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Fourier Analysis

Fourier Analysis. 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 a Fourier series representation

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Fourier Analysis

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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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