A periodic square-wave voltage of amplitude and period takes the value for and for in…

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 Delta functions, distributions and generalised functions

Problem

A periodic square-wave voltage

V(t)

of amplitude

A = 3.0\,\mathrm{V}

and period

L = 4.0\times10^{-3}\,\mathrm{s}

takes the value

+A

for

0 < t < L/2

and

-A

for

-L/2 < t < 0

in each period. Written with Chapter 18's time components,

V(t)=\sum_n c_n\,e^{-i2\pi nt/L}

, its complex coefficients are

c_n = 2iA/(\pi n)

for odd

n

and

c_n = 0

for even

n

. A periodic function's Fourier transform is the spike train

\tilde V(\omega) = 2\pi\sum_n c_n\,\delta(\omega - 2\pi n/L).

((a)) Give the angular frequency

\omega_1

at which the

n=1

spike sits, and the magnitude of its weight

|2\pi c_1|

. ((b)) Do the same for the

n=3

spike.

Browse the problem library