I'm not particularly experienced in numerical analysis, and so I recently had quite a massive shock when I discovered that sampling a smooth function and computing the FFT of the result does not return a correct$^1$ list of it's Fourier coefficients, despite theory saying that it ought to!

As an example, let $f(x)=x^2$, and suppose we want to compute it's Fourier coefficients $c_m=\int_0^{2\pi}f(x)e^{imx}\,dx$ on the interval $[0,2\pi]$, with $m\approx50$ or thereabouts. Since $f$ is smooth, it naively seemed reasonable to assume that by sampling $f$ with $n\approx300$ datapoints, computing the FFT would return a fairly accurate list of the $c_m$ for $-50\leq m\leq50$.

As it turns out, this is complete garbage except for the 3 or 4 largest coefficients, and in general the FFT cannot be used to reliably compute Fourier coefficients.

To give a feel for just how mindbogglingly terrible the FFT performs, in order to obtain $c_{50}$ to just two significant digits of accuracy for the simple case of $f(x)=x^2$ in the interval $[0,2\pi]$, one has to sample $f$ with $n\geq 16000$ datapoints! In Mathematica:

f[x_] := x^2;
expr = Integrate[f[x] Exp[I m x], {x, 0, 2 \[Pi]}];
n = 16000;
\[CapitalDelta] = (2 \[Pi])/n;
X = Fourier[
    Table[f[x], {x, 0, 2 \[Pi] - \[CapitalDelta], \[CapitalDelta]}], 
    FourierParameters -> {1, 1}][[2 ;; 51]];
 Table[Abs[(\[CapitalDelta] X[[m]] - N@expr)/N@expr] 100, {m, 50}], 
 PlotLabel -> "Percent Error"]

enter image description here

The absurdity increases when we go to simple 2D functions, such as $f(x,y)=x\sin(x+y)$, graphed below on $[0,2\pi]\times[0,2\pi]$:

enter image description here

The function shown above has to be sampled on a $160000\times160000$ grid in order to achieve 3 significant digits of accuracy for $c_{50,50}$ and similar coefficients, a computation which would require over 200 gigabytes of RAM just to hold the raw data, let alone compute its FFT! Thus the use of the FFT to compute Fourier coefficients of smooth multivariate functions is unfeasible, unless you're interested in the largest coefficients, or you have access to a cluster-computing facility.

As a result, I was quite relieved when I read Section 13.9 of "Numerical Recipes in C", which describes how bad the FFT is for computing Fourier integrals, and provides a correction which uses cubic splines to hugely increase the accuracy, making the FFT a viable method of computing Fourier coefficients. However, I am having trouble deriving the formulas in the paper (the derivation is not given), and would love some help.

In the case of the cubic correction, the article uses $\psi$ functions which is a cubic Lagrange interpolating approximation of the Kronecker delta function. The $W(\theta)$ function is then given by $$W(\theta)=\int_0^{2\pi}\psi(s)e^{i s\theta}\,ds=\frac{\left(\theta ^2+6\right) (-4 \cos (\theta )+\cos (2 \theta )+3)}{3 \theta ^4}.$$

I tried to figure out what exactly the author meant by "cubic Lagrange interpolating" by rederiving the result in Mathematica (copy/paste into Mathematica and hit Ctrl-Shift-N to convert to a more readable form):

CubicLagrangeInterpolation[X_, Y_, x_] := 
  Sum[Y[[j]]*Product[If[j == k, 1, (x - X[[k]])/(X[[j]] - X[[k]])], 
           {k, 1, 4}], {j, 1, 4}]; 
\[Psi][x_] := 
     x < -1}, {CubicLagrangeInterpolation[{-2, -1, 0, 1}, {0, 0, 1, 
       0}, x], -1 <= x <= 0}, 
         {CubicLagrangeInterpolation[{-1, 0, 1, 2}, {0, 1, 0, 0}, x], 
     0 <= x <= 1}, {0, x > 1}}]; 
 Integrate[\[Psi][s]*Exp[I*\[Theta]*s], {s, -Infinity, Infinity}]]

which yielded $\frac{\theta ^2-2 \left(\theta ^2+3\right) \cos (\theta )-2 \theta \sin (\theta )+6}{\theta ^4}$, which when plotted appears somewhat similar to, but is not quite the same as the $W(\theta)$ given in the paper.

My question is:

  • Does anyone with experience in this sort of spline-correction procedure know how the author proceeded to get his expressions for the $W,\alpha_0,\alpha_1,\alpha_2$ and $\alpha_3$ functions? Basically, I just want to know what expression he used for the $\psi$ and $\varphi$ basis functions.

Alternately, am I simply implementing Lagrange interpolation incorrectly?

$^1$: From a relative (or $\%$) error perspective.

  • $\begingroup$ As an aside, the reason this integrand is hard to estimate is because it is oscillatory. $\endgroup$ – Hurkyl Aug 11 '14 at 20:10
  • $\begingroup$ @Hurkyl: Yes, Fourier integrals are a subset of the more general class of highly-oscillatory integral problems, which generally tend to be ill-conditioned; the main point was that for this particular class, there exists a useful spline correction, and I am curious what the basis functions are (and possibly how to extend the spline correction to multidimensional FFTs). $\endgroup$ – DumpsterDoofus Aug 11 '14 at 20:13
  • $\begingroup$ @Hurkyl: Or more to the point, there exists a numerical method which preserves the use of the FFT, while at the same time drastically reduces the RAM requirements by several orders of magnitude, for computing large numbers of Fourier coefficients. In contrast, the non-FFT approach would essentially entail the numerical integration of several thousand highly-oscillatory functions, a far nastier procedure from a computational cost perspective. $\endgroup$ – DumpsterDoofus Aug 11 '14 at 20:31
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    $\begingroup$ In the second paragraph, the argument "since f is smooth.." doesn't make much sense, since the periodic extension of $f$ to $\mathbb{R}$ is discontinuous. $\endgroup$ – Patrick Sanan Aug 21 '14 at 8:21
  • $\begingroup$ For cubic interpolation there are four piecewise parts, not two as you have used. On the additional interval $[-2, -1]$ interpolate $\{-3,-2,-1,0\},\{0,0,0,1\}$ and on $[1,2]$ use $\{0,1,2,3\},\{1,0,0,0\}$. $\endgroup$ – WimC Jul 13 '18 at 20:34

For cubic interpolation $\psi$ has four piecewise parts, not two as you have used. On the additional interval $[-2, -1]$ use $(x+3)(x+2)(x+1)/6$ and on $[1,2]$ use $-(x-1)(x-2)(x-3)/6$. So yes, a mistake in your Lagrange interpolation code.

As noted in the numerical recipes reference, the correction terms $\varphi_k$ are a bit tedious. Let me restrict to the example of cubic interpolation. So in this case we want to approximate some function $f$ on $[0, M]$ (where I'll take $M$ large enough to circumvent pathological situations) piecewise by Lagrange polynomials. The basic idea is on the interval $[k, k+1]$ to interpolate based on the values of $f$ at $k-1, k, k+1, k+2$. This is what is achieved by the $\psi$ kernel:

$$f(x) \approx \sum_{k=0}^M f(k)\ \psi(x - k).$$

Now this works fine except for intervals near the end point. I'll discuss the left end point situation only, the right one is similar but kernels are reflected by $x \leftarrow -x$.

On the interval $[0, 1]$ we actually want to interpolate based on values at $0, 1, 2, 3$ instead of at $-1, 0, 1, 2$ as is done in the sum above (where $f$ is understood to vanish at negative arguments). In particular:

  • $f(0)\ \psi(x)$ has an incorrect contribution on the interval $[-2, 1]$
  • $f(1)\ \psi(x-1)$ has an incorrect contribution on $[-1, 1]$
  • $f(2)\ \psi(x-2)$ has an incorrect contribution on $[0, 1]$
  • $f(3)\ \psi(x-3)$ has no contribution on $[0,1]$

So we need $\varphi_0, \ldots, \varphi_3$ to correct for this as follows (where every $\varphi_k$ vanishes outside the specified intervals):

  • $\varphi_0(x)$ equals $-\psi(x)$ on $[-2, 0]$ and $-(x-1)(x-2)(x-3)/6 - \psi(x)$ on $[0, 1]$
  • $\varphi_1(x)$ equals $-\psi(x)$ on $[-2, -1]$ and $(x+1)(x-1)(x-2)/2 - \psi(x)$ on $[-1, 0]$
  • $\varphi_2(x)$ equals $-(x+1)(x+2)(x-1)/2-\psi(x)$ on $[-2, -1]$.
  • $\varphi_3(x)$ equals $(x+1)(x+2)(x+3)/6$ on $[-3, -2]$

Then the left end point corrected approximation of $f$ reads:

$$f(x) \approx \sum_{k=0}^M f(k)\ \psi(x-k) + \sum_{k=0}^3 f(k)\ \varphi_k(x-k).$$

And a similar correction follows for the right end point.

  • $\begingroup$ On that note, the simplest way to obtain these polynomials in Mathematica is to use the InterpolatingPolynomial command. For instance, InterpolatingPolynomial[{{0,1},{1,0},{2,0},{3,0}},x] yields the aforementioned cubic polynomial on [1,2]. (Of course, in this case the polynomial can be constructed by inspection since at most one of the interpolation points is nonzero over any integer-integer interval.) $\endgroup$ – Semiclassical Jul 16 '18 at 13:24
  • $\begingroup$ As a follow-up, do you understand how Numerical Recipes constructs their endpoint kernels i.e. the $\varphi_j(x)$? $\endgroup$ – Semiclassical Jul 16 '18 at 18:46
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    $\begingroup$ @Semiclassical Do you believe me if I say "yes"? ;-) $\endgroup$ – WimC Jul 16 '18 at 18:59
  • $\begingroup$ I'd love to believe you, but that doesn't get me any closer to knowing it myself :P. In the case of linear interpolation, I've found it works to use $\varphi_0(x)=-1-x$ on the $[-1,0]$ segment and $\varphi_0=0$ otherwise; additionally, the last endpoint kernal is $\varphi_M(x)=\varphi_0(-x)$ and all other $\varphi_j(x)$ vanish. Based on the remarks in the text, it seems as if the 'true' endpoint kernel is then $\varphi_0(x)+\psi(x),$ which equals $\psi(x)$ on $[0,1]$ and vanishes otherwise. But I haven't managed to reconstruct the far more involved cubic case... $\endgroup$ – Semiclassical Jul 16 '18 at 19:16
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    $\begingroup$ @Semiclassical I feared as much. Currently adding it to my answer. Hold on... $\endgroup$ – WimC Jul 16 '18 at 19:17

The FFT is no worse at computing the "Fourier Integral" (what we call in the DSP world the "Fourier Transform") than the Riemann summation is at computing an integral of a "non-pathological" function. Make the FFT larger and larger, sampling the continuous signal with finer and finer sampling precision (and extending outwardly the limits of the integral), and this corresponds directly to the Riemann integral with uniformly-spaced sample instances.

However, the FFT (DFT) is great at computing coefficients for the Fourier series of a bandlimited and sampled periodic function as long as exactly one period of the function becomes the total "span" (I dunno if that's the right word) of the DFT. The constant term is $X[0]$. The first harmonic (magnitude and phase) are in $X[1]$. The second harmonic in $X[2]$.

So maybe what you have to do is interpolate the input to the FFT rather than interpolate the output.

  • $\begingroup$ The cited method is about interpolating the input. $\endgroup$ – WimC Jul 13 '18 at 20:45

$$\mathbf{\color{green}{Preliminary\ remarks.}}$$ First, the discrete Fourier transform has a complex spectrum at the output, the phase of which changes abruptly with a minimal frequency shift of the signals. Therefore, it is assumed that the energy spectrum of the signal or another similar quantity enters the input of the cubic spline.

Secondly, the discrete Fourier transform usually generates parasitic oscillations, the cause of which is the discrepancy between the initial and final points of the sample. The discrete Fourier transform uses the assumption of the periodicity of the signal, and this discrepancy generates a broadband interference at the output of the FFT. To combat this phenomenon, weight processing of the signal would be used.

$$\mathbf{\color{green}{About\ splines.}}$$

Interpolation by a cubic spline of $16\,000$ points leads to a three-diagonal SLAE with $16000-2$ variables, and this is a serious problem for any computational method. At the same time, the requirement of connectivity of $16\,000$ points of the spectrum is clearly redundant.

The simplest way out of the situation is to split the output data array into fragments of a reasonable size, with independent spline interpolation for each fragment.

  • $\begingroup$ The question is not how to handle such large amounts of interpolation points but rather about a technique to improve approximations of fourier integrals based on relatively few interpolation points. $\endgroup$ – WimC Jul 18 '18 at 14:34
  • $\begingroup$ @WimC I'm trying to convey the idea that even three-diagonal SLAUs of only 16,000 points create problems regardless of the origin of the data. The need for spline interpolation on such a base is also questionable, since the impact of distant points on the result is minimal. Thus, the OP actually solves the only problem - the study of the limits of the efficiency of interpolation algorithms. $\endgroup$ – Yuri Negometyanov Jul 18 '18 at 15:19

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