Lab 8.21 — CSS dechirp and FFT detector¶

Goal¶

Turn the waveform from Lab 8.20 into a reproducible symbol detector and quantify its limits:

build a complete symbol bank for the selected spreading factor;
derive the noiseless symbol-to-FFT-bin mapping;
detect random symbols after dechirping;
measure symbol error rate (SER) versus SNR;
measure sensitivity to carrier-frequency offset (CFO);
report peak-to-second-peak separation as a detector confidence metric.

Detector structure¶

flowchart LR
    RX[received CSS symbol] --> MIX[dechirp with conjugate reference]
    REF[reference upchirp] --> MIX
    MIX --> FFT[FFT]
    FFT --> ARGMAX[maximum-magnitude bin]
    ARGMAX --> MAP[inverse bin-to-symbol mapping]
    MAP --> OUT[detected symbol]

Run¶

python blocks/block_08_modulation_and_synchronization/python/lab_8_21_css_dechirp_fft.py

Generated artifacts¶

docs/assets/lab821_css_ser_vs_snr.png
docs/assets/lab821_css_ser_vs_cfo.png
docs/assets/lab821_css_example_fft.png
docs/assets/lab821_css_detector_metrics.json

Default experiment¶

spreading factor: SF=7;
bandwidth: 125 kHz;
800 random symbols per sweep point;
SNR sweep: -18 ... 0 dB;
normalized CFO sweep: -0.45 ... +0.45 FFT-bin spacings.

The normalized CFO axis is useful because one FFT-bin spacing equals BW / 2^SF. It makes the experiment portable across bandwidth and spreading-factor choices.

Acceptance criteria¶

The executable lab checks or exposes enough data to verify that:

the noiseless symbol-to-bin mapping is a permutation;
the example symbol is detected correctly;
SER at 0 dB is zero for the deterministic default seed;
the lowest-SNR point has a clearly nonzero SER;
fractional-bin CFO degrades the detector before the FFT peak crosses into a neighbouring bin.

Why SER, not only SNR¶

A spectrum or SNR estimate does not prove that a digital link works. This lab therefore reports the actual symbol decisions. The same principle should later be extended to BER, packet error rate (PER), missed detections and false alarms.

Follow-on work¶

The next packet-level CSS laboratory should add:

a repeated-chirp preamble;
packet-start detection;
coarse and fine CFO estimation;
sample-rate offset;
sync-word and downchirp handling;
BER/PER and false-alarm measurements.

Report checklist¶

[ ] Include the SER-versus-SNR curve.
[ ] Include the normalized-CFO sensitivity curve.
[ ] Include one dechirped FFT example.
[ ] State the number of compared symbols at every point.
[ ] Explain why peak-to-second-peak ratio is useful but does not replace SER/PER.