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Using Noise Waves for Simulation and Measurement of Array SNR Penalty due to Passive Impedance Match
Arne Reykowski1, Christian Findeklee2, Paul Redder1, Tracy Wynn1, Tim Ortiz1, Randy Duensing2, and Scott B King1

1Invivo Corporation, Gainesville, FL, United States, 2Philips Research, Hamburg, Germany

### Synopsis

Active impedance matching versus passive impedance matching of array coils is a concept well understood when designing transmit arrays. Lesser known however is that this concept also applies to receive arrays. Even though it appears that preamplifiers are noise matched to the passive port impedance (usually 50 Ohms), preamplifier noise coupling creates active noise match impedances which are mode dependent. In this context, a mode is defined by a signal vector and the corresponding weighting factors for optimum combined SNR. We use coupled noise waves to explain by simple concepts how the weighted and combined coupled noise changes the active noise match impedance.

### Introduction

Receive array noise matching is a well-studied field [1-13], but the implications for MRI array coil design are often overlooked. We present a novel approach using noise waves and scattering parameters to explain matching to active array impedances. Active noise matching takes into account preamplifier noise via a modified noise covariance matrix. In case of using equal preamplifiers at all channels in a reciprocal antenna, the resulting noise match impedances are the equivalent of active array impedances for TX arrays. This complete noise model can also be used to demonstrate that preamplifier decoupling is mainly a convenience for tuning arrays, but has no impact on maximizing array SNR [6,7,10]. In cases of high Q coils and strong mutual coupling, coil designers can achieve significant SNR gains by matching to the active impedance of the dominant coil mode (typically the bird cage mode of volume arrays) to improve SNR. Our formulation provides solutions for active impedance match with LNAs having different input impedances Zin, and different noise figures Fmin, and we assume that optimum noise match impedance Zopt can be adjusted individually.

### Methods

We are using a preamplifier noise model involving noise waves to simulate the noise coupling in the receive array [3, 13, 14, 15, 16]. This model places two noise waves at the input of a noise free amplifier. One wave travels toward the LNA, while the other travels into the coil array, represented by a scattering matrix S. The noise waves are uncorrelated when normalized to the optimum noise match impedance. It is also possible to normalize the noise waves to any other impedance, in our case the LNA input impedance. This formulation results in correlated noise waves, but has the advantage of avoiding reflection at the LNA inputs, therefore leading to a more compact formulation for the problem.

Figure 1 introduces the concept of active noise match for the example of two LNAs connected to an array. For the sake of the argument, we focus only on the noise emanating from LNA 1 and normalized the LNA gains to unity by including them in the weighting factors wi.

The weighted and combined noise from LNA 1 can be written as

$$\frac{1}{w_{1} } c^{n} =a_{1}^{n} +\underbrace{\left(S_{11} +\frac{w_{2} }{w_{1} } S_{21} \right)}_{\Gamma _{act} } b_{1}^{n}$$

Where w1 and w2 are the weighting factors used for the signal combination. It can be easily shown that noise contribution from LNA 1 is minimized for

$$\Gamma _{act} =\Gamma _{opt} =S_{11} +\frac{w_{2} }{w_{1} } S_{21}$$

This means, that optimum noise match for LNA 1 is achieved if the active reflection coefficient is equal to the optimum reflection coefficient. This is in contrast to traditional noise matching to passive reflection coefficient S11.

Extending this concept to large arrays, the active reflection coefficients become:

$$\Gamma _{opt}^{i} =\frac{1}{w_{i} } \sum _{k=1}^{N}w_{k} \cdot S_{ki} =\frac{1}{w_{i} } S_{i} w^{T}$$

Optimum weighting factors for a given signal vector Vcoil can be found by using the formula given by [17, 18]

$$w_{opt} =\left(\left[R_{coil} +R_{LNA}^{min} \right]^{-1} \cdot V_{coil} \right)^{*}$$

Rcoil and RLNAmin are noise covariance matrices for coil array and LNAs.

$$R_{LNA}^{min} =4kBR_{in} \left(T_{min} -S\cdot T_{min} \cdot S^{H} \right)$$

Tmin is a diagonal matrix containing the LNA minimum noise temperatures; Rin is the LNA input resistance. Optimum noise match coefficients are

$$\Gamma _{opt} =diag\left(w_{opt} \right)^{-1} S\cdot w_{opt}$$

Noise figure for a set of signals Vcoil can be calculated as

$$F_{dB} \left(V_{coil} \right)=10\cdot \log _{10} \left(\frac{V_{coil}^{H} \left(R_{coil} \right)^{-1} V_{coil} }{V_{coil}^{H} \left(R_{coil} +R_{LNA} \right)^{-1} V_{coil} } \right)$$

RLNA is the complete LNA noise covariance matrix

### Results

For initial simulation we used a two channel array with both elements tuned to 50 Ohms, quality factor Q=50, resistive coupling kr=20%, magnetic coupling km=2%, minimum LNA noise figure NF_min=0.4dB and impedance matrix:

$$Z_{coil} =\left[\begin{array}{cc} {50\Omega } & {\left(10+j50\right)\Omega } \\ {\left(10+j50\right)\Omega } & {50\Omega } \end{array}\right]$$

Maps were created showing combined array noise figure for different signal ratios S2/S1 and different matching conditions.

Fig. 2 shows a noise figure map for traditional passive match to 50Ω. The average NF for passive noise match was 0.8dB and minimum NF = 0.6dB.

Fig 3 and 4 show active match to the signal mode (+1,+1) with Zopt=60Ω+j50Ω and signal mode (+1,-1) with Zopt=40Ω-j50Ω. For these two cases the average NF was 0.84dB and minimum NF = 0.4dB, the actual minimum NF for the preamps.

### Conclusions

Coil coupling can have severe consequences for combined array SNR. If decoupling is not an option, active matching to the dominant mode can improve SNR. The preamplifier input impedance has no effect on combined SNR when all active channels and proper noise statistics are applied.

### Acknowledgements

In memoriam Dr. Charles Saylor, PhD. A great scientist and good friend. Gone too soon.

### References

1. Reykowski A and Wang J, Rigid signal-to-noise analysis of coupled MRI coils connected to noisy preamplifiers and the effect of coil decoupling on combined SNR, Proc 8th ISMRM, Denver, Colorado, USA, April 2000, p. 1402.
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### Figures

Fig 1: Noise waves a1n and b1n emanating form LNA 1 are scattered across the two channel array and result in output noise waves c1n and c2n . The weighted combination cn of these output noise waves gives rise to the active mode impedance.

Fig 2: Noise figure map for traditional passive noise match to Zopt=50 Ohms. The average NF for passive noise match was 0.8dB and minimum NF = 0.6dB.

Fig 3: Noise figure map for active match to the signal mode (+1,+1) with Zopt=60+j50 Ohms. Average NF was 0.84dB and minimum NF was 0.4dB, the actual minimum NF for the LNAs.

Fig 4: Noise figure map for active match to the signal mode (+1,-1) with Zopt=40-j50 Ohms. Average NF was 0.84dB and minimum NF was 0.4dB, the actual minimum NF for the LNAs.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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