Fourier Transform

The Fourier transform of a cosine is the sum of two Diracs of opposite frequencies of identical amplitudes, consequence of the Euler formula \[ \cos(2\pi ft) = \frac{1}{2}\left( e^{i2\pi ft} + e^{-i2\pi ft}\right) \]

Modifying the amplitudes and the frequency of the Diracs demonstrate

• modulation, by moving the Diracs along the frequency axis,
• that zeroing one of the diracs yields a complex exponential,
• that a sine can be obtained, by giving opposite imaginary amplitudes to the Diracs: \[ \sin(2\pi ft) = \frac{1}{2i}\left( e^{i2\pi ft} - e^{-i2\pi ft}\right) \]