How to calculate the energy spectrum of a signal?

0
votes

I know by theory that the energy spectrum of a given signal is the sum of the squared fourier coefficient.

What if I have the real and imaginary part of the corresponding fourier coefficient, can I say that energy spectrum of a given signal is equal to sum of (real part + imaginary part)^2

2
signal-processingfftspectrumcoefficients
you probably need to take the sum of absolute value square of the coefficient, i.e. \sum_i |fourier_coefficient_i|^2. However, afaik, the Fourier coefficients of a signal give you the energy density at that frequency (i.e. the spectral density over the energy domain), and summing their absolute value give you, by Parseval's theorem, the total energy. – vsoftco
I'm voting to close this question as off-topic because belongs on math.stackexchange.com – user41871
I'm voting to close this question as off-topic because this is math, not programming. Furthermore, OP's definitions are questionable to plain wrong. – Marcus Müller

2 Answers

1
votes

Not quite. You want:

sum of fft_result_magnitudes^2

which is:

sum of (sqrt(real_part^2 + imaginary_part^2)^2

which is:

sum of (real_part^2 + imaginary_part^2)

to get the sum of the squared magnitude of a complex FFT's results.

As for a fuller statement of Parseval's theorem, see:

http://en.wikipedia.org/wiki/Parseval%27s_theorem

0
votes

If result is a column vector with N elements, the energy spectrum is also a vector with N elements.

powerSpec = abs(result).^2;

The total energy can be calculated by

totalPower = sum(powerSpec);

or

totalPower = result' * result;

If result is a row vector you have to use

totalPower = result * result';