A Algorithm statements
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A.1 Exact polyphase filtering
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Given \(x\), \(h\), and \(N\):
Form \(x_s[n]=x[Nn+s]\) for \(0\le s<N\).
Form every transfer kernel \(h_{ks}[m]=h[Nm+k-s]\).
Compute
\[ y_k[n]=\sum_s\sum_mh_{ks}[m]x_s[n-m]. \]Interleave the \(y_k\) channels.
The algorithm is algebraically identical to direct convolution; its purpose is to expose independent and interacting packets for block computation and refinement.
A.2 Conditional known-edge reconstruction
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Given a decomposition \(f=s+j\):
Apply the selected smooth approximation \(Q_m\) to \(s\) only.
Retain \(j\) exactly.
Return \(Q_ms+j\).
The error certificate is \(\|s-Q_ms\|\).
A.3 Limited positive-contrast estimator
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The included finite estimator does the following:
Split the observed values into two classes by an Otsu threshold.
Keep coherent connected components and fill holes.
Harmonically fill the selected packet to estimate the missing background.
Smooth the filled background spectrally.
Estimate one packet amplitude by a robust median residual.
Recombine the estimated smooth and edge packets.
This procedure is useful for the supplied separated synthetic case. It assumes a positive-contrast two-class packet and is not promoted to a general edge theorem.