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Uncertainty/Variability/Noise on the measured transfer function is quantified with respect to its input and output signal, i.e. the output voltage and the duty.

Five sets of experiments were designed where the system was perturbed with different perturbation amplitudes. In each of the experiments, the output voltage and the duty signal is measured by injecting a multi-tone sinusoid of frequency 𝐹_𝑃 while the amplitude 𝐴′_𝑃 (expressed in LSB) is varied according to

𝐴′𝑃 = 𝑡. 𝐴𝑝 ; 𝑡 = 2𝑜 ; 𝑜 = −2, −1, 0, +1, +2 _(6-4) where 𝐴_𝑝 defines the amplitude defined in Equation (3-3). In other words, the perturbation amplitude is varied from a quarter of an LSB, to half of an LSB to a quad of an LSB in every consecutive experiment. Successively, five sets of the transfer function are accessible based on five sets of experiments.

Prior to uncertainty analysis, each transfer function measurement is averaged to obtain a high SNR. The uncertainty on the transfer function magnitude |H| illustrated in Figure 6-3 a) and phase φ(𝐻) in Figure 6-3 b) is plotted with respect to the peak amplitude of the output voltage. The uncertainty on the magnitude response is expressed as relative uncertainty (Standard Deviation Ratio) in percentage and as uncertainty (Standard Deviation) in degrees on the phase response.

Careful examination shows low uncertainty/noise on the magnitude and phase response as the amplitude on the output voltage is increased. This is an expected result while extracting a signal buried in the Additive White Gaussian Noise (AWGN). Conversely, it is the characteristic behaviour of a signal suppressed in AWGN of the quantisers and random analog noise of the system.

Similarly, the relative uncertainty on the magnitude (Figure 6-4 a)) and phase (Figure 6-4 b)) response is quantified with respect to the peak amplitude of the duty signal. The relative standard deviation on the magnitude plot varies inversely

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to the amplitude of the duty signal. This analysis evidently depicts the source of AWGN in the system and its influence on the transfer function magnitude and phase response.

Figure 6-3 Relative Standard Deviation

a) |H| with respect to Peak Output Voltage Amplitude b) φ(H) with respect to Peak Output Voltage Amplitude

Figure 6-4 Relative Standard Deviation a) |H| with respect to Peak Duty Amplitude b) φ(H) with respect to Peak Duty Amplitude

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From the above analysis, it can be expressed that the relative uncertainty (standard deviation) on the |𝐻| and the uncertainty on 𝜑(𝐻) response with respect to the peak output voltage amplitude or the duty shadows a power law of the form

𝜎|𝐻|= 𝑎. (𝑣𝑜)𝑏 (6-5)

𝜎𝜑(𝐻)= 𝑐. (𝑣𝑜)𝑑 (6-6)

𝜎_|𝐻| = 𝑒. (𝑑𝑢𝑡𝑦) (6-7)

𝜎_𝜑(𝐻) = 𝑔. (𝑑𝑢𝑡𝑦)ℎ _(6-8)

whereby the regression coefficients a, c, e, g are the scaling factors and b, d, f, h in Table 6-1 define the dependency of uncertainty on the transfer function in relation to the peak amplitude of the output voltage/duty.

Table 6-1 Regression Coefficients of Relative Standard Deviation on |H| and φ(H) at Frequencies of Interest

In other words, the exponent b, d, f and, h signifies that the uncertainty on transfer function is inversely proportional to the amplitude of the output voltage/duty. This result satisfies the inverse interdependence between the input amplitude and the variability on the transfer function.

Frequency (Hz) a b c d e f g h 97 3.502 -7.0 0.360 -1.5 10.997 -1.3 0.012 -1.3 0.514 -1.2 195 3.594 -12.8 0.256 -1.1 9.666 -1.2 0.017 -1.0 0.580 -1.1 391 3.757 -28.0 0.280 -1.3 9.205 -1.2 0.013 -1.1 0.566 -1.0 781 3.609 -71.4 0.280 -1.2 8.533 -1.1 0.015 -1.1 0.579 -1.0 1560 1.395 -127.0 0.103 -1.1 3.843 -1.1 0.020 -1.0 0.708 -1.0 3100 0.361 -143.1 0.039 -1.1 2.025 -1.1 0.030 -1.0 1.543 -1.0 6250 0.120 -145.2 0.030 -1.0 1.788 -1.1 0.078 -0.9 4.550 -1.0 12500 0.043 -149.9 0.028 -1.1 1.575 -1.1 0.197 -1.0 11.050 -1.0 25000 0.019 -172.3 0.028 -1.3 2.033 -1.1 0.743 -1.2 29.850 -1.0 50000 0.011 131.9 0.026 -1.1 1.699 -1.0 0.899 -1.0 47.237 -1.0

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Pictorially, the shape of the regression coefficients with respect to the plurality of frequencies of interest depicts that the SNR on the transfer function is not consistent across the bandwidth under consideration. In other words, there is large uncertainty on both the magnitude and phase response at lower frequencies from 97 Hz to 781 Hz, compared to relatively low noise variance at higher frequencies, typically 3.1 kHz to 50 kHz. The possible source of uncertainty can result from the quantisation noise of the ADC/DPWM, analog noise, etc. However, it is difficult to describe the origin of this uncertainty. Nevertheless, it is important to quantify this noise source because the SNR on the transfer function influences the estimate of uncertainty on the model coefficients.

To recognise the source of this uneven SNR on the transfer function magnitude and phase, firstly, open-loop characteristics of the system are determined. By superimposing a sinusoidal noise source of amplitude equivalent to 1 % variation of the output voltage (12 mV), the noise spectral density of the quantiser (ADC) is analysed at a frequency band of interest.

The noise spectral density is extracted by performing FFT on single acquisition of the measured output voltage. Since noise is not constant/flat, this method yields biased results especially for lower frequencies. Therefore, for accurate estimation, the square root of the sum of squares of the individual noise spectral densities over several acquisitions of the output voltage is calculated. Mathematically, the total noise spectral density over 𝑁 acquisitions expressed in 𝑉/√𝐻𝑧 is 𝜎_𝑣𝑜 = √1 𝑛∑ (𝜎𝑣𝑜_𝑛) 2 𝑁 𝑛=1 (6-9)

Similarly, the noise spectral density of the quantisers is computed for each perturbation frequency. Figure 6-5 a) depicts the amplitude noise spectral density of the output voltage of an ADC including the DC components and the harmonics.

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Figure 6-5 Amplitude Noise Spectral Density on

a) Output Voltage Including DC and Harmonic Components b) Output Voltage Excluding DC and Harmonic Components

Neglecting DC components associated with each perturbation frequency as shown in Figure 6-5 b), two inferences can be formulated.

First, the amplitude noise spectral density over the measured Nyquist bandwidth represents a flat-band noise with average noise spectral density three times the variance of the quantisation noise. Secondly, the noise spectral density decreases linearly with frequency from 10 Hz to approximately 3 kHz. The result advocates the existence of flicker noise in the system with a corner frequency of 3 kHz and flat in-band quantisation white noise with noise spectral density equivalent to three times the variance of the ADC quantiser. Now both these noise sources may reduce the SNR on the transfer function. However, theoretical justification is required. Therefore, a simulation model is created in SIMetrix to validate the above results.