From: Robust widely linear beamforming using estimation of extended covariance matrix and steering vector
Symbols | Definitions |
---|---|
(·)^{∗} | Conjugate operator |
(·)^{T} | Transpose operator |
(·)^{H} | Hermitian transpose operator |
(·)^{−1} | Matrix inversion operator |
〈·〉 | Time-averaging operation |
E[·] | Expectation operator |
I_{N} | N×N identity matrix |
diag{·} | Diagonalization operator |
|·| | Absolute value operator |
∥·∥ | Euclidean norm |
∩ | Set intersection |
eig_{max}(·) | Maximum eigenvalue of a matrix |
N | Number of array antennas |
M | Number of narrowband signals |
k | Time index of snapshots |
x | Array observation vector |
a | Steering vector |
s | Complex waveform |
v | Whole interference-plus-noise vector |
γ | Noncircularity coefficient |
σ^{2} | Time-averaged power |
|γ| | Noncircularity rate |
ϕ | Noncircularity phase |
R | Covariance matrix |
C | Pseudo covariance matrix |
\(\breve {\mathbf {x}}\) | Extended observation vector |
\(\breve {\mathbf {v}}\) | Extended interference-plus-noise vector |
\(\mathbf {R}_{\breve {x}}\) | Extended covariance matrix |
s^{⊥} | Orthogonal decomposition of s |
\(\breve {\mathbf {a}}\) | Extended steering vector |
\(\breve {\mathbf {v}}_{\gamma }\) | Global noise vector |
y | WLB output |
\(\breve {\mathbf {w}}\) | WLB weight vector |
SINR | Output signal-to-interference-plus-noise ratio |
K | Number of observed snapshots |
\(\bar {\delta }\) | Minimum eigenvalue of \(\mathbf {R}_{\hat {\breve {x}}}\) |
θ | Signal direction |
Θ | Signal angular sector |
α_{i} | Eigenvalue of \(\mathbf {R}_{\hat {x}}\) |
g_{i} | Eigenvector of \(\mathbf {R}_{\hat {x}}\) |
G | Subspace projection matrix |
Γ | Diagonal matrix |
β | Adaptive uncertainty level |
η | Lagrange multiplier |
ϱ | Threshold constant |
λ_{i} | Eigenvalue of \(\mathbf {R}_{\hat {\breve {x}}}\) |
q_{i} | Eigenvector of \(\mathbf {R}_{\hat {\breve {x}}}\) |
Q | Matrix containing eigenvectors |
Λ | Diagonal matrix containing eigenvalues |
f | Projection value |
Q | Number of principal eigenvectors |
F | Extended subspace projection matrix |
ε | A predefined constant |
Π | Extended subspace |
π | Subspace coefficient vector |
p(θ) | Spatial power spectrum |
μ_{l} | Eigenvalue of \(\breve {\mathbf {R}}_{s}\) |
u_{l} | Eigenvector of \(\breve {\mathbf {R}}_{s}\) |
U | Matrix containing eigenvectors |
U | Number of principal eigenvectors |
ζ | A predefined constant |
Ω | Diagonal matrix containing eigenvalues |