Bingo.
In loose terms, the largest high power transceiver nodes would be perpetually broadcasting a signal whose information was enough to recover the updated blockchain by interpolation given a measurement of the transmission of sufficient duration commensurate with the degree to which the user's hard copy is outdated. Having more up to date information in digital form will assist interpolation of the entire blockchain given a capture that is of finite length (in time).
+ compressed sensing comes to mind http://en.wikipedia.org/wiki/Compressed_sensing
++ in the end, it may not be necessary to use the transmission to interpolate a full digital render of the updated blockchain. After researching I now feel one can exploit the advantages of continuous signals to speed up the acquisition time by using a weight function to decrease emphasis on unknown parts of the chain for security. Still need to start somewhere, so I'm going to continue with the model that a user wants to "recapture" perfect information. If that can be done it could serve as a launchpad as a proof of concept, as things would only get "easier" RF wise through such techniques.
+++ CS and RF people don't always mean exactly the same thing when we say "bandwidth" so careful there (somebody can teach me about CS usage though and we can all have epiphanies together about how its the same I'm sure). Here by bandwidth I mean an interval of frequencies measured in cycles per second of an electromagnetic wave traveling in space. Make no mistake, by frequency I mean f in the relevant equations f = c / λ and c is the speed of light (in a medium such as the ionosphere). In physics, we tend to favor f as a description of a band e.g. 800 MHz. In radio, the wavelength λ is often used e.g. "13 cm band". These descriptions are identical. c =~ 3e8 meters per second for quick ref.
Never ended up completing the text that follows, will be ongoing I suppose
A simple scenario - a gal wants to check her balance
In practice what will happen is a user with an outdated copy of the blockchain will try at an arbitrary time to listen to the transmission and will interpolate the updated blockchain using a measurement of duration Δt and recollect what transactions occured in the blocks that the user's old copy of the blockchain do not include. . How long this takes, I'm not exactly sure but for a first stupid guess take the inverse bandwidth (1/BW). If I end up renaming any common bitcoin conventions lmk
Correspondence times
I havent written this theory down yet but I can already tell a stumbling block for someone trying to understand what I mean is fourier transforms are all over the place. e.g. the time intervals over which the principal transmission is measured by a passive bitcoin user will not correspond linearly to events that occur (transactions are attempted, tx is verified, etc etc) in an intuitive way, unless you're already familiar with signal processing. If not, its OK... fourier space (as digital to analog talk normally) involves, roughly describes the notion that every single moment of the transmission is going to describe the entire blockchain in some regard. Fourier transforms reverse the notions of global and local characteristics of a signal. What a mouthful.
I'll have mathematics for this soon...I hope. Anyone who knows image processing will know what I'm trying to say. Hopefully a theorist will finish my sentences, I'm actually not a theorist but I'll start things off to describe the notion I have going right now.
The Idea:
at t = 0 the blockchain was born with block # N = (0 or 1? nbd right now). N can reference the blockchain age [is this called "block height" in jargon?]
the user's hardcopy of the blockchain is characterized by its age. Also the user's age will refer to the age of the hardcopy the user is holding when they connect to the bitcoin network
a bitcoin user U with an M block hard copy that is not being updated at all can be called UM
+ could imagine a situation in which a user with an active device could be updating their copy of the blockchain all the time...this is sort of like what happens today except it will be labeled instead by a time t so that user would be called a UM(t) because they have a time dependent copy of the blockchain.
A simple situation would be UM tunes in and starts interpolating at some arbitrary time at which the main chains age is M + K.
An important parameter the number of blocks K which I'll call the block deficit. This will determine the acquisition time needed to check the balance.
+++++++++++++
I'm going to scan literature and try to adopt conventions for the variables I named. I will update that when I read them.
I think I can see this but I'm running into problems with discrete variables hanging around. it's cool it can be done, I need to brush up on how to encode digital signals too. Going to read modulation schemes https://en.wikipedia.org/wiki/OFDM I was told this one scales well with bandwidth. It's likely for a full blkchain recovery the block deficit K will determine which band the users tunes to. A large K would naturally have greater needs for bandwidth. Now going to bed guys, i'm fried. not even sure what to do right now... write down the equations or just start testing things. I think some equations would be good because math is much better at communicating this stuff than words. anyway, i have it up there though it's churning. gonna talk to John R. Klauder... he is good at this stuff (he's also very old). He theorized chirp radars http://www.lucent.com/bstj/vol39-1960/articles/bstj39-4-809.pdf