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Year 2 of the natural sciences course of the University of Cambridge

Mathematics course of the year 2 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge

2020

Paper 1

For real $f$, <complex conjugation> commutes with its <absolutely convergent> Fourier <integral>:
$$
[\widetilde f(-\omega)]^*
=\left[\int_{-\infty}^\infty f(t)e^{i\omega t}\,dt\right]^*
=\int_{-\infty}^\infty f(t)e^{-i\omega t}\,dt.
$$
Thus
$$
\boxed{[\widetilde f(-\omega)]^*=\widetilde f(\omega).}
$$
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.

Solution

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

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

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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

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