3. Signal Processing (Continuous to Discrete) · Concept 9 of 11
Parseval's Theorem
It says the total energy of a sound is the same whether you measure it as a waveform in time or as a list of frequencies.
Square it and add it up in time, or square it and add it up in frequency: the total energy lands on the same number.
What it is
Total signal energy is identical whether you add it up over time (the waveform) or over frequency (the spectrum).
Key facts
Statement: sum of |x(t)|^2 over all time = sum of |X(f)|^2 over all frequency. Energy in = energy out, regardless of view.
Symbols: x(t) = signal amplitude at time t; X(f) = Fourier transform (how much of each frequency f is present); |.|^2 = value squared (= power).
Continuous form: integral of |x(t)|^2 dt = integral of |X(f)|^2 df. Discrete (DFT): sum |x[n]|^2 = (1/N) * sum |X[k]|^2, N = number of samples.
Energy is proportional to amplitude SQUARED. Double the amplitude = 4x energy (+6 dB).
+6 dB = 2x amplitude (voltage); +3 dB = 2x power/energy; -3 dB = half power (the 'half-power point' defining filter bandwidth); +10 dB ~ twice as loud.
dB (power): dB = 10*log10(P/Pref). dB (amplitude): dB = 20*log10(A/Aref).
RMS (root-mean-square) = sqrt(average of x(t)^2) is the time-domain energy measure; Parseval guarantees RMS^2 matches total spectrum power.
Named after Marc-Antoine Parseval (1799). Frequency version is Rayleigh's energy theorem.
Sine wave: RMS = 0.707 x peak; peak/RMS = 1.414 (sqrt 2); crest factor of a sine = 3 dB.
Reference levels: 0 dBFS = digital clip ceiling; 0 dBu = 0.775 V RMS; 0 dBV = 1 V RMS. Speed of sound = 343 m/s at 20 C.
How it works
Take the waveform x(t) and square every sample to get instantaneous power.
Add all those squared values together: that is total energy in the TIME view.
Run an FFT to get X(f), the amount of each frequency present.
Square each frequency bin's magnitude and add them all up (with the 1/N scaling).
Both totals come out the SAME number: that is Parseval's Theorem.
Result: a frequency display and an RMS level meter read the same energy, two ways.
Real examples
Spectrum analyser RTA total energy matches the SPL meter's RMS reading on the same signal.
Pink noise has equal energy per octave, so its time-domain RMS lines up with a flat-per-octave spectrum.
A loud cymbal: energy seen as a sharp time-domain spike equals the energy smeared across 5-15 kHz in the spectrum.
EQ check: boost 2 kHz by +6 dB and the extra energy shows up identically in waveform RMS and in the spectrum.
FFT crest-factor test: a 0 dB sine shows 0.707 RMS in time and the same total power summed across its bin.
How it helps in live sound
Trust your RTA: if the spectrum says energy is piling up at 120 Hz, your subs really are working that hard, the meter agrees.
Use RMS (not peak) metering for loudness/limiter decisions, it is the energy number Parseval ties to the spectrum.
Watch crest factor: peak minus RMS. Big gap = transient-heavy (drums); set limiter attack accordingly.
FFT looks hot but SPL meter looks fine? Suspect a window/scaling error, not magic, the energy must reconcile.
Hunting feedback: a single ringing frequency dumps real energy, it spikes the RTA AND jumps the meter.
Gain-stage by energy: +6 dB at the desk = 4x the energy hitting the amps, confirmed in both time and frequency views.
Everyday analogy
Counting your money as coins in your pocket or as a number in your bank app gives the same total: switching views never creates or destroys a cent.
Watch out
Myth: 'a tall thin spike on the spectrum is harmless because it's narrow.' Wrong, energy is height squared summed; a narrow but tall peak can carry as much energy (and heat your driver) as a wide low hump.
Fun fact
Parseval proved this in 1799, decades before the Fourier transform existed and over 150 years before the FFT, yet it is the exact rule that lets a modern Smaart or REW display reconcile with a humble VU meter.
Key takeaways
Total energy is conserved between the time view and the frequency view.
Energy = amplitude squared, so it always adds as |x|^2 or |X|^2, never raw amplitude.
+6 dB = 2x amplitude = 4x energy; +3 dB = 2x power; -3 dB = half power.
RMS in time equals total power in the spectrum, which is why your meters and analyser agree.
Remember the 1/N scaling factor on the discrete (FFT) side or the two totals won't match.