Power Spectral Density S(f) [log-log]
Time Series x(t)
Autocorrelation C(τ)
White (α=0)
Pink (α=1)
Red/Brownian (α=2)
Current
Colored noise is generated by shaping white noise in frequency domain:
X(f) = W(f) / f^(α/2), then iFFT.
White: S(f)=const; Pink (1/f): equal energy per octave, ubiquitous in nature (music, heartbeat, financial returns);
Red (1/f²): Brownian motion (integrated white noise).
Hurst exponent H = (α-1)/2 for 1<α<3. Long-range correlations: C(τ) ~ τ^(2H-2).