Cross Correlation Theorem

3 Key excerpts on "Cross Correlation Theorem"

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  eBook - PDF

  Introduction to Applied Statistical Signal Analysis

  Guide to Biomedical and Electrical Engineering Applications

  • Richard Shiavi(Author)
  • 2010(Publication Date)
  • Academic Press

    (Publisher)

  9 THEORY AND APPLICATION OF CROSS CORRELATION AND COHERENCE 9.1 INTRODUCTION T he concept of cross correlation was introduced in Chapter 6. It was defined in the context of linear systems and described a relationship between the input and output of a system. In general a relationship can exist between any two signals whether or not they are intrinsically related through a system. For instance, we can understand that two signals such as the temperatures in two cities or the noise in electromagnetic transmissions and atmospheric disturbances can be related; however, their relationship is not well defined. An example is demonstrated in Figure 9.1 in which the time sequence of speed measurements made at different locations on a highway are plotted. It is apparent that these signals have similar shapes and that some, such as V 1 and V 2 , are almost identical except for a time shift. How identical these signals are and the time shift which produces the greatest simularity can be ascertained quantitatively with the cross correlation functions. This same information can be ascertained in the frequency domain with cross spectral density and coherence functions. In addition, the latter functions can be used to ascertain synchronism in frequency components between two signals. They will be described in the second half of this chapter. There are several comprehensive references treating cross correlation and coherence. Good treatments with applications are Bendat and Piersol (1980) and Jenkins and Watts (1968). References which focus on other developments and give detailed procedures are Carter (1988), Silvia (1987), and the special issue of the IEEE Transactions on Acoustics, Speech and Signal Processing in 1981. 331 332 CHAPTER 9 THEORY AND APPLICATION OF CROSS CORRELATION AND COHERENCE FIGURE 9.1 Records of average speeds at four different locations on a highway.

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  eBook - ePub

  Neural Communication and Control

  Satellite Symposium of the 28th International Congress of Physiological Science, Debrecen, Hungary, 1980

  • Gy. Székely, E. Lábos, S. Damjanovich(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Let us now return to the topics touched upon in the introduction. 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 ψcy (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 xi (t) preceding it and all these input signals xi (t) have the common property that it gives rise to the initiation of one nerve spike. If we compute a correlation function from the PSTH, we effectively add (or average ) all these signals xi

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  Exploiting Seismic Waveforms

  Correlation, Heterogeneity and Inversion

  • Brian L. N. Kennett, Andreas Fichtner(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  Multi-channel cross-correlations also provide an effective means of estimating relative time delays between stations for arrivals from a single source as, e.g., for teleseismic arrivals across an areal array. An alternative procedure, when waveforms are less consistent, is provided by adaptive stacking. In the two-station method for the analysis of surface-wave dispersion, the fundamental mode portion of the seismograms at two stations lying close to a common great-circle path are correlated to estimate the phase difference in multiple frequency bands for the traverse between the two stations. 97 98 Correlations and Transfer Functions 5.1.1 Cross-Correlation for Time Shifts Consider the cross-correlation of two segments of time series u(t), v(t). If v(t) is derived from u(t) by scaling and a time-shift, v(t) = cu(t - t o ), and the cross-correlation C u,v (t) (3.1.1) will itself be a scaled and time-shifted version of the auto-correlation of u(t) with a peak at time t o . When the resemblance in the time series is strong, but not a direct reproduction, there will still be a maximum in the cross-correlation at the time shift with the closest correspondence between the time segments. The cross-correlation time shift T is defined as the time when the cross-correlation between u(t) and v(t), C u,v (τ) = Z t dt u(t)v(t + τ), (5.1.1) attains its global maximum. Thus T > 0 when v(t) is delayed compared to u(t), and T < 0 when v(t) is advanced. This approach allows time shifts to be directly linked to a reference trace, by suitable choice of v(t). The reference trace can be constructed by, e.g., modal summation for a specified Earth model with the extraction of a suitable seismogram segment, or by a simpler process concentrating on an individual phase. Thus, Bolton & Masters (2001) have constructed reference pulse forms for teleseismic arrivals by convolving a delta function at the appropriate time with the impulse response of the instruments and an attenuation operator.

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