Crosswavelet and Wavelet Coherence

This Matlab package for performing crosswavelet and wavelet coherence analysis is free for non-profit use. An example of what you can do with the wavelet package is given here…

Wavelet coherence between the winter Arctic Oscillation and sea ice extent in the Baltic

Wavelet coherence between the winter Arctic Oscillation index and the sea ice extent in the Baltic.

Click to show/hide each section; right-click on images to view the full size version. Please note that the usual NERC disclaimer applies to any information given.

  1. Theory


Our paper describes the theory behind cross wavelet and wavelet coherence, starting from the basics of the Continuous Wavelet Transform: "Application of the cross wavelet transform and wavelet coherence to geophysical time series."

The Frequently asked questions section below contains some very good information, and we highly recommend the paper by Torrence and Compo, "A practical guide to wavelet analysis", if you're completely new to wavelet analysis. (Also see Links for other resources.)



Grinsted, A., Moore, J.C., Jevrejeva, S. (2004) Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Processes Geophys., 11, 561–566, doi:10.5194/npg-11-561-2004 [pdf]

Jevrejeva, S., Moore, J.C.,  Grinsted, A. (2003) Influence of the Arctic Oscillation and El Niño-Southern Oscillation (ENSO) on ice conditions in the Baltic Sea: The wavelet approach, J. Geophys. Res., 108(D21), 4677, doi:10.1029/2003JD003417 [pdf]

Torrence, C., Compo, G.P. (1998) A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc., 79, 61–78

Torrence, C., Webster, P. (1999)  Interdecadal changes in the ESNO-Monsoon System, J.Clim., 12, 2679–2690

  2. Example of how to use the package


This example illustrates how simple it is to do continuous wavelet transform (CWT), Cross wavelet transform (XWT) and Wavelet Coherence (WTC) plots of your own data.

The time series we will be analysing are the winter Arctic Oscillation index (AO) and the maximum sea ice extent in the Baltic (BMI).

Load the data

First we load the two time series into the matrices d1 and d2.

seriesname={'AO' 'BMI'};
Change the probability density function (pdf)

The time series of Baltic Sea ice extent is highly bi-modal and we therefore transform the timeseries into a series of percentiles. The transformed series probably reacts 'more linearly' to climate.

Example 1
Continuous wavelet transform (CWT)

The CWT expands the time series into time frequency space.

tlim=[min(d1(1,1),d2(1,1)) max(d1(end,1),d2(end,1))];


Example 2
Cross wavelet transform (XWT)

The XWT finds regions in time frequency space where the time series show high common power.

title(['XWT: ' seriesname{1} '-' seriesname{2} ] )


Example 3
Wavelet coherence (WTC)

The WTC finds regions in time frequency space where the two time series co-vary (but does not necessarily have high power).

title(['WTC: ' seriesname{1} '-' seriesname{2} ] )


  3. Download the wavelet coherence package


The MatLab wavelet coherence package and recent changes can be downloaded here [from] (designed for MatLab 6 and above).

Please include an acknowledgement to Aslak Grinsted if these programs are used in any publication.

Note that most of the routines included in this package are under the following licence: This software may be used, copied, or redistributed as long as it is not sold and this copyright notice is reproduced on each copy made. This routine is provided as is without any express or implied warranties whatsoever.

However, not all the routines are published under these terms, and before redistributing them in any form we advise you to ask permission from the authors.


We would like to thank the following people for letting us include their programs in our package.

  4. Tips


Cross wavelet analysis and wavelet coherence are powerful methods for testing proposed linkages between two time series.

  • Check the histograms of the time series to ensure that they are not too far from normally distributed. Consider transforming the time series, if the pdfs of the time series are far from Gaussian. When choosing a transformation, it is preferable to choose an analytic transformation such as taking the logarithm if the data is log-normally distributed. In other cases a simple 'percentile' transformation might be useful. An advantage of using that particular transformation is that it does not have any outliers.
  • Consider what the expectations are for the outcome of the analysis given the proposed linking mechanism. We caution against blindly applying these methods to randomly chosen data sets. Like other statistical tests some data set sets will display highly statistically significant links simply by chance.
  • When a wavelet has been chosen the CWTs of both time series are calculated. We suggest a scale resolution of 10 scales per octave and use of the Morlet wavelet unless there are good grounds to do otherwise. For geophysical time series an AR1 red noise assumption is often suitable and can be used to calculate the significance level of the wavelet power. Remember to take special care not to misinterpret results inside the COI.
  • From the two CWTs the XWT is calculated. The XWT exposes regions with high common power and further reveals information about the phase relationship. If the two series are physically related we would expect a consistent or slowly varying phase lag that can be tested against mechanistic models of the physical process. The circular mean of the phase angles can be used to quantify the phase relationship.
  • Also, from two CWTs the WTC can be calculated which can be thought of as the local correlation between the time series in time frequency space. Where XWT unveils high common power, WTC finds locally phase locked behavior. The more desirable features of the WTC come at the price of being slightly less localized in time frequency space. The significance level of the WTC has to be determined using Monte Carlo methods.

  5. Frequently asked questions

a. How do I know whether AR1 noise is an appropriate null hypothesis to test against?



It is usually an appropriate null hypothesis if the theoretical AR1 spectrum is ‘a good model’ for the power decay in the observed spectrum. I recommend simply comparing the two power spectra visually:

legend('observed',sprintf('Theoretical AR1=%.2f',aa),'location','best')


b. When is the probability distribution of the data important?


The null-hypothesis in the significance tests for WT, XWT and WTC is normally distributed AR1 noise. The AR1 coefficient and process variance is chosen so that it best fits the observed data. It is therefore quite important that the data is close to normal and is reasonably well modeled by a Gaussian AR1 process. Otherwise we can trivially reject the null-hypothesis and the significance level calculated by the program is not appropriate. However, the Central Limit Theorem tells us that the distribution tends towards normality as we convolute with longer and longer wavelets (in the absence of long-range persistence). This means that the data distribution is only really important on the shortest scales. So, if we are primarily looking at longer scales we do not need to worry so much about the distribution. However, for the WT and XWT the color of the noise is very important and a very non-normal distribution will affect the performance of the ar1 estimators (ar1.m & ar1nv.m). The WTC is relatively insensitive to the colour of the noise in the significance test (see next question).

c. How important is the AR1 coefficient for WTC significance levels?



The definition of Wavelet coherence (WTC) effectively normalizes by the local power in time frequency space. Therefore WTC is very insensitive to the noise colour used in the null-hypothesis (see Grinsted et al. 2004). It can easily be demonstrated by an example:

figure('color',[1 1 1])
set(gcf,'pos',get(gcf,'pos').*[1 .2 1 2]) %make high figure
wtc(X2,Y2) %make input data more red, by moving averages of the data.
wtc(X2,Y2,'ar1',[0 0]) %Test the red series against white noise.


orig_arcoefs =
     0.038365     0.058916
smoothed_arcoefs =
      0.84525      0.79124

The three figures are very similar.


d. What does a peak in XWT tell us?


You have to be very careful interpreting XWT peaks. If you take the WTC of a signal with pure white noise then the XWT will look very similar to the WT of the signal. The same problem exists in ‘normal' power spectral analysis. If you calculate the cross Power spectral density of a periodic signal with a white noise signal then you will get a peak. It does not mean that the series have any kind of connection just because there is a peak. I recommend examining the WTC and the phase arrows. If there is a connection then you would expect the phenomena to be phase-locked –  that is, the phase-arrows point only in one direction for a given wavelength. So, if they vary between in-phase and anti-phase then it is a clue that they probably not are linked.

e. How should the phase arrows be interpreted?



The phase arrows show the relative phasing of two time series in question. This can also be interpreted as a lead/lag. How it should be interpreted is best illustrated by example:

figure('color',[1 1 1])
Y=sin(t-1); %X leads Y.
xwt([t X],[t Y]); % phase arrows points south east


Phase arrows pointing

  • right: in-phase
  • left: anti-phase
  • down: X leading Y by 90°
  • up: Y leading X by 90°

Note: interpreting the phase as a lead(/lag) should always be done with care. A lead of 90° can also be interpreted as a lag of 270° or a lag of 90° relative to the anti-phase (opposite sign).


f. How do I convert a phase-angle to a time lag?


This can not always be done and when it can, it should be done with care. A 90° lead might as well be a 90° lag to the anti-phase. There is therefore a non-uniqueness problem when doing the conversion. A phase angle can also only be converted to a time lag for a specific wavelength. This equation works best for determining the time lag when the series are near in-phase.



timelag =


A visual inspection of the time series at the wavelength in question should make it clear if the time lag is right. I also recommend calculating the time lag with other methods for support.

g. How do I calculate the average phase angle?


You can use anglemean.m provided with the package. Here is a small example that calculates the mean angle at the period closest to 11:

[Wxy,period,scale,coi,sig95]=xwt([t X],[t Y]);
[mn,rowix]=min(abs(period-11)); %row with period closest to 11.


ChosenPeriod =
meantheta =
anglestrength =
sigma =


If you want to restrict the mean to be calculated over significant regions outside the COI then you can do like this:



meantheta =
anglestrength =
sigma =


h. How do I determine if a point is inside the COI or not?


Here is an example that does just that:

[Wx,period,scale,coi,sig95]=wt([t X]);
p=[100 64; 100 10; 50 64]; %are these points in the COI?


ispointincoi =


i. How do I avoid the slow Monte Carlo significance test in WTC?



You can do that by simply specifying the MonteCarloCount to be zero. Example:

figure('color',[1 1 1])
wtc([t X],[t Y],'mcc',0); %MCC:MonteCarloCount

Note that the significance contour can not be trusted with out running the Monte Carlo test.


j. How do I change the Y-axis to frequency instead of period?



Here is a short example that does just that. The sampling frequency is 100 MHz, and the signal is 5Mhz.

figure('color',[1 1 1])
wtc([t X],[t Y])
freq=[128 64 32 16 8 4 2 1]*1e6;
ylabel('Frequency (MHz)')


k. Why is something missing from my figures on screen or when I try to save them?



This is usually caused by an incompatibility bug between Matlab and your graphics driver? There is unfortunately not any single method to resolve this issue, since it depends on your system. However, the problems can in some cases be resolved by changing the renderer property on the figure. In some cases it is caused by the shaded rendering of the COI. Here are some options you may try



Further reading on how to resolve this issue: – no longer available


  6. Published applications making use of this Toolbox


The wavelet coherence toolbox has been used in a wide variety of fields. Here is an incomplete list of papers:

  1. Yang L., Wong C.M., Lau E.H.Y., Chan K.P., Ou C.Q., Peiris J.S.M. (2008) Synchrony of clinical and laboratory surveillance for influenza in Hong Kong, PLoS ONE 29 2807-994
  2. Nezlin N.P. (2007) Seasonal and interannual variability of remotely sensed chlorophyll, Handbook of Environmental Chemistry, Volume 5: Water Pollution 98 991-2549
  3. Valdes-Galicia J.F., Velasco V.M. (2007) Variations of mid-term periodicities in solar activity physical phenomena, Advances in Space Research (5) 2537-2116
  4. Ramp S.R., Bahr F.L. (2007) Seasonal evolution of the upwelling process south of Cape Blanco, Journal of Physical Oceanography 98 (11) 2110-613
  5. Donner R., Thiel M. (2007) Scale-resolved phase coherence analysis of hemispheric sunspot activity: A new look at the north-south asymmetry, Astronomy and Astrophysics 35 (5) 609-636
  6. Casty C., Raible C.C., Stocker T.F., Wanner H., Luterbacher J. (2007) A European pattern climatology 1766-2000, Climate Dynamics 98 (07/08/2008) 625-5866
  7. Zhou W., Wang X., Zhou T.J., Li C., Chan J.C.L. (2007) Interdecadal variability of the relationship between the East Asian winter monsoon and ENSO, Meteorology and Atmospheric Physics 19 (9) 5847-111
  8. Ostlund N., Suhr O.B., Wiklund U. (2007) Wavelet coherence detects non-autonomic heart rate fluctuations in familial amyloidotic polyneuropathy, 29th Annual International Conference of IEEE-EMBS, Engineering in Medicine and Biology Society, EMBC'07 3 (15) 79-803
  9. Mor Y., Lev-Tov A. (2007) Analysis of rhythmic patterns produced by spinal neural networks, Journal of Neurophysiology 4 (12) 793-
  10. Kemp D.B., Coe A.L. (2007) A nonmarine record of eccentricity forcing through the Upper Triassic of southwest England and its correlation with the Newark Basin astronomically calibrated geomagnetic polarity time scale from the North America, Geology 51 (1) -596
  11. Castellanos N.P., Malmierca E., Nunez A., Makarov V.A. (2007) Corticofugal modulation of the tactile response coherence of projecting neurons in the gracilis nucleus, Journal of Neurophysiology 23 (8) 588-168
  12. Moore T.S., Nuzzio D.B., Deering T.W., Taillefert M., Luther III G.W. (2007) Use of voltammetry to monitor O2 using Au/Hg electrodes and to control physical sensors on an unattended observatory in the Delaware Bay, Electroanalysis 28 (7) 151-203
  13. Xiao C.J., Wang X.G., Pu Z.Y., Ma Z.W., Zhao H., Zhou G.P., Wang J.X., Kivelson M.G., Fu S.Y., Liu Z.X., Zong Q.G., Dunlop M.W., Glassmeier K.-H., Lucek E., Reme H., Dandouras I., Escoubet C.P. (2007) Satellite observations of separator-line geometry of three-dimensional magnetic reconnection, Nature Physics 112 (7) 193-
  14. Cazelles B., Chavez M., De Magny G.C., Guegan J.-F., Hales S. (2007) Time-dependent spectral analysis of epidemiological time-series with wavelets, Journal of the Royal Society Interface 51 -221
  15. Sato J.R., Morettin P.A., Arantes P.R., Amaro Jr. E. (2007) Wavelet based time-varying vector autoregressive modelling, Computational Statistics and Data Analysis 83 (2) 213-
  16. Sweeney-Reed C.M., Nasuto S.J. (2007) A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition, Journal of Computational Neuroscience 243 (12) -14
  17. Sammer G., Blecker C., Gebhardt H., Bischoff M., Stark R., Morgen K., Vaitl D. (2007) Relationship between regional hemodynamic activity and simultaneously recorded EEG-theta associated with mental arithmetic-induced workload, Human Brain Mapping 34 1-862
  18. Ruessink B.G., Coco G., Ranasinghe R., Turner I.L. (2007) Coupled and noncoupled behavior of three-dimensional morphological patterns in a double sandbar system, Journal of Geophysical Research C: Oceans 57 (9) 854-1611
  19. Reznikova V.E., Melnikov V.F., Su Y., Huang G. (2007) Pulsations of microwave flaring emission at low and high frequencies, Astronomy Reports 112 1605-754
  20. Mendoza B., Garcia-Acosta V., Velasco V., Jauregui E., Diaz-Sandoval R. (2007) Frequency and duration of historical droughts from the 16th to the 19th centuries in the Mexican Maya lands, Yucatan Peninsula, Climatic Change 338 (5) 737-
  21. Zolotova N.V., Ponyavin D.I. (2007) Synchronization in sunspot indices in the two hemispheres, Solar Physics 54
  22. Zong Q.-G., Zhou X.-Z., Li X., Song P., Fu S.Y., Baker D.N., Pu Z.Y., Fritz T.A., Daly P., Balogh A., Reme H. (2007) Ultralow frequency modulation of energetic particles in the dayside magnetosphere, Geophysical Research Letters 55 (3) -88
  23. Zhang Q., Chen J., Becker S. (2007) Flood/drought change of last millennium in the Yangtze Delta and its possible connections with Tibetan climatic changes, Global and Planetary Change 25 (5) 79-274
  24. Yates T.T., Si B.C., Farrell R.E., Pennock D.J. (2007) Time, location, and scale dependence of soil nitrous oxide emissions, soil water, and temperature using wavelets, cross-wavelets, and wavelet coherency analysis, Journal of Geophysical Research D: Atmospheres 34 (4) 265-272
  25. Kang S., Lin H. (2007) Wavelet analysis of hydrological and water quality signals in an agricultural watershed, Journal of Hydrology 34 (1) 257-173
  26. Brittain J.-S., Halliday D.M., Conway B.A., Nielsen J.B. (2007) Single-trial multiwavelet coherence in application to neurophysiological time series, IEEE Transactions on Biomedical Engineering 14 161-167
  27. Plett M.I. (2007) Transient detection with cross wavelet transforms and wavelet coherence, IEEE Transactions on Signal Processing 333 (4) 157-42
  28. Heilig B., Luhr H., Rother M. (2007) Comprehensive study of ULF upstream waves observed in the topside ionosphere by CHAMP and on the ground, Annales Geophysicae 55 (2) 33-746
  29. Pisaric M.F.J., Carey S.K., Kokelj S.V., Youngbut D. (2007) Anomalous 20th century tree growth, Mackenzie Delta, Northwest Territories, Canada, Geophysical Research Letters 28 (2) 737-1545
  30. Xu Y., Watts D.R., Wimbush M., Park J.-H. (2007) Fundamental-mode basin oscillations in the Japan/East Sea, Geophysical Research Letters 27 (1) 1527-152
  31. Divine D.V., Godtliebsen F. (2007) Bayesian modeling and significant features exploration in wavelet power spectra, Nonlinear Processes in Geophysics 17 145-
  32. Zhang Q., Xu C.-y., Jiang T., Wu Y. (2007) Possible influence of ENSO on annual maximum streamflow of the Yangtze River, China, Journal of Hydrology 373 (7) -809
  33. Rong Z., Liu Y., Zong H., Cheng Y. (2007) Interannual sea level variability in the South China Sea and its response to ENSO, Global and Planetary Change 28 (1) 805-
  34. Rowley A.B., Payne S.J., Tachtsidis I., Ebden M.J., Whiteley J.P., Gavaghan D.J., Tarassenko L., Smith M., Elwell C.E., Delpy D.T. (2007) Synchronization between arterial blood pressure and cerebral oxyhaemoglobin concentration investigated by wavelet cross-correlation, Physiological Measurement 50 (4) -516
  35. Zhou W., Chan J.C.L. (2007) ENSO and the South China Sea summer monsoon onset, International Journal of Climatology 111 (5) 505-208
  36. Zhang Z., Zhang Q., Jiang T. (2007) Changing features of extreme precipitation in the Yangtze River basin during 1961-2002, Journal of Geographical Sciences 409 (17) 192-4268
  37. Karimova L., Kuandykov Y., Makarenko N., Novak M.M., Helama S. (2006) Fractal and topological dynamics for the analysis of paleoclimatic records, Physica A: Statistical Mechanics and its Applications 33 (4) 4265-1222
  38. He Y., Guo X., Si B.C. (2006) Detecting grassland spatial variation by a wavelet approach, International Journal of Remote Sensing 199 1217-
  39. Su M.F., Wang H.J. (2006) Relationship and its instability of ENSO - Chinese variations in droughts and wet spells, Science in China, Series D: Earth Sciences 72 -584
  40. Fauria M.M., Johnson E.A. (2006) Large-scale climatic patterns control large lightning fire occurrence in Canada and Alaska forest regions, Journal of Geophysical Research G: Biogeosciences (0.16667) 577-458
  41. Mokhov I.I., Bezverkhnii V.A., Eliseev A.V., Karpenko A.A. (2006) Interrelation between variations in the global surface air temperature and solar activity based on observations and reconstructions, Doklady Earth Sciences 36 (22) 441-614
  42. Moore J., Grinsted A., Jevrejeva S. (2006) Is there evidence for sunspot forcing of climate at multi-year and decadal periods?, Geophysical Research Letters 33 (5) 598-
  43. Schroder B., Seppelt R. (2006) Analysis of pattern-process interactions based on landscape models-Overview, general concepts, and methodological issues, Ecological Modelling 13 (5)
  44. Rossetti F., Rodrigues M.C.A., Oliveira J.A.C.d., Garcia-Cairasco N. (2006) EEG wavelet analyses of the striatum-substantia nigra pars reticulata-superior colliculus circuitry: Audiogenic seizures and anticonvulsant drug administration in Wistar audiogenic rats (War strain), Epilepsy Research 27 (5)
  45. Sakkalis V., Oikonomou T., Pachou E., Tollis I., Micheloyannis S., Zervakis M. (2006) Time-significant wavelet coherence for the evaluation of schizophrenic brain activity using a graph theory approach, Annual International Conference of the IEEE Engineering in Medicine and Biology - Proceedings 42 (3)
  46. Jose M., Bolzan A., Vieira P.C. (2006) Wavelet analysis of the wind velocity and temperature variability in the Amazon forest, Brazilian Journal of Physics 111 (17) -1397
  47. Camayo R., Campos E.J.D. (2006) Application of wavelet trasnform in the study of coastal trapped waves off the west coast of South America, Geophysical Research Letters 111 (9) 1392-296
  48. Wei H.L., Billings S.A. (2006) An efficient nonlinear cardinal B-spline model for high tide forecasts at the Venice Lagoon, Nonlinear Processes in Geophysics 111 (15) 287-2934
  49. Rodo X., Rodriguez-Arias M.-A. (2006) A new method to detect transitory signatures and local time/space variability structures in the climate system: The scale-dependent correlation analysis, Climate Dynamics 111 2916-
  50. Mokhov I.I., Smirnov D.A. (2006) Study of the mutual influence of the El Nin~o-Southern Oscillation processes and the North Atlantic and Arctic Oscillations, Izvestiya - Atmospheric and Ocean Physics 646 (3) -376
  51. Wei J., Dickinson R.E., Zeng N. (2006) Climate variability in a simple model of warm climate land-atmosphere interaction, Journal of Geophysical Research G: Biogeosciences 13 (12) 373-330
  52. Girardin M.P., Tardif J., Flannigan M.D. (2006) Temporal variability in area burned for the province of Ontario, Canada, during the past 200 years inferred form tree rings, Journal of Geophysical Research D: Atmospheres 19 (5) 319-22
  53. Jevrejeva S., Grinsted A., Moore J.C., Holgate S. (2006) Nonlinear trends and multiyear cycles in sea level records, Journal of Geophysical Research C: Oceans 33 (233) 9-4
  54. Moore J., Kekonen T., Grinsted A., Isaksson E. (2006) Sulfate source inventories from a Svalbard ice core record spanning the industrial revolution, Journal of Geophysical Research D: Atmospheres 2 (2) 1-9
  55. Li K.J., Gao P.X., Qiu J. (2006) Does high-latitude solar activity lead low-latitude solar activity in time phase?, Astrophysical Journal 233 (1) 1-536
  56. Palus M., Novotna D. (2006) Quasi-biennial oscillations extracted from the monthly NAO index and temperature records are phase-synchronized, Nonlinear Processes in Geophysics 19 (22) 523-
  57. Mendoza B., Velasco V., Jauregui E. (2006) A study of historical droughts in southeastern Mexico, Journal of Climate 32 (11)
  58. Baas A.C.W. (2006) Wavelet power spectra of aeolian sand transport by boundary layer turbulence, Geophysical Research Letters 41 (5)
  59. Xiao C.J., Song L.T., Pu Z.Y., Wang J.X., Liu Z.X., Glassmeier K.-H., Balogh A., Reme H. (2005) Interaction between CME and magnetosphere observed by cluster on Nov. 6, 2001: (1) Waves excitation, Proceedings of the International Astronomical Union 41 (24) 349-
  60. Mendoza B., Velasco V.M., Valdes-Galicia J.F. (2005) Mid-term periodicities in the solar magnetic flux, Solar Physics 86 305-
  61. Mendoza B., Maravilla D., Jauregui E. (2005) Main periodicities of the minimum extreme temperature of three stations near the Mexican Pacific coast, Atmosfera 1 28-
  62. Ramos da Silva R., Avissar R. (2005) The impacts of the Luni-Solar oscillation on the Arctic oscillation, 3
  63. Bing C.S., Zeleke T.B. () Wavelet coherency analysis to relate saturated hydraulic properties to soil physical properties, Water Resources Research 2 805-
  64. Mokhov I.I., Bezverkhny V.A., Karpenko A.A. () 2008, 41 1 293-
  65. Moore J.C., Grinsted A., Jevrejeva S. () 2008, 38 3 4662-

  7. Links


Wavelet analysis

Links to authors' institutions
Univ. Oulu logo         Thule logo             Nessling logo
NOC logo                       Univ. Lapland logo         Arctic Centre logo


Aslak Grinsted, Arctic Centre, University of Lapland, Finland
John C. Moore, Arctic Centre, University of Lapland, Finland & Thule Institute, University of Oulu, Finland
Svetlana Jevrejeva, National Oceanography Centre, Liverpool, UK

This software © Copyright 2002–2004 Aslak Grinsted, may be used, copied, or redistributed as long as it is not sold and this copyright notice is reproduced on each copy made. This routine is provided as is without any express or implied warranties whatsoever. Published with MATLAB® 7.0


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