Cross Correlation Theorem
The Cross Correlation Theorem states that the cross-correlation of two signals in the time domain is equivalent to the multiplication of their Fourier transforms in the frequency domain. This theorem is widely used in signal processing, communication systems, and image processing to measure the similarity between two signals or to detect the presence of a signal within another signal.
4 Key excerpts on "Cross Correlation Theorem"
eBook - ePub- Frank J. Fahy(Author)
- 2000(Publication Date)
- Academic Press(Publisher)
...Appendix 4 Coherence and Cross-Correlation A4.1 Background The terms ‘coherence’ and ‘cross-correlation’ as descriptors of the time-average relation between two time-dependent quantities, or signals, are used rather indiscriminately in much of the literature dealing with acoustic and vibrational fields. This causes confusion and uncertainty in the minds of readers, which has been known to cause incorrect expressions to appear in the literature [ A4.1 ]. Although rigorous mathematical definition of these terms, and the distinction between them, depends upon knowledge of signal analysis beyond the elementary material presented in Appendix 2, an attempt is made in the following section to explain the distinction in qualitative terms. The term ‘signal’ may be taken to represent any physical quantity of concern. A4.2 Correlation The ‘cross-correlation function’ (also known as the ‘cross co-variance function’) quantifies the time-average relation between two time-dependent, time-stationary signals in the time domain: each must be reduced to zero mean prior to the operation. It is formed by time shifting one of the signals by τ with respect to the other, and estimating the time-average product of the two. This average is normalized by square root of the product of the mean square values of the two signals, to produce the cross-correlation coefficient, which can take values between plus and minus unity. This coefficient is evaluated as a function of the time shift τ. As a simple example, consider the propagation of a very broadband, random, plane wave past two microphones separated by distance d. Provided that the signal from the microphone that is reached first is positively shifted with respect to the other, the cross-correlation coefficient will peak at a value of unity at a time delay τ = d/c, and take values close to zero at all other time delays...
eBook - ePub- Yong Sang(Author)
- 2020(Publication Date)
- De Gruyter(Publisher)
...It can be used for pattern recognition, single particle analysis, averaging, electron tomography, cryptanalysis and neurophysiology. The cross-correlation function is a method to measure self-similarity between two waveforms x (t) and y (t). The cross-correlation functions in the case of infinite duration waveforms can be defined as (5.25) R x y (τ) = l i m T → ∞ 1 T ∫ − T / 2 T / 2 x (t) y (t + τ) d t The cross-correlation functions in the case of finite duration waveforms can be defined. as (5.26) R x y (τ) = ∫ − ∞ ∞ x (t) y (t + τ) d t Question 5.4: Find the cross-correlation function of the x (t) and y (t). x (t) = x 0 sin (ω 0 t + θ), y (t) = y 0 sin (ω 0 t + θ − ϕ) and ϕ is a random variable. Solution: The cross-correlation function is defined. by (5.27) R x y (τ) = l i m T → ∞ 1 T ∫ 0 T x (t) y (t + τ) d t = 1 T 0 ∫ 0 T 0 x 0 sin (ω 0 t + θ) y 0 sin [ ω 0 (t + τ) + θ − ϕ ] d t = 1 2 x 0 y 0 cos (ω 0 τ − ϕ)[. --=PLGO-SEPARATOR=--]where T 0 = 2 π / ω 0. It can be seen that R x y (τ) is periodic with period 2 π / ω 0 and is independent of the phase θ. 5.2.3.1 Properties of cross-correlation function (5.28) ρ x y (τ) = R x y (τ) − μ x μ y σ x σ y R x y (τ) = R y x (− τ), R x y (τ) is not necessarily an even. function. If τ → ∞, ρ x y (∞) → 0 (5.29) ρ x y (∞) = R x y (∞) − μ x μ y σ x σ y → 0 (5.30) ⇒ R x y (∞) → μ x μ y A maximum value of R x y (τ) occurs at τ = τ 0, where τ 0 is the time lag. between x (t) and y (t). If R x y (τ) = 0 for all τ, then x (t) and y (t) are said to be uncorrelated. R x y (τ) ranges from μ x μ y − σ x σ y to μ x μ y + σ x σ y. Two periodic signals with the same frequency are correlated, and two periodic signals with the different frequencies are not correlated. 5.2.3.2 Determining time delays Cross-correlation is useful for determining the time delay between two signals...
eBook - ePubNeural Communication and Control
Satellite Symposium of the 28th International Congress of Physiological Science, Debrecen, Hungary, 1980
- Gy. Székely, E. Lábos, S. Damjanovich, Gy. Székely, E. Lábos, S. Damjanovich(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
...The technique of using correlation functions can be viewed from at least two different angles. For once, it can be seen as a method to obtain quantitative parameters for a model – in our case the impulse responses of linear filters. This is a straightforward application and it does not need much discussion. Alternatively, however, a cross-correlation function can be regarded as a means for detecting common properties or a common origin of two signals. As explained above, the correlation function effectively measures the degree of similarity of two signals but it does so only for those frequency bands that are common to the signals. This consideration reveals more fundamental aspects. The ‘degree of similarity’ is a mutual property, it does not matter which signal is considered as primary. Common procedure is to consider the input signal x(t) as the primary signal, in fact, as the signal causing the second signal y(t) to appear. It is then logical to compare x(t) with y(t) taken at a later time and this illustrates why the most important lobes of the cross-correlation function ψ cy (τ) occur at negative values of τ. There is a second possibility, we can consider y(t) as the primary signal for observation and investigate the properties of the x(t) signal preceding it. In a way this is a fairly abstract procedure but in our case fully justified. We have been using an accumulated PSTH as the output signal of our system, this signal is built up from individual firings of the nerve fibre under test and it is well possible to consider each one of these firings as an independent, primary observation. To each of these observations there belongs one input signal x i (t) preceding it and all these input signals x i (t) have the common property that it gives rise to the initiation of one nerve spike...
eBook - ePubSignal Processing for Neuroscientists
An Introduction to the Analysis of Physiological Signals
- Wim van Drongelen(Author)
- 2006(Publication Date)
- Academic Press(Publisher)
...We have seen this principle applied in Chapters 2 and 7 when evaluating the effects of sampling and truncation of continuous functions (Sections 2.3 and 7.1.1, and Figs. 2.6 and 7.5). 8.4 AUTOCORRELATION AND CROSS-CORRELATION 8.4.1 Time Domain 8.4.1.1 Continuous Time Correlation between two time series or between a single time series and itself is used to find dependency between samples and neighboring samples. One could correlate, for instance, a time series with itself by plotting x n versus x n ; it will be no surprise that this would result in a normalized correlation equal to 1. Formally the autocorrelation R xx of a process x is defined as (8.13) Here the times t 1 and t 2 are arbitrary moments in time, and the autocorrelation demonstrates how a process is correlated with itself at these two different times. If the process is stationary, the underlying distribution is invariant over time and the autocorrelation therefore only depends on the offset τ = t 2 – t 1 : (8.14) Further, if we have an ergodic process, we may use a time average to define an autocorrelation function over the domain τ indicating a range of temporal offsets: (8.15) In some cases where the process at hand is not ergodic or if ergodicity is in doubt, one may use the term time autocorrelation functions for the expression in (8.15)...
