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Fourier Transform Properties

Fourier Transform Properties. t. F ( w ). w. - 6 p. - 4 p. - 2 p. 2 p. 4 p. 6 p. t. t. t. t. t. t. 0. Duality. Forward/inverse transforms are similar Example: rect(t/ t )  t sinc( w t / 2) Apply duality t sinc(t t /2)  2 p rect(- w / t )

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Fourier Transform Properties

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  1. Fourier Transform Properties

  2. t F(w) w -6p -4p -2p 2p 4p 6p t t t t t t 0 Duality • Forward/inverse transforms are similar • Example: rect(t/t)  t sinc(wt / 2) Apply duality t sinc(t t/2)  2 p rect(-w/t) rect(·) is even t sinc(t t /2)  2 p rect(w/t) f(t) 1 t -t/2 0 t/2

  3. Scaling • Given and that a 0 |a| > 1: compress time axis, expand frequency axis |a| < 1: expand time axis, compress frequency axis • Extent in time domain is inversely proportional to extent in frequency domain (a.k.a bandwidth) f(t) is wider  spectrum is narrower f(t) is narrower  spectrum is wider

  4. Shifting in Time • Shift in time Does not change magnitude of the Fourier transform Shifts phase of Fourier transform by -wt0(so t0 is the slope of the linear phase) • Derivation Let u = t – t0, so du = dt and integration limits stay same

  5. Sinusoidal Amplitude Modulation

  6. F(w) 1 w -w1 w1 0 Y(w) 1/2 F(w+w0) 1/2 F(w-w0) 1/2 w -w0 - w1 -w0 + w1 w0 - w1 w0 + w1 0 -w0 w0 Sinusoidal Amplitude Modulation • Example: y(t) = f(t) cos(w0 t) f(t) is an ideal lowpass signal Assume w1 << w0 • Demodulation (i.e. recovery of f(t) from y(t)) is modulation followed by lowpass filtering • Similar derivation for modulation with sin(w0 t)

  7. Frequency-shifting Property

  8. Conditions f(t)  0 when |t|  f(t) is differentiable Derivation of property:Given f(t) F(w) Time Differentiation Property

  9. Time Integration Property • Example:

  10. Summary • Definition of Fourier Transform • Two ways to find Fourier Transform Use definition Use transform pairs and properties

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