Arne Reykowski^{1}, Christian Findeklee^{2}, Paul Redder^{1}, Tracy Wynn^{1}, Tim Ortiz^{1}, Randy Duensing^{2}, and Scott B King^{1}

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.

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 *Z _{in}*, and different noise figures

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 *w _{i}*.

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 *w _{1} *and

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

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 *S _{11}.*

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 *V_{coil}
*can be found by using the formula given by [17, 18]

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

**R _{coil}** and

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

**T _{min} **is a diagonal matrix containing the LNA minimum noise
temperatures;

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

Noise figure for a set of signals **V _{coil}** can be calculated
as

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

**R**_{LNA }is the complete LNA noise covariance matrix

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 *S _{2}/S_{1}* 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.

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Fig 1: Noise waves a_{1}^{n} and b_{1}^{n} emanating form LNA 1 are scattered across the two channel array and result in output noise waves c_{1}^{n} and c_{2}^{n }. The weighted combination *c*^{n} of these output noise waves gives rise to the active mode impedance.

Fig 2: Noise figure map for traditional passive noise match to *Z*_{opt}=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.