Refine a signal and account for a missing packet

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Explore the interactive diagram to reshape the signal, refine its address packets, and inspect what is lost when a packet is discarded.

Question: What changes when a signal is described by finer address classes? This example uses list indexing and integer arithmetic.

1. Keep the original addresses

Take \(x=(1,2,3,4,5,6,7,8)\), indexed from zero. Separating addresses by their remainder modulo two gives the compact lists

\[x^{(2)}_0=(1,3,5,7),\qquad x^{(2)}_1=(2,4,6,8). \]

These lists retain all eight entries. Interleaving them recovers the original signal. Example 5 also expresses each list as a projector acting in the original eight-dimensional space.

2. Refine from two classes to four

Remainder modulo four	Original addresses	Values
0	0, 4	\((1,5)\)
1	1, 5	\((2,6)\)
2	2, 6	\((3,7)\)
3	3, 7	\((4,8)\)

Let \(P_0,P_1\) be the even and odd selectors, and let \(F_r\) select remainder \(r\) modulo four, all acting on the original eight coordinates. Then

\[P_0=F_0+F_2,\qquad P_1=F_1+F_3. \]

Each coarse packet is exactly the sum of its fine children. To reconstruct from the compact lists, put their entries back at the recorded addresses. Python's fine[i % 4][i // 4] implements this for address i in this example.

3. Discard one class as a control

Dropping remainder 2 produces

\[\widehat x=(1,2,0,4,5,6,0,8),\qquad r=x-\widehat x=(0,0,3,0,0,0,7,0). \]

The residual is original minus reconstructed. Its squared Euclidean norm is \(3^2+7^2=58\). Refinement with every packet retained is an exact coordinate description. Deleting a packet is an additional operation with a measurable loss in this example.

4. Run the calculation

Follow the download or checkout instructions. From the collection root, run:

python -B examples/komorebi_demonstrations.py refinement

Expected output:

Input: (1, 2, 3, 4, 5, 6, 7, 8)
Two packets: ((1, 3, 5, 7), (2, 4, 6, 8))
Four packets: ((1, 5), (2, 6), (3, 7), (4, 8))
Interleaved: (1, 2, 3, 4, 5, 6, 7, 8)
Without remainder 2: (1, 2, 0, 4, 5, 6, 0, 8)
Residual: (0, 0, 3, 0, 0, 0, 7, 0)
Squared residual norm: 58
PASS: refinement

The checks compare fine children with both coarse packets, reconstruct the list and test the discarded-packet residual. All displayed values and residuals use exact integer arithmetic, including the reconstruction after a packet is discarded.

5. Follow the result and its evidence

Continue to	Route and scope
Paper	Paper II's readable chapters.
Exact claim	Nested residue refinement.
Written proof	Readable statement and proof.
Lean	Paper II's recorded coverage and setup. Individual claim mappings are not transcribed in this claim index. Consult the formal guide for the exact scope.
Program	Read or download the complete program.
Signal applications	Paper V and the library's signal realization guide.
More hand calculations	Residue refinement and Haar averages and differences.

This is a finite address decomposition. It does not establish stability or accuracy for a filter bank, an infinite signal or a numerical implementation.

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