5. Refine a signal into smaller packets
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Follow the complete refinement demonstration for runnable code, expected output and a direct route to the written theorem.
Take the finite list
\[x=(1,2,3,4,5,6,7,8).
\]
Index its entries from zero through seven. Thus \(x_0=1\) and \(x_7=8\). We only rearrange this finite list. No boundary extension or convolution is needed.
Separate even and odd addresses
For remainder \(r=0,1\), collect the entries \(x_{2k+r}\):
\[x^{(2)}_0=(1,3,5,7),\qquad x^{(2)}_1=(2,4,6,8).
\]
These are compact coordinate lists. To view them as vectors in the original eight positions, put zeros in the other slots:
\[P_0x=(1,0,3,0,5,0,7,0),\qquad
P_1x=(0,2,0,4,0,6,0,8).
\]
Their sum is \(x\).
Refine from two address classes to four
Now collect entries \(x_{4k+r}\):
| Remainder \(r\) | Original indices | Compact list |
|---|---|---|
| 0 | 0, 4 | \((1,5)\) |
| 1 | 1, 5 | \((2,6)\) |
| 2 | 2, 6 | \((3,7)\) |
| 3 | 3, 7 | \((4,8)\) |
Let \(F_r\) select those positions in the original eight-dimensional space. The even positions split into remainders 0 and 2. The odd positions split into remainders 1 and 3. Therefore
\[P_0=F_0+F_2,\qquad P_1=F_1+F_3.
\]
For example,
\[F_0x=(1,0,0,0,5,0,0,0),\qquad
F_2x=(0,0,3,0,0,0,7,0).
\]
Adding these gives \(P_0x\). Interleaving the four compact lists recovers \((1,2,3,4,5,6,7,8)\).
What happens if one packet is dropped?
Discard remainder 2 and retain the other three. The reconstructed list is
\[\widehat x=(1,2,0,4,5,6,0,8).
\]
The residual, defined here as original minus reconstructed, is
\[r=x-\widehat x=(0,0,3,0,0,0,7,0),\qquad
\|r\|_2^2=3^2+7^2=58.
\]
Rearranging all packets is exact. Discarding a nonzero packet changes the signal, and the residual records the missing values.
Try it
Keep only remainders 0 and 2.
Answer: The retained signal is \((1,0,3,0,5,0,7,0)\). Its residual is \((0,2,0,4,0,6,0,8)\), whose squared Euclidean norm is \(4+16+36+64=120\).
Continue reading: Refinement, residue class, and the signal realization guide.
Papers, proofs, Lean, and code
Terms in this example: Refinement · Coarsening · Polyphase decomposition · Residue class · Compatibility
Read Paper II for refinement and Paper V for signal and polyphase constructions.
Paper II: Refinement Towers, Weighted Transfer, and Integral Separation of Operator Packets
Read · Written proofs · Lean coverage · Code and setup · Evidence
Paper V: Polyphase Signal Calculus
Read · Written proofs · Lean coverage · Code and setup · Evidence
Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: refinement(). This checks the displayed rational, modular or finite-set calculations exactly.
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