Is One Error in a Million Good Enough?
M5-L2 gave you the formulas. This lesson asks what the Q in them is, how steep the curve is, and what BER is actually acceptable. The answer to the last one is not what you expect.
A Probability, Not a Count
BER is expected bit errors divided by bits sent — so a measurement only estimates it. Poisson statistics say you need about 100 errors for 10% confidence, because 1/√100 = 0.1.
At 1 Mbps the same test takes 27.8 hours. Below 10−9, BER is extrapolated from the curve — never measured directly.
1.2% of Packets Are Gone
A 1500-byte frame is L = 12,000 bits. At p = 10−6, Lp = 0.012 and PER = 1 − e−0.012 = 1.19 × 10−2. “One in a million” loses one packet in eighty-four.
The Gaussian Tail
Q(x) is the probability a standard normal exceeds x — and x is a distance in standard deviations, exactly the dmin/2σ of M5-L3.
Three more decades of reliability cost only 1.245 more in x — 26% further from the decision boundary.
Q Has No Closed Form
The identity is exact; the approximation runs +9.4% high at x = 3 and +2.6% at x = 6. Q(0) = 0.5 and Q(x) + Q(−x) = 1 come free from symmetry.
The Waterfall Plotter
Slide the operating point along both curves, set a target, and watch the 3 dB gap stay constant.
What the Link Must Deliver
- 10−3: Q⁻¹ = 3.0902 → 4.775 → 6.79 dB
- 10−6: 4.7534 → 11.298 → 10.53 dB — M6-L4’s 10.5 dB ✓
- 10−9: 5.9978 → 17.987 → 12.55 dB
- OOK/BFSK: exactly +3.01 dB at every target
The Waterfall Is the Cliff
Near 10−6 one decibel is worth a whole decade of BER. Read that backwards: lose 1 dB and your errors multiply by ten; lose 3 dB and they multiply by a thousand.
There is no separate cliff mechanism. Analog loses 3 dB and gets 3 dB noisier; digital loses 3 dB and gets a thousand times wronger.
Nobody Optimises Raw BER
- Classical FEC: 5–8 dB of gain at 10−6
- LDPC and turbo (M6-L1): 10–11 dB
- Rate-1/2 ceiling at 10−9: 12.55 − (−0.82) = 13.37 dB
- Paid for in bandwidth, rate and delay
- LTE and 5G target ~10% first-try BLER and let HARQ repair it
A link that never errs was running below capacity. Raw BER is an input to a code, not a goal.
What you learned
- BER is a probability; 10−9 needs 1011 bits to measure
- PER ≈ 1 − e−Lp: 10−6 costs 1.2% of packets
- Q(4.753) = 10−6, Q(5.998) = 10−9
- Eb/N0 = ½[Q⁻¹(P)]²: 10.53 dB at 10−6
- The waterfall’s steepness is the cliff effect