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Let f(t) be a continuous time signal and let F(ω) be its Fourier Transform defined by F(ω) = ∫ from -∞ to ∞ f(t)e^-jωt dt. Define g(t) by g(t) = ∫ from -∞ to ∞ F(u)e^-jut du. What is the relationship between f(t) and g(t)?

1. g(t) would always be proportional to f(t).
2. g(t) would be proportional to f(t) if f(t) is an even function.
3. g(t) would be proportional to f(t) only if f(t) is a sinusoidal function.
4. g(t) would never be proportional to f(t).

Correct answer: g(t) would be proportional to f(t) if f(t) is an even function.

Solution

Using e^{-jut} in both transforms, g(t)=2pi f(-t). This is proportional to f(t) precisely when f is even (f(-t)=f(t)). So g is proportional to f only for even f.

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