Fourier uncertainty principle

Heisenberg’s uncertainty principle says there is a limit to how well you can know both the position and momentum of anything at the same time.  The product of the uncertainties in the two quantities has a lower bound.

There is a closely related principle in Fourier analysis that says a function and its Fourier transform cannot both be localized. The more concentrated a signal is in the time domain, the more spread out it is in the frequency domain.

There are many ways to quantify how localized or spread out a function is, and corresponding uncertainty theorems. This post will look at the form closest to the physical uncertainty principle of Heisenberg, measuring the uncertainty of a function in terms of its variance. The Fourier uncertainty principle gives a lower bound on the product of the variance of a function and the variance of its Fourier transform.

Variance

When you read “variance” above you might immediately thing of variance as in the variance of a random variable. Variance in Fourier analysis is related to variance in probability, but there’s a twist.

If f is a real-valued function of a real variable, its variance is defined to be

V(f) = \int_{-\infty}^\infty x^2 \, f(x)^2 \, dx

This is the variance of a random variable with mean 0 and probability density |f(x)|². The twist alluded to above is that f is not a probability density, but |f(x)|² is.

Since we said f is a real-valued function, we could leave out the absolute value signs and speak of f(x)² being a probability density. In quantum mechanics applications, however, f is complex-valued and |f(x)|² is a probability density. In other words, we multiply f by its complex conjugate, not by itself.

The Fourier variance defined above applies to any f for which the integral converges. It is not limited to the case of |f(x)|² being a probability density, i.e. when |f(x)|² integrates to 1.

Uncertainty principle

The Fourier uncertainty principle is the inequality

V(f) \, \, V(\hat{f}) \geq C\, ||f||_2^2 \,\,||\hat{f}||_2^2

where the constant C depends on your convention for defining the Fourier transform [1]. Here ||f||2² is the integral of f², the square of the L² norm.

Perhaps a better way to write the inequality is

\frac{V(f)}{||\phantom{\hat{f}}\!\!f\,||_2^2} \; \frac{V(\hat{f})}{||\phantom{\hat{.}}\hat{f}\,\,||_2^2} \geq C

for non-zero f. Rather than look at the variances per se, we look at the variances relative to the size of f. This form is scale invariant: if we multiply f by a constant, the numerators and denominators are multiplied by that constant squared.

The inequality is exact when f is proportional to a Gaussian probability density. And in this case the uncertainty is easy to interpret. If f is proportional to the density of a normal distribution with standard deviation σ, then its Fourier transform is proportional to the density of a normal distribution with standard deviation 1/σ, if you use the radian convention described in [1].

Example

We will evaluate both sides of the Fourier uncertainty principle with

    h[x_] := 1/(x^4 + 1)

and its Fourier transform

    g[w_] := FourierTransform[h[x], x, w]

We compute the variances and the squared norms with

    v0 = Integrate[x^2 h[x]^2, {x, -Infinity, Infinity}] 
    v1 = Integrate[w^2 g[w]^2, {w, -Infinity, Infinity}]
    n0 = Integrate[    h[x]^2, {x, -Infinity, Infinity}]
    n1 = Integrate[    g[w]^2, {w, -Infinity, Infinity}]

The results in mathematical notation are

\begin{align*} V(h) &= \frac{\pi}{4 \sqrt{2}} \\ V(\hat{h}) &= \frac{5\pi}{8\sqrt{2}} \\ || h ||_2^2 &= \frac{3\pi}{4 \sqrt{2}} \\ ||\hat{h}||_2^2 &= \frac{3\pi}{4 \sqrt{2}} \end{align*}

From here we can calculate that the ratio of the left side to the right side of the uncertainty principle is 5/18, which is larger than the lower bound C = 1/4.

By the way, it’s not a coincidence that h and its Fourier transform have the same norm. That’s always the case. Or rather, that is always the case with the Fourier convention we are using here. In general, the L² norm of the Fourier transform is proportional to the L² norm of the function, where the proportionality constant depends on your definition of Fourier transform but not on the function.

Here is a page that lists the basic theorems of Fourier transforms under a variety of conventions.

Related posts

[1] In the notation of the previous post C = 1/(4b²). That is, C = 1/4 if your definition has a term exp(± i ω t) and C = 1/16π² if your definition has a exp(±2 π i ω t) term.

Said another way, if you express frequency in radians, as is common in pure math, C = 1/4. But if you express frequency in Hertz, as is common in signal processing, C = 1/16π².

Fourier transform definitions also have varying constants outside the integral, e.g. possibly dividing by √2π, but this factor effects both sides of the Fourier uncertainty principle equally.

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