When samples tell a different story
Sample a 7 Hz sine at 10 Hz. Predict the apparent frequency, then increase the sampling rate.
First, picture it
The solid curve is the continuous sine; circles are the actual samples. The dashed curve is one lower-frequency candidate that passes through exactly the same samples. At f=7 Hz and fₛ=10 Hz it has signed frequency −3 Hz: sin(2π·7·k/10+φ)=sin(2π·(−3)·k/10+φ). Its apparent oscillation rate is 3 Hz. The sign preserves phase; simply drawing a positive 3 Hz sine with the same phase would be wrong.
What the model says
Unique ideal reconstruction requires a known band limit below fₛ/2. A finite set of samples alone does not prove that the source met it. Here the source is deliberately known, so we can expose the ambiguity. Raise fₛ to 20 Hz: the 7 Hz sine now fits below the 10 Hz Nyquist frequency, and the candidate overlaps it. The dashed curve is an analytic candidate, not a reconstruction algorithm.
x(t)=sin(2πft+φ), tₖ=k/fₛ · f_alias = f − round(f/fₛ)fₛ (principal band)Delay changes when a value becomes available. Quantization rounds its amplitude. Aliasing makes different continuous frequencies indistinguishable after sampling. This bench isolates sampling and phase: there is no noise, quantizer, delay or anti-alias filter. A digital low-pass filter after sampling cannot generally recover a component already folded into the signal band. Filter unwanted analog frequencies before sampling and choose a rate for both measurement and feedback dynamics.
A measurement is a set of times
The solid curve is the continuous sine; circles are the actual samples. The dashed curve is one lower-frequency candidate that passes through exactly the same samples. At f=7 Hz and fₛ=10 Hz it has signed frequency −3 Hz: sin(2π·7·k/10+φ)=sin(2π·(−3)·k/10+φ). Its apparent oscillation rate is 3 Hz. The sign preserves phase; simply drawing a positive 3 Hz sine with the same phase would be wrong.
What the sampling condition assumes
Unique ideal reconstruction requires a known band limit below fₛ/2. A finite set of samples alone does not prove that the source met it. Here the source is deliberately known, so we can expose the ambiguity. Raise fₛ to 20 Hz: the 7 Hz sine now fits below the 10 Hz Nyquist frequency, and the candidate overlaps it. The dashed curve is an analytic candidate, not a reconstruction algorithm.
Exactly two samples per cycle is a trap
Set f=5 Hz, fₛ=10 Hz and phase=0°. Every sample is zero even though the sine has amplitude one. Change phase to 90°: the samples alternate +1 and −1. At the boundary, phase changes what is visible; do not use equality as a practical design margin. Real anti-alias filters need a transition band below half the sampling rate.
Aliasing is not delay or quantization
Delay changes when a value becomes available. Quantization rounds its amplitude. Aliasing makes different continuous frequencies indistinguishable after sampling. This bench isolates sampling and phase: there is no noise, quantizer, delay or anti-alias filter. A digital low-pass filter after sampling cannot generally recover a component already folded into the signal band. Filter unwanted analog frequencies before sampling and choose a rate for both measurement and feedback dynamics.
Make it concrete
Sample a 7 Hz sine at 10 Hz. Predict the apparent frequency, then increase the sampling rate.
Open this experimentDelay changes when a value becomes available. Quantization rounds its amplitude. Aliasing makes different continuous frequencies indistinguishable after sampling. This bench isolates sampling and phase: there is no noise, quantizer, delay or anti-alias filter. A digital low-pass filter after sampling cannot generally recover a component already folded into the signal band. Filter unwanted analog frequencies before sampling and choose a rate for both measurement and feedback dynamics.
Check your understanding+
Sample a 7 Hz sine at 10 Hz. Predict the apparent frequency, then increase the sampling rate.
Set f=5 Hz, fₛ=10 Hz and phase=0°. Every sample is zero even though the sine has amplitude one. Change phase to 90°: the samples alternate +1 and −1. At the boundary, phase changes what is visible; do not use equality as a practical design margin. Real anti-alias filters need a transition band below half the sampling rate.