Summary

Fourier analysis represents signals as sums of sinusoids. The fourier_transform converts time_domain data to the frequency_domain showing spectrum, amplitude, and phase. Periodic signals use a fourier_series. Digital work uses the dft, computed efficiently by the fft with O(N log N). Key properties include linearity, orthogonality, and the convolution_theorem. Sampling needs sampling_rate above twice the highest frequency, the nyquist_frequency rule. Outputs often use complex_exponentials with real and imaginary parts.