This note explains how to enhance signal resolution for high-bandwidth frontends in the time domain, viewed from a statistical perspective. The resolution improvement will also be illustrated using a low-pass filter.

Can an ADC with a 1 mV resolution distinguish a 1 µV signal? With a single sample, the answer is no—the weak signal’s information is lost. But what if the signal is noisy and consists of a series of samples over time?

Consider a simple example: a quantizer with a resolution of 0.5 and an analog signal with an amplitude of 0.2. Because the signal amplitude is smaller than the quantizer’s resolution, the quantized output should be 0. This holds true in an ideal, noiseless system. However, all ADCs introduce electronic and thermal noise. In this case, the signal seen by the quantizer isn’t exactly 0.2; it is 0.2 plus noise. If the noise level is 0.1, for example, the combined signal (0.2 + noise) can exceed a half of the quantizer’s resolution, resulting in an output of 0.5.

In the attached plot, the blue line represents the noisy signal with a fixed mean value, and the orange dots are the outputs from a quantizer with a 0.5 resolution step. In the top panel, the mean value is 0.2; in the bottom panel, the mean value is -0.2. As shown, the quantizer output doesn’t remain constantly at 0; instead, it fluctuates due to the input noise. This fluctuation indicates that the signal’s finer details are not entirely lost, and by analyzing the time series statistically, we can resolve smaller signal components even with a large quantization step.

In this case, the distribution of the quantizer output is shown in the figure below. The distinction between the blue and orange distributions is clearly visible, with their mean values reflecting the true value of the input signal. 

The mean value can be extracted by low-pass filtering, as this effectively removes the quantization noise introduced by the quantizer. Generally, if a signal contains random Gaussian noise with a standard deviation exceeding one-third (1/3) of the quantization step, the noise will largely randomize the quantization process. Consequently, quantization noise will resemble white noise in the frequency domain, which can be removed with a low-pass filter.

In the attached figure, the orange trace shows the power spectral density (PSD) of the quantized signal, while the blue trace represents the analog, noisy input (non-quantized). The orange line has a higher noise level than the blue line due to quantization noise. After low-pass filtering both the quantized and original signals, we obtain the purple and yellow lines, respectively. The low-pass filter reduces not only the input noise but also effectively removes the quantization noise.

Here, we can compare the low-pass filtered signal with the raw signal. It is clear that the quantization noise is significantly reduced, allowing for a more accurate recovery of the original signal compared to the non-filtered version. This improvement is due to the substantial reduction of quantization noise introduced by the quantizer.

This note demonstrates that signals with amplitudes smaller than the quantization resolution can still be recovered by averaging the quantized data from both time-domain and frequency-domain perspectives. In other words, the effective resolution of the ADC can be improved by low-pass filtering the digitized signal, as the quantization noise is expected to be evenly distributed across the frequency spectrum.