- #1
wolski888
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Homework Statement
S(t) = S(0)e^{-i \pi f_{o}t} e^{-t/T^{*}_{2}}, 0 \leq t < \infty
S(t) = 0, t < 0
Show that the spectrum G(f) corresponding to this signal is given by:
G(f) = S(0) { \frac{T^{*}_{2}}{ 1 + [2 \pi (f- f_{o} )T^{*}_{2}]^{2}} + \frac{i2 \pi (f- f_{o} ) (T^{*}_{2})^{2}}{ 1 + [2 \pi (f- f_{o} )T^{*}_{2}]^{2}} }
Homework Equations
G(f) = \int^{\infty}_{- \infty} S(t) e^{i 2 \pi f t} dt
The Attempt at a Solution
G(f) = \int^{\infty}_{- \infty} S(t) e^{i 2 \pi f t} dt
= \int^{\infty}_{- \infty} S(0) e^{-i 2 \pi f_{o}t} e^{-t/T^{*}_{2}}e^{i 2 \pi f t} dt
= S(0) \int^{\infty}_{0} e^{i 2 \pi (f - f_{o})t} e^{-t/T^{*}_{2}} dt * S(t) is 0 when t is less than 0, so took integral from infity to 0. Took out S(0) and combined the exponents.
= S(0) \int^{\infty}_{0} e^{i 2 \pi (f - f_{o})t - t/T^{*}_{2}} dt *Combined the exponents
= S(0) \int^{\infty}_{0} e^{t[i 2 \pi (f - f_{o}) - 1/T^{*}_{2}]} dt *factored out t. Now all the stuff in the square brackets are basically a constant.
= S(0) [ \frac{e^{t[i 2 \pi (f - f_{o}) - 1/T^{*}_{2}]}}{i 2 \pi (f - f_{o}) - 1/T^{*}_{2}} ]^{\infty}_{0} *This is what I get after integrating. But as t goes to infinity, so does the fraction. Which I think makes sense since its a continuous spectrum. So maybe integrating from 0 to T2*? That way it looks like I am getting closer to the answer. Is my integration wrong?