Here is the Nyquist explanation rewritten using pure ASCII diagrams (no Unicode).

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Nyquist Theorem (ASCII Version)

The Nyquist theorem says:

To sample a signal correctly, you must sample

at least twice its highest frequency.

    sample_rate >= 2 * signal_frequency

This minimum rate is the Nyquist rate.

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1. Sine wave (conceptual ASCII)

A sine wave at some frequency f:

   /``\    /``\    /``\ 

  /    \  /    \  /    \

 /      \/      \/      \

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2. Proper sampling (sample rate >= 2f)

Two or more samples per cycle:

   /``\    /``\  

  / *  \  /  * \

 /      \/     *\

        *       

You can reconstruct the original wave.

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3. Undersampling (sample rate < 2f)

Too few samples per cycle → aliasing:

   /``\    /``\  

  /    \  /    \

 *     * *     *

    *        *

The samples no longer match the real wave.
The system thinks the signal is a different, lower frequency.

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4. Summary

Correct:     sample_rate >= 2 * frequency

Aliasing:    sample_rate < 2 * frequency

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If you want, I can also produce:

• A fully ASCII diagram showing aliasing (folded spectrum)
• An ASCII discrete-time version (stem plot)
• An ASCII spectrogram-style explanation

Just tell me!

You’re looking for the Nyquist theorem (often the Nyquist–Shannon sampling theorem) and how it describes a sine wave in sampling terms. Here’s a clear explanation:

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Nyquist Theorem Explained (with a Sine Wave Example)

The Nyquist theorem states:

To sample a signal accurately, you must sample at at least twice its highest frequency.

This minimum required rate is called the Nyquist rate.

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Applying it to a sine wave

Suppose you have a sine wave:

x(t) = A \sin(2\pi f t)

with frequency .

Nyquist condition:

You must sample at:

f_s \ge 2f

Otherwise, the wave will appear at the wrong frequency — this error is called aliasing.

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Examples

1. Sine wave at 1 kHz

To sample properly:

• Nyquist rate: 2 kHz
• Good practice: sample at 5–10× the frequency (e.g. 10 kHz)

2. Sample too slowly (e.g. 1.5× the frequency)

The signal will appear to be a completely different sine wave or even look distorted. The system cannot distinguish between the original frequency and a lower “aliased” frequency.

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Intuition

A sine wave has a smooth oscillation.
To capture its shape, you need at least two sample points per cycle:

• One point to show the rising direction
• One point to show the falling direction

More samples give a better representation.

Imagine drawing a wave with only one point per cycle—that point could fit an infinite number of possible waveforms. With two or more points, the shape becomes constrained.

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Visual Summary (conceptual)

Sampling ≥ 2× frequency

✓ You capture the real sine wave.

Sampling < 2× frequency

✗ The sine wave “folds over” and appears at the wrong frequency → aliasing.

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