This article summarizes spectral analysis.
Theoretical preparation to spectral density estimation: Random-Process#power-spectral-density
Intensities over a frequency spectrum (such as power spectral density) is often plot on logarithmic scales, reflecting the dynamic range (ratio of the maximum and the minimum) of spectral components. Some common log scale units: decade is a factor of 10, octave is a factor of 2; 10 decibels (dB) is 1 decade. Octave is such named because a frequency band with dynamic range 2 is called an octave (纯八度) in music, where the eighth note (音符) denotes a harmonic twice the frequency of the first. Given a reference value, these dimensionless units become absolute units. For signal power, decibel watt (dBW) and decibel milliwatt (dBm) take watt (W) and milliwatt (mW) as reference value respectively.
White noise is a random signal with power uniformly distributed over the frequency spectrum. In general, power-law noise is a random signal with power spectral density being a power-law function, $S_X(f) = f^{\beta}$. On log-log scale, power spectral density of a power-law noise is linear, with slope $\beta$ (or $10 \beta$ in dB/decade). Power-law noise with specific values of $\beta$ have color names: -1 for pink noise, -2 for brown/Brownian noise; 1 for blue noise, 2 for violet noise. The color names correspond to the apparent color if the random signal is an electromagnetic wave on the visible spectrum. All power-law noises are theoretical models, because actual signals can never have a frequency spectrum of infinite dynamic range.
Spectral density estimation is estimating the power spectral density of a random signal from a time-series sample.
Figure: A voice waveform over time and its broad power spectrum.Spectral analysis involves a trade-off between resolving comparable strength components with similar frequencies and resolving disparate strength components with dissimilar frequencies. The following explains the cause and trade-off strategy in detail.
Sampling: the reduction of a continuous signal to a discrete signal, typically equal-spaced in time.
Aliasing (effect of sampling on the frequency spectrum): periodic repetition of the entire spectrum.
Figure: continuous Fourier transform, Fourier transform of periodic summation, discrete time Fourier transform (DTFT), and discrete Fourier transform (DFT). Notice that sampling always introduces periodic spectral repetition.Window function: a function of finite support that gets multiplied with a waveform before estimating its spectral density (on a finite time interval).
Spectral leakage (effects of windowing on the frequency spectrum): Windowing causes the true spectrum S(f) to be convoluted with the Fourier transform of the window function F{w}.
Choice of window function for spectral density estimation: