Lab 8.22 — Packet-level CSS synchronization and PER

Goal

Extend the signal-level CSS detector from Labs 8.20/8.21 into a deterministic packet receiver:

• detect a repeated-upchirp preamble and estimate packet start;
• verify a two-symbol sync word and two downchirps;
• estimate coarse integer-bin and fine fractional-bin CFO;
• correct CFO before payload decisions;
• measure SER, PER, missed detections and false alarms;
• expose sample-rate-offset sensitivity.

The waveform remains CSS/LoRa-like, not LoRa-compatible. It deliberately omits the LoRa header, whitening, coding, interleaving and CRC.

Packet structure

noise prefix | 8 upchirps | sync symbols 18,52 | 2 downchirps |
16 payload symbols | noise suffix

The packet-start metric dechirps each candidate preamble window, accumulates FFT-bin power over all repeated chirps and compares the strongest bin with the mean bin power. The sync word and downchirps reject a high timing metric that does not have the expected packet structure.

CFO estimation

The preamble FFT peak supplies the integer-bin CFO. Phase advance of that peak across consecutive repeated chirps supplies the fractional-bin CFO:

CFO = integer FFT-bin shift + phase_step / (2π)

The default experiment injects 1.25 bin spacings. Without correction every payload packet fails; after synchronization and correction the reference -6 dB point reaches zero PER for the deterministic seed.

Run

python blocks/block_08_modulation_and_synchronization/python/lab_8_22_css_packet_sync_per.py

Generated artifacts

docs/assets/lab822_css_timing_metric.png
docs/assets/lab822_css_per_vs_snr.png
docs/assets/lab822_css_sro_sensitivity.png
docs/assets/lab822_css_packet_metrics.json

Statistical scope

• 1000 packets at every SNR point;
• 16,000 compared payload symbols per SNR point;
• 1000 independent noise-only false-alarm trials;
• 200 packets per sample-rate-offset point.

The zero-error points are finite tests, not proof of an arbitrarily low PER. With 1000 clean packets, the usual zero-failure 95% upper bound is approximately 3/1000.

Acceptance criteria

• the example packet start, sync word and downchirps are recovered;
• normalized CFO error is below 0.05 bin;
• corrected PER is lower than uncorrected PER at the reference point;
• at least 1000 packets are evaluated per SNR point;
• missed-detection and false-alarm statistics are present;
• the SRO sweep shows both degradation and the benefit/limit of resampling correction.

What to expect

Packet results against SNR (1000 packets per point, 1.25-bin CFO injected):

SNR	Missed detection	PER uncorrected	PER corrected	SER corrected
−15 dB	0.948	1.0	1.0	0.609
−12 dB	0.419	1.0	0.93	0.139
−9 dB	0.013	1.0	0.045	0.004
−6 dB	0	1.0	0	0
−3 dB	0	1.0	0	0

Sample-rate offset (200 packets per point): up to 100 ppm nothing fails; at 250 ppm every uncorrected packet fails and correction recovers all of them; at 500 ppm even the corrected receiver loses 20 % of packets. No false alarms in 1000 noise-only trials.

• Without CFO correction nothing gets through at any SNR: a 1.25-bin offset shifts every symbol by one bin plus a quarter, which is exactly the failure Lab 8.21 predicts.
• Missed detection dominates below −9 dB. The packet is lost before the payload detector sees it, so PER rises faster than SER.

Exercises

1. What is the 95 % upper bound on PER at −6 dB given 0 failures in 1000 packets? What would you need to claim PER below 1e-4?
2. Convert 250 ppm of sample-rate offset into how far a 16-symbol payload drifts, in samples, by its last symbol. Why does even the corrected receiver start losing packets at 500 ppm?

Answers (checked by running the lab script)

1. With 0 failures in 1000 packets the 95 % upper bound is about 3/1000 = 3·10⁻³ (rule of three). To claim PER < 10⁻⁴ you need about 30 000 packets without a failure.
2. From the packet start to the end of the payload there are (8 + 2 + 2 + 16) × 128 = 3584 chips, so 250 ppm drifts by 0.9 chip and 500 ppm by 1.8 chips; over the 16 payload symbols alone, 0.5 and 1.0 chip. In CSS a timing slip of one chip moves the dechirped peak by one bin, like a CFO of one bin. The corrected receiver removes a constant offset estimated on the preamble, but a sample-rate offset keeps growing through the payload, so at 500 ppm the last symbols are about a bin off again and some packets fail.