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.
VarianceWhen 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
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 principleThe Fourier uncertainty principle is the inequality
where the constant C depends on your convention for defining the Fourier transform [1]. Here ||f||2^2 is the integral of f^2, the square of the L^2 norm.
Perhaps a better way to write the inequality is
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 , then its Fourier transform is proportional to the density of a normal distribution with standard deviation 1/, if you use the radian convention described in [1].
ExampleWe 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
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 L^2 norm of the Fourier transform is proportional to the L^2 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/(4b^2). That is, C = 1/4 if your definition has a term exp( i t) and C = 1/16^2 if your definition has a exp(2 i 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 = 1/16^2.
Fourier transform definitions also have varying constants outside the integral, e.g. possibly dividing by 2, but this factor effects both sides of the Fourier uncertainty principle equally.
The post Fourier uncertainty principle first appeared on John D. Cook.