator for the spatial covariance matrix is the sample spatial covariance matrix, which is obtained by averaging rank-one data matrices of the form (xn xHn ), i.e.:

where K is the total number of snapshots of data available from the sensors.

Although the discussion so far has focused on the uniform linear array, the principles of signal and noise subspaces also apply to other array geometries such as the uniform planar and the semispherical arrays.

2.2ANTENNA BEAMFORMING BASICS

In this section, a brief introduction to beamforming is provided. Classical direction of arrival estimation algorithms are based on beamforming concepts; therefore, this discussion is included to give the reader the necessary background to understand Sections 3.1.1 and 3.1.2.

If a weighted linear combination of the output of each element is taken, the array output can be computed by

where wk are the complex weights of the beamformer, which are shown in Figure 2.2. In vector notation, the output can be written as:

where the N × 1 complex vector w contains the beamformer weights wk, k = 0, 1, …, N.

The array’s response to the incoming signals can be controlled by adjusting the elements of the weight vector wn. This process is referred to as spatial filtering or beamforming. Many methods exist to design or compute wn such that it produces a desired pattern. A beampattern is a plot of the gain of the beamformer in each possible direction.

Figure 2.3 shows a plot of the beampattern of a 10-element array where all the weights are equal to 1, for angles of arrival ranging from −90° to +90°.

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Figure 2.3: Polar plot of beampattern for a 10-element uniform linear array; D = λ /2.

is the vector containing the beamformer weights. The steering vector is defined as

The beamformer output can also be written as a vector inner product as follows:

The beamformer gain of the signal s0[k] is the beampattern evaluated at the angle of arrival corresponding to s0[k]. The beampattern gain can be evaluated as the vector inner product of the weight vector w and a(θ). If w and a(θ) are orthogonal, then wHa(θ) = 0 and hence the signal s0[k] is cancelled or nulled. Now, suppose two digitally modulated signals, s0(t) and s1(t), are present with angles of arrival θ0 and θ1, respectively. The beamformer output is given by

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For a uniform linear array (Figure 1.3), the relationship between θ and ω is given in (2.24). Note that in the case where only one signal is present (r = 1) and neglecting noise, the beamformer output using the conventional beamformer is:

In this case, the beamformer is said to be phase-aligned with the signal of interest and the signal of interest appears undistorted at the output. The signal-to-noise ratio (SNR) gain of the conventional beamformer can be computed by comparing the SNR at the output of a single element with the overall beamformer output. Assuming E[|vn[k]|2] = σ 2 and knowing |an(θ)|2 = 1, the SNR of the signal received by the kth element can be computed by first considering the signal model for element k.

The SNR of the received signal by a single element is (Es /En)=(Es /σ 2).

Assuming E[vkvkH] = σ 2I and knowing a(θ)Ha(θ) = N, the SNR of the signal at the beam-

Hence, the SNR for the overall beamformer output is (Es /En)=(NEs /σ 2). The SNR has increased by a factor of N over the SNR at the output of a single element.