a) Show that the convolution of two functions, f(t) and g(t) has a fourier transform which is just the product of the in
Posted: Fri May 06, 2022 6:29 am
a) Show that the convolution of two functions, f(t) and
g(t) has a fourier transform which is just the product of the
individual fourier transforms. That is, if the convolution is
defined as h(t) = ∫f(s)g(t-s)ds, then H(ω)=F(ω)G(ω) where H(ω) is
the fourier transform of h(t). As you have seen, this is a result
with enormous practical applications in both diffraction theory and
in image processing.
b) First, a definition: if f(t) is a function of time and
F(ω) is its fourier transform, then the power spectrum is S(ω)
=|F(ω)| 2 Find the power spectrum of the AM wave f(t) =
(A+Bcos(2πqt)) cos(2πpt) where p is the carrier frequency and q is
the modulating frequency. So, for example, p is the high frequency
you dial up on your AM radio station (e.g. 1430 kHz) and q is the
slow frequency of speech and song (audio frequencies between 10Hz
to 10kHz) that are superimposed on it
g(t) has a fourier transform which is just the product of the
individual fourier transforms. That is, if the convolution is
defined as h(t) = ∫f(s)g(t-s)ds, then H(ω)=F(ω)G(ω) where H(ω) is
the fourier transform of h(t). As you have seen, this is a result
with enormous practical applications in both diffraction theory and
in image processing.
b) First, a definition: if f(t) is a function of time and
F(ω) is its fourier transform, then the power spectrum is S(ω)
=|F(ω)| 2 Find the power spectrum of the AM wave f(t) =
(A+Bcos(2πqt)) cos(2πpt) where p is the carrier frequency and q is
the modulating frequency. So, for example, p is the high frequency
you dial up on your AM radio station (e.g. 1430 kHz) and q is the
slow frequency of speech and song (audio frequencies between 10Hz
to 10kHz) that are superimposed on it