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.

In this paper, we consider a linear multiple-input-port and multiple-output-port (MIPMOP) device having

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:

the MIPMOP device is made of

the ports of the

the ports of the

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 Figure

An array of

A conventional wireless receiver front-end uses a MIPMOP device consisting of

A type of nonconventional wireless receiver front-end uses a MIPMOP passive matching network to obtain a nondiagonal

Another type of nonconventional wireless receiver front-end comprises a MIPMOP LNA presenting a nondiagonal

In both nonconventional front-end designs,

In their theory of linear noisy networks, Haus and Adler presented many results which are applicable to MIPMOP devices [

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 [

We use rms values throughout the paper. We use

In the small bandwidth

For the case

Equivalent circuit for a noisy MIPMOP device.

The second-order statistics of the noise produced by the MIPMOP device is determined by the covariance matrix of the open-circuit noise voltages

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 [

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 is

If

The noise figures defined above are identical to the one defined by Randa [

Let us first assume that the impedance matrix of the source is

By (

Using (

Let us use

Thus, the matrix of the self- or cross-complex powers which the

The available power of the

Consequently,in the impedance representation, the noise figure

Alternatively, we can use

Proceeding as above to obtain (

The matrix of the self- or cross-complex powers which the

We therefore have

The available power of the

Thus, in the admittance representation, the noise figure

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 that

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.

In the case

For

We note that for

In the case

We use (

Let us use

Using (

The Property

If a MIPMOP device B is made of a cascade consisting of a lossless passive input network having

This is the direct consequence of the following facts:

the lossless passive output network does not change the total real power delivered to the

the lossless passive input network does not change the available power of the

In the configuration considered in the Property

Each lossless passive two-port does not change the real power delivered to each port of the

The Property

Notwithstanding the fact that

for

for

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.

We are going to introduce a definition of the

If the

The covariance matrix

As pointed out in [

If we use (

Thus,

Using (

If the

If we use (

If the

In the laboratory, it is possible to use a nearly ideal noisy passive network at the temperature

The following property relates to a special case where the natural noise figures are easier to determine using standard measuring instruments. Let us use

In the case where the

All single-port sources and loads being assumed uncoupled and uncorrelated to each other, let us use

In the case defined above, we have

So that we obtain

Using (

Note that in the case where the contribution of the single-port loads to the self-power spectral density at the output

In [

We will now apply our results to the 1880 MHz MIPMOP LNA shown in Figure

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 0.3

For a zenithal angle

The powers

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 number

Let us also define the transmission matrix

After some derivation [

If we assume that four identical 54 mm-long interconnections are used, having, at

We note that if we compare (

The MIMO-SSFA consists of

The MIMOP LNA shown in Figure

The comparison of (

The powers

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: (

In this manner we successively determined:

the covariance matrix of the open-circuit voltages

the covariance matrix

the noise figures of the MIMO-SSFA, using (

the matrices

the noise figures of the MIPMOP LNA, using (

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 Property

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

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 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 to

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 Section

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 [

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, Figure

In the example treated in Section

A linear MIPMOP device having

The real power

If we assume that the linear MIPMOP device provides hermitian matching at the input ports, by (

If we now assume that the MIPMOP device is lossless, by (

If we compare (

We note that this theorem does not use reciprocity.