A triangular pulse for

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Fourier transforms, spectra and convolution

Problem

A triangular pulse

f(x) = 1 - |x|/b

for

|x| \le b

, and zero elsewhere, is repeated with period

L > 2b

to make a pulse train, with complex coefficients

c_n = \frac1L\int_{-L/2}^{L/2}f(x)e^{-ik_nx}\,\mathrm{d}x

k_n = 2\pi n/L

. ((a)) Show that

Lc_n = \dfrac{4\sin^2(k_nb/2)}{bk_n^2}

for

n \ne 0

and

Lc_0 = b

, and say where the condition

L > 2b

is used. ((b)) Take

b = 1.0\,\mathrm{mm}

. The point

k = \pi/b

is one of the

k_n

both for

L = 4b

and for

L = 6b

. For each period, find which

n

lands there and evaluate

c_n

and

Lc_n

. ((c)) By letting

L \to \infty

, identify the Fourier transform

\tilde f(k)

of a single triangle, and give its value at

k = 0

and its first zero for

k > 0

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