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Introduction to ECE 802

Introduction to ECE 802. Selin Aviyente Associate Professor. Introduction. Classical signal analysis focuses on two domains Time Domain: Amplitude, Energy, Correlation Frequency Domain: Power, Spectrum, Bandwidth Frequency domain analysis is done through FOURIER TRANSFORM.

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Introduction to ECE 802

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  1. Introduction to ECE 802 Selin Aviyente Associate Professor ECE 802-603 Spring 2010

  2. Introduction • Classical signal analysis focuses on two domains • Time Domain: Amplitude, Energy, Correlation • Frequency Domain: Power, Spectrum, Bandwidth • Frequency domain analysis is done through FOURIER TRANSFORM ECE 802-603 Spring 2010

  3. Time Domain Analysis • In the time domain, any signal can be expressed as: • Expanded in terms of an infinite number of delta functions over the real axis with expansion coefficients equal to the amplitude. ECE 802-603 Spring 2010

  4. Spectrum Analysis • Fourier transform is defined to be: • Decompose a given time signal in terms of eternal sinusoids (complex exponentials) • Expansion Coefficients=Spectral Power • What is wrong with this?? ECE 802-603 Spring 2010

  5. Disadvantages • Not all signals are eternal Most signals are time-limited. • Gives us the global picture, no local information • Most real life signals are not combinations of sinusoids. • Fourier transform is sufficient for stationary signal analysis and LTI systems. • How does the frequency change with time? ECE 802-603 Spring 2010

  6. Concept of Transients • Transient: Any phenomena that is not stationary, has finite duration • The human sensory system concentrates on the transients rather than the stationary for selecting important information. • Examples: • A word pronounced at a particular time • Epilepsy pattern in brain waves • An object located in the corner of an image • Most of signal processing is devoted to analyzing stationary signals and LTI systems. ECE 802-603 Spring 2010

  7. Example Notemporal localization ECE 802-603 Spring 2010

  8. Example ECE 802-603 Spring 2010

  9. Wavelets Expand signals in terms of localized functions (wavelets)  expansion coefficients  set of decomposing functions Linear Compare this to Fourier series Time-Frequency Distributions Expand the signal’s time-varying auto-correlation function in terms of sinusoids Fits sinusoids locally instead of globally Nonlinear Time-Frequency and Wavelet Analysis ECE 802-603 Spring 2010

  10. Properties • We will study how to choose these functions • Are there some particular functions to choose? • Is it signal adaptive? • General properties • How to design wavelets? ECE 802-603 Spring 2010

  11. Tools • In order to determine, the ‘best’ set of functions to expand our signals, we need to define a space for our signals Vector space • After the ‘signal space’ is defined, we need to define a measure for quantifying how good any approximation is Inner product • Different sets of expansion functions Basis ECE 802-603 Spring 2010

  12. Applications • Biomedical Signal Processing • Speech Processing • Radar Analysis, Target Detection • Blind Source Separation • Image Compression (JPEG2000) • Object Recognition ECE 802-603 Spring 2010

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