Polyphase additive route packets

Read side by side

Perfect reconstruction in polyphase and filter-bank routing is additive and depends on cancellation. Convex or maximum-based branch aggregation generally changes that reconstruction. For the normalized Haar bank and input (1, 0), the two branch contributions to the first reconstructed sample are 1/2 and 1/2, while those to the second are 1/2 and -1/2. Sum semantics reconstruct (1, 0); max/convex semantics would produce (1/2, 1/2).

A route packet therefore pairs its continuation state with a declared carrier algebra. See carrier-change obstructions for the conditions under which an aggregation law can be replaced.

For causal FIR polyphase routing the carrier is additive. If E_h is the polyphase route packet of h, with

E_h[target, source][q] = h[N q + target - source],

then a cascade satisfies

E_(g*h)[target, source]
  = sum_middle E_g[target, middle] * E_h[middle, source],

where * on scalar entries is convolution in the coarse-lag variable. Intermediate phase/tap histories can therefore be summed exactly into a finite matrix of lag kernels. This is the standard polyphase matrix multiplication law expressed as a route-packet contract.

komorebi_math.signals.route_packets.PolyphaseRoutePacket implements this finite additive routing law.