PSI - Issue 80

Guangxiao Zou et al. / Procedia Structural Integrity 80 (2026) 93–104 Author name / Structural Integrity Procedia 00 (2023) 000–000

100

8

R xy =⃗ e xy

xy +

| ˆ x ( t ) | 2 dt⃗ e H

H xy ( t ) dt =⃗ e xy R s⃗ e

H xy + R n

n xy ( t )⃗ n

(21)

t ∈ excitation window

t ∈ excitation window⃗

where R s represents the variance of the excitation signal x ( t ) within the time window, and R n is the noise correlation matrix. This matrix is typically modeled as R n = σ 2 I , where σ 2 is the noise power and I is the identity matrix, which assumes the noise is uncorrelated across all channels. Since the spatiotemporal correlation matrix R xy is separable into signal and noise components, its eigen decomposition can also be partitioned into corresponding signal and noise subspaces:

R xy = Q S Σ Q H

H N

S + Q N Σ Q

(22)

In this equation, Q S Σ Q H N corresponds to the noise component R n . Because this signal component is rank-one (assuming a single scatterer), the signal subspace Q S is spanned by a single eigenvector, [⃗ q 1 ] which corresponds to the largest eigenvalue. Consequently, the noise subspace, Q N is spanned by the remaining M − 1 eigenvectors, [⃗ q 2 , ...,⃗ q M ] . What’s more, the signal eigenvector⃗ q 1 should be the same as the steering vector⃗ e xy ,as Q S corresponds to the⃗ e xy .And because the signal is assumed to be uncorrelated with the noise, the signal subspace is mathematically orthogonal to the noise subspace. Consequently, the steering vector must also be orthogonal to the noise subspace at the damage position:⃗ S corresponds to the signal component⃗ e xy R s⃗ e H xy and Q N Σ Q H

e H

xy Q N =⃗ 0

(23)

But in practice, while perfect orthogonality may not always occur, a very high degree of orthogonality can still be expected at the damage location. This property enables the MUSIC imaging algorithm to calculate a steering vector for each pixel in the inspection area. It then uses the squared norm of the vector⃗ e H xy Q N to measure the degree of orthogonality between that steering vector and the noise subspace (A high degree of orthogonality implies that the squared norm of the vector⃗ e H xy Q N will be small): ⃗ e H xy Q N 2 =⃗ e H xy ( Q N Q H N )⃗ e xy (24) Inverting the squared norm creates a peak in the final image. The location of this peak corresponds to the location of the damage. Therefore, the MUSIC pixel value should be

1⃗

(25)

P x , y =

H N )⃗ e xy

xy ( Q N Q

e H

3. Experimental setup and results

3.1. Experimental setup

The experimental investigation was conducted on a 400 mm x 400 mm quasi-isotropic composite plate. The lami nate was constructed with a symmetric ply sequence of [45 , − 45 , 0 , 0 , 90 , 0] s .