Complex Morlet Wavelets and Extracting Power and Phase

JJ Morrow2.18.2014

The Wavelet Complex

• Morlet wavelet is only 2D • EEG data has 3 dimensions that need

analysis (time, power, phase)• Key conceptual point: we need a third

dimension… and it is imaginary

Matlab Visuals

• Figure 13.1– plot3(time,real(wavelet),imag(wavelet),'m')

• Figure 13.2– plot3(time,real(wavelet),imag(wavelet),'k')

How is the analysis real… if our kernel is imaginary?!

Imaginary Math

• Complex numbers have two parts– Real part and imaginary

part (eg a+ib)• Square root of negative

numbers• Can still be thought of as the

dimensions of a Cartesian plot (x and y axes) Electronics-tutorial.ws

Polar Notation• Way to characterize circular and spherical

data• Magnitude – hypotenus, point from origin to

complex number• θ (theta) – angle formed by hypotenus and

positive real axis• So θ gives the direction to point and M gives

the length– Together this gives the

location (like rectangular notion)

Conversion between Polar and Rectangular Notation

• M = sqrt(real2 +imag2)– Same as a2 + b2 = c2

• θ = arctan(imag/real)• OR

– cart2pol OR pol2cart in Matlab

Mathisfun.com

Euler’s Formula

• Meiθ = M[cos(θ)+isin(θ)]• Remember that a+ib = M[cos(θ)+isin(θ)]• So Euler’s formula allows representation

of complex numbers as points on a circle!• Help with conceptualizing this

relationship?• Matlab 13.4

– plot( (-.35+cos(time)/5) + 1i*(.35+sin(time)/5) ,'m'

Complex Morlet Wavelet

• Ae-t2/2s2 ei2πft =Complex Morlet Wavelet• ae-t2/2s2 is the Gaussian• ei2πft Euler’s combined with sine wave

– eiθ except θ is 2πft part of a sine wave

• So we have all the pieces here– Gaussian plus a complex sine wave!

• Additional points?

Creating a CMV• % parameters...• srate = 500; % sampling rate in Hz• f = 10; % frequency of wavelet in Hz• time = -1:1/srate:1; % time, from -1 to 1 second in steps of 1/sampling-rate• s = 6/(2*pi*f);• % and together they make a wavelet• wavelet = exp(2*pi*1i*f.*time) .* exp(-time.^2./(2*s^2)); • figure• plot(221)• % show the projection onto the real axis• plot3(time,real(wavelet),imag(wavelet),'r')• xlabel('Time (ms)'), ylabel('real axis')• view(0,90)• title('Projection onto real and time axes')

Complex Wavelet

Convolution•The result of the real-valued wavelets map onto the real axis (fig 12.7)•The imaginary value is ignored in these

Extracting Information from the Complex Dot Product

• “When you compute the dot product between a complex wavelet and a signal the result of the dot product is a complex number”

• We can use Euler’s formula to represent this number in polar space!

• Compare complex dot product with real-valued dot product (ignored the imaginary part of complex value)

Extracting Information from Complex Dot Product

• Bandpass-filter signal– The projection on the real axis– This was the information from chapter 12– Dependent on the phase relationship

Extracting Power from Complex Dot Product

• The length of M is defined by the result of the dot product– Length of vector (M) = amplitude– Amplitude2 = power

• The amount of similarity in the kernel and signal dictates the length

• When the EEG data contain abundant energy at thewavelet frequency the power will be high

Bme240.eng.uci.edu

Extracting Phase from Complex Dot Product

• The angle of the vector in relation to the real axis gives an estimate of the phase angle

• Estimated as the point in time correspond-ing to the center of the wavelet and at the peak frequency ofthe wavelet

• Can anyone unpack this statement?

Time Points to Time Series• Extracting the information from a series of

dot products gives us time series data• Gives information across 3-dimensions for a

recording • This can then be used to create plots in up to

3 dimensions (e.g. time series of power for a frequency band

• Mteiθt Euler’s formulavarying over time

• Figure 13.9

Parameters of Wavelets

• Basically they will vary depending on what you are looking for

• Frequencies– At least a few cycles per epoch– Nyquist frequency upper limit– 20-30 wavelets at varying

frequencies is usually a good number– Higher frequency = more data points = good

Parameters: Scaling

• If you want to highlight the low frequencies use logarithmic scaling

• If you want to emphasize high frequencies use linear scaling

• Keep an eye on what you use!

Logarithmic frequency scaling

0 500 10002

4.2

8.7

18.3

38.3

80Linear frequency scaling

0 500 1000

10

20

30

40

50

60

70

Wavelet Length

• Long enough so that the ends taper to zero• Definitely better to overshoot here

– Avoid edge effects and there is no negative affect of multiplying data by more zeros

-0.5 0 0.5-1

0

1Good.

-0.5 0 0.5-1

0

1Not good.

-0.5 0 0.5-1

0

1

Time

Still works.

Cycles for the Gaussian Taper• Altering this number creates a trade off

between temporal and frequency precisions

• Larger number of cycles gives better frequency precision

• Smaller yields better temporal precision

-0.5 0 0.5-1

-0.5

0

0.5

1

Time

Wavelet at 10 Hz with 3 cycles

-0.5 0 0.5-1

-0.5

0

0.5

1

Time

Wavelet at 10 Hz with 7 cycles

Cycles for the Gaussian Taper

• Smaller number of cycles: – Better for detecting transient activations– More precise at localizing dynamics

• Larger number of cycles – More sensitive to longer activations – More precise at determining the frequency of the

dynamic

Plaza.ufl.edu

0 10 20 30 40 500

50

100

150

200

250

300

Frequency (Hz)

3 cycles7 cycles

Frequency Smoothing

• FWHM for a Gaussian distribution• Reports the influence of neighboring

frequencies on the result of wavelet convolution

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Wavelet spectrum and FWHM

Frequency (Hz)

Nor

mal

ized

pow

er

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70FWHM by Frequency

Frequency (Hz)

FWH

M (

Hz)

3103-10

Smoothing Continued

• Normalize power spectrum • Identify the points around peak• Then subtract these points

terpconnect.umd.edu/~toh/spectrum/iSignal.html

Tips

• FFT only needs to be computed once before looping over frequencies– Using conv function causes FFT redundancy

• Use an FFT that is a power of two– Zero-padding to get this

• Concatenate those trials!– Only need to perform one convolution

Thank you!

http://www.cs.nyu.edu/~dodis/