Consider the down-converter chain of a satellite ground station: the run of modules that takes the antenna's signal, amplifies it, mixes it down and hands an intermediate frequency to the demodulator. Eight replaceable items, 102.5 failures per 10⁶ hours between them. Every value is an illustrative teaching figure; the method is the standard's.
| Item | λ per 10⁶ h | Test | Kind | |
|---|---|---|---|---|
| F1 · Low-noise amplifier | 8.0 | T1 · RF input power detector | online | |
| F2 · Image filter | 1.5 | T2 · LO lock detect | online | |
| F3 · Mixer | 6.0 | T3 · IF output level | online | |
| F4 · Local oscillator | 22.0 | T4 · PSU rail monitors | online | |
| F5 · IF amplifier | 12.0 | T5 · Controller watchdog | online | |
| F6 · Power supply | 30.0 | T6 · Noise-figure check | off line | |
| F7 · Controller board | 18.0 | |||
| F8 · Output isolator, drift | 5.0 |
The matrix, and what it says
Reading the columns under the online tests alone, T1 to T5:
| Item | Signature | Verdict |
|---|---|---|
| F1 | 1 0 1 0 0 | isolated |
| F2 | 0 0 1 0 0 | ambiguous |
| F3 | 0 0 1 0 0 | ambiguous |
| F4 | 0 1 1 0 0 | isolated |
| F5 | 0 0 1 0 0 | ambiguous |
| F6 | 1 1 1 1 1 | isolated |
| F7 | 0 0 0 0 1 | isolated |
| F8 | 0 0 0 0 0 | undetected |
FD = (102.5 − 5.0) / 102.5 = 95.1 per cent
FI(1) = (102.5 − 5.0 − 19.5) / 97.5 = 80.0 per cent of what is detected
Two findings, and they are different problems. The isolator's slow degradation trips nothing: the IF level check never crosses its threshold because the loss creeps. And the image filter, the mixer and the IF amplifier are indistinguishable, because the only online test that sees any of them is the one that sees all three. That group is 19.5 per 10⁶ hours arriving as one symptom.
Two candidate tests, two different purchases
T6, an off-line noise-figure check. It responds to F1, F2 and the isolator's degradation. Adding it:
FD = 100 per cent and FI(1) = 82.4 per cent, with the group reduced to F3 + F5
It is the only thing on the list that finds F8 at all, so it is worth 4.9 points of detection. It splits F2 out of the group, which is worth 2.4 points of isolation, because F2 is the smallest rate in it.
T7, a mixer bias current monitor. It responds to F3 and nothing else. Adding it instead:
FD = 95.1 per cent, unchanged, and FI(1) = 86.2 per cent
It detects nothing new whatsoever and is worth 6.2 points of isolation, more than twice what the off-line check delivers there, because it splits the largest item out of the group.
With both, every fault is detected and every fault is isolated to one item: FD = 100 per cent, FI(1) = 100 per cent.
The two tests are not competing. One answers "can we see it", the other answers "can we name it", and a requirement written as a single testability percentage cannot express which one the programme is short of.
What the ambiguity costs at the bench
A group of n costs (n+1)/2 removals per fault. Over 1,000 operating hours:
| Configuration | Removals | Genuine faults | Unnecessary |
|---|---|---|---|
| Online BIT only | 0.117 | 0.098 | 16.7% |
| With both extra tests | 0.102 | 0.102 | 0% |
One removal in six is a good module, pulled to find out whether it was the faulty one. That is the number the support contract feels, and it is invisible in a detection figure of 95 per cent.
Reading the strategy out of the same matrix
Test order is an optimisation, and the matrix contains the answer. T3 first, because it responds to six of the eight items and halves the problem in one measurement: if it passes, the candidates are the controller and the isolator; if it fails, the LO lock and the RF input detector split the rest. Starting instead with the PSU rail monitors, which many designs do because they are cheap and always present, resolves one item and leaves seven.
What the analysis leaves open
The report says 100 per cent detection and 100 per cent isolation once both tests are added, and that claim is worth exactly as much as the false alarm rate that is not in it. If the bias monitor trips on a cold start, technicians will learn to ignore it, and the isolation figure it bought becomes a number on a page. The two figures this analysis produces and the one it cannot are quoted together or not at all, which the testability concept page works through in detail.