# computing delta power of EEG signal

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Michael on 6 Mar 2012
Commented: Waqas Rasheed on 9 Mar 2021
I wish to compute the “delta power” of EEG recorded over the course of a night’s sleep. From the raw EEG signal (in microvolts), I’m attempting to do the short-time Fourier transform (STFT) in small windows of the raw signal, then analyze the outputs in the range of 0.5-4Hz (delta).
I have lots of questions about 1) how STFTs should be ideally computed; 2) how Matlab computes STFTs; and 3) how to average results across frequency bands and time ranges of interest.
I apologize for the length of this post. It is clear that I should take a course in signal processing, but life is short, I’m in the middle of my career, and there is no time to learn things from 1st principles. Any help relieving my ignorance about all this will be greatly appreciated.
As background, I am running Matlab 7.5.0.342 with the Signal Processing Toolbox on a WinXP machine (64 bit).
1) My understanding is that I could take a spectrogram of my raw EEG signal as a way to implement STFTs. Does this seem reasonable? As opposed to snipping out small time windows of the raw signal, hanning windowing them, computing FFT, calculating power of output, moving on to next window, lather rinse repeat, etc…
2) For some reason, I have two spectrogram functions: “spectrogram” and “specgram”. The latter may be 3rd party. When I try to call the spectrogram, it gives me the error: Undefined function or method 'gencoswin' in ‘hamming’ at 14. Is one of specgram / spectrogram not included in the signal processing toolbox? Is this a licensing error? Is there any great difference between these two functions?
3) The few papers I’ve read on the process of extracting delta power breaks the signal into 4s windows (=nfft), computes the STFT, then averages across several of those windows (e.g., to get avg power in 20s windows). Some authors also average across frequency bin in order to get a single estimate of power in the range 0.5-4Hz. It is usually not specified how this averaging is done: since abs, square and log are non-linear, it matters when I average across frequency and time bins. Any suggestions at what step averaging should be done (e.g., should I average the log PSD or take the log PSD of the average?)
4) Related to 3), is the output b from specgram (or s in spectrogram) in squared units?
5) Right now, I’m using a window of the size of nfft. In this case, I assume there can be no overlap, is that true? Is there a problem not overlapping the signal?
6) Let's say I take a 20s snippet of my raw signal. Then I calculate a spectrogram with a window size (nfft) of 4s. I would expect there to be 5 time bins and Nyquist*4 (+1) frequency bins. I do get this result (e.g., f = Nyquist*4+1; t = 5), but when I plot the output (as surf), it goes from 0-16 sec, not 0-20 sec. That is, it is plotting 4 time windows from 0-4,4-8,8-12,12-16s. If I increase my time resolution (but worsen my spectral resolution), I get something closer to 20s windows. Is there an ideal trade-off? If I go with nfft=4s, then should I start the next 20s time window one 4s window behind, in order to make up for the lost window from 16-20s?
7) In older papers, the window is a boxcar the length of nfft. I think specgram takes a hanning window. Is there a way to specify boxcar for specgram?
8) Finally, another approach would be to do all this in the time domain: generate a, say, Chebyshev Type I bandpass filter between 0.5 and 4Hz; filter the raw EEG signal with these coeffs (using filtfilt to avoid phase shift errors); then simply compute 1og power on the filtered values of the form 10*log10(x2.^2 + 1) (the +1 is to remove fractions in the signal, which will produce negative values after logging). Should this produce the same thing as the spectrogram version above?
Thanks for the help

Kosai on 7 Mar 2012
Hi Michael, I am going to answer the part of your question about the calculation of Delta-Power : at first you don't need to use the STFT to extract the Delta-Band-Information, but simply you have to use the Wavelet-Decompostion-Analysis(Look at the Wavelet-ToolBox), the most important point by wavelet analysis is to determine the Sampling-Freqeny(Fs) in your EEG-Data-Row, i will assume that your (Fs = 1000Hz as example) , so we need then an Wavelete-Decompostion-Analysis with (8 Levels) to achive the Band-Frequency = 4Hz(Delta-Band), simply type these commands :
waveletFunction = 'db8' OR 'sym8' OR 'coif5' OR 'db4';
[C,L] = wavedec(S,8,waveletFunction);
%%Calculation The Coificients Vectors
cD1 = detcoef(C,L,1); %NOISY
cD2 = detcoef(C,L,2); %NOISY
cD3 = detcoef(C,L,3); %NOISY
cD4 = detcoef(C,L,4); %NOISY
cD5 = detcoef(C,L,5); %GAMA
cD6 = detcoef(C,L,6); %BETA
cD7 = detcoef(C,L,7); %ALPHA
cD8 = detcoef(C,L,8); %THETA
cA8 = appcoef(C,L,waveletFunction,8); %DELTA
%%%%Calculation the Details Vectors
D1 = wrcoef('d',C,L,waveletFunction,1); %NOISY
D2 = wrcoef('d',C,L,waveletFunction,2); %NOISY
D3 = wrcoef('d',C,L,waveletFunction,3); %NOISY
D4 = wrcoef('d',C,L,waveletFunction,4); %NOISY
D5 = wrcoef('d',C,L,waveletFunction,5); %GAMMA
D6 = wrcoef('d',C,L,waveletFunction,6); %BETA
D7 = wrcoef('d',C,L,waveletFunction,7); %ALPHA
D8 = wrcoef('d',C,L,waveletFunction,8); %THETA
A8 = wrcoef('a',C,L,waveletFunction,8); %DELTA
POWER_DELTA = (sum(A8.^2))/length(A8);
vinay kulkarni on 17 Jul 2020
@sunidhi gupta I was facing same error. I added following line:
Data_Name=Data_Name(:);
And it works in my code.

Shitanshu Mishra on 21 Mar 2012
Thanks Kosai for such a favour.... My EEG Data had 256 Hz Sampling frequency, And I want to extract signals of all bands from given main signal. Please tell me that in how many levels should I decompose my Signal for Fs=256? Also tell me that how/where my final all 5 bands signals are returned, how can they be retrieved seperately? (I mean in what variable or function)?
Thanks for your help till now, Please do a little more favour. Regards. Shitanshu Mishra

Shitanshu Mishra on 21 Mar 2012
Also:
Delta up to 4
Theta 4 – 8
Alpha 8 – 13
Beta >13 – 30
Gamma 30 – 10
Pleas help me how can I extract all 5 five bands Signals. I am new to Wavelet Decomposition.
Piyush kant on 20 Oct 2017
Thanks for great explanation. I have a doubt, as you said Delta is 0-4Hz but in documentation it is given as 0.5-4Hz. the question is about that lower frequency, wouldn't that 0.0-0.5 Hz signal effect the remaining part of the same?

Salaheldin on 10 May 2012
Hi Kosai,
Thank you very much for this very helpful code. I have two questions: (1) why do we need the cD=detcoef() and cA=appcoef() commands? We don't use cD or cA variables in any operations. (2) How does one calculate the power in the each of the decomposition levels, more specifically, bands of interest (gamma, beta, alpha, delta, and theta)?
one last point, if my sampling frequency is 2000, then I would need 9 decomposition levels to capture the bands of interest in the above expample, no?
thank you very much!!!
Kosai on 7 Jun 2012
Hi Salaheldin,
actually we dont need the (cD,...,cA) but i have included them in code to have a better vision about Coificients and Details vectors in wavelet.
About the Power , simply calculate the Energy of every band (Energy = Power = abs(sum(myBand.^2)))
If your Fs is 2000 yes you need 9 Decomposition levels .

Riheen on 26 Feb 2013
Edited: Riheen on 26 Feb 2013
Hi KOSAI, Great explanation.I have a question. Suppose, my data is sampled with 500Hz. So, according to Nyqusit theorem there is 250 hz spectrum. I filtered the signal 0-64 Hz. I need 0-8Hz band. Now how much decomposition level is required 3 or 5?? Sampling freuency always 500 Hz.

Shishi on 27 Mar 2014
Thank you Kosai, your piece of code and explanation was really helpful to me. As Riheen mentioned I'd like to add that the sampling frequency (Fs) is twice frequency spectrum based on Nyquist's theorem. Accordingly, if the sample frequency is 1000Hz then the frequency spectra is 500Hz and we need 7 levels of decomposition to get the delta band. You also can refer to the paper in below wherein the authors use the same approach. There is a figure of discrete wavelet packet transform (DWPT)which really shows the decomposition levels.
The sampling frequency of their EEG is 128 Hz.

VENKATA PHANIKRISHNA B on 15 May 2017
Edited: VENKATA PHANIKRISHNA B on 15 May 2017
decomposition level decision depends
on smaple frequency. it is ok. after decomposition
[C,L] = wavedec(mySignal,8,waveletFunction);
cD8 = detcoef(C,L,8); %THETA
cA8 = appcoef(C,L,waveletFunction,8); %DELTA
D8 = wrcoef('d',C,L,waveletFunction,8); %THETA
A8 = wrcoef('a',C,L,waveletFunction,8); %DELTA
among two which one is correct for getting Theta and Delta sub-bands of EEG signal. To calculate power and energy of theta and gamma sub-bands which signal is correct 'CD' or 'A'. Please provide the clarity.
Thank you

Jae Moon on 1 Dec 2017
I've got an EEG data set that used a sampling freq of 512. I am trying to extract only the beta band (16-32Hz). What level of decomposition do I need?
Waqas Rasheed on 9 Mar 2021
You may need 6, according to Kosai's logic.
Kosai! I tried to write code to find levels of decomposition following your logic. Please correct me if I'm wrong.
fs = 512; L = wmaxlev(fs,'db8'); LEVELS = L+round(L/4)
Gives 6 for 512, 8 for 1000, 9 for 2000, 5 for 256. Interestingly, it gives 4 for 128, and 3 for 64.
Now I have a question for Kosai, or anybody who can answer this. I have a signal at 5000 Fs, and from the above logic, it requires 10 levels of decomposition (if I am not wrong). Please let me know if teh following source is correct to extract delta, theta, alpha, beta, and gamma waves from any signal (say inputsignal).
waveletFunction = 'db8';
[C,L] = wavedec(inputsignal,10,waveletFunction);
GAMMA = wrcoef('d',C,L,waveletFunction,7);
BETA = wrcoef('d',C,L,waveletFunction,8);
ALPHA = wrcoef('d',C,L,waveletFunction,9);
THETA = wrcoef('d',C,L,waveletFunction,10);
DELTA = wrcoef('a',C,L,waveletFunction,10);
... and 1 - 6 are NOISE, right?