# Why is generating $1/f^\alpha$ noise so complicated?

Why can't I just take the square root of the power spectrum $P(f) = Cf^{-\alpha}$, multiply with $e^{i\theta_n}$ ($\theta_n$ being $N/2$ random phases in $[0, 2\pi]$), and then do an inverse Fourier transform? This can produce arbitrarily long time series with a $1/f^\alpha$ spectrum. Quite obviously, I'm missing something.

EDIT: To be honest, I'm actually looking at methods to generate fractional Brownian motions (fBm). An fBm is characterized by a power spectrum $P(f) = Cf^{-(2H + 1)}$ with $0 < H < 1$ being the Hurst parameter. For example, a popular book on chaos theory by A.A. Tsonis (pp. 62-53) suggests the exact same method for generating an fBm trace:

How To Generate an fBM Trace with a Desired Spectral Density Function. A time series $X_i$, $i = 1, ..., N$, whose spectral density function satisfies the relation $S(f) = Cf^{-a}$ where $a = 2H + 1$, is obtained via the relation

$$x_i = \sum^{N/2}_{k=1} \left[Ck^{-a}\left(\frac{2\pi}{N}\right)^{1-a}\right]^{1/2}\cos{\left(\frac{2\pi ik}{N} + \phi_k\right)}$$

where $C$ is a constant and $\phi_k$ are $N/2$ random phases randomly distributed in $[0, 2\pi]$.

While this method is very simple, almost no papers on fBm generation discuss it.

• Well, for simulation purposes this could be good. – Moti Jul 12 '15 at 6:17
• Could you explain what you mean by "complicated"? Normally you would just filter white noise with an appropriately shaped filter, which isn't that complicated, is it? – Matt L. Jul 12 '15 at 9:46
• @MattL. I've updated the question. – StrangeAttractor Jul 12 '15 at 10:26
• OK, if that's the simple method, what is the complicated method then? It would be interesting to know what the other papers on fBm generation suggest. – Matt L. Jul 12 '15 at 10:51
• @MattL. This review paper discusses quite a few. – StrangeAttractor Jul 12 '15 at 11:01