Parseval’s Theorem Demo

Parseval’s theorem demo (DFT / FFT)

Build a length-N signal, optionally apply a window, and compute its energy two ways: directly in time as Σ|x[n]|², and from the FFT. The DFT has two common scalings, and the demo shows both: the ordinary (unnormalized) DFT needs a 1/N factor, the unitary DFT (divide by √N) does not. The ratio E_time/E_freq should be 1 to rounding error.

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Signal Real-valued test signal. Energy is computed on the signal after the window (if any).

Samples N Powers of two so a radix-2 FFT applies. Bin spacing is fs/N.

Window Multiplies x[n] by w[n] before the FFT. Parseval then holds for the windowed sequence.

Plot |X[k]| using Changes only the lower plot and its height. Both energies are always reported below.

Pulse width = 0.25 × N Only for Rectangular pulse. Fraction of the record that is 1 (starts at N/4).

Tone frequency = 8.00 bins Only for Tone. Integer = exactly on a bin; a fraction (e.g. 8.5) leaks across bins but Parseval still holds.

Gaussian σ = 0.08 × N Only for Gaussian pulse. Width of the pulse, centered at N/2. Narrow in time means wide in frequency.

Noise seed Only for White noise. Same seed gives the same realization (shareable).

Time domain: E_time = Σ |x[n]|² —

Ordinary DFT: (1/N) Σ |X[k]|² —

Unitary DFT: Σ |X_u[k]|², X_u = X/√N —

Ratio E_time / E_freq (ordinary, with 1/N) —

Ratio E_time / E_freq (unitary) —

Worst relative error |ratio − 1| —

Common bug: forget 1/N, E_time / Σ|X[k]|² —

Average power P = E/N and RMS = √P —

Window ENBW = N Σw² / (Σw)² (bins) —

|x[n]| after windowing

|X[k]|

Try this: leave everything on defaults and note the ratio is 1.000000000000. Switch the plot between ordinary and unitary: the spectrum height changes by √N, yet both energies match E_time. Then read the amber “common bug” row: without the 1/N the answer is off by exactly a factor of N.

Windows: set Signal = Tone, frequency 8.5, then toggle Hann. Leakage drops, E_time drops too (the window removes energy), but Parseval still balances on the windowed data.

Background articles: Parseval’s theorem derivation · Power and energy of a signal · Significance of RMS value.

Controls explained

Signal: Rectangular pulse — a block of ones (spectrum is a sampled sinc). Tone — cos(2πfn/N); a bin-centered tone puts all energy in two bins (k = f and N−f). Gaussian pulse — smooth, its spectrum is also Gaussian. White noise — independent N(0,1) samples; energy is about N and spreads evenly over bins.
Samples N: Record length. Energy grows with N for a fixed signal shape (more samples to sum), and the unnormalized |X[k]| grows too. The 1/N in Parseval is exactly what reconciles them.
Window: Rectangular leaves x[n] untouched. Hann multiplies by w[n] = 0.5(1 − cos(2πn/N)). The reported energies are those of x[n]·w[n], not of x[n]. ENBW (equivalent noise bandwidth) is NΣw²/(Σw)²: 1.00 bin for rectangular, 1.50 bins for Hann.
Plot |X[k]| using: Ordinary is what numpy.fft.fft, MATLAB fft, and most libraries return. Unitary divides by √N (the norm="ortho" option in NumPy), so that Σ|X_u|² = Σ|x|² with no extra factor. Only the lower plot is affected.
Pulse width / Tone frequency / Gaussian σ / Noise seed: Signal-specific shape parameters. Each is hidden when it does not apply to the selected signal. None of them can break Parseval; they only change how the same energy is distributed over time and frequency.
Results panel: The two ratios should equal 1 up to floating-point rounding (relative error near 1e−15). The “common bug” row shows what you get if you sum |X[k]|² and forget the 1/N: the ratio is exactly 1/N.

How to use this demo

Pick a signal, choose N, and read the results panel. The top rows give the time-domain energy Σ|x[n]|² and the frequency-domain energy computed from the FFT under both common scalings: (1/N)Σ|X[k]|² for the ordinary DFT and Σ|X_u[k]|² for the unitary DFT. Both ratios sit at 1 up to rounding. The amber row shows the classic mistake of summing |X[k]|² without the 1/N.

Quick checks: white noise, N = 1024 (the forgotten-1/N error is now 1/1024), and a leaky tone with a Hann window.

FAQ

Why is my energy N times too big? You summed |X[k]|² from an unnormalized FFT. Divide by N, or use the unitary (norm="ortho") transform.

Why doesn’t the ratio hit exactly 1? Floating-point rounding. Expect a relative error around 1e−15 to 1e−13, growing slightly with N.

Does a window break Parseval? No. Parseval applies to whatever sequence you hand to the FFT. After windowing, that sequence is x[n]·w[n], so its energy is lower than the original signal’s.

Why do fractional tone frequencies still satisfy Parseval? Leakage spreads the energy over many bins, but the total over all N bins is unchanged. Leakage affects where the energy shows up, not how much there is.