Aliasing in 1∕f^α noise spectra: Origins, consequences, and remedies

The scaling exponent of a 1∕f^α noise time series is commonly estimated from the power-law slope of its Fourier power spectrum. Here I show that because 1∕f^α noises typically have significant power above the Nyquist frequency, measurements of their power spectra will often be severely distorted by aliasing, not only near the Nyquist frequency, but also far below it. I show that spectral aliasing typically leads to large systematic biases in the scaling exponents, and thus the fractal dimensions, that are estimated from the power-law slopes of 1∕f^α noise spectra. I describe a simple spectral filtering method that corrects the distortions introduced by spectral aliasing, and recovers the broadband spectrum of 1∕f^α noises. Like a Wiener filter, this filtering method does not require that the correct spectrum is known in advance. I illustrate this filtering technique using two environmental noise spectra that are distorted by aliasing.