Efficient tracking of the cross-correlation coefficient

Abstract

Abstract-In many (audio) processing algorithms, involving manipulation of discrete-time signals, the performance can vary strongly over the repertoire that is used. This may be the case when the signals from the various channels are allowed to be strongly positively or negatively correlated. We propose and analyze a general formula for tracking the (time-dependent) correlation between two signals. Some special cases of this formula lead to classical results known from the literature, others are new. This formula is recursive in nature, and uses only the instantaneous values of the two signals, in a low-cost and low-complexity manner; in particular, there is no need to take square roots or to carry out divisions. Furthermore, this formula can be modified with respect to the occurrence of the two signals so as to further decrease the complexity, and increase ease of implementation. The latter modification comes at the expense that not the actual correlation is tracked, but, rather, a somewhat deformed version of it. To overcome this problem, we propose, for a number of instances of the tracking formula, a simple warping operation on the deformed correlation. Now we obtain, at least for sinusoidal signals, the correct value of the correlation coefficient. Special attention is paid to the convergence behavior of the algorithm for stationary signals and the dynamic behavior if there is a transition to another stationary state; the latter is considered to be important to study the tracking abilities to nonstationary signals. We illustrate tracking algorithm by using it for stereo music fragments, obtained from a number of digital audio recordings.

Low-complexity tracking and estimation of frequency and amplitude of sinusoids

“…While the analysis is similar to that of an other algorithm [6], the present papers differs in two aspects. First, the former method tracks the correlation coefficient of two signals, while the present method tracks the frequency of a single signal.…”

“…We do this by reformulating the recursion in Eq. (1) so that it assumes the same form as the recursion in [6]. It appears to be convenient to introduce the new variables…”

“…., with γ a small positive parameter and δ k , β k bounded sequences with 0 β k 1. In [6] it was shown how to obtain the limiting behavior ofr(k) as k → ∞ when γ > 0 is small. This was done under an assumption (slightly stronger than required) that the mean values (denoted by…”

“…The final tracking formula differ in both methods. The analysis in [6] was facilitated considerably by switching from (discretetime) difference equations as in Eq. (1) to (continuous-time) differential equations, and the same approach shall be followed here.…”

An Efficient Signature Recognition Based on Histogram Using Statistical Characteristics

“…특징기반 방법은 영상들로부터 추출된 특징에 바탕을 두며, 영역기반 방법은 영상의 전체 영역을 기반으로 한다 [10]. 일반적으로 후자에 바탕을 둔 기 법이 특징검출에 소요되는 비용부담이 적어 널리 이용되며, 주로 거리 계수나 상관성 계수를 척도로 이용한다 [7][8][9][10][11][12][13] NxN의 픽셀크기를 가지는 2차원 참조영상과 입력영상 각각의 번째 픽셀의 명암값 와 가 주어질 때, 축과 축 방향으로 각각 와 번 이동된 번째 픽셀 위치에서 두 영상사이의 NCC는 식 (7)과 같다. …”