Frequency mixing

Why is that if one multiplies a function say g(t) by exp(jwt) in matlab, the frequency resulting signal is upconverted rather for it to be downconverted. mathematically by Fourier analysis; FT-means Fourier transform say g(t)=sin(wt) if FT[g(t)]=G(wo) then FT[g(t)exp(jwt)]= G(wo+w), but in matlab the plotting abs(FT[g(t)exp(jwt)]) gives a result that presents FT[g(t)exp(jwt)]=G(wo-w) Pls, kindly shed light on this!

Answer 1

[Wayne King]

Let X(\omega) = G(\omega-\omega_0)

Then

X(\omega_0) = G(0)

if \omega_0 is positive, then X(\omega) is shifted to the right of G(\omega)

Answer 2

[Wayne King]

Multiplying a signal g(t) by e{j\omega_0 t} shifts the spectrum to the right by \omega_0

G(\omega-\omega_0)

not G(\omega+\omega_0)as you indicate above.

The Fourier transform of g(t)e{j\omega_0 t} is G(\omega-\omega_0)

[Wayne King]

G(\omega+\omega_0) is the Fourier transform of g(t)e{-j\omega_0 t}

[Bayus]

Yeah, that was a mistake!

FT[g(t)exp(jwt)]= G(wo-w) as you mentioned. Also I agree that it shifts the spectrum to the right by w in matlab, but why the negative btw G(wo-w). if wo=100, w=80, then G(wo-w) =G(20), which means the spectrum is shifted to the left. pls explain