Forbes, Barbara J.; Sharp, David B.; Kemp, Jonathan A. and Li, Aijun
Singular system methods in acoustic pulse reflectometry.
Acta Acustica united with Acustica, 89(5)
Acoustic pulse reflectometry is a method of non-invasive duct profiling that is based on solution of the acoustical inverse problem through inverse-scattering algorithms that refer indirectly to waveforms measured at the duct mouth. More specifically, the duct reconstruction algorithm assumes a transient 'input impulse response' that is derived by inversion of the Fredholm equation relating the experimentally-measurable input and reflected waves. In the context of finite-bandwidth signals, the Fredholm inversion is an ill-posed problem for which regularisation procedures are required. Whereas previous work has assumed a frequency-domain approach, the present study examines the inversion within the temporal framework of singular systems theory. In particular, the singular-function basis set of the experimental system is examined, and the truncated singular value decomposition (TSVD) is proposed for the elimination of out-of-bandwidth components and the conditioning of the system matrix. In agreement with theoretical predictions, denoised solutions have been found from TSVD regularisation that show improved characteristics over those obtained by Fourier-domain methods. It is demonstrated that the axial resolution in the duct reconstruction may decrease from as much as 3.5 cm to 2.3 cm. For the experimental conditioning of the system matrix, the maximum length sequence (MLS) is then proposed as a development of the standard pulse-reflectometry technique. An increase of more than 50% in the number of singular functions above the noise threshold is found, leading to further improvements of up to 0.8 cm in the axial resolution. Tikhonov regularisation is shown to be effective in reducing Gibbs oscillations in the reconstructed radius and the residual error falls to within the measurement uncertainty of ±0.05 mm.
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