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Fourier Transform Example

Module by: Don Johnson

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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

Pf=∫-∞∞ptⅇ-ⅈ2πftdt=∫0Δⅇ-ⅈ2πftdt=1-ⅈ2πfⅇ-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-ⅈπfΔsinπfΔπf 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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