Walsh Sampling with Incomplete Noisy Signals

Abstract

With the advent of massive data outputs at a regular rate, admittedly, signal processing technology plays an increasingly key role. Nowadays, signals are not merely restricted to physical sources, they have been extended to digital sources as well.Under the general assumption of discrete statistical signal sources, we propose a practical problem of sampling incomplete noisy signals for which we do not know a priori and the sampling size is bounded. We approach this sampling problem by Shannon's channel coding theorem. Our main results demonstrate that it is the large Walsh coefficient(s) that characterize(s) discrete statistical signals, regardless of the signal sources. By the connection of Shannon's theorem, we establish the necessary and sufficient condition for our generic sampling problem for the first time. Our generic sampling results find practical and powerful applications in not only statistical cryptanalysis, but software system performance optimization.

Efficient estimation of Pauli channels

“…We also note that a better run-time may be achieved in this setting using methods for reconstructing a sparse signal from few measurements [68][69][70]. Specifically, the present variant can be cast in a similar form to sparse Fourier and Hadamard transforms [71][72][73][74]. However, it is an open question if the methods in those references can be adapted to the symplectic structure of the Pauli group to achieve near-optimal sparse reconstruction.…”

Practical tera-scale Walsh-Hadamard Transform

“…In signal processing, noisy (sparse) WHT can be very efficiently solved recently (see [3,8]) with SNR > 0 dB. Further, it has been identified that SNR > 10 log 10 8 log 2 2 n dB (4) has greatest significance in cryptography (and coding theory) (see [2,10,14]).…”

“…[1,17]). In particular, the topic of (noisy) sparse WHT has attracted most academia attention from various areas: signal processing [3,8,12], cryptography (and coding theory) [2,7,10,14]. Informally speaking, sparse WHT refers to the case that the number of nonzero Walsh coefficients is much smaller than the dimension; the noisy version of sparse WHT refers to the case that the number of large Walsh coefficients is much smaller than the dimension while there exists a large number of small nonzero Walsh coefficients.…”

“…In cryptography (and coding theory), two long-term research themes exist: 1) researchers study super-sparse WHT of super-large dimension in the noisy setting with emphasis on better time and memory complexities than general Walsh-Hadamard Transform (e.g., [2,7]). 2) researchers aim at studying practical implementation of noisy WHT with both largest possible dimension and strongest possible noise, and somehow can tolerate with solving part of those large Walsh coefficients and index positions (e.g., [10,14]). Clearly, general Walsh-Hadamard Transform is a first solution to the noisy sparse WHT, which can obtain all Walsh coefficients larger than a given threshold and the index positions.…”