Lab 8.12 — GFSK: BT, occupied bandwidth and discriminator BER

Goal

Build a continuous-phase GFSK link and study:

• Gaussian pulse shaping and the BT bandwidth/time trade-off;
• constant-envelope transmission and approximately 0 dB PAPR;
• quadrature-discriminator decisions;
• occupied bandwidth and BER versus Eb/N0;
• differences from OFDM peak back-off and CSS processing gain.

Why this lab matters

Not every radio can afford a linear power amplifier. GFSK (the modulation behind Bluetooth and many low-power sub-GHz links) keeps a constant envelope, so a cheap, efficient, nonlinear amplifier can transmit it without spectral regrowth. The price is paid in two places that this lab measures: the Gaussian filter's BT product trades occupied bandwidth against inter-symbol interference (ISI), and the simple discriminator receiver is far less sensitive than a coherent one.

Run

python blocks/block_08_modulation_and_synchronization/python/lab_8_12_gfsk_bt_ber.py

Artifacts

docs/assets/lab812_gfsk_bt_waveforms.png
docs/assets/lab812_gfsk_bandwidth.png
docs/assets/lab812_gfsk_ber.png
docs/assets/lab812_gfsk_metrics.json

What to expect

BT sets the 99 % occupied bandwidth (as a fraction of the sample rate, 8 samples per symbol): 0.079 at BT = 0.3, 0.099 at BT = 0.5 and 0.123 at BT = 1.0, i.e. 0.63, 0.79 and 0.98 of the symbol rate. The constant-envelope error is at floating-point noise level (2e-16) and the PAPR is 0 dB for every BT, which is the whole point of the modulation.

The BER sweep (BT = 0.5, about 40 000 bits per point) reads:

Eb/N0	BER
0 dB	0.466
8 dB	0.330
16 dB	0.130
24 dB	0.080
32 dB	0.034
40 dB	0.0084

Read this curve for what it is:

• It needs an absurd 40 dB to reach 1 %. A good GFSK receiver reaches that at roughly 10-12 dB. This lab's receiver has no channel filter: it sums raw per-sample phase differences over each symbol, so it integrates the noise of the whole sample-rate bandwidth, and it decides from a symbol-aligned sum that leaves very little margin for worst-case ISI patterns. The curve shows the trend (BER falls as Eb/N0 rises) and the relative cost of this receiver, not the performance a real GFSK link achieves.
• ISI is visible without any noise. With no noise at all this discriminator makes no errors at BT = 0.5 or 1.0, but at BT = 0.3 it already decides about 20 % of the bits wrongly. The narrow filter smears each symbol into its neighbours so much that this simple detector cannot recover them; real receivers use a matched (Gaussian) receive filter or sequence detection for low BT.
• The BER ends at zero only around 60 dB in a quick check, because worst-case bit patterns leave a small decision margin; that is a property of this detector, not of GFSK.

Exercises

1. Reproduce the noiseless result: run discriminator_demodulate on a noiseless waveform for BT = 0.3, 0.5 and 1.0. What BER do you get, and what does that say about the choice of BT?
2. Add a moving-average (symbol-length) filter to the received waveform before the discriminator. Does the Eb/N0 needed for BER 1e-2 fall? Try 2, 4 and 8 samples.
3. Compare the occupied bandwidth against BT. Where is the knee, and how does it relate to the ISI you measured in exercise 1?

Answers (checked by running the lab script)

1. Over 20 000 bits: BER 0.199 at BT = 0.3, and 0 at BT = 0.5 and 1.0 (0.187 at BT = 0.4, 0.280 at BT = 0.2). Below BT ≈ 0.5 the Gaussian filter spreads each symbol so far into its neighbours that this detector errs without any noise. BT is a bandwidth knob with a cliff behind it for a simple receiver.
2. No. A symbol-length (8-sample) average lowers the BER at low Eb/N0 (0.33 → 0.15 at 8 dB) but adds its own ISI, which creates a floor near 0.1, so 1e-2 is never reached; 2–3 samples leave the 1e-2 point at 40–42 dB, and from 5 samples errors appear even without noise. The reason is the detector: the sum of the phase differences over a symbol telescopes to the phase difference between its first and last samples, so it averages nothing, and its errors come from a few bit patterns with very little margin. A filter long enough to cut noise also shrinks that margin. A real fix is a matched Gaussian receive filter with a decision at the symbol centre, or sequence detection.
3. The 99 % bandwidth grows smoothly with BT: 0.066, 0.079, 0.090, 0.098, 0.111 and 0.123 of the sample rate for BT = 0.2, 0.3, 0.4, 0.5, 0.7 and 1.0 (×8 for the symbol rate: 0.53 to 0.98). The knee is not in the bandwidth but in the ISI of exercise 1, which jumps between BT = 0.4 and 0.5. Going from 0.5 to 0.3 saves only about 20 % of bandwidth and costs 20 % bit errors, which is why standards that use BT = 0.3 pair it with better receivers.

Interpretation

Lower BT narrows the occupied spectrum but spreads transitions across more neighbouring symbols. GFSK keeps a constant envelope and is friendly to nonlinear transmitters, unlike OFDM, but its simple discriminator receiver pays a sensitivity penalty. The BER curve therefore reports actual decisions rather than inferring link quality from spectrum alone.