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  "slug": "aliasing",
  "title": "Sampling & Aliasing Simulator + AAF",
  "category": "signal-processing",
  "category_label": "Signal processing",
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  "description": "The Aliasing & Sampling Workbench visualizes the Nyquist-Shannon sampling theorem in real time. Pick a signal frequency f and a sample rate Fs, and watch what happens above and below Fs/2. The workbench computes the aliased frequency, the Sinc vs ZOH reconstruction error, SNDR, SFDR, THD+N, and ENOB, so you can see exactly how close your ADC choice is to the textbook limit. Use it to size an anti-aliasing filter, verify a sample-rate budget, or debug a \"the signal folded back\" bug in production.",
  "keywords": [
    "aliasing",
    "sampling theorem",
    "Nyquist sampling",
    "anti-alias filter",
    "ADC",
    "signal reconstruction"
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    "formulas": [
      "Nyquist-Shannon: f_Nyquist = Fs / 2 (no aliasing if f_signal < f_Nyquist)",
      "Aliased frequency: f_alias = |f_signal − n·Fs| for the nearest n",
      "Anti-aliasing filter: stopband starts at Fs/2, passband ripple < 0.1 dB, stopband attenuation > 60 dB",
      "SNDR = 6.02·N + 1.76 dB (ideal N-bit ADC, full-scale sine)",
      "ENOB = (SNDR − 1.76) / 6.02 (effective number of bits)",
      "Sinc reconstruction: x̃(t) = Σ x[n]·sinc((t − n·T)/T)",
      "ZOH step: x̃(t) = x[n] for t ∈ [n·T, (n+1)·T)"
    ],
    "citations": [
      "Oppenheim, A. V., Willsky, A. S., \"Signals and Systems,\" 2nd ed., Prentice Hall, 1997, Ch. 7.",
      "Nyquist, H., \"Certain Topics in Telegraph Transmission Theory,\" Trans. AIEE, 1928.",
      "Shannon, C. E., \"Communication in the Presence of Noise,\" Proc. IRE, 1949.",
      "GATE ECE 2017 Q.25, 2019 Q.42 (aliasing in discrete-time sampling)."
    ],
    "faq": [
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        "q": "Why does my 9 kHz tone appear as 1 kHz after sampling at 10 kHz?",
        "a": "Because 9 kHz exceeds the Nyquist rate (5 kHz for Fs = 10 kHz). The aliased frequency is |9 − 10| = 1 kHz. The original frequency cannot be recovered."
      },
      {
        "q": "What is the minimum anti-aliasing filter order?",
        "a": "Depends on your stopband attenuation spec. For 60 dB stopband at 1.5·(signal bandwidth) away from the passband edge, a 5th-order Butterworth is typical. Bessel is better for time-domain (less pulse ringing)."
      },
      {
        "q": "How do I increase ENOB without changing the ADC?",
        "a": "Reduce input frequency (less aperture jitter), reduce noise on V_ref (better PSRR), avoid clipping (use full-scale input). The workbench shows how each affects the SNDR."
      }
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  "last_updated": "2026-07-05"
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