Lab 8.11 — 16-QAM bridge: BER, EVM and implementation limits

Goal

Bridge QPSK to higher spectral efficiency with normalized Gray 16-QAM:

• map/demap four bits per symbol with thresholds at {-2, 0, +2}/√10;
• measure BER and EVM versus Eb/N0;
• expose gain/phase imbalance sensitivity;
• quantify fixed-point quantization and saturation;
• use 16-QAM as an alternative OFDM payload constellation.

Run

python blocks/block_08_modulation_and_synchronization/python/lab_8_11_16qam_tradeoffs.py

The default sweep compares 200,000 bits per Eb/N0 point. The OFDM bridge uses 52 active subcarriers: QPSK carries 104 bits per OFDM symbol and 16-QAM carries 208.

Why this lab matters

QPSK is the workhorse of this course, but links that need more throughput in the same bandwidth move to higher-order QAM. The trade is simple to state and easy to underestimate: 16-QAM carries 4 bits per symbol instead of 2, but the points are closer together, so every impairment (noise, gain/phase imbalance, quantization, clipping) eats a larger fraction of the decision margin. This lab measures that price in three places: against noise, against analog imbalance, and against fixed-point word length, so you can decide when the extra bit density is worth having.

Artifacts

docs/assets/lab811_16qam_ber.png
docs/assets/lab811_16qam_imbalance.png
docs/assets/lab811_16qam_fixed_point.png
docs/assets/lab811_16qam_metrics.json

What to expect

Bit error rate and EVM against Eb/N0 (200 000 compared bits per point). The last column is the closed-form Gray-coded 16-QAM approximation BER ≈ (3/8)·erfc(sqrt(0.4·Eb/N0)):

Eb/N0	errors	BER	EVM	theory
0 dB	28 371	1.42e-1	50.1 %	1.39e-1
4 dB	11 749	5.87e-2	31.6 %	5.86e-2
8 dB	1 825	9.13e-3	19.9 %	9.25e-3
12 dB	31	1.55e-4	12.5 %	1.39e-4
16 dB	0	0 (< 5e-6)	7.9 %	6e-9

• The simulation matches theory to within Monte Carlo noise, which is what validates the Gray mapping and the decision thresholds at {-2, 0, +2}/sqrt(10).
• BER = 0 at 16 dB means "fewer than about 1 error in 200 000 bits", not "error free": the theory value is about 6e-9, far below what 200 000 bits can resolve.
• EVM tracks Eb/N0 smoothly (50 % → 8 %) while BER falls by orders of magnitude; that is why EVM is the better early-warning metric (compare Lab 8.7).

The imbalance and word-length sweeps are noiseless, so what they show is the pure effect of the impairment, and BER stays 0 in every row while EVM moves. Gain/phase imbalance from (−2 dB, 0°) to (+2 dB, 12°) gives EVM 23.4 %, 11.8 %, 6.2 %, 14.9 %, 26.9 %: the constellation gets visibly worse long before a decision flips, and there is no noise yet to push the points across a threshold. For fixed-point, EVM falls from 23.8 % (4 bits) to 11.0 % (6), 8.6 % (8) and 8.1 % (10 bits) and then stops improving. The reason is in the saturation counter, which reads 50 144 for every word length: the sweep drives the signal at 1.15x, so the outer constellation levels (0.95 x 1.15 = 1.09) exceed the +1.0 range and are clipped whatever the word length, which sets an EVM floor near 8 %. BER stays 0 only because a clipped outer point is still on the right side of its decision threshold, which is exactly why the interpretation below says extra bits cannot repair clipping caused by insufficient headroom.

Exercises

1. Find the Eb/N0 at which 16-QAM reaches BER 1e-3, and the Eb/N0 at which QPSK does (0.5·erfc(sqrt(Eb/N0))). How many dB does the doubled bit density cost?
2. In fixed_point_sweep, change the drive factor 1.15 to 1.0 and then to 0.9. The saturation counter drops (with drive 1.0 it reaches zero from 8 bits upward) and EVM now keeps falling with word length: with drive 1.0 it is about 0.6 % at 8 bits and 0.06 % at 10 bits. Explain why headroom, not word length, was the limit before.
3. The imbalance sweep is noiseless. Add channel noise at Eb/N0 = 12 dB on top of the (+1 dB, 8°) imbalance. Does BER stay near the noise-only value of about 1.5e-4? Predict first, then run.

Answers (checked by running the lab script)

1. QPSK reaches 1e-3 at Eb/N0 = 6.79 dB. For Gray 16-QAM the nearest-neighbour approximation (3/8)·erfc(√(0.4·Eb/N0)) gives 10.52 dB, and the lab's simulation agrees (1.02e-3 at 10.5 dB over 1.6 million bits). Doubling the bits per symbol costs about 3.7 dB.
2. With drive 1.15 the signal was pushed past full scale, so the saturation counter was busy and clipping, not rounding, set the error floor: adding bits refined values that were already wrong. With drive 1.0 the peaks fit, saturation disappears from 8 bits up and EVM falls about 10× per two bits, the rounding behaviour you expect. Headroom first, then word length.
3. No: it gets about 20 times worse. The imbalance alone (EVM 14.8 %) causes no errors in a noiseless channel, but at Eb/N0 = 12 dB the BER is 2.8e-3 instead of 1.35e-4 with noise only. The imbalance eats into the distance to the decision boundaries, and noise that would otherwise be harmless now crosses them. A distortion that "costs nothing" without noise is a lost SNR margin with it.

Interpretation

16-QAM doubles payload density relative to QPSK but reduces decision margin. EVM therefore exposes gain/phase imbalance and numeric saturation before BER necessarily becomes large. Increasing word length reduces quantization error, while extra bits cannot repair clipping caused by insufficient headroom.