Lab 2.1 — Sampling Axis and Interpretation

Goal

Verify that the same captured signal can produce a correct-looking but incorrect engineering conclusion if the sampling rate is interpreted incorrectly.

Why this matters

A spectrum plot is only meaningful together with the correct Fs. If Fs is wrong, every FFT bin is mapped to the wrong physical frequency, even when the plot shape itself still looks plausible.

Experiment

The script generates a complex tone with:

• correct sample rate Fs = 1.0 MHz;
• intentionally wrong interpretation Fs = 0.8 MHz;
• non-zero DC offset;
• additive noise;
• a known expected tone near 123.456 kHz.

The lab compares:

• time-domain I/Q preview;
• FFT magnitude with the correct frequency axis;
• the same FFT magnitude with a wrong frequency axis;
• measured peak frequency error for both interpretations.

Run

From the repository root:

python blocks/block_02_signals_and_sampling/python/sampling_analysis.py

Or run the representative lab pack:

python tools/run_all_labs.py

Expected artifacts

Artifact	Meaning
docs/assets/lab21_sampling_time_domain.png	I/Q preview in time domain
docs/assets/lab21_sampling_frequency_axis.png	correct vs wrong Fs interpretation on the FFT axis
docs/assets/lab21_sampling_metrics.json	expected tone, measured peaks and interpretation errors

Interpretation checks

• The correct interpretation should place the tone close to the expected 123.456 kHz.
• The wrong interpretation should shift the measured peak significantly, even though the spectral shape still looks reasonable.
• The metrics JSON should show a small error for the correct axis and a much larger error for the wrong axis.
• The time-domain preview should also help detect DC offset and clipping risk before moving to frequency-domain conclusions.

What to expect

Expected tone: 123456.000 Hz
Measured peak with correct Fs: 123474.121 Hz
Measured peak with wrong Fs: 98779.297 Hz
Correct interpretation error: 18.121 Hz
Wrong interpretation error: -24676.703 Hz

• The 18 Hz "error" with the correct Fs is not an error. It is the FFT's bin quantization: the peak is read at the nearest bin centre. With 16 384 samples at 1 MS/s the bins are 61 Hz apart, so the error stays below half a bin.
• With the wrong Fs every frequency is scaled by 0.8: 123 474 x 0.8 = 98 779 Hz. The spectrum shape is identical, which is exactly why the mistake is easy to miss.
• A 20 % sample-rate mistake produced a 24.7 kHz frequency error. The metadata field Fs is therefore as important as the samples themselves.

Exercises

1. The same file is read with Fs = 1.2 MHz by mistake. Predict the reported tone frequency before running.
2. How many samples do you need so that the bin quantization error is below 1 Hz at 1 MS/s?
3. A tone is known to be at 200.0 kHz, and your analysis reports 160.0 kHz. What sample rate did the analysis assume if the true rate is 1 MS/s?

Answers (checked by running the lab script)

1. Every frequency is scaled by 1.2 / 1.0: 123 474.1 × 1.2 ≈ 148 169 Hz, an error of about +24.7 kHz.
2. The peak is read at the nearest bin, so the error is at most half a bin, Fs / (2N). For less than 1 Hz at 1 MS/s, N > 500 000; the next power of two is N = 524 288 (bin 1.9 Hz). Interpolating between bins gets there with far fewer samples.
3. The reported frequency scales with the assumed rate: 160 / 200 × 1 MS/s = 0.8 MS/s, the same mistake as in this lab.

Report checklist

• Record the assumed Fs, FFT size and tone frequency.
• Explain why the same FFT magnitude can support both a correct and a wrong conclusion.
• Attach the time-domain preview and the frequency-axis comparison plot.
• Quote the correct and wrong frequency errors from the metrics JSON.
• State which metadata field must be protected in a real IQ capture workflow.