Behaviour of the Spectral Weight Function $\frac{\sin^2{(\pi f t)}}{(\pi f)^2}$

Question

I'm looking into the properties of the so called spectral weight function $W_0 = \frac{\sin^2{(\pi f t)}}{(\pi f)^2}$. While not important for the question, this function is is encountered in the context of power spectral densities; it is basically a function that describes which parts of the power spectral density have the most weight in determining the mean squared value of a random variable at a certain point in time.

In any case, I stumbled upon it in the paper Decoherence of a superconducting qubit due to bias noise by Martinis et al. found at http://journals.aps.org/prb/abstract/10.1103/PhysRevB.67.094510 (equation 11, for those with access to the paper). There they write that the spectral weight function is constant for frequency $f \leq 1/t$ and decreases like $1/f^2$ at higher frequencies. At first I trusted this, but then I started looking into the function.

So first I plotted it for $t = 2$.

If the paper is to be believed, the function should be more or less constant up to f = 0.5. But it is not at all, not even close. Clearly something is off, but I do not understand what. Seeing as this is a published and peer reviewed paper, I am inclined to believe the mistake is on my side. My question is therefore if you can help me figure out what I am doing wrong.

The above plot is generated with a small piece of Mathematica code, which can be used to look at different $t$ values

Manipulate[
 Plot[Sin[Pi*f*t]^2/(Pi*f)^2, {f, 0, 10}, 
  PlotRange -> {{0, 1}, {0, 4}}, GridLines -> {{1/t}, {}}, 
  ImageSize -> 500, FrameLabel -> {"f", "W(f)"}], {{t, 2}, 0, 10}]

Answer 1

Your function is just the Fourier transform of the "triangle function" (Heaviside Lambda) $$ f(x) = \left(1-\left|x\right|\right)\cdot\mathbb{1}_{(-1,1)}(x) $$ that is compact supported. By the Paley-Wiener theorem, $\widehat{f}$ is an entire function: since it is not constant (by just comparing the behaviour at $0$ and at $\pm\infty$), it cannot be constant on any open subset of $\mathbb{C}$.

The paper you mention probably uses constant in a (very!) improper way, by meaning smooth and non-negative.