Paper V: One-dimensional numerical study

8 One-dimensional numerical study

On this page · Visual guide: Why a Fourier cutoff cannot reproduce this jump pointwise · Permalink · Source line 761

The one-dimensional signal contains a smooth non-bandlimited background and a positive rectangular packet. Three reconstructions are compared:

Permalink · Source line 763

global Fourier truncation of the complete signal;
Fourier truncation of the known smooth packet plus the exact jump packet;
a two-class estimated packet followed by harmonic background filling, spectral smoothing, and jump-amplitude estimation.

Permalink · Source line 765

One-dimensional signal reconstructions and absolute errors for global, known-packet and estimated-packet methods. — One-dimensional reconstruction and absolute error. The known packet removes the edge from the global spectral approximation. The estimated packet is visibly better than the global reconstruction in this limited test but retains smooth-background estimation error.

Numerical certificate 8.1: One-dimensional errors

For \(1024\) samples on \([-1,1)\) and Fourier cutoff \(m=18\), the retained frequency indices are \(-18,\ldots,18\): 37 bins including the zero-frequency bin. The measured values were

\[ \begin{array}{lrr} \toprule \text{method}&{\operatorname{RMSE}}&\text{maximum absolute error}\\ \midrule \text{global Fourier truncation}&5.4023\times10^{-2}&3.4968\times10^{-1}\\ \text{known edge packet}&1.3692\times10^{-4}&1.4503\times10^{-3}\\ \text{estimated edge packet}&2.4029\times10^{-2}&5.9530\times10^{-2}\\ \bottomrule \end{array} \]

The simple estimator recovered the binary mask exactly in this deliberately separated two-class example. That result is a property of this test, not a universal guarantee.

Permalink · Source line 778 · Claim record · Lean coverage · Code · Worked examples

Reconstruction errors as spectral resolution increases, including the estimated-packet error floor. — Error as spectral resolution increases. The known packet follows only the smooth approximation error. The estimated reconstruction reaches a floor determined by background and amplitude estimation.

Source and conversion record

Source SHA-256: 5d3c6b2e5709c37ea551eb8fd1d823e5249c5fbc411d36b806241af4df888f36

Unsupported prose commands stop the build. The mathematical expression sequence is checked against the source, with manuscript macros expanded for display.

Conversion records

Read and explore

Proofs and implementation

About this work

Other resources

8 One-dimensional numerical study

Numerical certificate 8.1: One-dimensional errors