1. Compressed Sensing Applied to Modeshapes Reconstruction

Modal analysis classicaly used signals that respect the Shannon/Nyquist theory. Compressive sampling (or Compressed Sampling, CS) is a recent development in digital signal processing that offers the potential of high resolution capture of physical signals from relatively few measurements, typically well below the number expected from the requirements of the Shannon/Nyquist sampling theorem. This technique combines two key ideas: sparse representation through an informed choice of linear basis for the class of signals under study; and incoherent (eg. pseudorandom) measurements of the signal to extract the maximum amount of information from the signal using a minimum amount of measurements. We propose one classical demonstration of CS in modal identification of a multi-harmonic impulse response function. Then one original application in modeshape reconstruction of a plate under vibration. Comparing classical \( {\ell_2} \) inversion and \( {\ell_1} \) optimization to recover sparse spatial data randomly localized sensors on the plate demonstrates the superiority of \( {\ell_1} \) reconstruction (RMSE).