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### Coherence of two signals

Trial software. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. How to calculate frequency coherence between two signals. Vote 0. Edited: dark lion on 6 Dec I have two signals of equal length, and I have plotted the power spectral density plots of the two signals, I need to calculate the coherence between the two signals in terms of frequency.

I need to plot coherence with frequency on x axis. I have added the matlab codei am attaching eeginput and resampledeegoutput files which are loaded into matlab 'for reference'. Answers 0. See Also. Tags coherence between two signals. Start Hunting!Documentation Help Center. This example shows how to use wavelet coherence and the wavelet cross-spectrum to identify time-localized common oscillatory behavior in two time series.

The example also compares the wavelet coherence and cross-spectrum against their Fourier counterparts. Many applications involve identifying and characterizing common patterns in two time series.

In some situations, common behavior in two time series results from one time series driving or influencing the other. In other situations, the common patterns result from some unobserved mechanism influencing both time series. For jointly stationary time series, the standard techniques for characterizing correlated behavior in time or frequency are cross-correlation, the Fourier cross-spectrum, and coherence.

However, many time series are nonstationary, meaning that their frequency content changes over time. For these time series, it is important to have a measure of correlation or coherence in the time-frequency plane.

You can use wavelet coherence to detect common time-localized oscillations in nonstationary signals. In situations where it is natural to view one time series as influencing another, you can use the phase of the wavelet cross-spectrum to identify the relative lag between the two time series. For the first example, use two signals consisting of time-localized oscillations at 10 and 75 Hz.

The signals are six seconds in duration and are sampled at Hz. The Hz oscillation in the two signals overlaps between 1. The overlap for the Hz oscillation occurs between 0. Both signals are corrupted by additive white Gaussian noise. Obtain the wavelet coherence and display the result.

Enter the sampling frequency Hz to obtain a time-frequency plot of the wavelet coherence. In regions of the time-frequency plane where coherence exceeds 0. Phase is indicated by arrows oriented in a particular direction. The two time-localized regions of coherent oscillatory behavior at 10 and 75 Hz are evident in the plot of the wavelet coherence. The phase relationship is shown by the orientation of the arrows in the regions of high coherence.

The white dashed line shows the cone of influence where edge effects become significant at different frequencies scales. Areas of high coherence occurring outside or overlapping the cone of influence should be interpreted with caution. The Fourier magnitude-squared coherence obtained from mscohere Signal Processing Toolbox clearly identifies the coherent oscillations at 10 and 75 Hz. In the phase plot of the Fourier cross-spectrum, the vertical red dashed lines mark 10 and 75 Hz while the horizontal line marks an angle of 90 degrees.

You see that the phase of the cross-spectrum does a reasonable job of capturing the relative phase lag between the components. However, the time-dependent nature of the coherent behavior is completely obscured by these techniques. For nonstationary signals, characterizing coherent behavior in the time-frequency plane is much more informative. The following example repeats the preceding one while changing the phase relationship between the two signals. Plot the wavelet coherence and threshold the phase display to only show areas where the coherence exceeds 0.In signal processingthe coherence is a statistic that can be used to examine the relation between two signals or data sets.

It is commonly used to estimate the power transfer between input and output of a linear system. If the signals are ergodicand the system function is linearit can be used to estimate the causality between the input and output.

The coherence sometimes called magnitude-squared coherence between two signals x t and y t is a real -valued function that is defined as: [1] [2]. The magnitude of the spectral density is denoted as G. Given the restrictions noted above ergodicity, linearity the coherence function estimates the extent to which y t may be predicted from x t by an optimum linear least squares function. For an ideal constant parameter linear system with a single input x t and single output y tthe coherence will be equal to one.

However, in the physical world an ideal linear system is rarely realized, noise is an inherent component of system measurement, and it is likely that a single input, single output linear system is insufficient to capture the complete system dynamics. If C xy is less than one but greater than zero it is an indication that either: noise is entering the measurements, that the assumed function relating x t and y t is not linear, or that y t is producing output due to input x t as well as other inputs.

If the coherence is equal to zero, it is an indication that x t and y t are completely unrelated, given the constraints mentioned above. The coherence of a linear system therefore represents the fractional part of the output signal power that is produced by the input at that frequency.

This leads naturally to definition of the coherent output spectrum:. Consider the two signals shown in the lower portion of figure 2. There appears to be a close relationship between the ocean surface water levels and the groundwater well levels. It is also clear that the barometric pressure has an effect on both the ocean water levels and groundwater levels. As expected, most of the energy is centered on the well-known tidal frequencies.

Likewise, the autospectral density of groundwater well levels are shown in figure 4. It is clear that variation of the groundwater levels have significant power at the ocean tidal frequencies.

To estimate the extent at which the groundwater levels are influenced by the ocean surface levels, we compute the coherence between them. Let us assume that there is a linear relationship between the ocean surface height and the groundwater levels. We further assume that the ocean surface height controls the groundwater levels so that we take the ocean surface height as the input variable, and the groundwater well height as the output variable.

However, one must exercise caution in attributing causality. If the relation transfer function between the input and output is nonlinearthen values of the coherence can be erroneous. For example, it is clear that the atmospheric barometric pressure induces a variation in both the ocean water levels and the groundwater levels, but the barometric pressure is not included in the system model as an input variable. We have also assumed that the ocean water levels drive or control the groundwater levels.

In reality it is a combination of hydrological forcing from the ocean water levels and the tidal potential that are driving both the observed input and output signals.

## Compare Time-Frequency Content in Signals with Wavelet Coherence

Additionally, noise introduced in the measurement process, or by the spectral signal processing can contribute to or corrupt the coherence. If the signals are non-stationaryand therefore not ergodicthe above formulations may not be appropriate. For such signals, the concept of coherence has been extended by using the concept of time-frequency distributions to represent the time-varying spectral variations of non-stationary signals in lieu of traditional spectra.

For more details, see. Coherence has been found a great application to find dynamic functional connectivity in the brain networks. Studies show that the coherence between different brain regions can be changed during different mental or perceptual states.Documentation Help Center.

If x and y are both vectors, they must have the same length. If one of the signals is a matrix and the other is a vector, then the length of the vector must equal the number of rows in the matrix. The function expands the vector and returns a matrix of column-by-column magnitude-squared coherence estimates. If x and y are matrices with the same number of rows but different numbers of columns, then mscohere returns a multiple coherence matrix. The m th column of cxy contains an estimate of the degree of correlation between all the input signals and the m th output signal.

See Magnitude-Squared Coherence for more information. To obtain a multiple coherence matrix, append 'mimo' to the argument list. You must use at least two segments. Otherwise, the magnitude-squared coherence is 1 for all frequencies.

In the MIMO case, the number of segments must be greater than the number of input channels. This syntax can include any combination of input arguments from previous syntaxes. To input a sample rate and still use the default values of the preceding optional arguments, specify these arguments as empty, []. Valid options for freqrange are 'onesided''twosided'and 'centered'. To create the first sequence, bandpass filter the signal.

Design a 16th-order filter that passes normalized frequencies between 0. Specify a stopband attenuation of 60 dB. Filter the original signal. To create the second sequence, design a 16th-order filter that stops normalized frequencies between 0.

Specify a passband ripple of 0.This value determines the frequencies at which the coherence is estimated. Any arguments that you omit from the end of the parameter list use the default values shown below. If x or y is complex, cohere estimates the coherence function at both positive and negative frequencies, and Cxy has length nfft. Specify nfft as a power of 2 for fastest execution. If you supply a scalar for windowcohere uses a Hanning window of that length. The length of the window must be less than or equal to nfft ; cohere zero pads the sections if the window length exceeds nfft.

You can use the empty matrix [] to specify the default value for any input argument except x or y. For example:. The ' dflag ' parameter must appear last in the list of input arguments. Examples Compute and plot the coherence estimate between two colored noise sequences x and y :. Diagnostics An appropriate diagnostic message is displayed when incorrect arguments are used:.

Algorithm cohere estimates the magnitude squared coherence function [2] using Welch's method of power spectrum estimation see references [3] and [4]as follows:. See Also csdpwelchtfe. References [1] Stoica, P. Introduction to Spectral Analysis.

Modern Spectral Estimation. Theory and Application of Digital Signal Processing. Audio Electroacoust. AU June Documentation Help Center. Wavelet coherence is useful for analyzing nonstationary signals. The inputs x and y must be equal length, 1-D, real-valued signals. The coherence is computed using the analytic Morlet wavelet. You can use the phase of the wavelet cross-spectrum values to identify the relative lag between the input signals.

The duration ts is used to compute the scale-to-period conversion, period. The duration array period has the same format as specified in ts. The sampling frequency fs is in Hz. If you specify the sampling frequency, fsthe cone of influence is in Hz. This syntax may be used in any of the previous syntaxes.

Due to the inverse relationship between frequency and period, a plot that uses the sampling interval is the inverse of a plot the uses the sampling frequency.

For areas where the coherence exceeds 0. The arrows are spaced in time and scale. The direction of the arrows corresponds to the phase lag on the unit circle. The corresponding lag in time depends on the duration of the cycle. Use default wcoherence settings to obtain the wavelet coherence between a sine wave with random noise and a frequency-modulated signal with decreasing frequency over time. The default coherence computation uses the analytic Morlet wavelet, 12 voices per octave and smooths 12 scales.

The default number of octaves is equal to floor log2 numel x -1which in this case is 9. Obtain the wavelet coherence data for two signals, specifying a sampling interval of 0. Both signals consist of two sine waves 10 Hz and 50 Hz in white noise. The sine waves have different time supports.

Set the random number generator to its default settings for reproducibility. Then create the two signals. Use wcoherence x,y,seconds 0. This plot includes the phase arrows and the cone of influence. Obtain the wavelet coherence for two signals, specifying a sampling frequency of Hz. Set the random number generator to its default settings for reproducibility and create the two signals.

Obtain the wavelet coherence. The coherence plot is flipped with respect to the plot in the previous example, which specifies a sampling interval instead of a sampling frequency. Obtain the wavelet coherence for two signals.

Use the default number of scales to smooth. This value is equivalent to the number of voices per octave. Both values default to Then, create the two signals and obtain the coherence. Set the number of scales to smooth to The increased smoothing causes reduced low frequency resolution.Documentation Help Center. This example shows how to measure signal similarities. It will help you answer questions such as: How do I compare signals with different lengths or different sampling rates?

How do I find if there is a signal or just noise in a measurement? Are two signals related? How to measure a delay between two signals and how do I align them?

How do I compare the frequency content of two signals? Similarities can also be found in different sections of a signal to determine if a signal is periodic.

Consider a database of audio signals and a pattern matching application where you need to identify a song as it is playing. Data is commonly stored at a low sampling rate to occupy less memory. The first and the second subplot show the template signals from the database.

The third subplot shows the signal which we want to search for in our database. Just by looking at the time series, the signal does not seem to match to any of the two templates. A closer inspection reveals that the signals actually have different lengths and sampling rates. Different lengths prevent you from calculating the difference between two signals but this can easily be remedied by extracting the common part of signals.

Furthermore, it is not always necessary to equalize lengths. Cross-correlation can be performed between signals with different lengths, but it is essential to ensure that they have identical sampling rates. The safest way to do this is to resample the signal with a lower sampling rate.

The resample function applies an anti-aliasing low-pass FIR filter to the signal during the resampling process. We can now cross-correlate signal S to templates T1 and T2 with the xcorr function to determine if there is a match.