# 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