how is realation between time and frequency domain?

[lekhoi]

Hi all,
I can show that convoluntion in time domain is multiplation in frquency domain. However, Can anyone help me show that multiplation in time domain is convolution in frequency domain? like this : v(t)w(t) -> V(f)*W(f)
thanks in advance!

 

[nishantp]

hi lekhoi

it works as under

F{v(t)w(t)} = -∞∫∞v(t) w(t) e-jωtdt

= -∞∫∞[-∞∫∞V(ω’)ejω’t dω] w(t)e-jωt dt (Replacing v(t) as
inverse fourier transform)

= -∞∫∞ V(ω’) dω -∞∫∞ w(t)e-jωt ejω’t dt (Interchanging the order of
integration)

= -∞∫∞ V(ω’) dω -∞∫∞ w(t) e-jt(ω- ω’)dt

= -∞∫∞ V(ω’) W(ω-ω’) dω

This is the convolution in frequency domain we set out to prove.

 

[petrelflying]

the convolution in time domain is a multiply in frequency domain,
and the convolution in frq domain is a multiply in time domain
the preperty and advantage is the most obvious