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### Abstract

A few years back, our lab developed a signal averaging technique that greatly reduces the number of scans required to achieve a comparable signal-to-noise ratio to that of conventional signal averaging for continuous wave magnetic resonance measurements. We utilize an adaptive filter in a signal averaging scheme without any prior knowledge of the signal under observation. We termed this technique adaptive signal averaging (ASA). The technique was successful in reducing the noise variance by a factor of at least 10 in a single trace and is shown to converge in time by the same factor. ASA can also be useful in many other applications where signal averaging is utilized, such as medical imaging, electrocardiography, or electroencephalography. The purpose of this paper is to describe the advancements made to the technique, present a derivation of its performance enhancement, and illustrate the power of the technique through a set of simulations.

This work has been primarily supported by the US Army Research Laboratory, with additional support from Intel Corporation and General Electric under the U.S. Department of Commerce under Award No. NIST60NANB10D109.

I. INTRODUCTION

II. ADAPTIVE SIGNAL AVERAGING

III. PERFORMANCE OF ADAPTIVE SIGNAL AVERAGING

IV. SIMULATION

V. CONCLUSIONS

### Key Topics

- Magnetic resonance
- 17.0
- Absorption spectra
- 5.0
- 1/f noise
- 4.0
- Electron paramagnetic resonance spectroscopy
- 4.0
- Probability theory
- 4.0

## Figures

Simplified block diagram of adaptive signal averaging.

Simplified block diagram of adaptive signal averaging.

Adaptive linear predictor used in the adaptive signal averaging technique. Note that the desired signal is replaced with the unfiltered, conventional average.

Adaptive linear predictor used in the adaptive signal averaging technique. Note that the desired signal is replaced with the unfiltered, conventional average.

(a) Reduction in noise power for both the unfiltered and filtered averages as a function of scan for different values of the exponential weighting factor. The range of λ is 0.55 ≤ λ ≤ 0.95 with 0.05 increments. (b) Reduction in noise power for both the unfiltered and filtered averages as a function of scan for different values of the turn on time *L*. The turn on time was varied from 2 ≤ *L* ≤ 40 at increments of 4 scans.

(a) Reduction in noise power for both the unfiltered and filtered averages as a function of scan for different values of the exponential weighting factor. The range of λ is 0.55 ≤ λ ≤ 0.95 with 0.05 increments. (b) Reduction in noise power for both the unfiltered and filtered averages as a function of scan for different values of the turn on time *L*. The turn on time was varied from 2 ≤ *L* ≤ 40 at increments of 4 scans.

(a) The top plot illustrates the desired simulated signal and the bottom plot illustrates a representative noise sequence *u*(*n*) that was added to create *x*(*n*). The three smaller signals to the left are enlarged in a few of the plots in the figure to better visualize results of the simulation. (b) The bottom plot illustrates the 100th individual unfiltered scan and the top figure illustrates its filtered version using the average of 100 unfiltered scans as the desired signal in the ALP. Note that only two of six line shapes are apparent in the individual unfiltered trace, while five of the six line shapes are revealed in an individual filtered trace. (c)The top plot illustrates the average of 85 filtered scans and the bottom figure illustrates the average of 100 unfiltered scans. Note that all 6 line shapes are almost completely resolved in the filtered average. (d) The top plot illustrates the average of 985 filtered scans and the bottom figure illustrates the average of 1000 unfiltered scans. Note that even now, the conventional average still does not match the SNR achieved when averaging 85 filtered scans.

(a) The top plot illustrates the desired simulated signal and the bottom plot illustrates a representative noise sequence *u*(*n*) that was added to create *x*(*n*). The three smaller signals to the left are enlarged in a few of the plots in the figure to better visualize results of the simulation. (b) The bottom plot illustrates the 100th individual unfiltered scan and the top figure illustrates its filtered version using the average of 100 unfiltered scans as the desired signal in the ALP. Note that only two of six line shapes are apparent in the individual unfiltered trace, while five of the six line shapes are revealed in an individual filtered trace. (c)The top plot illustrates the average of 85 filtered scans and the bottom figure illustrates the average of 100 unfiltered scans. Note that all 6 line shapes are almost completely resolved in the filtered average. (d) The top plot illustrates the average of 985 filtered scans and the bottom figure illustrates the average of 1000 unfiltered scans. Note that even now, the conventional average still does not match the SNR achieved when averaging 85 filtered scans.

(a) Histogram of the 85th sequence of prediction errors *v*(*n*) with a superimposed Gaussian fit. This figure justifies the assumption made earlier about the prediction error distribution being Gaussian. (b) Histogram of the averaged (985 scans) prediction errors *v* _{985}(*n*) with superimposed Gaussian fit. This close correspondence is expected as suggested by the central-limit theorem.

(a) Histogram of the 85th sequence of prediction errors *v*(*n*) with a superimposed Gaussian fit. This figure justifies the assumption made earlier about the prediction error distribution being Gaussian. (b) Histogram of the averaged (985 scans) prediction errors *v* _{985}(*n*) with superimposed Gaussian fit. This close correspondence is expected as suggested by the central-limit theorem.

The top plot illustrates the reduction in noise power for both unfiltered and filtered averages as a function of scan. The bottom plot illustrates the noise reduction factor between these two variance levels as a function of scan. Note that the filtered average maintained a noise power level that was consistently 10 times lower than that of the unfiltered average. This would correspond to a SNR improvement by a factor of more than 3.

The top plot illustrates the reduction in noise power for both unfiltered and filtered averages as a function of scan. The bottom plot illustrates the noise reduction factor between these two variance levels as a function of scan. Note that the filtered average maintained a noise power level that was consistently 10 times lower than that of the unfiltered average. This would correspond to a SNR improvement by a factor of more than 3.

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