Table 1 The definitions of the symbols used in the paper

(·)^∗

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

σ²

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