SNR simulations including coupled preamplifier noise
Matthias Malzacher1,2, Markus Vester3, Robert Rehner3, Christopher Stumpf1, and Patrick Korf1

1Friedrich-Alexander University Erlangen-Nuremberg, Erlangen, Germany, 2Computer Assisted Clinical Medicine, University Heidelberg, Medical Faculty Mannheim, Mannheim, Germany, 3Siemens Healthcare GmbH, Erlangen, Germany

### Synopsis

Using reciprocity, available SNR from a receive coil array can be calculated by maximizing B1- at the target voxel for unit input power. However, for strongly coupled or lightly loaded coil elements, the noise figure degradation due to coupled preamplifier noise becomes significant. It is shown here that this effect can be modeled by power loss in a resistive attenuator at each coil port. Thus, it is now possible to simulate any coil configuration, including those where coil coupling cannot be neglected.

### Purpose

Using reciprocity, available SNR from a receive coil array can be calculated by maximizing $B_1^-$ at the target voxel for unit input power [1-2]. However, for strongly coupled or lightly loaded coil elements, the noise figure degradation due to coupled preamplifier noise (or equivalent eigenmode mismatch) becomes significant [3-8]. We will show here that this effect can be modeled by power loss in a resistive attenuator at each coil port.

### Theory

Receiver noise is generally described by noise voltage and current sources $(u_n, i_n)$ at the preamplifier input. Choosing a reference impedance $Z_0=Z_{opt}$, the noise figure is:

$$F(\Gamma) = F_{min} + \frac{4r_n |\Gamma^2|}{(1-|\Gamma^2|)}[9]$$

Using two uncorrelated directional noise waves $T_1=F_{min}-1$ (inwards) and $T_2=4r_n-T_1$ (outwards), it can also be expressed as

$$F(\Gamma) = 1 + \frac{T_1 + T_2 |\Gamma^2|}{1-|\Gamma^2|}$$

In our new approach, we replace the noise sources by a room-temperature attenuator $a$ $(a=\frac{1}{s_{21}^2}, Z_0=Z_{opt})$, which reproduces the $\Gamma$-dependence in the power calculation, and an additional constant global SNR scaling factor $b$. The power efficiency of an attenuator in the presence of load mismatch is

$$\eta(\Gamma) = \frac{P_{out}}{P_{in}} = \frac{s_{21}^2(1-|\Gamma^2|)}{1-s_{21}^4|\Gamma^2|}$$

equivalent to a noise figure

$$F_a(\Gamma) = \frac{1}{\eta} = \frac{a-\frac{|\Gamma^2|}{a}}{1-|\Gamma^2|}$$

Setting $b\cdot F_a(\Gamma)$ equal to $F(\Gamma)$, we get

$$a\cdot b = 1 + T_1 ; \; \; \; \;\frac{b}{a} = 1 - T_2$$

and thus

$$a = \sqrt{\frac{1+T_1}{1-T_2}} ; \; \; \; \; b = \sqrt{(1+T_1)\cdot (1-T_2)}$$

In typical low-noise amplifiers, $u_n$ and $i_n$ are not uncorrelated, implying $T_2=T_1$ and $r_n=\frac{F_{min}-1}{2}$. Then

$$a = \frac{1+T_1}{\sqrt{1-T_1^2}} \sim F_{min} ;\; \; \; \; b = \sqrt{1-T_1²} \sim 1$$

For example, 0.5 dB preamp noise figure (T1=0.1220) can be modeled by a 0.5326 dB attenuator and -0.0326 dB scaling factor.

### Simulation

Transmission efficiency $\frac{B_1}{\sqrt{P}}$ was calculated using MATLAB (The Mathworks Inc.), using $B_1^-$ fields and S-parameters from FDTD simulation (CST AG Darmstadt), and measured coil resistance and preamp noise figure (NF). Absolute SNR was determined using a calibration factor [10].

### Measurement

Setup was a pair of identical coils (diameter=16cm) in a lightly loaded setup (phantom diameter=9cm, saline solution σ=0.5S/m) and adjustable coupling by changing coil distance (Figure 1). Matching and preamplifier decoupling was adjusted for each coil without the presence of the other coil. The adjustment was then maintained for each coil distance. SNR was acquired using a GRE sequence on a 3T MAGNETOM Skyra (Siemens Healthcare, Erlangen Germany).

### Results and Discussion

Measured and simulated SNR evaluated at a single pixel in the phantom center and the measured kQ product over the coil distance are plotted in Figure 2. SNR simulation was performed for different preamp noise figures (0.1dB, 0.3dB, 0.5dB) including noise coupling, and for 0.5dB neglecting coupling. With decreasing distance, one might expect that the SNR improves, but measured SNR decreases due to coupled preamplifier noise. Neglecting noise coupling is significantly overestimating SNR. Though our model predicts the general behavior of SNR degradation due to preamplifier noise coupling, the degradation seems to be overestimated, probably due to a systematic error in the simulated k*Q factor.

### Conclusion

We showed that SNR degradation due to coupled preamplifier noise can be modeled by power loss in a resistive attenuator. Thus it is now possible to simulate any coil configuration, including those where coil coupling cannot be neglected.

### Acknowledgements

No acknowledgement found.

### References

[1] Hoult, J. Magn. Res vol 24, p.71-85, 1976 [2] Schnell, IEEE Antennas and Propagation, vol 48 (3) p.418-426 [3] Reykowski, PhD Thesis, Texas A&M University, 1996 [4] Reykowski, ISMRM 2000 p.1402 [5] Findeklee, ISMRM 2011 p.1883 [6] Vester, ISMRM 2012 p.2690 [7] Wiggins, ISMRM 2012 p.2689 [8] Sodickson, ISMRM 2014 p.0618 [9] Pozar, Microwave Engineering 3. Ed., eq. (11.57), Wiley 2005 [10] Stumpf, ISMRM 2012 p.2684.