 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/8/a/ii/solution

For real

f

[f (- ω)]^{*} = [\int_{- \infty}^{\infty} f (t) e^{iω t} d t]^{*} = \int_{- \infty}^{\infty} f (t) e^{- iω t} d t .

Thus

[f (- ω)]^{*} = f (ω) .

This Hermitian symmetry pairs the positive- and negative-frequency components of a real signal; its real part is even and its imaginary part is odd.

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