An Interesting Fourier Transform - 1/f Noise
Power law functions are common in science and engineering. A surprising property is that the Fourier transform of a power law is also a power law. But this is only the start- there are many interesting features that soon become apparent. This may even be the key to solving an 80-year mystery in physics.
It starts with the following Fourier transform:
The general form is tα ↔ ω-(α+1), where α is a constant. For example, t2 ↔ ω–3 and t -0.75 ↔ ω–0.25. Unfortunately, there are additional terms that distort this simple relation. First, the left side contains the term, u(t), the unit step function. This is defined as u(t) = 0 for t < 0, and u(t) = 1 for t ≥ 0. In other words, this makes the time domain a one-sided power law. Second, the power law in the frequency domain only pertains to the magnitude; there is a phase term that doesn't resemble a power law at all. Third, there is a scaling factor in the frequency domain, Γ(α +1). This is the Gamma function, which is essentially a continuous version of factorials. A graph of the Gamma function is shown below. Don't worry too much about this strange function. Think of it simply as a constant that scales the amplitude of the frequency domain, depending on the value of α.
The table below shows nine cases of this Fourier transform, with alpha running from -2.0 to 2.0, and rough sketches of the curves. The frequency domain also shows a rough sketch of the magnitude graphed on a log-log plot, which turns out to be is a straight line with a slope of -(α+1). Take a few minutes to examine this figure, especially noting the symmetry between the time and frequency domains.
One of these cases should be familiar to you, where α=0. This is the Fourier transform of the unit step function, with a magnitude of 1/ω, and a phase of -π/2. As you probably recall, this describes the impulse and frequency response of the perfect integrator. Now consider the case of an integrator followed by another integrator. The impulse response of this two stage combination is the unit step response convolved with itself. In the frequency domain the magnitude becomes 1/ω × 1/ω, and the phase becomes 2 × (-π/2). This two-integrator cascade is shown on the graph for α=1, where the impulse response is a linearly increasing line, and the frequency spectrum is 1/ω2 , with φ = -π. Likewise, α=2 represents a cascade of three integrators, and so on. It is interesting that these Fourier transforms are so well behaved, in spite of both domains containing nasty features (such as: t2 as t → ∞ , and ω-3 as ω → 0 ).
Here is something even more interesting. As you approach α = -1, the time domain approaches a shape of t-1, and the frequency domain approaches a flat magnitude with a zero phase. However, a flat magnitude and zero phase corresponds to a delta function, δ(t), in the time domain. That is, in the limit as α → -1, u(t)t-α = δ(t). This is because the sharp point of t-α grows rapidly as α → -1, dominating the entire function. But what happens exactly at α = -1? How could the sharp point ever completely negate the seemingly finite width of t-1? The mathematics tells you not to ask this question. Recall that the frequency domain has a scaling factor of Γ(α+1). For α = -1, the gamma function is undefined.
Now we come to a feature that I find absolutely fascinating. In general, we have seen that a power law in one domain corresponds to a power law in the other domain. Further, there is an inverse relationship; if the time domain decays faster, then the frequency domain decays slower, and vice-versa. This means that there must be a certain decay rate that is unique, where both domains are equal. This occurs for α = -0.5, where the time domain is t-0.5 and the frequency domain is ω-0.5. Interesting, but what does this mean?
Now look at the figure below, a graph of the measured noise that originates within a common electronic amplifier. The flat section above 100 Hz is called white noise , and is well understood. However, the sloping portion below 100 Hz is not well understood at all. This is 1/f noise, a mystery that has resisted explanation for over 80 years. 1/f noise has been observed in the strangest places- electronics, traffic density on freeways, the loudness of classical music, DNA coding, and many others.
Many of the properties of noise are directly related to the amount of power in a signal, that is, to the square of the amplitude. Accordingly, most of those working with noise think in terms of power spectra, not amplitude spectra. 1/f noise gets its name because its power spectrum has a shape that is close to 1/f. However, if we look at the amplitude spectrum for 1/f noise it has a shape of 1/f1/2. As can be seen above, on a log-log plot of amplitude, 1/f noise has a slope of -0.5. Now you can see where I’m going.
At least in a limited sense, 1/f noise is its own Fourier transform, with ω-1/2 in the frequency domain, and t-1/2 in the time domain. For instance, a single pulse given by u(t) t-1/2 has a 1/f power spectrum. Likewise, a randomly occurring sequence of such pulses has a 1/f power spectrum, at least over a wide range frequencies. Further, 1/f noise can be created by passing white noise through a filter with an impulse response of u(t) t-1/2. Unfortunately, none of these scenarios seems to have a physical interpretation that explains the widespread observation of 1/f noise. It’s clear that something is still missing.
There is also another issue: u(t) t-1/2 is the transform pair of Mag = ω-1/2 , φ = -π/4. However, no one knows what the phase of 1/f noise is, or even if it has a defined phase. If the phase happens to be different from φ = -π/4, then the corresponding time domain signal will also be different. That is, it may be that the characteristic time domain waveform of 1/f noise is simply not t-1/2.
Nevertheless, the idea that 1/f noise is its own Fourier transform is very compelling. Consider the Gaussian curve, the most important waveform associated with random events. The Central Limit Theorem tells us why the Gaussian is so commonly observed. However, it is also true that the Fourier transform of a Gaussian is a Gaussian. This seems more than coincidence– I think it is a critical clue in solving the mystery of 1/f noise.
Comments are certainly welcome! I especially appreciate suggestions for new directions in research and (god forbid) errors in math.
Next post by Steve Smith:
Waveforms that are their own Fourier Transform
Does anyone have a reference for the result that the FT of a power law is a power law? I have been hunting for one and can't find it. Thanks
Thanks Steve. This is a wonderful and stimulating article. I like this kind of observations/interpretations. This is what makes math 'beautiful'. The moment I read this, the first thing that strikes my mind is the theory of eigenvalues and vectors. So many problems become much easier when though-off as matrices. When the signal is sampled, FT is replaced with DFT which can simply be denoted by the multiplication of matrix and vector. If F is the DFT matrix and x is a vector, then xf=Fx, and xf is the DFT of x. We are looking at vectors (signals) x that satisfy Fx = \lamda x where \lamda is an eigenvalue of the transformation x. We know from matrix theory that there could be an infinite number of eigenvectors for a matrix A but then no of non-zero eigenvalues cannot exceed the rank of matrix A, and that the space spanned by the eigenvectors is the null space of A and its dimensionality is determined by the rank of A as well. In your other article 'waveforms that are their own fourier transform' , you clearly show that the no of eigenfuncions (functions who shape remains the same under FT) is infinite. The same conclusion can be achieved through linear algebra as well. For DFT, the eigenvectors would span a finite-dimensional space N and for continuous FT, the eigenfunctions would span an infinite dimensional space and so on.
Anyways, coming back to the interpretation of the 1/f noise's PSD being an eigenfunction of FT - since Gaussian function can be expanded into polynomials, it is not very surprising that power law functions satisfy the same property. I think the uniqueness of flicker noise is the following. While the probability distribution itself is an eigenfunction (Gaussian), it time correlation is also an eigenfunction. Dont know if this would help solve the mystery. But enjoyed it.
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