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## Fourier Transform

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**Fourier Transform**Comp344 Tutorial Kai Zhang**Outline**• Fourier Transform (FT) Properties • Fourier Transform of regular signals • Exercises**Period and Frequency**• Imagine a rod spinning around the center, taking 2 seconds for one round • Period • T = 2 (second) • Ordinary frequency • f = 0.5 (Hz or times/second) • Angular frequency • ω = 1 (degree/second) • Remember**FT Properties**• Property 1: time domain shifting (or delay) • Proof**FT Properties**• Property 2: frequency domain shifting • Proof**FT Properties**• Property 3: scaling • Proof**FT Properties**• Property 4: time domain differentiation • Proof • Question: what about**FT Properties**• Property 5: Symmetry • Proof:**Some common FT-pairs**• Impulse function**Some common FT-pairs**• Complex exponential function • Using the frequency domain shifting property, and symmetry property**Some common FT-pairs**• Sine function • By using the Euler formula and delay property**Some common FT-pairs**• Rectangular function**Some common FT-pairs**• Gaussian function**More Example**• Let f(t)-F(w) be a FT-pair. Now compute the FT of g(t) = f(t)cos(t). • using Euler’s Formula and FT frequency shifting property**Exercise**• Compute the FT of the following signals • u(t)cos(ω0t) • u(t)sin(ω0t)e-at • e-|a|t • u(t)e-at • u(t)te-at Here u(t) is the heavyside step function