Presented on 7/8/2014, at 2014 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS/URSI 2014), Memphis, TN

1. Minimum Complexity Decoupling Networks for Arbitrary
Coupled Loads
Ding Nie, Bertrand Hochwald and Erik Stauffer
University of Notre Dame
Broadcom Cooperation
nding1@nd.edu
bhochwald@nd.edu
eriks@broadcom.com
July 8, 2014
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 1 / 13

2. Overview
1 Introduction to Decoupling Networks
2 Systematic Design of Decoupling Networks
3 Minimum Complexity Decoupling Networks
4 Summary
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 2 / 13

3. Coupled Loads
RF mutual coupling
Coupled antennas in MIMO
communications
RFIC coupled microstrip lines
…
Coupling is undesirable
Introduces power reflection
Mixes useful signals with unwanted signals
Solution: decoupling network
Achieves perfect impedance matching at the design frequency
Maximizes the power efficiency of the RF system
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 3 / 13

4. Introduction to Decoupling Networks
Two-port matching network is used to match single source to a single
load
Decoupling network is used to match uncoupled sources to coupled
loads
Transforms the coupled impedance of the loads into the uncoupled
characteristic impedance of the sources
2N-port
matching
network
N coupled
loads
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0Z
0Z
Two-port
matching
network
LZ 0Z0Z
0Z
0Z
.
.
.
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 4 / 13

5. Complexity of Decoupling Networks
The decoupling networks are
complicated in general
But the realization is not unique
...
...
1
2
3
1N −
N
1N +
2N +
3N +
2 1N −
2N
Our Contribution
Systematic and unified decoupling
network design for arbitrary coupled
loads
Decoupling network design method with
minimum complexity
...
...
1
2
3
1N −
N
1N +
2N +
3N +
2 1N −
2N
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 5 / 13

6. Examples of Minimum Complexity Decoupling Networks
Two and Three Dipoles (at 2.4 GHz)
0.17 + 0.48j 0.36 − 0.36j
0.36 − 0.36j 0.17 + 0.48j


0.18 + 0.45j 0.42 − 0.34j −0.10 − 0.10j
0.42 − 0.34j 0.16 + 0.60j 0.16 − 0.26j
−0.10 − 0.10j 0.16 − 0.26j 0.37 + 0.35j


10
λ
2
λ
5
λ
2
λ
10
λ
(a) (b)
1.39 nH
2.74 pF
2.15 pF
1.09 nH
3.81 pF
1.61 nH
10.03 nH
1
2
3
4
11c
14c 44c
12c
13c
33c
23c 22c
25c
55c
36c
66c
1
2
3
4
5
6
c11 32.06 pF
c12 9.20 pF
c13 2.24 nH
c14 29.66 pF
c22 3.46 pF
c23 4.26 pF
c25 3.96 pF
c33 3.13 pF
c36 5.41 pF
c44 0.29 nH
c55 2.12 nH
c66 2.14 nH
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 6 / 13

7. Properties of Decoupling Networks
Definition of Decoupling Network
A decoupling network for N
dissipative reciprocal loads with
S-matrix SL is a lossless, reciprocal,
2N-port network S that satisfies
SLM = 0, where
SLM = S11 + S12SL(I − S22SL)−1
S21
0Z
2N-port
matching
network
0
Z
.
.
.
LSLMS
N-port
loads
1a

1b

2a

2b

.
.
.
Outputports
N+1~2N
Inputports
1~N
11 12
21 22
S S
S S
 
 
 
0
Non-uniqueness of Decoupling Networks
Set of decoupling networks for SL
S := {S ∈ C2N×2N
: S22 = SH
L , SH
S = I, ST
= S}
S has N2 degrees of freedom
All S-matrix in S has the same performance, but different realization
complexity
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 7 / 13

8. Network Synthesis with Generalized Π-Network
Generalized Π-Network
...
...
1
2
3
1N −
N
1N +
2N +
3N +
2 1N −
2N
11c
22c
33c
44c
12c
13c
24c
34c
14c
23c
1
2
3
4
(b)
22c
21
12c
(a)
(c)
11c
Π-network:
Y =
c11 + c12 −c12
−c12 c12 + c22
Generalized 2N-port Π-network:
Y =




2N
i=1 c1i −c12 · · · −c1(2N)
−c12
2N
i=1 c2i · · · −c2(2N)
...
...
...
...
−c1(2N) −c2(2N) · · · 2N
i=1 ci(2N)





Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 8 / 13

9. Systematic Decoupling Networks Design Steps
Systematic Decoupling Networks Design Steps
1 Find a S-matrix S that belongs to the set of decoupling networks S,
such that the number of impedance is minimized
2 Compute the admittance matrix of the decoupling network using
Cayley transform
Y =
1
Z0
(I − S)(I + S)−1
3 Realize Y using generalized Π-network
The minimum number of impedances needed to realize a decoupling
network is N2 + N
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 9 / 13

10. Minimum Complexity Decoupling Networks Design
Minimum Complexity Decoupling Networks Design
For arbitrary coupled loads, we obtain the following decoupling network
structure with N2 + N impedances, which is the minimum number
achievable.
Y =
























× × × · · · × × × 0 0 · · · 0 0
× × × · · · × × 0 × × · · · ×
× × × · · · × × 0 0 × · · · × ×
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× × × · · · × × 0 0 0 · · · × ×
× × × · · · × × 0 0 0 · · · 0 ×
× 0 0 · · · 0 0 × 0 0 · · · 0 0
0 × 0 · · · 0 0 0 × 0 · · · 0 0
0 × × · · · 0 0 0 0 × · · · 0 0
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0 × × · · · × 0 0 0 0 · · · × 0
0 × · · · × × 0 0 0 · · · 0 ×
























...
...
1
2
3
1N −
N
1N +
2N +
3N +
2 1N −
2N
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 10 / 13

11. Minimum Complexity Decoupling Networks for Symmetric
Loads
Symmetric loads are SL that has the form
SL =





µL + ξL ξL · · · ξL
ξL µL + ξL · · · ξL
...
...
...
...
ξL ξL · · · µL + ξL





Apply the systematic design method, we get
...
...
1
2
3
1N −
N
1N +
2N +
3N +
2 1N −
2N
...
2
3
1N −
N
1N +
2N +
2 1N −
2N
3N +
1
3c
4c2c
1c
Only 4N impedances are needed
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 11 / 13

12. Summary
Systematic design of decoupling networks for arbitrary coupled loads
Decoupling networks realization using N2 + N components, the
minimum possible
Examples of two-, three-antennas and symmetric loads
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 12 / 13

13. References
J. B. Anderson and H. H. Rasmussen, “Decoupling and descattering
networks for antennas,” IEEE Transactions on Antennas and
Propagation, vol. 24, no. 6, pp. 841-846, Nov. 1976.
J. C. Coetzee and Y. Yu, “Design of decoupling networks for circulant
symmetric antenna arrays,” IEEE Antennas and Wireless Propagation
Letters, vol. 8, pp. 291-294, 2009.
D. M. Pozar, Microwave Engineering 4th ed., John Wiley & Sons,
2011.
D. Nie, B. Hochwald and E. Stauffer, “Systematic design of large-scale
multiport decoupling networks,” IEEE Transaction on Circuits and
Systems I: Regular Papers, vol. 61, no. 7, pp. 2172-2181, July 2014.
Ding Nie, Bertrand Hochwald and Erik Stauffer (University of Notre Dame)Minimum Complexity Decoupling Networks July 8, 2014 13 / 13