# Phase Correction-Adaptive Line Enhancement for Noise Reduction of Low-Field NMR

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## Abstract

The amplitude of low-field nuclear magnetic resonance (NMR) is weak, and the echo is buried in the noise. The reduction of noise is critical to accurately extract echo amplitude. Phase correction-adaptive line enhancement (PC-ALE) is proposed to noise suppression based on the principle of ALE and NMR spin-echo characteristics. The echo amplitude is calculated after two-stage processes; phase shift from time-delay and filter tap would be compensated effectively in frequency domain. Simulation and experiments show that PC-ALE has prominent performance on noise suppression, envelope recovery, as well as the correction of the phase shift. The amplitude from the method of sample average nearby the middle of the echo is more accurate than the maximum peak when the PC-ALE is applied to noise suppression of spin-echo.

## Keywords

Nuclear Magnetic Resonance Noise Reduction Adaptive Algorithm Spin Echo Noise Suppression## 1 Introduction

Low-field nuclear magnetic resonance (NMR) is applied for NMR well logging, core analyses, fluid magnetic resonance imaging, etc. [1, 2]. Compared to medium-field (0.500–1 T) and high-field (1.500–2 T) NMR [3], spin echo is easily contaminated by noise and the amplitude is weak. The reduction of noise is critical to accurately extract echo, the amplitude, and the phase information.

Methods of noise suppression in low-field NMR are diverse, such as, finite impulse response (FIR) filter [4], wavelet transform [5, 6, 7], principal component analyses (PCA) [8, 9], etc. FIR filter (i.e., low-pass, band-pass, and high-pass) is generally based on a priori knowledge of the spin echo; the de-noising performance is awful if a priori knowledge is insufficient.

The signal-to-noise ratio (SNR) is improved by accumulation [10]; however, the method requires a large amount of data acquisition and is a time-consuming procedure. Despite of the good performance on noise reduction, the wavelet transform and PCA are not suitable for the real-time process due to the computational complexity, and are adopted to post-processing of echo trains.

An adaptive noise canceller (ANC) based on the adaptive algorithm is widely used for noise reduction and weak signal extraction in recent years [11, 12, 13, 14, 15]. Adaptive line enhancement (ALE) is the degenerate form of ANC [11], and needs one input channel to detect the signal interfused by background noise, which depends on the principle of different autocorrelations between the signal and noise after a time delay. A large number of papers discuss the different adaptive algorithms for the NMR noise suppression [12, 13, 14, 15]. In fact, the phase shift is a considerable factor for the NMR spectrum caused by the time delay of the ALE system, which results in the distortion of the position and amplitude of the NMR spectrum.

In this study, the two-stage adaptive de-noising with phase correction is proposed. Two-stage processes serve to reduce noise of the spin echo, and the phase correction in the frequency domain is achieved after the ALE process.

## 2 Principle of PC-ALE

*E*represents the amplitude of the spin echo,

*f*is the hydrogen nuclei precession frequency (called the Larmor frequency), \( \phi_{0} \) is the initial phase,

*v*(

*t*) is the background noise. The noisy echo is filtered by a low-pass filter to eliminate high-frequency components, and is discreted by an analog–digital converter for digital processing.

*x*(

*n*) and its delay version. The delay

*m*represents the prediction depth of the ALE. The input signal after delay

*m*is processed by a transversal FIR filter and the output of the FIR filter

*y*′(

*n*) is given by

*W*

_{ k }(

*n*) represents the tap-weight of the adaptive filter at the time

*n*,

*L*is the filter order. The error signal

*e*(

*n*) is defined as the difference between the noisy echo and the ALE output. The error signal is employed to update the weights of the transversal filter with the adaptive algorithm. Development on adaptive algorithm is mainly based on Widrow and Hoff’s least-mean-square (LMS) algorithm [16]. The tap-weight of LMS is given by

*e*(

*n*) is the adaptation error at time

*n*,

*μ*is the fixed step size which is limited by \( 0 < \mu < 1/\lambda_{\hbox{max} } \), the \( \lambda_{\hbox{max} } \) is the maximum eigenvalue of autocorrelation matrix for the input data

*x*(

*n*).

However, several significant elements for noise reduction of low-field NMR should be taken into account. First, the improvement of SNR is essential for the noisy spin echo. The accurate amplitude and phase information come from the high SNR data. Second, numerous computations from the matrix inversion of adaptive algorithm are not suited to fast measurement of low-field NMR tools. For example, the echo time of NMR well logging tools is generally few microseconds to measure the short relaxation of porous media that reflects the contents of clay bound water [17, 18]. Measurements are not precise if the computing time is larger than the echo time. Thirdly, the convergence rate and the stability are very important for the noise reduction of the spin echo.

With regard to the convergence rate, the variable step size has the better performance than the fixed step of LMS. The instantaneous mean square error can be minimized through the normalized least mean square (NLMS) algorithm without the evaluation of the matrix correlation [19, 20]. Moreover, the NLMS algorithm is stable with the low computational complexity. Another approach to improve the convergence rate is the affine projection (AP) algorithm [21, 22, 23], which is associated with the correlation of the spin echoes. The better SNR of signal, the faster convergence rate is. Consequently, both algorithms (NLMS and AP) are selected to the PC-ALE for the noise reduction of variety SNR data.

*b*is a small constant that is used to avoid the excessive step size.

The second part is related to the phase correction. To verify the phase shift from the time-delay and filter tap, a linear correction algorithm can be performed. The algorithm is based on the fact that a delay in the time domain is equivalent to a phase shift in the frequency domain. Thanks to a priori knowledge of delay numbers of ALE system and the clock frequency, the phase shift can be dealt with in three steps as follows.

*Y′*(

*k*) denotes the spectrum of the NMR signal,

*j*is the imaginary unit,

*y′*(

*n*) is the output of ALE system,

*N*is the number of sampled data.

*m*is the delay number of the ALE system, it can be measured by a counter that operates on the clock signal the same as an analog-to-digital converter (ADC).

*Y*(

*k*) is inversely transformed to the time domain to give the corrected spin echo

The whole process of the PC-ALE outlines as follows: first, the narrow-band echo envelope and wideband noise are separated by the NLMS because of its excellent trace performance to sine wave, and correlation of spin-echo would be enhanced. Second, noise can be removed with the AP owing to the improvement of the spin echo correlation by the NLMS. Thirdly, the phase shift from the time delay and filter tap would be compensated in the frequency domain. At last, the spin echo in the frequency domain is transformed to the time domain with the inverse discrete Fourier transform.

## 3 Numerical Simulation

^{−3}T, sample frequency is 1 MHz, the SNR is −12.352 dB (the noise is with mean of zero). Here, SNR is given by

*E*

_{s}and

*E*

_{n}represent the energy of the signal and noise, respectively.

Results of the noise reduction using a variety of algorithms

Algorithm | SNR (dB) | Time (s) | MA (mV) | SA-TE (mV) |
---|---|---|---|---|

Echo model | 0.995 | 0.995 | ||

Noisy echo | −12.352 | 2.826 | 2.304 | |

LMS | −5.973 | 0.131 | 2.303 | 1.741 |

BLMS | −5.999 | 0.036 | 2.240 | 1.712 |

ADJLMS | −5.599 | 0.049 | 2.498 | 1.756 |

DLMS | −5.998 | 0.032 | 2.299 | 1.512 |

NLMS | −2.716 | 0.029 | 2.147 | 1.322 |

AP | −2.217 | 0.072 | 2.216 | 1.385 |

NLMS | 1.054 | 0.109 | 1.928 | 1.223 |

SNR and computing time are listed in columns 2 and 3; NLMS is the most efficient algorithm as compared to the other adaptive algorithms. The computing time and the improvement of SNR are obviously better than those for the DLMS and ADJMS algorithms. In addition, the extreme increase in SNR can be achieved combining with the NLMS and AP, the increase of 13.406 dB in SNR can be implemented after the ALE system.

Two conventional methods for peak extractions of the spin-echo envelope are listed in columns 4 and 5. Column 4 is the maximum peak (MA) of the echo envelope after processing. Due to the noise preserved to peak, the difference of amplitude between noisy echo and model is relatively much larger. Column 5 represents the sample average near the middle of the echo (SA-ME). The maximum amplitude appears in the middle of the echo on the basis of the NMR principle regardless of the noise [1, 3]. However, the value of SA-ME is not equal to the MA value for the low SNR of the NMR spin echo because of the intense noise. The best result is the proposed two-stage processes (NLMS + AP) among the several algorithms. Compared to other algorithms, the MA and the SA-ME are close to the model after the PC-ALE processing.

However, there is a very important fact about the amplitude extraction with the PC-ALE algorithm. The difference between the MA and model are much larger than that for the SA-ME because the convergence rate is in conflict with the offset of the algorithm. When the convergence is implemented, a part of noise preserving to peak of the spin echo would be processed as signal, which leads to the increase in the offset. Furthermore, noise is a random distribution and is not related with each other. It can be decreased through averaging the several sample points near the middle of the echo. Consequently, the SA-ME plays the role of a smoothing filter; the amplitude for the SA-ME is more accurate than the maximum peak, and the SNR is improved as well. The amplitudes are close to that for the model after two-stage processes, and the noise is efficiently reduced by the ALE system.

## 4 Implementation of PC-ALE

Comparison of the noise reduction performance with PC-ALE

Number | Noisy spin echo | Output of PC-ALE | ||||||
---|---|---|---|---|---|---|---|---|

SNR (dB) | Peak (mV) | Half-TE (mV) | Phase (°) | SNR (dB) | Peak (mV) | Half-TE (mV) | Phase (°) | |

1 | 5.899 | 1.380 | 1.210 | 139.590 | 7.0604 | 1.351 | 1.198 | 140.249 |

2 | 5.663 | 1.373 | 1.064 | 135.809 | 6.879 | 1.133 | 1.063 | 133.652 |

3 | 8.412 | 2.009 | 1.836 | 173.190 | 22.544 | 1.819 | 1.692 | 175.897 |

4 | 9.043 | 1.510 | 1.177 | −158.463 | 20.787 | 1.246 | 1.151 | −161.199 |

5 | 9.559 | 2.032 | 1.465 | −89.898 | 20.381 | 1.964 | 1.791 | −87.550 |

6 | 9.092 | 2.094 | 1.781 | 28.678 | 22.087 | 1.869 | 1.680 | 27.365 |

7 | 9.033 | 1.818 | 1.724 | −147.785 | 14.338 | 1.722 | 1.551 | −146.204 |

8 | 11.275 | 1.938 | 1.678 | 13.774 | 21.382 | 1.837 | 1.688 | 12.286 |

9 | 12.998 | 2.035 | 1.857 | 128.137 | 22.571 | 1.991 | 1.794 | 129.202 |

10 | 14.228 | 1.855 | 1.521 | 126.703 | 15.368 | 1.811 | 1.642 | 123.922 |

## 5 Summary and Conclusions

Simulations and experimental investigations were used to evaluate the PC-ALE algorithm with various SNRs of spin echoes. The PC-ALE algorithm incorporates the advantages of the NLMS and the AP in the convergence rate and the correlation of the spin echo. The amplitude of the spin echo can be extracted accurately after two-stage processes of the PC-ALE, and the background noise keeping in the spin echo can be removed significantly. The phase shift from the time delay can be corrected in the frequency domain. The experiments show that PC-ALE has the prominent performance on the noise suppression, the envelope recovery, as well as the correction of the phase shift.

Moreover, the simulation shows that the amplitude calculation with SA-ME is more accurate than that with MA for the PC-ALE. The interference of the peak is decreased with the effect of the smoothing filter of the SA-ME. The phase shift is caused by multiple elements such as the inhomogeneous static field, the unstable radio frequency, the time delay, etc. Here, one factor is corrected by PC-ALE, others will be taken into account in a forthcoming work.

## Notes

### Acknowledgments

This work was supported by National 863 Plan Projects (grant No. 2013AA064503), National Natural Science Foundation of China (NSFC) (grant No. 41130417 and 41074102) and Chongqing National Natural Science Foundation (cstc2012jjb9007), and Program for Changjiang Scholars and Innovative Research Team in University.

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