ECE 210 Fourier Quantities Explorer

Explore what X(0), the energy spectrum, total energy, and 3-dB bandwidth really mean. The applet deliberately separates amplitude spectrum from energy spectrum, and it keeps the Parseval normalization visible so students do not treat these as interchangeable pictures.

X(0) = signed time-domain area
W = ∫|x(t)|²dt = (1/2π)∫|X(ω)|²dω
3-dB means half power, not half amplitude
Time shift changes phase, not energy spectrum
Presets
Signal controls
Amplitude A
Width parameter
Extra parameter
Analysis mode
The active view controls which spectrum panel gets the main crosshair and threshold emphasis. The 3-dB definition always comes from the energy or power level. On the amplitude plot, that same point sits at 1/√2 of the peak, not at 1/2 of the peak.

1. Time-domain signal x(t)

Shaded signed area illustrates why X(0) = ∫x(t)dt.

DC readout
signal x(t) signed area contributing to X(0)
Positive and negative regions both count. X(0) is a signed area, not an absolute area.

2. Fourier transform magnitude |X(ω)|

Use this view to compare half-height with the correct 3-dB amplitude level |X|/|X|max = 1/√2.

view

3. Energy spectrum |X(ω)|²

This is the object that integrates to total energy after multiplying by 1/(2π).

view

4. Readout box

Core quantities and the current Parseval check.

X(0)
Signed area ∫x(t)dt
This should match X(0) exactly for the current signal.
Total energy in time
Wt = ∫|x(t)|²dt
Total energy in frequency
Wf = (1/2π)∫|X(ω)|²dω
3-dB bandwidth
Half-height vs half-power

5. Explanation panel with current reasoning

Use this as the “why” panel during teaching or self-study.

Instructor notes

Suggested classroom sequence.

  • Start with rectangle. Ask students what X(0) should be before showing the readout. The right answer is “the shaded signed area”, not “x(0)”.
  • Then switch to shifted pulse. Ask what changes in frequency. Students often guess the spectrum magnitude shifts. It does not. Only the phase changes, so the energy spectrum stays the same.
  • Use low-pass spectrum to attack the common 3-dB mistake. On the amplitude plot, the half-height point is wrong. The correct 3-dB point is where the energy drops to one-half, equivalently where the amplitude drops to 1/√2 of the peak.
  • Use sinc to show that some spectra make the usual “3-dB bandwidth” idea awkward. A brick-wall spectrum has an abrupt edge, so the smooth half-power crossing is not the right mental model.
  • Use modulated band to show that band-pass signals can have nearly zero DC area and yet plenty of energy. This helps separate “DC content” from “total energy”.
  • Keep returning to the Parseval line. Students need repeated exposure to the fact that the energy spectrum contributes only after the 1/(2π) normalization.

Good prompts to ask live.

  • “If I double the amplitude, which quantities double and which quadruple?”
  • “If I shift the signal in time, what happens to X(0), |X(ω)|, and |X(ω)|²?”
  • “At the 3-dB point, what is halved: amplitude, power, or energy density?”
  • “Why can a band-pass signal have X(0) close to zero but still carry a lot of energy?”
Misconceptions list
  • Confusing X(0) with x(0). X(0) is the zero-frequency Fourier coefficient, equal to the signed area under x(t). It is not the value of the signal at t = 0.
  • Thinking the energy spectrum is just the amplitude spectrum with a different label. The energy spectrum is |X(ω)|², so its shape weights large Fourier components more strongly.
  • Forgetting the 1/(2π) factor in Parseval. Without that normalization, the frequency-domain energy is wrong.
  • Using half-height instead of half-power for 3-dB bandwidth. On the amplitude plot the correct threshold is peak/√2, not peak/2.
  • Thinking time shift changes energy spectrum. A pure time shift multiplies X(ω) by a phase factor e-jωt₀, leaving |X(ω)| and |X(ω)|² unchanged.
  • Thinking zero DC means zero energy. A band-pass signal can have X(0) ≈ 0 while still having substantial total energy.