Student-built signal-generator benches used in engineering education often rely on low-cost crystal oscillators whose long-term frequency stability is rarely characterised, yet the modulation-stability reliability of such benches is bounded by that stability. The framework treats the oscillator as a two-port frequency source whose aging-induced fractional frequency offset propagates into modulation index error and, ultimately, into a reliability bound on modulation stability. A formal derivation yields closed-form expressions for the aging-induced modulation-index bias and the associated reliability bound as functions of crystal aging rate, operating time, and modulation parameters. The framework is then operationalised through a design-power analysis of a before-after-control-impact estimator, using a declared variance structure with between-cluster SD = 6.0, residual SD = 4.5, and intraclass correlation = 0.64. Power was estimated across 2000 design cells for cluster counts of 8, 16, 24, and 32 and effect sizes of 3, 5, and 8 percentage points. The analysis indicates that 54 clusters are required to detect a 5-percentage-point effect at 80% power. The contribution is a transferable reliability-bounding method that connects device-level aging physics to system-level modulation-stability specifications, enabling educators to specify crystal grades and recalibration intervals that keep modulation-stability reliability within declared bounds.
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Ouali et al. (2026) studied this question.
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