Mechanical Engineering

Summary of the topics covered 1) Notes on The Fourier transform (FT) Impulse response and transfer function The time-frequency domain convolution theorem Shift and scaling properties, Hermitian symmetry Digital signal processing continuous vs discrete time signals • Sampled and windowed functions, • Nyquist-Shannon sampling theorem, • Time sampling and frequency sampling, observation time, aliasing, rippling effect, Discrete Fourier transform • Frequency content of a signal, narrow and wideband signals • Example of aliasing in MEMS ACC Worked exercises with Matlab: • FT of Sequence of unit and equally shifted impulses • FT of a rectangular function • FT of a cosine/sine function • FT of the Dirac delta 2) Introduction to Time-Frequency Analysis Mathematical and physical meaning of the FT Range of applicability and main limitations of the FT Worked exercises in Matlab: transient harmonic disturbance and frequency shift 3) The Short Time Fourier Transform (STFT) Mathematical formulation and physical meaning of the STFT The role of the window function Time and frequency characterization of a waveform Heisenberg - Gabor Uncertainty Principle Time frequency resolution, Heisenberg box Gabor transformation of trigonometric functions with large frequency fluctuations Range of applicability and main limitations of the STFT Worked exercises with Matlab: • STFT of a chirp • STFT of superposition of windowed trigonometric components • Worked examples in Matlab: Range of applicability and main limitations of GT 4) Wavelet transform (WT) Mathematical formulation and physical meaning Relation between scale and frequency The wavelet transform as a convolution integral Variable time-frequency resolution Range of applicability and main limitations Worked exercises with Matlab: • Time-frequency characterization of Mexican hat and Morlet wavelet • Analogies and differences between STFT and Wavelet transform, comparison between Morlet wavelet and Gabor atom • Filtering property of the wavelet transform • Identification of coherent structures • Edge detection • Analysis of a signal with “wavemenu” toolbox Lecture notes on: Structural health monitoring of a plate excited by ambient load by wavelet transform. 5) The Hilbert transform (HT) and analytic signal (AS) The need for analytic signal: its role on time-frequency analysis AS and HT: Mathematical formulation and physical meaning Bedrosian theorem Phasor representation of AS The instantaneous frequency (IF) Worked exercises with Matlab: • HT of a harmonically decaying function • AS of harmonic component with DC offset • AS of composition of harmonics • AS of a Chirp • AS of Harmonic amplitude modulated signal 6) Empirical mode decomposition (EMD) and Hilbert transformation Monocomponent and multicomponent signals, Intrinsic Mode Functions (IMF) Basic concepts of the EMD, main properties of IMF with examples Physical meaningfulness of IMFs: the length-of-day data Main advantages, range of applicability and main limitations of EMD+HT comparison with other time-frequency methods Inter-wave and Intra-wave frequency modulation Worked exercise with Matlab: • The sifting process • Introduction to the HHT-package code by Huang • EMD+HT of the Stokes wave • Analysis of damped oscillations with EMD+HT Lecture notes on: Damage detection in structures under traveling loads by Hilbert–Huang transform.

Basic knowledge of Analysis of Mathematical, Geometry, General Physics and Mechanical Vibrations.

Lecture Notes of the course. E. O. Bringhan , The Fast Fourier Transform, Prentice Hall Inc , Englewood Cliffs, New Jersey A. Papoulis, The Fourier Integral and Its Applications, McGraw HiII Book Co., New York K. Grochenig , Time Frequency Analysis, Springer Science, 2001 Cohen, L. 1995 Time frequency analysis. Englewood Cliffs, NJ: Prentice Hall P. Flandrin , Time Frequency/Time Scale Analysis, Academic Press, 1999 G.Kaiser , A Friendly Guide To Wavelets, New York, Birkhäuser , 1994 A Wavelet Tour of Signal Processing, 3rd ed. Stéphane Mallat . Academic Press, dec. 2008 P. Addison, The Illustrated Wavelet Transform Handbook , IoP Hahn S., Hilbert transforms in signal processing. Artech House, 442 pp., 1995. N.E. Huang, Hilbert Huang transform and its application, World Scientific

The student will be guided to analyze real life signals with the aid of Matlab software in weekly meetings, whose attendance is warmly advised.

The student will be guided to analyze real life signals with the aid of Matlab software in weekly meetings, whose attendance is warmly advised.

Written report concerning a time-frequency analysis of a given signal caried out with a MATLAB code written by the student.

Boashash B., Estimating and interpreting the instantaneous frequency of a signal. I.Funndamentals . Proc. IEEE 1992; 80, 520 538. Huang N.E., Shen Z., Long S.R., Wu M.C., Shih H.H., Zheng Q., Yen N., Tung C.C., Liu H.H., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non stationary time series analysis, Proc. R. Soc. London A 1998; 454, 903 995. N.E. Huang, Z. Wu, S.R. Long, K.C. Arnold, X. Chen, K. Blank, On the instantaneous frequency, Advances in Adaptive Data Analysis, 1(2), 177 229 (2009).

The course is delivered through frontal lectures and computer-based practical sessions, aimed at introducing and experimenting with the main techniques of numerical signal analysis in the MATLAB environment. Throughout the semester, example cases and guided exercises are proposed and later serve as the reference model for the final exam, which is conducted in presence on a computer. The final assessment consists of a practical test lasting approximately four hours, during which the student develops and discusses a technical report similar in structure to those carried out during class activities. The test includes: the numerical analysis of one or more real or synthetic signals using MATLAB; the description of the adopted signal-processing procedures; the critical discussion of the obtained results and their physical interpretation. Lectures, supported by multimedia materials and shared MATLAB scripts, are integrated with guided exercises to make the exam a natural continuation of the learning experience developed during the course.