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Understanding Fourier Analysis in Signal Processing

Learn about Fourier analysis, periodic signals, Fourier series, transforms, Laplace transform, and Fourier transform properties. Discover how to interpret signal frequency content through transformation. Explore fundamental concepts and applications in signal processing.

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Understanding Fourier Analysis in Signal Processing

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  1. Fourier Analysis

  2. Periodic Signals • For all t,x(t + T) = x(t) x(t) is a periodic signal Smallest value of T is the fundamental period Fundamental frequency 1/T • Periodic signals have aFourier series representation Fourier series coefficient Cm quantifies the strength of the component of x(t) at frequency m/T • Fourier transforms (defined next) are for both periodic and aperiodic signals

  3. Fourier Integral • Conditions for Fourier transform of x(t) to exist x(t) is single-valued with finite maxima and minima in any finite time interval x(t) is piecewise continuous; i.e., it has a finite number of discontinuities in any finite time interval x(t) is absolutely integrable • Conditions not obeyed for cos(t), sin(t) and u(t) We’ll find ways to define Fourier transforms for them

  4. Laplace Transform • Generalized frequency variable s = s + j w • Laplace transform consists of an algebraic expression and a region of convergence (ROC) • For substitution s = j w or s = j 2 p f to be valid, ROC must contain the imaginary axis Laplace transform of u(t) is 1/s with ROC of Re{s} > 0 This ROC does not include the imaginary axis

  5. Fourier Transform • What system properties does it possess? • Memoryless (in fact requires infinite memory) • Causal • Linear • Time-invariant (doesn’t apply) • What does it tell you about a signal? Answer: Measures frequency content • What doesn’t it tell you about a signal? Answer: When those frequencies occurred in time

  6. F(w) f(t) t F 1 w t -t/2 0 t/2 -6p -4p -2p 2p 4p 6p 0 t t t t t t Fourier Transform Pairs

  7. From the sifting property of the Dirac delta, Consider a Dirac delta in the Fourier domain Using linearity property, F{ 1 } = 2pd(w) x(t) = 1 1 t 0 F Fourier Transform Pairs X(w) = 2 p d(w) (2p) w 0 (2p) means that the area under the Dirac delta is (2p)

  8. F Fourier Transform Pairs F(w) f(t) (p) (p) t w 0 -w0 w0 0

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