Improving resolution with averaging

This section has been published [142, 143] in a modified form in conference proceedings.

A Monte Carlo model was constructed which can test the expected improvement in the signal level error for an idealised receiver. In order to understand the model, it is necessary to understand how signal resolution can be improved through averaging. If a measurement, xi, can be repeated

N times, where the timing of the signal is known (i.e. it always appears at the same point in time),

then repeated measurements can be averaged using an arithmetic mean,x¯: ¯ x= 1 N N X i=1 xi (3.4)

The population mean is the underlying signal if no noise were present, and the sample mean is an approximation to it. Since the (white) noise is uncorrelated between measurements (i.e. uncor- related over time, albeit a long time is acceptable in this case due to the gap between consecutive

deviation of the noise distribution.

In the case of an ADC, the values xi do not form a continuous distribution, and instead are

put into discrete levels defined by the ADC parameters. This introduces a subtlety into the usual relationship for the standard deviation of the sample mean, the usual relationship being:

σx¯=σ/

√

N (3.5)

The error observed will be a function of where the true signal level is within the discretised level, the size of the discretised level, the noise distribution, and the number of averages, N. For

example, if the true signal level is right in the middle of a discretised level, and the size of the discretised level is far larger than the noise in the signal (10σfor example), then the unaveraged

signal will be equivalent to the averaged signal. Although it is probable that an analytic model for this additional constraint could be constructed, a Monte Carlo model of the situation can be constructed in very little time, and for that reason, was considered preferable.

The model assesses the error expected on a signal at any given noise level, by simulating AWGN inserted onto a DC signal that is measured multiple times and averaged (arithmetic mean averaging) in subsequent processing. A single discretisation bin of the ADC is chosen in the centre of the discretisation range. The top bin is centred at 0.5V and the bottom bin is centred at -0.5V. Therefore, any given bin covers a range of 3.92mV for 8bit (256 levels) and 0.0153mV for 16bit (65536 levels). The actual choice of discretisation bin is irrelevant as long as the noise does not push the signal above the maximum bin or below the minimum bin, as doing so causes a non-linear response. A series of DC levels are chosen within the selected discretisation bin, ranging from exactly on the input level the bin represents, to the edge of the range the bin represents. Where within the bin the correct value sits, changes the output of the model, as if the input is initially exactly correct, the error can only go up once the noise exceeds the bin range. If the the input is initially at the edge of the range, the initial error is equal to half the size of the bin, but the smallest amount of random noise causes the ADC to switch between the two bins at that edge, and the error quickly decreases before rising again. Most interesting is a point between these extremes, which initially has some error, that slowly decreases as the noise allows the ADC to jump between two discretisation levels, before reaching a minimum error at a threshold noise and then increasing again as the noise increases.

Both 8bit and 16bit ADCs are simulated, for single-shot operation and averaged operation, over 50 captures; the output (figure 3.10) is the difference between the level the ADC outputs and the actual DC input after averaging the ADC outputs over the number of captures (1 and 50 respectively). The experiment is repeated multiple times (averaging the magnitude of the error), until statistical fluctuation due to the random nature of the noise is minimised.

The results for when the DC input is a quarter-way through the discretisation range are shown in figure 3.11. It is assumed that noise never pushes the ADC outside of its maximum range, as otherwise a linear response cannot be maintained. Some statistical fluctuations due to the method of modelling mean that the lines are not perfectly smooth, but this should not detract from the conclusions drawn.

For this input range and taking 50 averages, there is no significant improvement in accuracy using 16bit over 8bit for AWGN with a standard deviation greater than 2mV. However, for single

INPUT NOISE ADC DIFF ERROR

Figure 3.10: The model takes an input level that sits within one discretisation bin as a DC level and adds a user defined amount of AWGN. The combined signal is digitised in a simulated (ideal) ADC with either 8bit or 16bit resolution. The value output is compared to the exact value input (considering a value with 64bit resolution as effectively exact for this purpose) by simply taking the difference, which is reported as an error. If averaging is being used, the average of multiple outputs from the simulated ADC are averaged with 64bit resolution before the difference is taken and the error reported. This process is itself averaged over multiple iterations to smooth out the variation due to the random nature of the AWGN added.

(a) 0 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 Noise Amplitude (mV) Error (mV) 8bit 16bit Difference (b) 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 Noise Amplitude (mV) Error (mV) 8bit 16bit Difference

Figure 3.11: Averaging improves the error and resolution. An input value a quarter of the way through a discretisation level is chosen as representative of the worst case averaging improvement in the presence of AWGN. There is a minimal difference (dashed-dotted line) between 16bit (dashed line) and 8bit (solid line) above the threshold noise of approximately 2mV when averaging 50 collections (a), but 16bit is superior in single shot mode (b). Switching to starting centred exactly on a discretisation point, the 16bit ADC never shows any advantage for 50 averages or single shot, but that is a best case and hence not representative. Bin size is 3.92mV for 8bit and 0.015mV for 16bit.

laboratory environment, no real advantage can be gleaned from a 16bit ADC over an 8bit ADC, as averaging can be used. In industrial use, single shot may be necessary, and this would potentially require the use of a 16bit ADC. If the input signal was ideal and therefore had no noise, averaging would confer no benefit. It is the random noise that allows the improvement via averaging to occur, and if the noise level were actually below the noise level that produced the minimum error for any given number of averages and signal level, then there would be a benefit from artificially adding AWGN to the signal, most likely after amplification but of course prior to digitisation.

For an input range of 400mV, the noise was experimentally tested to be 1.8mV (to one decimal place). On the previous charts, this scales to 4.6mV (to one decimal place). This is easily above the threshold noise, and hence no change in ADC resolution is required if the data is averaged over 50 captures. Data was captured during the preparation laser shots, and this was used to optimise the dynamic range of the ADC by altering the gain and bias such that the signal covered as large an input range as possible, hence using as much of the available resolution as possible.