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Fourier Transforms of Generalised Functions

Fourier Transforms of Generalised Functions. Existence of FT. The FT of a function exists provided the function satisfies the Dirichlet conditions:. and any discontinuities are finite. Generalised functions - Do not satisfy the Dirichlet conditions - Often used in signal processing

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Fourier Transforms of Generalised Functions

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  1. Fourier Transforms of Generalised Functions

  2. Existence of FT • The FT of a function exists provided the function satisfies the Dirichlet conditions: • and any discontinuities are finite • Generalised functions • - Do not satisfy the Dirichlet conditions • - Often used in signal processing • - We can still find their FTs

  3. The Delta Function • Properties Let h0 1/h h • Graph becomes a ‘spike’ • Models eg a very large current applied for a very short time

  4. The Delta Function • Further properties... a 0

  5. FT of the Delta Function In general…

  6. Fourier Transform of 1 If Symmetry property: Delta function is even

  7. FT of Cosine and Sine Symmetry: Similarly

  8. FT of Powers of t Multiplication property: Similarly

  9. FT of sgn(t) Definition Result from Homework 1… Let and

  10. FT of the Heaviside Function Hence

  11. Now look at Tutorial 3

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