Permutation Entropy for Signal Analysis

TL;DR

This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and includes results on a classification task.

Abstract

Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given signals (represented as time series) by considering random variables which sample from specified subsequences. In this paper, we will discuss how an entropy variant, the \textit{permutation entropy} can be used to study and classify radio frequency signals in a noisy environment. The permutation entropy is the entropy of the random variable which samples occurrences of permutation patterns from time series given a fixed window length, making it a function of the distribution of permutation patterns. Since the permutation entropy is a function of the relative order of data, it is (global) amplitude agnostic and thus allows for comparison between signals at different scales. This article is intended to describe a permutation patterns approach to a data driven problem in radio frequency communications research, and includes a primer on all non-permutation pattern specific background. An empirical analysis of the methods herein on radio frequency data is included. No prior knowledge of signals analysis is assumed, and permutation pattern specific notation will be included. This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and includes results on a classification task.

Figures (7)

• Figure 1: Example 6 possible permutation patterns for $t=3$.
• Figure 2: An example time series $\mathbf{x}$ and histogram of permutation patterns $\Pi_{3,2}(\mathbf{x})$. The time series is of length $t=9$ with example subsequence $A^{2}_{3,2}$ with $k=2$, $n=3$, and $\tau=2$ and corresponding permutation pattern $\pi_1$.
• Figure 3: An example to illustrate Binary Phase Shift Key modulation. Figure $(a)$ represents an input binary signal. Figure $(b)$ shows the input signal with the bits of the signal encoded by the phase of a carrier signal. Figure $(c)$ shows the output of a demodulator, which interprets the phases of the modulated signal as positive or negative. Figure $(d)$ reconstructs the original binary signal from the demodulated signal.
• Figure 4: A length $t=14$ time series and length $k=5$ windows with overlap proportion $\alpha = .4$. Since $\alpha k = 2$, consecutive windows overlap by two.
• Figure 5: Confusion matrix for CNN trained on 25 dB raw signals (a) compared to the MSPE (b) and spectrogram results (c).

Theorems & Definitions (2)

• Definition 1: Vector Pattern
• Definition 2: Permutation Entropy