Fourier Transform Example 2.8 2000/08/14 2009/01/30 16:13:43.330 US/Central Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Doug Daniels ddaniels888@gmail.com Fourier pulse series signal transform This module calculates the Fourier transform of the pulse signal. Let's calculate the Fourier transform of the pulse signal, p t . P f t p t 2 f t t 0 Δ 2 f t 1 2 f 2 f Δ 1 P f f Δ f Δ f Note how closely this result resembles the expression for Fourier series coefficients of the periodic pulse signal.

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

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 t t has a special name, the sinc (pronounced "sink") function, and is denoted by sinc t . Thus, the magnitude of the pulse's Fourier transform equals Δ sinc f Δ .