Matthias Malzacher^{1,2}, Markus Vester^{3}, Robert Rehner^{3}, Christopher Stumpf^{1}, and Patrick Korf^{1}

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

### Synopsis

**Using
reciprocity, available SNR from a receive coil array can be calculated by maximizing B**_{1}- 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 (**T**_{1}=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.