Modulation, Frequency Shifting, and Low-Pass Filtering

This applet is built to train communication-systems intuition. Students can move back and forth between time-domain multiplication and frequency-domain spectral copying, then test what a low-pass filter actually keeps. The notation follows the standard Chapter 8 modulation story: multiplying by cos(ωct) creates two shifted copies with a factor of 1/2, while cos²(ωct) creates a DC term plus a double-frequency term.

ECE 210 intuition builder Shifted replicas LPF survival test Prediction before reveal

Controls

Change the mixer, input spectrum, and LPF cutoff. Any change hides the final answer again so students must predict first.
Use this to switch between a lowpass-like baseband spectrum and a bandpass-like two-band input.
The squared-cosine mode is there on purpose. Students often miss the DC term.
Increasing ωc pushes copies farther away from the origin. In cos² mode the shifted copies move to ±2ωc.
The ideal LPF passes |ω| ≤ ωL and rejects frequencies outside that band.

Presets

1. Input signal in time domain

The time waveform is reconstructed from the chosen input spectrum. It is not arbitrary decoration; it reflects the current spectral shape.

2. Mixer signal in time domain

This is the multiplying signal. Watch how the identity behind the mixer controls the spectral replicas created in frequency.

3. Spectrum before mixing

This is F(ω). Keep track of its width and where its energy sits. That determines what happens after shifting.

4. Spectrum after mixing

The mixed spectrum is built from shifted copies of F(ω). The colored overlays make the amplitude factors explicit.

5. Low-pass filter response

The LPF passes only the shaded center band. The actual question is: which shifted replicas overlap that passband?

6. Final output spectrum and output signal

This remains hidden until the student makes a prediction. That forces a real before-and-after comparison.

Predict first
Decide which replicas make it through the LPF, then click Reveal output in the challenge box below.

Prediction challenge: which replicas survive?

What this applet is trying to teach

f(t)cos(ωct) ↔ 1/2 F(ω-ωc) + 1/2 F(ω+ωc)
cos²(ωct) = 1/2 + 1/2 cos(2ωct)
Y(ω) = H(ω) X(ω)
Input spectrum F(ω)
Right-shifted copy
Left-shifted copy
Baseband/DC copy
Instructor notes and misconceptions list

Instructor notes

  • Start with the input spectrum and ask students to identify its bandwidth before any mixing. Make them say what counts as “near zero” and what does not.
  • Then move to the mixer panel and ask which identity applies: either cosine splitting into two exponentials, or squared cosine splitting into a DC term plus a double-frequency cosine.
  • Use the “show only spectral copies” box when students conflate the original spectrum with the shifted replicas. It isolates the mechanism.
  • Do not reveal the final output immediately. Have students choose the replicas that overlap the LPF passband first.
  • The preset called Coherent-demod intuition is good for connecting Chapter 8 modulation to the idea of mixing again and low-pass filtering to recover baseband.
  • For a quick board derivation, write the cosine case first, then contrast it with cos². That contrast is where many weak intuitions break.

Common misconceptions to attack directly

  • “Multiplying by cosine just moves the spectrum once.” No. It creates two mirrored shifted copies, each scaled by 1/2.
  • “The original baseband always stays.” Not in ordinary cosine mixing. The baseband copy disappears unless the shift creates overlap back near zero or the mixer itself includes a DC term.
  • “cos² is just a stronger cosine.” No. It is a different spectral operation because cos²(ωct)=1/2+1/2cos(2ωct).
  • “The LPF keeps the biggest lobe.” Wrong criterion. The LPF keeps whatever lies inside its passband, not whatever looks visually tallest elsewhere.
  • “Mirrored copies are optional artistic symmetry.” No. Real signals force conjugate symmetry, and cosine mixing creates left and right replicas together.
  • “The factor of 1/2 is bookkeeping only.” No. It changes actual spectral amplitude and matters after repeated mixing or filtering.
Designed as a polished communication-systems intuition builder rather than an exam spoiler. Frequencies are shown in normalized ω-units so the geometry of shifting and filtering stays visually clear.