Properties Of The Fourier Series

Question 1

Is the Fourier transform linear?

Answer 1

Yes, like the Laplace Transform and Fourier Series.
h(t) = af(t) + bG(t) => aF(w) + bG(w) = H(w)

Question 2

What is the time scaled form of F(w), the Fourier transform of f(t) (ie, what is f(at))?

Answer 2

f(at) —> 1/a F(w/a)

Question 3

What is the Heisenberg-Gabor principle?

Answer 3

If any function f(t) has time duration T, and it’s Fourier Transform F(w) has bandwidth B, the TB >= 1

Question 4

What is the FT of f(t - t0)?

Answer 4

f(t - t0) —> F(w) exp(-jwt0)

Question 5

What is the function associated with FT F(w - w0)?

Answer 5

f(t) exp(jw0t) —> F(w - w0)