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.

What to expect¶

SER against SNR (800 symbols per point, SF = 7):

SNR	−18 dB	−15 dB	−12 dB	−9 dB	−6 dB	−3 dB	0 dB
SER	0.853	0.636	0.245	0.010	0	0	0

SER against normalized CFO (fraction of one bin spacing):

CFO	−0.45	−0.30	−0.15	0	+0.15	+0.30	+0.45
SER	0.405	0.078	0.015	0.011	0.019	0.080	0.431

The detector works below the noise floor (SER 1 % at −9 dB SNR): dechirping and a 128-point FFT collect the whole symbol's energy into one bin, a processing gain of 10*log10(128) = 21 dB.
Half a bin of CFO nearly destroys it: at ±0.45 bin the energy splits between two bins and SER exceeds 40 %. This is why Lab 8.22 estimates and removes CFO before payload decisions.

Exercises¶

Repeat the SNR sweep with SF = 9. By how many dB does the curve move, and does it match 10*log10(512/128)?
At what CFO (in hertz) does a 125 kHz, SF = 7 link lose half a bin? Compare with a 0.3 ppm oscillator offset at 868 MHz.

Answers (checked by running the lab script)

With 800 symbols per point the SF 9 curve reaches the SF 7 value at −9 dB (SER 1.3 %) at about −14.4 dB: a shift of about 5.5–6 dB, close to 10·log10(512/128) = 6.0 dB. The longer symbol collects four times the energy into one bin.
One bin is 125 kHz / 128 = 977 Hz, so half a bin is 488 Hz. A 0.3 ppm offset at 868 MHz is 260 Hz, about 0.27 bin: already in the range where the table shows SER rising (0.078 at 0.30 bin and −9 dB). Two oscillators of that quality can differ by twice as much, which is why a CSS receiver estimates the CFO from the preamble before deciding symbols.

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.