Table 1 Summary symbols and notations

w(n)

Gaussian noise

s(n)

The signal transmitted by the PU

x(n)

The signal received by the SU

N

The number of sampling points

M

The number of SUs participating in CSS

H₁, H₀

PU exists, PU does not exist

p_f, p_d

False alarm probability and detection probability

x _i

The signal acquired by the ith SU

X

Signal matrix

R

Covariance matrix corresponding to X

X ^T

The transposition of X

Y_O−DAR, Y_I−DAR

A matrix after O-DAR and I-DAR

R^O, R^I

Covariance matrix corresponding to Y_O−DAR

and Y_I−DAR

x

An n-dimensional sample

θ

A parameter vector, a point on the manifold

Ω

A random variable

S

The probability distribution function family

Θ

Probability distribution space

R_w, R_s+R_w

Covariance matrix corresponding to X under

H₀ and H₁

q

The split parameter

s

The length of the split signal vector after splitting

\({\mathbf {R}}_{k}^{O}\), \({\mathbf {R}}_{k}^{I}\)

The kth noise signal covariance matrix after

O-DAR and I-DAR

τ

Iteration step size

l

Iteration step

\({\overline {\mathbf {R}}^{O}}\), \({\overline {\mathbf {R}}^{I}}\)

Riemann mean of \({\mathbf {R}}_{k}^{O}\) and \({\mathbf {R}}_{k}^{I}\)

d₁, d₂

Distance between the perceived signal and

reference point \({\overline {\mathbf {R}}^{O}}\) and \({\overline {\mathbf {R}}^{I}}\) on the manifold

D

Geodesic distance feature vector (GDFV)

\(\overline {\mathrm {D}}\)

Training set

Z _c

The set of training feature vectors belonging

to class c

Ψ _c

Center point of Z_c

u _cj

The membership degree

\({\overline {\overline {\mathbf {D}}}}\)

The feature extracted under the channel of the

unknown PU state

C

Number of clusters

ν

Smoothing index or fuzzy weighted index

m

Error metric

ε

Fault tolerance factor