Calculate energy of time domain data

Question

I am new to digital signal processing. I have the following sensor sample data

Time(milliseconds)            data
------------------    -------------------
0                     0.30865225195884705   
60                    0.14355185627937317   
100                  -0.16846869885921478   
156                  -0.2458019256591797    
198                  -0.19664153456687927
258                   0.27148059010505676   
305                  -0.16949564218521118   
350                  -0.227480947971344 
397                   0.23532353341579437   
458                   0.20740140974521637

Which means at time 0 I have the value 0.30865225195884705 and at time 60 I have the value 0.14355185627937317 and so on.

Data is taken from the sensor at each 10 milliseconds. So, I assume sampling rate should be set to 100 Hz.

I want to calculate the total energy of the time domain signal.

I read that it can be calculated using Parseval's theorem as following:

where X[k] is the DFT of x[n], both of length N.

Any suggestion, how can I calculate the total energy using MATLAB?

Answer 1

Parseval's theorem is useful in linking the time domain energy to the frequency domain. However, if you do not need to perform other computations in the frequency domain, you can compute the energy directly in the time domain with:

Energy = sum(abs(x).^2)

If on the other hand, you need to convert the signal to the frequency domain for other reasons, you may also compute the energy with (as per Parseval's theorem):

Xf = fft(x); % compute the DFT (using the Fast Fourier Transform)
Energy = sum(abs(Xf).^2) / length(Xf); % Get the energy using Parseval's theorem

Answer 2

Parseval's theorem and DFT analysis only apply to band-limited data sampled with regular equal spacings (constant sample rate above Fmax*2). Since your time stamps are not regularly spaced, you will need to use them to interpolate a vector of new evenly spaced samples before you can calculate the energy using Parseval's equation. Or you will have to do a numerical integration instead of a simple summation.