36 NARROWBAND DIRECTION OF ARRIVAL ESTIMATION FOR ANTENNA ARRAYS
λi = tan( d sin θi ) |
θi = tan− 1[(λi )/( d)] . |
(3.64) |
Note that the estimation of the matrix Es and the solutions (3.63) and (3.64) involve only real computations.
A summary of the algorithm is shown in Figure 3.11.
3.2.8 QR ESPRIT
TheTotal Least Squares (TLS) ESPRIT involves computing a singular value decomposition,a matrix inverse,a matrix product,and an eigenvalue decomposition of an r × r matrix.This is a heavy computational burden, especially if DOAs are to be tracked across time. An alternative to the above procedure is QR reduction to a standard eigenvalue problem [6]. One can start with the generalized eigenvalue problem that is associated with the ESPRIT algorithm, ExTΦ = EyT. In this case, Ex is a matrix
Figure 3.11: The unitary ESPRIT [22].
Nonadaptive Direction of Arrival Estimation 37
whose columns represent the signal subspace of the first subarray, and the matrix Ey contains the signal subspace for the second of the two subarrays. Next, premultiply the above equation for the generalized eigenvalue problem by the matrix THExH:
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(3.65) |
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According to Strobach [6], the matrix T can be chosen such that |
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TH ExH Ex T = I . |
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(3.66) |
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If T is chosen to satisfy (3.66), then |
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Ф= TH ExH EyT. |
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(3.67) |
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Now suppose that E |
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T. Then, (3.65) can be written as Q |
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TΦ = E |
R-1R T, |
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i.e., |
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Qx QΦ= ExHRx−1 Q Þ |
QΦ = QHx ExHRx−1Q , |
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(3.68) |
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or |
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QΦQ−1 = QHx |
ExH Rx−1. |
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(3.69) |
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Equation (3.69) is an eigenvalue problem in standard form. To summarize, the DOAs can be computed via a QR reduction, which corresponds to the solution of the ESPRIT problem but not the TLS ESPRIT. Figure 3.12 is a flowchart of what could be called the QR ESPRIT algorithm [6].
3.2.9 Beamspace DOA Estimation
Beamspace algorithms are efficient in terms of computational complexity. They use an N × P beamspace matrix, T, whose columns represent the beamformer weights. With this approach the data vectors are transformed to a lower dimensional space by the matrix T. This transformation is written as, zn = THxn. The P elements of the vector zn can be thought of as outputs of P beamformers. If zn = THxnH, then the DOA algorithm operates on the transformed data space contained in the columns of zn. If information on the incoming signal direction is available, the columns of T can be designed such that beams in the columns of T point in the general direction of the signals whose DOAs are to be estimated. In block DOA estimation, the eigendecomposition of an N × N matrix requires O(N 3) operations. If two beamspace processors are designed, T1 and T2, such that T1 covers from 0° to 90° and T2 covers from 0° to −90°, then the two beamspace processors can estimate the
38 NARROWBAND DIRECTION OF ARRIVAL ESTIMATION FOR ANTENNA ARRAYS
Figure 3.12: The QR ESPRIT algorithm [6].
DOAs in their respective sectors. Therefore, the number of computations can be reduced considerably as each beamspace processor will have complexity O((N/2)3). Several subspace-based DOA algorithms have been developed for beamspace processors such as the beamspace MUSIC [23], the beamspace root MUSIC [24], the beamspace ESPRIT [25], and the DFT beamspace ESPRIT [22]. In this section, the ESPRIT versions of the beamspace DOA algorithms are described.
3.2.10 The DFT Beamspace ESPRIT
Recall that for a uniform linear array, the conjugate symmetric steering vector has the form
a(θ) = [e − j( |
N − 1 |
)2 d sin θ . . . e− j 2 d sin θ 1 e j 2 d sin θ . . . e j( |
N −1 |
)2 d sin θ ]. |
(3.70) |
2 |
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Now consider the inner product [22] of the above steering vector with the mth row of the centrosymmetrized DFT matrix, which is given by:
wmH = e− j( |
N − 1 |
)m |
2 |
[1 e− j |
2 |
m e− j 2 |
2 |
m . . . e− j(N −1)m |
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]. |
(3.71) |
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Note that this is a scaled version of the mth row of the DFT matrix. The inner product of wm and a(θ) is
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(3.72) |
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Nonadaptive Direction of Arrival Estimation 39
Now let bN(θ) be an N × 1 vector containing the N samples of the centrosymmetrized DFT of the steering vector a(θ), i.e.,
bN (θ) = [b1(θ) b2(θ) . . . bN (θ)]T . |
(3.73) |
Notice that the numerator of bm(θ) and bm + 1(θ) are the negative of one another since the arguments of the sine waves are π radians apart. This observation leads to the following equation [22]
bm(μ) sin [ |
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μ − m |
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μ − (m + 1) |
2 |
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(3.74) |
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N |
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where μ = 2dsinθ. Now, using the trigonometric identity sin(θ − φ) = sin(θ)cos(φ) − sin(φ)cos(θ), the above equation can be written as
tan |
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(m + 1) |
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+ bm +1(μ) sin
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The previous two equations can be written in matrix notation but to do this, first it is necessary to relate b0(θ) to bN − 1(θ). Let us first write bN(θ) as follows:
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(3.76) |
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Now use the above equation along with the equation relating bm(θ) to bm + 1(θ) with m = N – 1 to establish the following relationship between b0(θ) and bN – 1(θ):
tan |
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bN −1 |
(µ) cos |
(N − 1) |
+ (−1)N −1b0(µ) cos( ) |
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(3.77) |
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As per [22], these equations can be used to write a matrix equation relating the first N − 1 elements of bN(θ) to the last N elements of bN(θ)., i.e.,