Normalized Cross-correlation Function Research Articles

Kreuzkorrelationsfunktionen akustisch evozierter Hirnstammpotentiale

A method for the analysis of brainstem acoustic evoked potentials (BAEP) is presented. The calculation of the normalized cross correlation function between BAEP recordings of the right and left side yields a quantitative and objective valuation of the symmetry of the electrical brainstem activity. Contrary to the usual interpretation of latencies and amplitudes the BAEP waveform is also considered. To demonstrate the regular properties of the normalized cross correlation function 20 healthy subjects were tested. Different stimulus and recording conditions, the influence of the BAEP signal length on which the calculation of the cross correlation function is based as well as filtering of the BAEP with variing cutoff frequencies are studied.

Cross-correlation technique for arrythmia detection using PR and PP intervals

A combined, cross-correlation procedure for computerized arrythmia detection is presented. It enables the classification of a wide range of arrythmia and is based upon the determination of both P and QRS waves. The electrocardiogram (ECG) signal is recorded from external leads, digitized (10 bits of resolution) at 1.28 kHz and processed with a CDC 6600 computer. A normal beat is stored as a reference signal and the onsets of the P wave and the QRS complex are measured for determining the PR, PP, and RR intervals. The combined cross-correlation procedure is carried out between the template and each successive ECG waveform over two time intervals, the first of which includes the P wave and the second, the QRS complex. Prior to this correlation procedure each of the ECG waveforms is filtered through a nonrecursive digital bandpass filter (10 and 100 Hz for low and high cutoff frequencies, respectively). The cross-correlation functions are then calculated by means of a cross spectrum and Fast Fourier Transform algorithm. The maximum value of the normalized cross-correlation function and the time shift providing that value are searched for, allowing (1) determination of the similarity between the present beat and the normal reference beat, allowing for the discrimination of abnormal P and QRS shapes; (2) accurate measurement of the RR, PP, and PR intervals of this subsequent waveform. Applying this method enables a reliable detection of a wide variety of arrythmia.

Three-dimensional image analysis of the complex of thin filaments and myosin molecules from skeletal muscle. V. Assignment of actin in the actin-tropomyosin-myosin subfragment-1 complex.

To assign the actin molecule in the three-dimensional image of the actin-tropomyosin-myosin subfragment-1 (actin-TM-S1) complex, the three-dimensional image of the actin-tropomyosin complex was correlated to that of actin-TM-S1. To assess the similarity of two structures in a quantitative manner, we used a normalized cross-correlation function ("similarity function"). The calculation of similarity indicated that domain A and domain B defined in (1, 2) correspond to actin-tropomyosin. This assignment indicates that one S1 molecule strongly interacts with only one actin molecule, but at least two regions of S1 contribute to the binding. Comparison of the reconstituted models of thin filaments with those of decorated thin filaments suggested a change in the shape of the actin molecule.

Investigation of the aerodynamic noise generating region of a jet engine by means of the simple source fluid dilatation model

An experiment was conducted on a full-scale jet engine to investigate the aerodynamic noise generating regions in the free jet. Cross-correlation measurements were made between the static pressure fluctuations and the farfield radiated sound. These measurements were made for two different static pressure probe positions and a large number of farfield positions (at various angles). In addition, each test geometry was run for four different jet exit velocities. The measured, normalized cross-correlation functions varied between 0.004 and 0.155. A new Q-function, based on the above normalized cross correlation is defined and plotted. This function represents the source strength per unit volume within the jet region. This Q-function shows dependence on the probe position, the angular position of the farfield microphone, and the jet exit Mach number. Third-octave analyses of both the probe signal and the farfield radiated sound were made. The results show that cross-correlation techniques are a valuable tool in the investigation of the aerodynamic noise generating regions of an actual jet engine.

Investigation of the Aerodynamic Noise Generating Region of a General Electric T-58 Jet Engine by Means of the Simple Source/Fluid Dilatation Model

Experimental tests were carried out at NASA Ames Research Center. Crosscorrelation measurements were made between the static pressure fluctuations, as measured with a B & K 14-in. microphone fitted with a nose cone, and the far-field radiated sound. These measurements were made for two different probe positions and a large number of far-field positions at various angles. In addition, each test geometry was run for jet exit velocities at four different Mach numbers. The measured, normalized cross-correlation functions varied between 0.155 and 0.004. These values depended upon the angular position of the far-field microphone, the jet exit velocity Mach number, and the position of the probe. In addition, the cross-correlation technique was employed to study the symmetry of the farfield radiated sound. Third-octave analysis of both the probe signal and the far-field radiated sound were made. The results of these measurements will be reported.

Interpretation of infinitely clipped speech properties

The paper is based upon the statistical theory of signals. It is shown that the variation of the clipped Gaussian noise spectrum is 5 percent, while the normalized cross-correlation function is 0.80; the results are extended to more general cases. A property of the conditional mean value of speech signals is derived. A typical value of the normalized intelligibility of clipped speech in the presence of noise is 0.70 for S/N = 3 dB; assuming the ear is a clipper, theoretical results are well in agreement. The existence of an intelligibility threshold is shown. The need for information on the envelope (or dynamics) of the speech wave to meet the requirement of naturalness is underlined.

Some Theorems on the Unimodular Complex Degree of Optical Coherence

The complex degree of coherence γ(r1, r2, t1 − t2) of a stationary optical field is defined as the normalized cross-correlation function (|γ| ≤ 1) of the light disturbances at two space-time points (r1, t1), (r2, t2), the disturbances being represented by means of Gabor's analytic signals. In the present paper the general form of γ is examined under the condition that |γ| takes on the extreme value unity for all possible time differences τ = t1 − t2 (− ∞ < τ < ∞). Several cases are distinguished, depending on whether this condition is satisfied for some points or for all points in some fixed domain of space. It is pointed out, that the previously published derivations of the relevant theorems contain serious errors. The methods employed in the present paper make use of the property of nonnegative definiteness which the complex degree of coherence is shown to obey. The results have a bearing on the important but as yet unsolved ``phase problem'' of optical coherence theory.

A theorem on cross correlation between noisy channels

Two channels carry noise waveforms, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_0 (t) + N_1 ( t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_0 (t) + N_2 (t)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_0 (t)</tex> is a common narrow-band Gaussian noise and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_1 (t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_2 (t)</tex> are independent narrow-band Gaussian noises associated with each channel. The outputs of each channel are sent through detectors whose outputs, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(x, y)</tex> , are identical homogeneous functions of the components, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> , of their inputs <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(t) = x(t) \cos \omega_0 t + y(t) \sin \omega_0 t</tex> . Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{12}(\tau)</tex> be the normalized cross-correlation function of the two detector outputs. It is shown that to determine <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{12}(\tau)</tex> it suffices to know the normalized auto-correlation function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_0(\tau)</tex> of the output of a single such detector when the input is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_0(t)</tex> ; i.e., if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_0(\tau) = G(\sigma_0^2, p(\tau))</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(\tau)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma_0</tex> are the normalized auto-correlation function and rms of either component of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_0</tex> , then it is shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{12}(\tau) = G(\sigma_0^2, Z_{\rho}(\tau))</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z = [\{1 + (\sigma_1^2/\sigma_0^2)\} \{(1 + (\sigma_2^2/\sigma_0^2)\}]^{-\frac{1}{2}}</tex> .