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Fourier Transform Example

Module by: Don Johnson. E-mail the author

Summary: This module calculates the Fourier transform of the pulse signal.

Let's calculate the Fourier transform of the pulse signal, pt p t .

Pf=pte(i2πft)dt=0Δe(i2πft)dt=1(i2πf)(e(i2πfΔ)1) P f t p t 2 f t t 0 Δ 2 f t 1 2 f 2 f Δ 1
(1)
Pf=e(iπfΔ)sinπfΔπf P f f Δ f Δ f
(2)
Note how closely this result resembles the expression for Fourier series coefficients of the periodic pulse signal.

Figure 1: The upper plot shows the magnitude of the Fourier series spectrum for the case of T=1 T 1 with the Fourier transform of pt p t shown as a dashed line. For the bottom panel, we expanded the period to T=5 T 5 , keeping the pulse's duration fixed at 0.2, and computed its Fourier series coefficients.
Spectrum
Spectrum (spectrum4.png)

Figure 1 shows how increasing the period does indeed lead to a continuum of coefficients, and that the Fourier transform does correspond to what the continuum becomes. The quantity sintt t t has a special name, the sinc (pronounced "sink") function, and is denoted by sinct sinc t . Thus, the magnitude of the pulse's Fourier transform equals |ΔsincπfΔ| Δ sinc f Δ .

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