Energy in time domain equals energy in frequency domain.
∫‖x(t)‖²dt = (1/2π)∫‖X(ω)‖²dω — energy is conserved between time and frequency domains; for periodic signals, average power equals the sum of ‖c_n‖². Lets you compute energy in a band or compare distortion power.
A filter that removes 10% of spectrum energy removes 10% of signal energy.
Parseval's theorem says energy is the same in both domains, so you can compute energy or power from either the time or the frequency view.
Energy in time domain equals energy in frequency domain.
Parseval's theorem says energy is the same in both domains, so you can compute energy or power from either the time or the frequency view.
The integral of the squared signal in time equals the integral of the squared magnitude of its Fourier transform, so energy is preserved.
An FFT that loses energy has a scaling or windowing error, so comparing the two sums quickly exposes many bugs.
Typical applications include checking an FFT result by comparing its energy with the time-domain energy, computing the power in a noisy measurement from its spectrum and estimating the total energy of a transient pulse for an impact test.
Study "Spectrum and bandwidth" first (topic 16 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
Power spectral density. Energy spectral density. Total harmonic distortion.
A filter that removes 10% of spectrum energy removes 10% of signal energy.
The next topic is "Laplace transform in systems", which builds directly on this one.
It is classed as Beginner in Signals, Control & Comms (topic 17 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.