Measurement of Analog Modulation Index with a Calibrated Radio Frequency Attenuator

This paper presents a method for the accurate and traceable measurement of the analog modulation index. A calibrated step attenuator was used as the main apparatus because it has a higher dynamic range and lower uncertainty than a spectrum analyzer or an oscilloscope. In amplitude modulation (AM), the modulation index is obtained from the amplitude difference between the carrier and the first sideband, as in the conventional method. The resolution and calibration uncertainties of the step attenuator were propagated to the measurement uncertainty of the modulation index. The uncertainty produced by the impedance mismatch and repeatability was also included. For frequency modulation (FM) and phase modulation (PM), the modulation index, β, was estimated (with the step attenuator) from the spectrum of each sideband through the nonlinear fitting of the Bessel function. Thus, the uncertainty of the fitting process was added to the uncertainty of the measurement. The three modulations, AM, FM, and PM, exhibited an expanded uncertainty (approximately 95% confidence level, k = 2) of 0.372% for 50% nominal depth of the AM, 88.8 Hz for the peak frequency deviation of 10 kHz, and 0.88 mrad for a 0.1 radian modulation index, respectively.