# The Noise Performance of a Multiple-Input-Port and Multiple-Output-Port Low-Noise Amplifier Connected to an Array of Coupled Antennas

Authors: Frédéric Broydé; Evelyne Clavelier
Journal: International Journal of Antennas and Propagation
 (2011)

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## Abstract

We address the noise characterization of a multiple-input-port and multiple-output-port (MIPMOP) device receiving signals from an array of antennas. A definition of the noise figures and natural noise figures of a MIPMOP device is provided, and the resulting properties are detailed in the impedance and admittance representations. We compute the natural noise figures of a low-noise MIPMOP amplifier comprising a MIMO series-series feedback amplifier (MIMO-SSFA), designed for a wireless receiver front-end.


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## Body

## 1. Introduction

In this paper, we consider a linear multiple-input-port and multiple-output-port (MIPMOP) device havingninput ports and m output ports, connected to an n-port source, such as an array ofncoupled antennas, and to an m-port load. The MIPMOP device could be passive, but our motivation lies in the case where it provides amplification. We want to characterize the noise performance of the MIPMOP device in a small bandwidth Δf around a given frequency f. The case n=m=1 is covered in detail in the literature since more than 50 years [1, 2] and will be referred to as the two-port case.The theory and measurement methods developed for the two-port case can be applied to a MIPMOP device only when all the following conditions are met:(i)
the MIPMOP device is made ofm=n uncoupled two-ports;(ii)
the ports of then-port source are uncoupled;(iii)
the ports of them-port source are uncoupled.The first condition is obvious. However, it is important to note that the theory of the two-port case is irrelevant when the second or the third conditions are not met.Let us for instance consider the multiport antenna array and the multiport wireless receiver shown in Figure1, which may be used in a multiple-input and multiple-output (MIMO) or single-input and multiple-output (SIMO) wireless communication system. In Figure 1, ZSant is the impedance matrix seen by the receiver, and ZLI is the input impedance matrix of the receiver. The physical size of the antenna array is often limited by the application, for example in the case of portable transceivers. Antenna coupling may in such a case be so large that the nondiagonal entries cannot be ignored when one computes the directivity patterns and correlations of the channels [3, 4]. A large coupling between the antennas will also impact the noise performance of the receiver front-end because the noise power delivered by an input port may excite other input ports via antenna coupling.Figure 1
An array ofn antennas connected to n uncoupled 2- conductor transmission lines connected to a wireless receiver having n input ports.A conventional wireless receiver front-end uses a MIPMOP device consisting ofnuncoupled two-port low-noise amplifiers (LNAs) presenting a diagonal ZLI. Even though the noise figure of each LNA as a two-port can be defined and measured, it gives no information on the true noise performance of the receiver front-end because it does not take the above-mentioned LNA noise coupled through the antenna array into account. This has been investigated by several authors [5–9].A type of nonconventional wireless receiver front-end uses a MIPMOP passive matching network to obtain a nondiagonalZLI.  For instance, a passive lossless MIPMOP device can be used to provide reflectionless matching to its antenna ports (i.e., ZLI=ZSant) and to uncoupled two-port LNAs at its output [10]. More importantly, a passive lossless MIPMOP device can be used to provide hermitian matching to its antenna ports (i.e., ZLI=ZSant* where the star denotes the hermitian adjoint) and hermitian matching to uncoupled two-port LNAs at its output [11–14]. This possibility is very interesting since hermitian matching provides a maximum power transfer and decorrelated output voltages in the case of 2-dimensional Rayleigh channels (because the matching network modifies the directional patterns such that they are orthogonal over the azimuth). This type of passive MIPMOP device can also theoretically be used to obtain noise matching [15, 16]. However, a passive MIPMOP device having a prescribed nondiagonal ZLI is a complex circuit in which losses must be taken into account to obtain a realistic design [17, 18]. Such losses degrade the noise performance.Another type of nonconventional wireless receiver front-end comprises a MIPMOP LNA presenting a nondiagonalZLI [19]. This scheme is new [20]. Also, it offers more flexibility than a design using a passive lossless MIPMOP device having its output ports connected to uncoupled two-port LNAs. Thus, better characteristics are expected with an optimized MIPMOP LNA.In both nonconventional front-end designs,ZSant and ZLI are nondiagonal, so that there is no possibility of applying the two-port noise theory to anything. Thus, a noise theory of MIPMOP devices must be used.In their theory of linear noisy networks, Haus and Adler presented many results which are applicable to MIPMOP devices [21]. However, they did not define an adequate figure of merit, since their purpose was the systematic description of the noise performance of a two-port linear amplifier. More recently, Randa [22] introduced a definition of the noise figures of a MIPMOP device. This definition generalizes the definition used in the two-port case, but it is not directly related to a signal-to-noise ratio.Randa expressed the noise figures using the S-parameter representation of the device. This formalism is adequate for RF measurements used for the noise characterization of a MIMOP device. However, a noise theory of MIPMOP devices, presented in the impedance and admittance representations, is more appropriate for circuit design. Basic elements of this theory and a validation are available [23]. In this paper, starting from the definitions in Sections 2 and 3, we further develop this noise theory in Sections 3 to 5. In Section 6, this theory is applied to the design of a MIPMOP LNA connected to an array of coupled antennas.


## 2. Notations

We use rms values throughout the paper. We useXt to denote the transpose of a matrix X,X̅ to denote the complex conjugate of X,X* to denote the hermitian adjoint X*=X̅t,  and tr X to denote the trace of X.In the small bandwidthΔf, we define the column-vector IIN of the input noise currents iIN1,…,iINn flowing into the positive terminals of the input ports of the MIPMOP device and the column vector VIN of the input noise voltages vIN1,…,vINn between the positive and negative terminals of the input ports. We use ISCIN and VOCIN to denote the column vectors of the short-circuit input noise currents iSCIN1,…,iSCINn  and of the open-circuit input noise voltages vOCIN1,…,vOCINn, respectively. We also define the column vector ION of the output noise currents iON1,…,iONm flowing into the positive terminals of the output ports and the column vector VON of the output noise voltages vON1,…,vONm between the positive and negative terminals of the output ports. We use ISCON and VOCON to denote the column vectors of the short-circuit output noise currents iSCON1,…,iSCONm and of the open-circuit output noise voltages vOCON1,…,vOCONm, respectively. Let us write(1)VN=(VINVON),VOCN=(VOCINVOCON),(2)IN=(IINION),ISCN=(ISCINISCON).For the casen=m=2, we obtain the equivalent circuit a of Figure 2, consisting of n+m=4 noise voltage sources and one noiseless  (n+m)-port. Such an equivalent circuit corresponds to the impedance representation, since(3)VN=ZIN+VOCN,
where Z is the impedance matrix of the MIPMOP device, of size (n+m)×(n+m). Alternatively, we obtain the equivalent circuit b shown in Figure 2, consisting of n+m=4 noise current sources and one noiseless (n+m)-port. Such an equivalent circuit corresponds to the admittance representation, since(4)IN=YVN+ISCN,
whereY=Z-1 is the admittance matrix of the MIPMOP device. We easily obtain(5)VOCN=-ZISCN,ISCN=-YVOCN.Equivalent circuit for a noisy MIPMOP device.
(a)(b)The second-order statistics of the noise produced by the MIPMOP device is determined by the covariance matrix of the open-circuit noise voltages〈VOCNVOCN*〉 or equivalently by the covariance matrix of the short-circuit noise currents 〈ISCNISCN*〉, where the brackets 〈〉 indicate an average over an ensemble of noise processes. The covariance matrices are related by(6)〈VOCNVOCN*〉=Z〈ISCNISCN*〉Z*,〈ISCNISCN*〉=Y〈VOCNVOCN*〉Y*.These relations are important for noise computations because they can be used to switch between the impedance and admittance representations to easily take into account additional series-connected or shunt-connected circuit elements [24, 25].


## 3. Noise Figures of MIPMOP Devices

In the definition of the noise figure of a two-port, the spectral density of the available power of the noise delivered by the source iskT0 where k is Boltzmann’s constant and T0=290 K. In the case of devices having multiple input ports, we have to consider that the noises delivered by the different ports of the n-port source may be correlated and may have different noise temperatures.Definition 1.
Ifα is an integer such that 1≤α≤m, the noise figure Fα of the MIPMOP device for the output α is the ratio of the self-power spectral density Re(〈vONαiONα¯〉)/Δf at the output α  to the portion of this self-power spectral density that is due to the input noise, evaluated for the case where (a) the available power spectral density of the n-port source is nkT0 and (b) the available power of the noise delivered by the load is zero.The noise figures defined above are identical to the one defined by Randa [22] using the scattering representation, except that the condition (a) is not required in his definition. We use this condition because it guaranties that our definition becomes the usual definition in the two-port case, as shown in Section 4.Let us first assume that the impedance matrix of the source isZS, that the column vector of the open-circuit noise voltages produced by the source is ESN, and that the impedance matrix of the noiseless load is ZL. We have(7)VN=(ESN0m1)-ZSLINwithZSL=(ZS0nm0mnZL),
where 0pq is used to denote the null matrix of size p×q. We use (3) and (7) to write(8)IN=(Z+ZSL)-1((ESN0m1)-VOCN).
Consequently, we have(9)〈ININ*〉=(Z+ZSL)-1(〈ESNESN*〉0nm0mn0mm)(Z+ZSL)-1*+(Z+ZSL)-1〈VOCNVOCN*〉(Z+ZSL)-1*.By (2), we directly obtain (10)〈ININ*〉=(〈IINIIN*〉〈IINION*〉〈IONIIN*〉〈IONION*〉),
so that(11)〈IONION*〉=(0mn1m)〈ININ*〉(0nm1m),
where 1p is the identity matrix of size p×p. Since by  (1), (2), and (7) we have VON=-ZLION, we can write(12)-〈VONION*〉=(0mnZL)〈ININ*〉(0nm1m).Using (9) in (12), we obtain the matrix of the self- and cross-complex powers received by the m-port load, in the impedance representation(13)-〈VONION*〉=(0mnZL)×{(Z+ZSL)-1(〈ESNESN*〉0nm0mn0mm)(Z+ZSL)-1*+(Z+ZSL)-1〈VOCNVOCN*〉(Z+ZSL)-1*(〈ESNESN*〉0nm0mn0nn)}×(0nm1m).Let us useVSN and ISN to denote the column vectors of noise voltages and noise currents of the source, respectively. In the case of hermitian matching, we have(14)ISN=(ZS+ZS*)-1ESN,VSN=ZS*(ZS+ZS*)-1ESN.Thus, the matrix of the self- or cross-complex powers which then-port source would deliver to a noiseless n-port load providing hermitian matching is(15)PCSN=〈VSNISN*〉=ZS*(ZS+ZS*)-1〈ESNESN*〉(ZS+ZS*)-1.
We therefore have(16)〈ESNESN*〉=(ZS+ZS*)ZS-1*PCSN(ZS+ZS*).The available power of then-port source, denoted by PASN, is equal to the real power which it delivers to a noiseless n-port load providing hermitian matching [12]. The available power of the n-port source is(17)PASN=Re(〈ISN*VSN〉)=Re(〈ESN*(ZS+ZS*)-1ZS*(ZS+ZS*)-1ESN〉),
where we have used (14). Thus,(18)PASN=12〈ESN*(ZS+ZS*)-1ESN〉=Re(tr⁡PCSN)=-tr⁡NZ,
where we have used the characteristic noise matrix of the source  in the impedance representation [21, chapter 3], defined as(19)NZ=-12(ZS+ZS*)-1〈ESNESN*〉.Consequently,in the impedance representation, the noise figureFα may be written(20)Fα=1+Re[{ZHLYTL〈VOCNVOCN*〉YTL*}n+αn+α]Re[{ZHLYTL(〈ESNESN*〉0nm0mn0mm)YTL*}n+αn+α],
where YTL=(Z+ZSL)-1, and  {X}ij is the entry of row i and column j of X, where 〈ESNESN*〉 is computed for  PASN = −tr NZ=nkT0Δf and where(21)ZHL=(0nn0nm0mnZL).Alternatively, we can useYS to denote the admittance matrix of the source, JSN to denote the column vector of the short-circuit noise currents produced by the source, and YL the admittance matrix of the noiseless load. We have(22)IN=(JSN0m1)-YSLVNwithYSL=(YS0nm0mnYL).
We use (4) and (22) to write(23)VN=(Y+YSL)-1((JSN0m1)-ISCN).Proceeding as above to obtain (13), we get the matrix of the self- and cross-complex powers received by the m-port load, in the admittance representation(24)-〈VONION*〉=(0mn1m)×{(Y+YSL)-1(〈JSNJSN*〉0nm0mn0mm)(Y+YSL)-1*+(Y+YSL)-1〈IOCNIOCN*〉(Y+YSL)-1*(〈JSNJSN*〉0nm0mn0nn)}×(0nmYL*).The matrix of the self- or cross-complex powers which then-port source would deliver to a noiseless n-port load providing hermitian matching is given by(25)PCSN=(YS+YS*)-1〈JSNJSN*〉(YS+YS*)-1YS.We therefore have(26)〈JSNJSN*〉=(YS+YS*)PCSNYS-1(YS+YS*).The available power of then-port source is(27)PASN=12〈JSN*(YS+YS*)-1JSN〉=Re(tr⁡PCSN)=-tr⁡NY,
where NY is the characteristic noise matrix of the source in the admittance representation, given by(28)NY=-12(YS+YS*)-1〈JSNJSN*〉.Thus, in the admittance representation, the noise figureFα is given by(29)Fα=1+Re[{ZTL〈ISCNISCN*〉ZTL*YHL*}n+αn+α]Re[{ZTL(〈JSNJSN*〉0nm0mn0mm)ZTL*YHL*}n+αn+α],
where ZTL=(Y+YSL)-1 and 〈JSNJSN*〉 is computed for PASN=-trNY=nkT0Δf and(30)YHL=(0nn0nm0mnYL).


## 4. Properties of the Noise Figures

The above-defined noise figures of a MIPMOP device do not depend on the particular representation used to express them, since the definition is based on a ratio of real powers. We also note thatPCSNand PASN do not depend on the representation chosen.The four following properties indicate that our definition of the noise figures of a MIPMOP device is an adequate generalization of the well-known definition for a two-port.Property 1.
In the casen=m=1, F1 is the noise figure F defined for the two-port, which does not depend on ZL or YL.Proof.
Forn=m=1,  Definition 1 is the standard definition of F for a two-port [22, Section  2.1].We note that forn=1, using (18), (27) and PASN=kT0Δf, we get the usual single-port noise source formulas(31)〈ESNESN*〉=〈ESNESN¯〉=4kT0ΔfRe(ZS),〈JSNJSN*〉=〈JSNJSN¯〉=4kT0ΔfRe(YS),
where the bar indicates the complex conjugate.Property 2.
In the casen=m, if the n-port source is made of n uncoupled and uncorrelated identical single-port sources of internal impedance ZS, if the MIPMOP device is made of n uncoupled identical two-ports having the noise figure FTP for the source impedance ZS, and if the n-port load is made of n uncoupled single-port loads of impedance ZL, then Fα=FTP for any α  such that 1≤α≤m. Thus, in this case, Fα does not depend on ZL or YL.Proof.
We use (15) and (19) to write
(32)PCSN=ZS¯4Re(ZS)2〈|ESN|2〉1n,NZ=-14Re(ZS)〈|ESN|2〉1n.
Consequently, for PASN=-trNZ=nkT0Δf, we have
(33)〈ESNESN*〉=4kT0ΔfRe(ZS)1n.
Let us usez to denote the 2×2 impedance matrix of one of said identical two-ports and zij to denote an entry of z. We have
(34)Z+ZSL=((z11+ZS)1nz121nz211n(z22+ZL)1n),(35)(Z+ZSL)-1=((z22+ZL)1n-z121n-z211n(z11+ZS)1n)ZLZS+z11ZL+z22ZS+z11z22-z12z21.
We also have
(36)〈VOCNVOCN*〉=(〈|vOCIN|2〉1n〈vOCINvOCON¯〉1n〈vOCONvOCIN¯〉1n〈|vOCON|2〉1n).
Using (20),  (33), (35) and (36), after some derivations, we get
(37)Fα=1+(-z21z11+ZS)t(〈|vOCIN|2〉〈vOCINvOCON¯〉〈vOCONvOCIN¯〉〈|vOCON|2〉)(-z21̅z11+ZS)4kT0ΔfRe(ZS)(-z21z11+ZS)t(1000)(-z21̅z11+ZS),
which can be shown to be equal to FTP.The Property2 indicates that the noise figures given by (20) and (29) are intensive quantities: each noise figure does not change when the number of identical uncoupled systems is changed.Property 3.
If a MIPMOP device B is made of a cascade consisting of a lossless passive input network havingn input ports and n output ports, of a MIPMOP device A having n input ports and m output ports and of a lossless passive output network, then the sum of the noise figures of the MIPMOP device B is equal to the sum of the noise figures of the MIPMOP device A.Proof.
This is the direct consequence of the following facts:(i)
the lossless passive output network does not change the total real power delivered to them-port load;(ii)
the lossless passive input network does not change the available power of then-port source because it is possible to prove (see the appendix) that hermitian matching occurs at its input ports if and only if hermitian matching occurs at its output ports.Property 4.
In the configuration considered in the Property3, if we also require that the lossless passive output network is made of m uncoupled lossless passive two-ports, each having an input port connected to the output port α  of the MIPMOP device A and an output port corresponding to the output port α  of the MIPMOP device B, then, for any α  such that 1≤α≤m, the noise figure of the MIPMOP device B for the output α  is equal to the noise figure of the MIPMOP device A for the output α.Proof.
Each lossless passive two-port does not change the real power delivered to each port of them-port load.The Property4 is a generalization to MIPMOP devices of a property of noisy two-ports which is a direct consequence of the Friis formula for cascaded noisy two-ports [1, 2]. We note that the configurations considered in the properties 3 and 4 can be used for computing the noise figures of a MIPMOP amplifier consisting of an active MIPMOP circuit inserted between two MIPMOP matching circuits having negligible losses.Notwithstanding the fact thatF is a vector, we have to keep in mind that there are three important differences between the usual definition of the noise figure for a two-port and the definition introduced in Section 3 for a MIPMOP device:(i)
forn≥2, each Fα depends on the correlations between the ports of the n-port source;(ii)
form≥2, each Fα depends on the m-port load, that is to say, on ZL or on YL in the case of a linear m-port load [23, Section 3];(iii)
since there is no obvious generalization of the concept of signal-to-noise ratio to MIPMOP devices, the noise figures of a MIPMOP device cannot be viewed as a direct measure of the degradation of a signal-to-noise ratio.


## 5. Natural Noise Figures of a MIPMOP Device

We are going to introduce a definition of thenatural noise figures of the MIPMOP device, based on two properties.Property 5.
If then-port source is a noisy passive linear network at the temperature T, then the available power spectral density of the n-port source is nkT. Additionally, we have
(38)〈ESNESN*〉=2kTΔf(ZS+ZS*),(39)〈JSNJSN*〉=2kTΔf(YS+YS*).Proof.
The covariance matrix〈ESNESN*〉 is hermitian and positive definite, and ZS+ZS* is hermitian. Consequently [26, Section  7.6.5], there exists a nonsingular matrix C such that C*(ZS+ZS*)C is real and diagonal and C*〈ESNESN*〉C=1n.
As pointed out in [21, page 29], this transformation may be obtained with a lossless imbedding. We may introduce the impedance matrix ZS′ of the resulting network, the column vector of its open-circuit noise voltages ESN′ and write
(40)ZS′+ZS′*=C*(ZS+ZS*)C=2diagn(R1,…,Rn),(41)〈ESN′ZSN′*〉=C*〈ESNESN*〉C=1n,
where diagn(x1,…,xn) denotes the diagonal matrix of size n×n having the entries x1,…,xn, and where R1,…,Rn are real. We then introduce the impedance matrix Zrem such that
(42)ZS′=diagn(R1,…,Rn)+Zrem.
If we use (42) in (40), we find that
(43)Zrem=-Zrem*.
Thus,Zrem is the impedance matrix of a lossless network. We now connect a lossless network of impedance matrix -Zrem in series with the network of impedance matrix ZS′ and obtain a network of impedance matrix
(44)ZS′′=ZS′-Zrem=diagn(R1,…,Rn),
this network having the column vector of the open-circuit noise voltages ESN′. We consequently havenuncoupled resistances at the temperature T, which proves that R1,…,Rn are positive and
(45)〈ESN′ESN′*〉=nkTΔfdiagn(R1,…,Rn).
Using (41), we obtain
(46)〈ESNESN*〉=C*-1〈ESN′ESN′*〉C-1=4nkTΔfC*-1diagn(R1,…,Rn)C-1.
Using (40) we then find that
(47)〈ESNESN*〉=2nkTΔf(ZS+ZS*),〈JSNJSN*〉=ZS-1〈ESNESN*〉ZS*-1=2nkTΔf(YS+YS*).Property 6.
If then-port source produces the noise of a noisy passive network at the temperature T0, the constraint (a) of the Definition 1 is automatically satisfied.Proof.
If we use (15), (18), and (38), we find that
(48)PCSN=2kT0ΔfZS*(ZS+ZS*)-1,PASN=-tr⁡NZ=nkT0Δf,
which proves the property. Alternatively, we could have used (25), (27), and (39), to obtain
(49)PCSN=2kT0Δf(YS+YS*)-1YS,PASN=-tr⁡NY=nkT0Δf,
and reached the same conclusion.Definition 2.
If then-port source produces the noise of a noisy passive network at the temperature T0, the noise figures of Definition 1 will be referred to as the natural noise figures.In the laboratory, it is possible to use a nearly ideal noisy passive network at the temperatureT0, and to measure natural noise figures in this manner. However, this parameter does not necessarily correspond to the application situation. For instance the noise collected by multiple antennas will in general not comply with (48) and (49).The following property relates to a special case where the natural noise figures are easier to determine using standard measuring instruments. Let us use(i)
Fαβ′ to denote the noise figure between an input β and an output α, defined as the usual noise figure of a two-port, evaluated in the case where all unused inputs and outputs are connected to uncoupled and uncorrelated single-port terminations for which the available power spectral density is kT0;(ii)
Fαα′′′  to denote the noise figure between an outputα′, used as an input, and an outputα, defined as the usual noise figure of a two-port, evaluated in the case where all unused inputs and outputs are connected to uncoupled and uncorrelated single-port terminations for which the available power spectral density iskT0.Property 7.
In the case where then-port source is made of uncoupled and uncorrelated single-port sources, and where the m-port load is made of uncoupled and uncorrelated single-port loads, the natural noise figures are given by
(50)Fα=1-∑α′=1,α′≠αm1/Fαα′′′∑β=1n1/Fαβ′.Proof.
All single-port sources and loads being assumed uncoupled and uncorrelated to each other, let us use
(i)
P1α to denote the self-power spectral density at the output α, evaluated for the case where (c) the inputs are connected to single-port sources, (d) the available power spectral density of each single-port source is kT0, (e) the outputs are connected to single-port loads, and (f) the available power spectral density of each single-port load is zero;(ii)
P2αβ to denote the self-power spectral density at the output α, evaluated for the case where (g) the MIPMOP device is noiseless, (c) the inputs are connected to single-port sources, (h) the available power spectral density of each single-port source is zero except the single-port source connected to the input β for which the available power spectral density is kT0, (e) the outputs are connected to single-port loads, and (f) the available power spectral density of each single-port load is zero.
In the case defined above, we have(51)Fα=P1α∑β=1nP2αβ,
so that
(52)Fα=1∑β=1n1/FαβwhereFαβ=P1αP2αβ.
Fαβ is not the usual noise figure of a two-port, evaluated in the case where all unused inputs and outputs are connected to single-port terminations, because, in this case, the unused single-port loads also contribute to the self-power spectral density at the output α. Let us use P3αα′ to denote the self-power spectral density at the output α, evaluated for the case where (g) the MIPMOP device is noiseless, (c) the inputs are connected to single-port sources, (j) the available power spectral density of each single-port source is zero, (e) the outputs are connected to single-port loads, and (k) the available power spectral density of each single-port load is zero except the single-port load connected to the output α′ for which the available power spectral density is kT0. We have
(53)Fαβ′=P1α+∑α′=1,α′≠αmP3αα′P2αβ,Fαα′′′′=P1α+∑α′=1,α′≠αmP3αα′P3αα′′.
So that we obtain(54)Fαβ=Fαβ′(1-∑α′=1α′≠αm1Fαα′′′).
Using (54) in (52) leads us to (50).Note that in the case where the contribution of the single-port loads to the self-power spectral density at the outputα is negligible, that is,(55)∑α′=1α′≠αm1Fαα′′′≪1,
we may assume that (56)Fα≈1∑β=1n1/Fαβ′.In [23], (56) was incorrectly presented as an exact formula. However, (56) was a good approximation of the exact formula (50) for the MIPMOP amplifier considered in [23].


## 6. A MIPMOP LNA Connected to an Antenna Array

We will now apply our results to the 1880 MHz MIPMOP LNA shown in Figure3, having n=4 signal input terminals and m=4 signal output terminals. It is intended to be used as a wireless receiver front-end. The core of this MIPMOP LNA is a new circuit called MIMO series-series feedback amplifier (MIMO-SSFA) described below [23, 27, 28]. Referring to the impedance matrices defined in Figure 1, the MIPMOP LNA is designed to provide hermitian matching between ZSant and ZLI.Figure 3
A low-noise MIPMOP amplifier comprising a MIMO-SSFA, 4 uncoupled input matching networks, and 4 uncoupled output matching networks.The antenna array is a circular array of four parallel half-wave dipole antennas (side-by-side configuration) for 1880 MHz. The radius of the array is 0.3λ=47.8 mm, so that it presents a 0.424 λ=67.7 mm spacing between the nearest array elements. This antenna array has been presented in [19]. The impedance matrix of the antenna array is computed as(57)Zant=(73.11.1-36.4j-23.3-15.9j1.1-36.4j1.1-36.4j73.11.1-36.4j-23.3-15.9j-23.3-15.9j1.1-36.4j73.11.1-36.4j1.1-36.4j-23.3-15.9j1.1-36.4j73.1)Ω.For a zenithal angleθ=π/2, and if we assume that the array sees four uncoupled scalar loads of 73.0 Ω, the power delivered by each antenna and the average power are shown in Figure 4 as a function of the azimuth for an incident field of 1 V/m peak. The average power oscillates between 3.3 μW and 4.2 μW as a function of the angle of arrival whereas, in the same field, a single dipole antenna in free space would deliver 4.4 μW to a 73.0 Ω load.Figure 4
The powersP0, P1, P2, and P3 delivered by each antenna (in μW) and the average power as a function of the azimuth (in degree) for a conventional front-end design receiving an incident plane wave of 1 V/m (peak).We assume that each antenna is connected to the MIPMOP amplifier via a two-conductor interconnection (for instance a coaxial cable) behaving as a two-conductor transmission line. The antenna numberj is connected to the near-end of the interconnection number j of lengthdj, characteristic impedance zCj, phase velocity cj, and attenuation constant αj. The far-end of the interconnection number j is connected to the input port number j of the MIPMOP amplifier. Let us define the characteristic impedance matrix ZC of the set of interconnections as(58)ZC=diag⁡n(zC1,…,zCn).Let us also define the transmission matrixT of the set of interconnections as(59)T=diagn(e-[α1+j(2πf0/c1))]d1,…,e-[αn+j(2πf0/cn)]dn).After some derivation [19], we get(60)ZSant={T+T-12Zant-T-T-12ZC}×{T+T-12ZC-T-T-12Zant}-1ZC.If we assume that four identical 54 mm-long interconnections are used, having, atf0=1880 MHz, a propagation velocity of 0.6 c0, a characteristic impedance of 80 Ω, and an attenuation of 1.667 dB/m, the resulting impedance matrix seen by the MIPMOP LNA at f0 is equal to(61)ZSant=(84.4+10.1j-18.7-32.5j-17.9+13.5j-18.7-32.5j-18.7-32.5j84.4+10.1j-18.7-32.5j-17.9+13.5j-17.9+13.5j-18.7-32.5j84.4+10.1j-18.7-32.5j-18.7-32.5j-17.9+13.5j-18.7-32.5j84.4+10.1j)Ω.We note that if we compare (57) and (61), the relative difference between diagonal entries is small, but the nondiagonal entries are thoroughly modified.The MIMO-SSFA consists ofn active subcircuits (ASC) and a feedback network. Each ASC comprises a packaged low-noise pseudomorphic HEMT (Q311 to Q314) and a drain resistor (R311 to R314) providing resistive loading to improve stability. The feedback network consists of the 4 inductors L411 to L414, coupled by mutual induction. This coupling is the only phenomenon leading to a nondiagonal input impedance matrixZLI. The MIPMOP LNA also comprises an input matching network (C211 to C214 and L221 to L224) and an output matching network (C521 to C524 and L511 to L514).The MIMOP LNA shown in Figure3 has been designed in such a way that when each output port is connected to a grounded 50 Ω resistor, the amplifier provides an impedance ZLI close to ZSant* [29, Section  5]. At f0=1880 MHz, the resulting voltage gain matrix GV of the MIPMOP amplifier and ZLI are(62)GV=(-2.80-3.32j0.06+0.86j0.24+0.37j0.06+0.86j0.06+0.86j-2.80-3.32j0.06+0.86j0.24+0.37j0.24+0.37j0.06+0.86j-2.80-3.32j0.06+0.86j0.06+0.86j0.24+0.37j0.06+0.86j-2.80-3.32j),(63)ZLI=(84.5-10.2j-18.7+32.5j-17.9-13.5j-18.7+32.5j-18.7+32.5j84.5-10.2j-18.7+32.5j-17.9-13.5j-17.9-13.5j-18.7+32.5j84.5-10.2j-18.7+32.5j-18.7+32.5j-17.9-13.5j-18.7+32.5j84.5-10.2j)Ω.The comparison of (61) and (63) shows that hermitian matching is almost exactly reached between the receiver’s end of the interconnections and the inputs of the MIPMOP LNA. We computed the power delivered to each 50 Ω resistor and the average power shown in Figure 5, in the configuration used for Figure 4 [19, Section  8.2].Figure 5
The powersP0, P1, P2, and P3 delivered by the outputs of the MIPMOP amplifier shown in Figure 3 (in μW) and the average power as a function of the azimuth (in degree) when the antenna array receives an incident plane wave of 1 V/m (peak).Assuming that the input and output matching networks are lossless, we have computed the noise figures for three configurations of the MIPMOP amplifier, in a worksheet of a generic calculation program. A technique used several times in this worksheet is the following: (6) being possibly used to obtain the appropriate representation, the covariance matrix of a series or parallel connection of two uncorrelated circuits is obtained as a sum of covariance matrices [24, 25].In this manner we successively determined:(1)
the covariance matrix of the open-circuit voltages〈VAVA*〉 of an ASC comprising a transistor and a series drain resistor, this matrix being denoted by
(64)〈VAVA*〉=(〈VA1VA1¯〉〈VA1VA2¯〉〈VA2VA1¯〉〈VA2VA2¯〉);(2)
the covariance matrix〈VBVB*〉 of the open-circuit voltages of the MIMO-SSFA, using
(65)〈VBVB*〉=(〈EFBEFB*〉〈EFBEFB*〉〈EFBEFB*〉〈EFBEFB*〉)+(〈VA1VA1¯〉1n〈VA1VA2¯〉1n〈VA2VA1¯〉1n〈VA2VA2¯〉1n),
and the assumption that the feedback network of the MIMO-SSFA behaves as a noisy passive network at the temperature T0, so that according to (38) the covariance matrix of the open circuit voltages at the terminals of the feedback network is
(66)〈EFBEFB*〉=2kT0Δf(ZFB+ZFB*),
where ZFB is the impedance matrix of the feedback network, the real part of which is determined by losses in the inductors L411 to L414;(3)
the noise figures of the MIMO-SSFA, using  (20) applied to 〈VAVA*〉 given by  (65);(4)
the matrices〈VOCNVOCN*〉 and 〈ISCNISCN*〉 of the MIPMOP LNA, once parallel and series circuit elements of the input and output matching networks are included;(5)
the noise figures of the MIPMOP LNA, using  (20) applied to 〈VOCNVOCN*〉 or (29) applied to 〈ISCNISCN*〉.Since we assume that the input and output matching networks are lossless, we expect that the steps (3) and (5) provide the same result, according to the Property4. We indeed always obtain the same noise figures, in the three configurations defined below.In the first configuration, we have canceled the mutual induction between the inductors L411 to L414, the resulting MIPMOP amplifier being driven by a source having a diagonal impedance matrix equal to(84.436+j10.124)1n Ω. Consequently, the four channels are uncoupled. All computed natural noise figures are equal to about 1.134 dB, irrespective of the choice of diagonal load impedance matrix. We checked that in line with the Property 2, this result is the same as the one obtained for one channel treated as a two-port.In the second configuration, we have kept zero mutual inductances between the inductors L411 to L414, but the input of the resulting MIPMOP amplifier sees the impedance matrixZSant given by (61). All computed natural noise figures are equal to about 1.218 dB, for a load impedance matrix equal to 50 1m Ω. Consequently, the couplings between the antennas increase the noise figures, as qualitatively explained in the introduction. We also observed that, as expected, these noise figures depend on the choice of the diagonal load impedance matrix.In the third configuration, the mutual inductances between the inductors L411 to L414 had their nominal values, the MIPMOP amplifier being driven by a source having an impedance matrix equal toZSant given by (61). This configuration is the MIPMOP LNA designed in [29] and used in [19], as said above. In this configuration, all natural noise figures are equal to about 1.896 dB, for a load impedance matrix equal to 50 1m Ω. We again observed that, as expected, these noise figures depend on the choice of the diagonal load impedance matrix.It must again be emphasized that, for the second and third configurations, there is no direct relationship between the computed natural noise figures and a measure of the degradation of a signal-to-noise ratio, as pointed out in the introduction and at the end of Section4.


## 7. Conclusion

In this paper, we have defined the noise figures of a MIPMOP device in a manner that extends the classical definition applicable to a two-port. This definition has an additional condition compared to the definition used by Randa [22]. When the n-port source connected to the input ports of the MIPMOP device is a passive multiport source at T0, our noise figures are referred to as natural noise figures, and the source impedance matrix determines the self- and cross-correlations of the noises applied to the input ports. We have also developed a theory of the noise figures of MIPMOP devices in the impedance and admittance representations.We have presented an example which shows how this theory can be applied to the design of a MIPMOP LNA connected to an array of coupled antennas and providing hermitian matching. It is worth noting that if mutual induction was not present, Figure3 would represent four independent single-input and single-output LNA with inductive source degeneration. This scheme is popular because it can simultaneously provide a high gain and a low noise figure [30–33]. Since introducing mutual induction between coils does not involve any additional circuit element, the 4-input and 4-output MIPMOP LNA shown in Figure 3 has the same number of circuit elements as four independent 2-port LNAs with inductive source degeneration.In the example treated in Section6, we observe that ZC given by (58) is quite different from ZLI given by (63) and Zant given by (57), so that multiple reflections will take place. This phenomenon is inherent to hermitian matching. It does not cause problems when the interconnections between the antennas and the MIPMOP LNA are sufficiently short.



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*Source: 438478-2011-11-01.xml*