Fig. 1(a) shows a block diagram and a photograph of the proposed measurement system for the analog modulation indices. The modulated signal generated by the device under test (DUT) was passed through the calibrated attenuator to the spectrum analyzer. The spectrum analyzer measured the strength of the carrier and sideband signals. Note that the spectrum analyzer had not yet been calibrated. The DUT and the spectrum analyzer were synchronized with 10 MHz, and each device was remotely controlled via a general purpose interface bus (GPIB, IEEE 488). We used a step attenuator with a resolution of 0.1 dB, which was calibrated at 0 dB to 100 dB [8].

Prior to measuring, we analyzed the effect of the impedance mismatch. The state of the impedance matching was adjusted by the insertion of an impedance stub tuner between the spectrum analyzer and the step attenuator (Fig. 2). The analog modulation index was measured by taking the relative amplitude difference between the carrier and sidebands (see a detailed explanation in Section III). Table 1 shows the measurement results. The absolute power of the carrier varied greatly on the basis of the status of the impedance stub tuner. The relative difference between the first sideband and the carrier, however, did not change regardless of the impedance matching status of the impedance tuner. Thus, the impedance mismatch did not significantly affect the analog modulation index measurement.

AM changes the amplitude of the carrier, c(t), on the basis of the modulating signal or message, s(t). If s(t) is a continuous wave (CW), similar to that of the carrier, the AM signal, y(t), can be easily expressed as follows:

where A is the amplitude of the carrier and m is the modulation index of AM. Thus, the amplitude of the sideband signal (f_c ± f_m) changes in proportion to the modulation index, and the modulation index is calculated as follows:

where V_C and V_SB are the amplitude of the carrier and sideband, respectively. The differences between the carrier and sideband signals on the dB scale are represented by ΔP, and it is easily obtained from the delta marker on the spectrum analyzer. In this paper, all the measured values were corrected with the calibrated attenuator for traceability to the measurement standard (details are provided in Section IV).

Fig. 3 shows the AM measurement results. For the DUT, the modulation index and the frequency of the modulating signal, f_m, were set to 50% and 50 kHz, respectively. Thus, the frequency interval between the carrier and sideband signals became 50 kHz. The nonlinear distortion was generated in the DUT because the large modulation index of 50% produced additional harmonics at f_c ± nf_m (n > 2). In theory, both of the first sidebands should have the same amplitude; however, they should exhibit asymmetric characteristics in the presence of FM distortion [9]. In this study, the average value of the two signals was used as the V_SB because the difference between them was very small, less than 0.005 dB. This value is counted as the standard uncertainty for the unbalanced sidebands, assuming a rectangular distribution of ±0.0025 dB.

FM is expressed as (6), where Δf_p indicates the peak frequency deviation. Therefore, the changes in the frequency of y(t) depend on s(t). The amplitude, A, is time invariant:

If s(t) is CW, as was the case for AM, (6) can be represented as follows:

where β = Δf_p/f_m indicates the FM modulation index. Usually, the peak frequency deviation, Δf_p, is measured for a given frequency of the modulating signal, f_m. Eq. (7) also represents PM given that s(t) = cos(2πf_mt) causes a phase change on y(t). Thus, when s(t) is a sinusoidal function, the FM and PM have the same form, and they are commonly referred to as angle modulation. Mathematically, (7) can be expressed as the sum of the Bessel function of the first kind J as follows:

where k is the order of the sidebands. For example, when s(t) is a CW, the amplitude values of the carrier, first sideband, and second sideband are AJ₀(β), AJ₁(β), and AJ₂(β), respectively. As stated in previous section, the calculation of analog modulation index requires the relative difference between the carrier and each sideband. Therefore, the amplitude of carrier AJ₀(β) can be normalized to 1.

Then, the angle modulation index β can be estimated by minimizing the difference between the measurement and estimated values through nonlinear least squares fitting:

Here y_meas(k) is the measured value of the k-th sideband. The nlinfit function in the MATLAB environment was applied as the estimator [10]. This is based on the Levenberg-Marquardt nonlinear least squares algorithm [11]. Nonlinear fitting problems often reach the local minima or slowly converge depending on the initial value. Robust results were obtained from the initial β, which is Δf_p divided by f_m (both are the values for the DUT setting).

Fig. 4 shows the measurement results. As was previously described, the number and shape of the generated sidebands depend on β. The fitted results (“x” marks) on the basis (9) agree well with the measured results (solid line). Estimation of β using 23 sideband measurements takes less than 0.05 seconds on an Intel Xeon E5 2.4 GHz CPU.

The measuring uncertainty of the AM modulation index is caused by the uncertainty introduced by the attenuator and the impedance mismatch between the DUT and the attenuator. The attenuator had a reading uncertainty of 0.1/(23) dB obtained from a resolution of 0.1 dB and a calibration uncertainty of 0.012 dB. If we actually correct the impedance mismatch by using the measured impedance values, the associated uncertainty will significantly increase due to the uncertainty of a vector network analyzer. To resolve this, we measured the difference the carrier and first sideband of the AM signal under an arbitrary matching condition in Table 1. To account for the uncertainty of the impedance mismatch for the scheme above, the standard deviation was obtained from the measured values in Table 1. Next, (4) was differentiated to propagate each uncertainty to the AM modulation index. The obtained sensitivity coefficient was log(10)×10ΔP+20 log 10(2)20/20. The coefficient was dependent on ΔP; thus, the large modulation index produced a large measurement uncertainty. Type A uncertainty was evaluated from 10 repeated measurements. The uncertainty budget is summarized in Table 2. The modulation index of 50% was measured, and the combined standard uncertainty, u_c(m) 0.186%, was obtained by taking the root sum of squares of the uncertainty contributions presented in Table 2. Since the effective degree of freedom is sufficiently large, the expanded uncertainty U(m) at approximately 95% confidence level is obtained by multiplying u_c(m) by 2.

Tables 3 and 4 show the uncertainty budget for FM and PM, respectively. In both cases, the angle modulation required additional uncertainty for estimating β from the measured amplitude of each sideband. Unfortunately, the fitting uncertainty for β could no longer be regarded as analytical. Instead, it was calculated from the residuals, which were calculated from the difference between the fitting model and measurements [12, 13]:

where l is the number of sidebands used in the measurement, including the carrier; thus, l is equal to 2n + 1. The number of unknown parameter β used in the fitting process, i.e., 1, is indicated by p. Therefore, the covariance for the fitting parameter σβ2 was calculated using (12) with the fitting residuals σresidual2. It must be noted that σ_β directly became the fitting uncertainty of β in PM. In the case of FM, the sensitivity coefficient was f_m because the peak frequency deviation was Δ f_p = f_mβ. Thus, the covariance of the peak frequency deviation σΔfp2 for FM was calculated as follows:

The uncertainty of the attenuator could not be analytically propagated to the uncertainty of the FM and PM modulation indices because of the fitting process. Thus, Monte Carlo simulations were used to account for the uncertainty of the attenuator. The Monte Carlo simulations were performed separately for the reading and the calibration uncertainty of the attenuator because of their rectangular and normal distribution, respectively. The results are summarized in Tables 3 and 4. In addition, the impedance mismatch and repeatability uncertainties were accounted for in the same way as was done for AM. As shown in Fig. 2, the uncertainty of the impedance mismatch was evaluated by the insertion of the impedance tuner between the spectrum analyzer and the attenuator and the adjustment of the state of the tuner. The expanded uncertainty combining the above-described fitting uncertainty, attenuator uncertainty, impedance mismatch uncertainty, and repeatability was observed at 88.8 Hz to 571.4 Hz for FM (k = 2) and 0.88 mrad to 5.70 mrad (k = 2) for PM.