ECE 210: Continuous-Time Sampling, Spectral Replication, Aliasing, and Reconstruction

This applet is built to make the replica picture concrete. Move the sampling period T, watch the replica spacing ωs = 2π/T change, and test whether an ideal low-pass filter can recover the original signal exactly.

The key idea is simple but deep: multiplying by an impulse train in time creates periodic spectral copies in frequency. Exact recovery works only when the baseband copy is isolated.

s(t) = x(t) · Σn=-∞∞ δ(t - nT)
S(ω) = (1/T) Σk=-∞∞ F(ω - kωs)
Ideal recovery: H(ω) = T for |ω| < ωc

Controls

Challenge answer hidden
T
fs = 1/T
ωs = 2π/T
Nyquist edge ωs/2
Chosen cutoff
Required to include baseband
Recoverable
original samples aliased alternative reconstructed

Challenge mode

Predict before revealing:
With the current preset, sampling period T, and cutoff ωc, can the original signal be exactly recovered?
Tip: exact recovery needs more than a low-pass filter. The baseband copy must be isolated, and the filter must include the whole baseband without admitting neighboring replicas.

1. Original continuous-time signal and samples

Time-domain sampling multiplies the signal by an impulse train. The stems are the sample values x(nT).

2. Original spectrum F(ω)

The original spectrum determines whether replicas will overlap after sampling.

3. Sampled spectrum S(ω) with replicated copies

Replica centers are spaced by exactly ωs = 2π/T. If copies overlap, aliasing is unavoidable.

4. Reconstruction filter H(ω)

Higher cutoff is not always better. A too-wide low-pass filter can pass parts of neighboring replicas.

5. Recovered spectrum and reconstructed signal

Top: Y(ω) = H(ω)S(ω). Bottom: reconstructed signal y(t). If the original is not recoverable, the aliased alternative can still match the samples.

Misconceptions addressed

  • "Higher cutoff is always better" is false. A cutoff that reaches into neighboring replicas destroys exact reconstruction.
  • "Sampling only affects the time domain" is false. Time-domain multiplication by an impulse train causes periodic spectral repetition.
  • "Aliasing is just noise" is false. It is a deterministic overlap of shifted spectral copies.
  • "If the samples match, the original must be unique" is false. When aliasing occurs, a different continuous-time signal can fit the same samples.

Quick instructor use

  • Start with single cosine below Nyquist and sweep T until the copies nearly touch.
  • Then switch to single cosine above Nyquist to show that the samples do not reveal the original frequency uniquely.
  • Use sinc-type low-pass to make the "isolated baseband copy" condition visually obvious.
  • Turn the cutoff too high on purpose to kill the "more bandwidth is always better" misconception.
Instructor notes

Suggested in-class flow

  1. Ask for a prediction first. Keep challenge mode hidden.
  2. Show that changing T changes the replica spacing, not the original spectrum.
  3. Point to the central copy and ask whether it is isolated.
  4. Only then reveal whether exact reconstruction is possible.
  5. Finally vary ωc to show that even a recoverable sampling setup can be ruined by a bad reconstruction filter.

How the verdict is decided

  • Recoverable means: the preset is theoretically bandlimited, ωs > 2Ωmax, the filter is ideal, ωc includes the full baseband, and ωc stays below the nearest neighboring copy.
  • Not recoverable means at least one of those conditions fails.
  • Non-bandlimited presets can look visually fine at high sample rate, but exact reconstruction still fails in principle.

Preset intent

  • Single cosine below Nyquist: clean success case.
  • Single cosine above Nyquist: clean alias case.
  • Two-tone: one more realistic multi-line spectrum.
  • Sinc-type low-pass signal: true bandlimited continuous signal.
  • Deliberately aliased case: overlap is obvious and dramatic.
  • Oversampled case: easy win with wide separation.
  • Gaussian pulse: non-bandlimited cautionary case.

What to emphasize verbally

  • Sampling does not "destroy" information by magic. Information is lost when the repeated copies overlap.
  • The low-pass filter is not guessing. It is selecting one spectral copy.
  • When aliasing occurs, reconstruction returns a different continuous-time signal that is consistent with the same samples.
  • The replica picture is the main story. Everything else follows from it.
Self-contained file. No external libraries required.