Cross-correlation on a single channel for resistance noise measurements

TL;DR

The paper tackles resistance-noise measurements limited by $1/f$ and amplifier white noise by introducing a single-channel cross-correlation scheme that uses two simultaneous carrier frequencies and dual software demodulators to create two uncorrelated copies of the DUT noise. The method reproduces conventional cross-correlation results with accuracy within ~14% relative to a standard single-reference approach and delivers a substantial SNR improvement of about $7\ \mathrm{dB}$, scalable with longer averaging. It eliminates the need for additional hardware, enabling routine cross-correlation in resistance-noise spectroscopy, and can be extended to many carrier frequencies for large-scale cross-correlation.

Abstract

Cross-correlation is an established tool to reduce the background in resistance noise measurements. However, the conventional method requires the amplifier, demodulator and digitizer channels to be duplicated, increasing the cost and complexity of the measurement circuit. We propose an alternating-current technique that allows cross-correlation with only a single channel by modulating the device under test with two carrier frequencies simultaneously. Using multiple software-based demodulators, we show that this method produces accurate amplitude measurements and noise spectra. The signal-to-noise-ratio is improved by 7 dB, and longer measurement durations increase this improvement.

Figures (5)

• Figure 1: (a) Conventional single-reference lock-in demodulator circuit. A sinusoidal current is passed through $R_\mathrm{DUT}$ and its voltage drop is amplified, recorded, and demodulated by the reference sine wave through mixing and low-pass filtering. The "x1" symbol indicates that the reference voltage usually does not need to be amplified, only converted from a differential/floating to a single-ended/ground-referenced signal. (b) New multi-reference circuit with two superimposed reference signals and two corresponding demodulator chains.
• Figure 2: Top: Background noise in the multiple-reference circuit. Vertical grey lines mark the reference frequencies, light-grey shaded areas show the pass bands that will be contained in the demodulated signals later. Bottom: Same data as top, zoomed in on the reference frequencies.
• Figure 3: Left: Good agreement of the normalized resistance noise power spectra between the single-reference and the multi-reference methods. Interference peaks have been removed in this data set. The multi-reference spectrum is the cross-PSD of the two output signals. Right: When the spectra on the left are smoothed (linear Savitzky-Golay filter, half a frequency decade wide), the error in the multi-reference spectrum relative to the single-reference spectrum is below $14%$ (horizontal bars).
• Figure 4: (a)/(b): Comparison of the voltage power density spectra for the conventional single-reference method (a) and the new multi-reference cross-correlation method (b). The measurement at non-zero excitation current $I > 0$ reflects the $1/f$-like resistance noise of the DUT, while the spectrum at $I = 0$ represents the background of the measurement setup. The mean low-frequency background PSD $S_V(f \leq 30Hz)$, shown by black horizontal lines, is far lower in the multi-reference setup thanks to cross-correlation. (c) Computing the signal-to-noise-ratio $\mathrm{SNR}(f)$ for (a) and for (b), and then subtracting the two for each frequency $f$ gives the SNR improvement spectrum shown here. In this case, averaging over $N_\mathrm{avg} = 50.0$ sections gave a mean improvement using the new cross-correlation method of $7.3dB$ across all frequencies below $30Hz$. (d) Performing the analysis in (c) for varying amounts of averaging sections increases the improvement in SNR, because the background cross-PSD $S_\mathrm{V}(f, I=0)$ scales as $1 / \sqrt{N_\mathrm{avg}}$.
• Figure 5: The same analysis as in Fig. \ref{['fig:signal-to-noise']} done for three different cross-correlation methods. The SNR improvement is relative to a single-channel, single-reference measurement.