While reproducing a research paper, I came across the following equation,
∂f/∂t−(H(f)(∂f/∂x)=0
where [H(f)] is hilbert transform of 'f.'
and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by
F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)},
where F(f(x,t) is...
Homework Statement
Show that the Hilbert transform of ##\frac{\sin(at)}{at}## is given by
$$\frac{\sin^2(at/2)}{at/2}.$$
Homework Equations
The analytic signal of a function is given by ##f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,## where ##F(\nu)## is the Fourier transform...
Homework Statement
For a real, band-limited function ##m(t)## and ##\nu_v > \nu_m,## show that the Hilbert transform of
$$h(t) = m(t) cos(2\pi \nu_c t)$$
is
$$\hat{h}(t) = m(t) sin(2 \pi \nu_c t),$$
and therefore the envelope of ##h(t)## is ##|m(t)|.##
Homework Equations
Analytic signal...