A digital clock produces a periodic voltage that alternates between and with period :…

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 digital clock produces a periodic voltage

V(t)

that alternates between

+5.0\,\mathrm{V}

and

-5.0\,\mathrm{V}

with period

L = 2.0\times10^{-3}\,\mathrm{s}

: a square wave of amplitude

A

which, written in Chapter 17's form

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

, has complex coefficients

c_n = -2iA/(n\pi)

for odd

n

and

c_n = 0

for every even

n

(including

n=0

). ((a)) Write the Fourier transform

\tilde V(\omega)

V(t)

, in this book's time convention, as a train of delta functions in angular frequency

\omega

, and state the spacing between neighbouring spikes. ((b)) Give the angular frequency and the weight (the coefficient multiplying each delta function) of the three lowest positive-frequency spikes. ((c)) The clock is re-biased to swing between

0

and

+10\,\mathrm{V}

with the same period. What single term does this add to

\tilde V(\omega)

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