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.

Modeling note

The cosine line spectra are rendered as narrow Gaussian peaks so they can be plotted numerically. The recoverability verdict uses the ideal Dirac-line model, not the finite-width drawing.