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.

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.