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# Harmonic Magnification by Time Reversal based on a Hammerstein Decomposition

Sébastien Ménigot, Jean-Marc Girault

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Sébastien Ménigot, Jean-Marc Girault. Harmonic Magnification by Time Reversal based on a Hammerstein Decomposition. IEEE International Ultrasonic Symposium 2016, Sep 2016, Tours, France. hal-01371266

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Submitted on 25 Sep 2016

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# Harmonic Magnification by Time Reversal based on a Hammerstein Decomposition

Sébastien Ménigot† and Jean-Marc Girault†

sebastien.menigot@univ-tours.fr, jean-marc.girault@univ-tours.fr

† Université François-Rabelais de Tours, Inserm, Imaging and Brain UMR U930, Tours, France

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UNIVERSITÉ
FRANÇOIS - RABELAIS
TOURS

Instituts
thématiques

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Inserm

Institut national
de la santé et de la recherche médicale

## Introduction

The medical ultrasound imaging systems have been improved by taking into account the nonlinear wave propagation. These improvements leads to increase the signal-to-noise ratio and the focusing in tissue harmonic imaging (THI). Since the backscattered nonlinearities are a function of the transmitted signal, enhancing these nonlinearities means to design the best wave [1]. Time reversal process is well-known to optimize the SNR and free itself to phase aberration, by combining a waveform design in space and in time thanks a physical matched filter [2]. However, it is not well-adjusted for THI, because the propagation of the time reversed signal destroys the harmonics.

**How can we optimize the SNR in tissue harmonic imaging ?**

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As we want to guarantee a SNR optimization with a good focusing, the method has to include time reversal. The solution firstly consists in extracting the harmonic component at $2f_0$ by a Hammerstein filter. Then this time reversed harmonic component at $2f_0$ is frequency shifted to the fundamental component at $f_0$ and retropropagated in the medium.

## Methods

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1. Sending a first standard excitation $x_{standard}(n)$

2. Harmonic extraction based on a Hammerstein model where the nonlinear function is a frequency shifting by modulation (since it is a bijective function on $\mathbb{R}$):

$$ \hat{y}(n) = \sum_{p=1}^{2} \sum_{m=1}^{M} h_p(m) x_{standard}(n-m) \cdot C_p(n), \quad (1) $$

$n$ the discrete time, $M$ the memory of the Hammerstein model and $C_p(n) = \cos(2\pi \frac{(p-1)f_0 n}{F_s})$ with $F_s$ the sample frequency. The model can be solved by a pseudo-inversion:

$$ \mathbf{h} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}. \quad (2) $$

$\mathbf{h} = [h_1(1), \dots, h_1(M), h_2(1), \dots, h_2(M)], \hat{\mathbf{y}} = [y(M+1), \dots, y(N)]^T$ with $N$ the sample quantity, the matrix of input signals $\mathbf{X} = [\mathbf{x}_1 \mathbf{x}_2]$ with

$$ \mathbf{x}_p = \begin{pmatrix} v_p(M) & v_p(M+1) & \dots & v_p(N) \\ v_p(M+1) & v_p(M+2) & & 0 \\ \vdots & \vdots & & \vdots \\ v_p(N) & 0 & & 0 \end{pmatrix}, \quad (3) $$

and $v_p(n) = x(n)C_p(n)$. Finally, the second harmonic signal is: $y_{NL} = x_2^T [h_2(1), \dots, h_2(M)]^T$.

3. Time reversing $y_{NL}$

4. Annihilation of the second harmonic effects of the second harmonics:

$$ x_{opt}(n) = A \cdot y_{NL}(n) \cdot \cos \left( 2\pi f_0 \frac{n}{F_s} + \phi \right), \quad (4) $$

A preserve the transmit power to $x_{standard}(n)$ and $\phi = \arctan(\phi(y_{NL})/\phi(\cos))$, with $\phi(y_{NL})$ the phase of the second harmonic signal $y_{NL}$ and the $\phi(\cos)$ the phase of a cosine at the frequency $f_0$.

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Program example of a Hammertein decomposition on Runmycode.org

## Simulation

* 2D nonlinear wave propagation in a cavity [3]

* 8-element probe centred at $f_c = 4$ MHz

* Initial Gaussian pulse $x_{standard}(n)$ centred at $f_0 = 22$ MHz and with a bandwidth of 50%

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## Results

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## Discussions and Conclusion

* Magnify the second harmonics and improve the SNR

* Hammerstein model using frequency shifting with the good phase → Annihilation of the harmonic effects before sending the time-reversed signal

* Matched filter for harmonic components → Extension of the time reversal principle to second harmonics

<footnote>[1] Ménigot S. & Girault, J.-M. Optimization of Contrast Resolution by Genetic Algorithm in Ultrasound Tissue Harmonic Imaging. Ultrasonics. 2016.</footnote>

<footnote>[2] Fink, M. Time-reversal of Ultrasonic Fields. 1. Basic Principles. IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control. 1992 (39).</footnote>

<footnote>[3] Anderson, M. E. A 2D Nonlinear Wave Propagation Solver Written in Open-Source MATLAB Code. Proceeding IEEE Ultrasonic Symposium. 2000.</footnote>