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Beamforming for Space Division Duplexing

Damith Senaratne and Chintha Tellambura

Department of Electrical and Computer Engineering,

University of Alberta, Edmonton, AB, Canada.

Email: {damith, chintha}@ece.ualberta.ca

Abstract—Eigenmode transmission in multiple-input multiple-

output (MIMO) systems is examined under space division duplex-

ing (SDD). The antennas of each full-duplex node are partitioned

to form two antenna banks – one for transmission, the other

for reception. Self-interference is suppressed by utilizing the

nullspace (or the left nullspace) of corresponding self-interference

channel for transmission (or reception). Simulation results are

provided on the error performance. Useful insights are obtained

on how finite computational precision and quantization errors

affect the feasibility of SDD.

Index Terms—MIMO, space division duplexing, eigenmode

transmission, null space

I. INTRODUCTION

In this paper, the use of multiple antennas and spatial

signal processing to make a wireless terminal full-duplex

is considered. Full-duplex wireless communication may be

achieved exploiting the degrees of freedom (DoFs) available

in time-, frequency- or any suitable dimension. Frequency

division duplexing (FDD) and time division duplexing (TDD)

techniques are proven; and their applications are ubiquitous.

Spectral efficiency being based on the resource utilization in

time- and frequency- dimensions, the use of other independent

dimensions to achieve duplexing has become attractive, despite

the practical challenges. Space division duplexing (SDD) for

single-antenna systems has been attempted [1], [2] in this

respect, however, with non-spatial techniques for interference

suppression. It is with multiple-input multiple-output (MIMO)

technology, which supports system nodes with multiple spatial

DoFs, that spatial interference suppression became possible.

Full-duplex MIMO repeaters [3] and relays [4], [5] are

already receiving the attention, evidently because of the

prospects relaying has on extending the coverage of exist-

ing/ emerging MIMO compliant cellular and wireless data

networks. In a SDD configuration, a given antenna may not

transmit and receive simultaneously over the same frequency

band. Therefore, the antennas at a node are partitioned to

form 2 banks - one dedicated for transmission, and the

other, for reception (e.g. Nttransmit antennas vs. Nrreceive

antennas, in Fig. 1). Duplexing is achieved through signal

processing techniques that suppress the self-interference the

node’s transmission causes on its own reception.

SDD in MIMO wireless channels resembles suppressing

near-end crosstalk in digital subscriber lines (DSLs) [6]. How-

ever, the spatial channels in wireless systems are virtual, and

arise as a result of transmit/ receive beamforming, whereas the

Tx

w

Rx

r

H1

G

self-interference channel

Nt

Nr

H2

receive channel

transmit channel

Fig. 1. A MIMO node transmitting and receiving over same frequency band.

wire-pairs of a DSL exist physically. This distinction makes

SDD more challenging than crosstalk cancellation.

SDD also pose significant practical challenges in the form

of its high amplifier dynamic range requirement, and the

high analog-to-digital converter (ADC) resolution requirement.

Inspired by new experimental evidence [7] on achieving over

45dB of spatial interference suppression, SDD techniques are

investigated with a renewed interest.

The simplest, and perhaps, the most obvious approach for

self-interference cancellation is temporal. It involves assessing

and subtracting self-interference from the received signal [4,

Sec. III]. Its variant for full-duplex relay nodes is regarding

self-interference a ‘feedback’ (as in a control system), and

optimizing the relay gain matrix for interference suppression

[3], [5], [8].

Spatial interference mitigation is an alternative, in which the

transmit precoding matrix (w, in Fig. 1) and/ or receiver recon-

struction matrix (r) are chosen such that the self-interference,

irrespective of the data being transmitted in either direction,

has zero (or negligible) effect at the input of the detector.

Such techniques are based on: (i) the additional spatial DoFs

transmit (or receive) antennas of a node has [4], [7], [9], or

(ii) the orthogonality of distinct spatial modes in the self-

interference channel [10]. Joint optimization of transmitter-,

relay- and receiver- processing for full-duplex relaying too has

been considered [11].

The aim of this paper is exploring spatial self-interference

mitigation techniques usable for MIMO SDD eigenmode

transmission. The paper is organized as follows: Section II

presents the mathematical framework. Numerical results on

the performance of selected MIMO SDD configurations are

provided in Section III. The conclusion follows, highlighting

certain limitations that need to be overcome to realize SDD.

978-1-61284-231-8/11/$26.00 ©2011 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Page 2

Notation: Given a matrix A, its transpose, conjugate trans-

pose, rank, nullity, nullspace, and left nullspace are denoted:

AT, AH, rank(A), nullity(A), Null(A), Null?AT?

formed with its first n columns, and first m rows, respectively.

{A}C(m:n)is the sub-matrix of A formed with its columns

m through n. Main diagonal of A is given by diag(A).

A ∈ Cm×ndenotes that A is an m×n matrix. The notation

[A1, A2] represents the concatenation of matrices A1and A2.

II. MATHEMATICAL FRAMEWORK

A. SDD through nullspace & left nullspace projection

The singular value decomposition (SVD) of a matrix

G ∈ Cm×nis of the form G = UΣVH, where (i)

Σ ∈ Cm×nis nonnegative real rectangular diagonal; and (ii)

U ∈ Cm×m,V ∈ Cn×nare unitary.

Suppose G does not have full column-rank (i.e. r =

rank(G) < n). The columns of V(0)= {V}C(r+1:n)span

Null(G), such that GV(0)= 0 ∈ Cm×(n−r). Similarly,

Null?GT?is spanned by the columns of U(0)= {U}C(r+1:m)

such that

not have full row-rank (i.e. r < m). The nullspace and the

left nullspace exist simultaneously iff G is rank deficient (i.e.

r < min(m,n)).

Suppose G corresponds to the self-interference channel of

the MIMO capable node shown in Fig. 1. Given x, the symbols

to be transmitted, self-interference component at the detector

input is given by the term rGwx. The interference can be

nullified irrespective of x, if either of the constraints:

respec-

tively [12]. {A}C(n)and {A}R(m)give the sub-matrices of A

?U(0)?HG = 0 ∈ C(m−r)×n, whenever G does

Gw = 0,

rG = 0,

(1a)

(1b)

can be enforced. The constraints (1a) and (1b) provide three

possibilities for implementing SDD at a node.

1) Transmit SDD: Forming w with columns of V(0)enforces

(1a). It makes transmitted signal wx to be orthogonal to G.

This approach requires G to not have full-column rank, a

sufficient condition for which is allotting more antennas for

transmission than for reception.

2) Receive SDD: Forming r using rows of?U(0)?Henforces

orthogonal to the row space of G. This approach requires

G to not have full-row rank, guaranteed if the majority of

antennas are set aside for reception.

3) Joint Transmit and Receive SDD: Simultaneously enforc-

ing (1a) and (1b) as in reference [4], requires1G to be

rank deficient. This may only be achieved through proper

antenna design and placement (e.g. by arranging a key-hole

channel to exist between the antenna banks).

Since G is not bidirectional, ‘Joint Transmit and Receive

SDD’ appears redundant. Moreover, it complicates beamform-

ing where two nodes implementing SDD communicate. Hence,

we focus only on ‘Transmit SDD’ and ‘Receive SDD’.

(1b). The desired received signal component is forced to be

1Reference [10] examines another possibility based on the orthogonality of

eigenvectors. It holds if numbers of transmit and receive antennas are equal.

Tx

Rx

Node-1

Rx

Tx

Node-2

H1

s1

H2

G1

G2

M1

N1

M2

N2

wireless channel

Node-1

M1

N1

w1

r1

x1

n1

Node-2

M2

N2

w2

r2

x2

n2

# transmit antennas

# receive antennas

transmit precoding matrix

receiver reconstruction matrix

transmitted signal

additive noise at reception

s2

Fig. 2.System model: eigenmode transmission over a MIMO SDD system.

B. Eigenmode transmission with SDD

Consider two MIMO capable nodes: Node-i, i ∈ {1,2}

(see Fig. 2), each having a subset of Mi antennas set aside

for transmission, and the remaining Niantennas dedicated for

reception. The transmit (or receive) antennas of a given node

need not be physically adjacent.

Suppose the forward MIMO channel from Node-i is Hi∈

CNj×Mifor i,j ∈ {1,2},i ?= j. Its self-interference MIMO

channel Gi∈ CNi×Mimay or may not be rank deficient2.

wi∈ CMi×Miand ri∈ CNi×Niare the transmit precoding

and receiver reconstruction matrices. xi∈ CMi×1denotes the

signal transmitted by Node-i, while yi∈ CNi×1is the signal

it receives. ni ∈ CNi×1is the additive noise component at

reception. The received signal at the detector input of each

Node-i is then given by

yi= ri(Hjwjxj+ Giwixi+ ni).

(2)

Suppose sispatial modes need to be facilitated from each

Node-i to the other. This requires

rank(Hi) ≥ si,

(3)

and, either of

nullity(Gi) ≥ si,

nullity

or(4a)

?

GiT?

≥ sj,

(4b)

to be satisfied for i,j ∈ {1,2},i ?= j.

1) Case: Transmit SDD implemented at both nodes:

Design requirements: A necessary, but not sufficient condition

for (3) is having Nj≥ si. The requirement (4a) can be met,

irrespective of rank(Gi), by ensuring that (Mi− Ni) ≥ si.

Where His are not rank-deficient, the requirements are satis-

fied for (Mi− si) ≥ Ni≥ sj.

Example 1: Having Mi = 4 and Ni = 2, for instance,

guarantees 2 spatial modes in each direction, provided

Hi,i ∈ {1,2} are not keyhole channels. If communica-

tions were only from Node-1 to Node-2, each node would

have had 6 DoFs; but SDD yields only 4 spatial modes.

2Rank deficiencies in Gis would lessen the spatial DoFs SDD costs.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Page 3

Beamforming matrices : Suppose the SVDs: Gi= UiΣiViH

hold for i

∈

{Vi}C(rank(Gi)+1:Mi)span Null(Gi). Define3 ˆ Hi= HiV(0)

for i ∈ {1,2}, and let their SVDs beˆ Hi= QiΛiWiH.

The choice of wi= V(0)

i

{Wi}C(si)and rj=

therefore produces the spatial modes in both directions.

Remarks:

- The effective MIMO channelˆ Hiis Nj× nullity(Gi), and

no longer Nj× Mi. This implies reduced diversity orders.

Since rank

≤ min(rank(Hi),nullity(Gi)) a loss of

multiplexing gain too is apparent.

- Under Rayleigh fading, each Hi would be a complex

Gaussian random matrix; HiViwould have the same dis-

tribution since Viis unitary. Therefore,ˆ Hiwould also be

complex Gaussian irrespective of the distribution of Gis.

This premise makes performance analysis of MIMO SDD

under Rayleigh fading straightforward.

- Channel estimation may be easily performed, for example,

by TDD the pilot signals, and estimating each Gi, Hjpair

while Node-i transmits the pilots, for i,j ∈ {1,2},i ?= j.

- Transmit SDD requires that each Node-i (i) receives channel

state information (CSI) for the forward channel Hi from

Node-j; (ii) computes wi,rj as outlined above; and (iii)

conveys rjand the gains diag(Λi) back to Node-j.

2) Case: Receive SDD implemented at both nodes:

Design requirements: Where His are not rank-deficient, the

requirements (3), (4b) are satisfied for (Ni− sj) ≥ Mi≥ si.

Beamforming matrices : Suppose the SVDs: Gi= UiΣiViH

hold for i ∈ {1,2}. Each U(0)

would span Null

. Defineˆ Hi=

{1,2},i ?= j; and let their SVDs beˆ Hi= QiΛiWiH.

Choosing wi= {Wi}C(si), and rj=

would yield the desired spatial modes.

Remarks:

- The effective channelˆ Hiis nullity

?ˆ Hi

diversity and multiplexing gains results in.

- Swapping the transmit/ receive role of each antenna should

convert a given Receive SDD configuration to a Transmit

SDD configuration exhibiting equivalent error performance,

and vice versa. Receive SDD appears simpler in practice,

since it requires only the wis to be exchanged over the

channel as an overhead.

3) Case: Transmit SDD implemented at one node, and

Receive SDD at the other:

Without a loss of generality, suppose that Node-1 implements

Transmit SDD, while Node-2 implements Receive SDD.

The requirements (3) and (4) are met if (M1−s1) ≥ N1≥

s2 and (N2− s1) ≥ M2 ≥ s2. The effective channel for

3Definingˆ Hi = Hi

V(0)

i

possible here. It would however yield lower diversity orders.

{1,2}. The columns of each V(0)

i

=

i

?

QiH?

R(si)

?ˆ Hi

?

i

= {Ui}C(rank(G)+1:Ni)

?

??

?

GiT?

U(0)

j

?H

U(0)

Hifor i,j ∈

?H?

jQi

R(si)

?

GjT?

×Mi. Moreover,

GjT??

rank

?

≤ min

?

rank(Hi),nullity

?

. A loss of

??

C(si), using si columns from V(0)

i

too is

−15−10−5051015

10

−4

10

−3

10

−2

10

−1

10

0

signal to noise ratio (dB)

symbol error rate

{4,2}1↔{4,2}1direction 1

{4,2}1↔{4,2}1direction 2

{4,2}1↔{2,4}1direction 1

{4,2}1↔{2,4}1direction 2

{2,4}1↔{2,4}1direction 1

{2,4}1↔{2,4}1direction 2

Fig.

{M1,N1}1↔{M2,N2}1MIMO SDD configurations.

3.SNRvs.averageSERcurvesineither directionof

eigenmode transmission from Node-1 to Node-2 would be

ˆ H1=

U(0)

2

H1V(0)

for the other direction.

Remark:

-ˆ H1 becomes nullity

remains unchanged as N1× M2for the opposite direction.

Notably, the excess spatial DoFs can help achieve SDD,

even if available only in one direction.

?

?H

1. The channel H2can be used as is

?

G2T?

× nullity(G1). Butˆ H2 = H2

III. NUMERICAL RESULTS

Let us denote by {M1,N1}s1↔{M2,N2}s2

SDD configuration having Mi transmit antennas and Ni re-

ceive antennas at Node-i; and si spatial modes in-use from

Node-i to Node-j, where i,j ∈ {1,2},i ?= j.

Fig. 3 depicts the signal-to-noise ratio (SNR) vs. average

symbol error rate (SER) curves corresponding to the first

spatial mode in either direction, for MIMO SDD configu-

rations: (a) {4,2}1↔{4,2}1, (b) {4,2}1↔{2,4}1, and (c)

{2,4}1↔{2,4}1. 106-point Monte-Carlo simulation is used.

Assumptions: Block fading (with 10 symbols, per spatial

mode, per channel realization), quadrature phase shift

keying (QPSK) modulation scheme, independent and iden-

tically distributed (i.i.d.) Rayleigh faded His, and i.i.d.

Rayleigh faded Gis are assumed. Elements of Gis have

100dB (= 1010) greater variance than those of His.

All three configurations show identical performance, which

is expected since the effective MIMO channelˆ Hi in either

direction is 2 × 2 complex Gaussian, for all three cases.

Fig. 4 illustrates more clearly the diversity and mul-

tiplexing gain reduction owing to SDD, considering the

{7,4}3↔{5,3}2MIMO SDD configuration. The assumptions

are as highlighted with respect to Fig. 3. Spatial modes

in ‘direction 1’ (i.e. from Node-1 to Node-2) exhibit error

performance identical to that of a 3×3 MIMO channel; while

a 4×2 MIMO channel is resembled in the opposite direction.

This observation confirms our premise that eachˆ Hi, although

of reduced dimensionality: Nj× nullity(Gi), represents i.i.d.

the MIMO

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Page 4

−20−15−10 −5

signal to noise ratio (dB)

05 1015 20

10

−4

10

−3

10

−2

10

−1

10

0

symbol error rate

SDD direction 1, k = 1

SDD direction 1, k = 2

SDD direction 1, k = 3

SDD direction 2, k = 1

SDD direction 2, k = 2

3 x 3 MIMO

4 x 2 MIMO

Fig. 4.

direction of {7,4}3↔{5,3}2 MIMO SDD configuration. Average SER

performance for eigenmode transmission over 3 × 3 MIMO (•), and 4 × 2

MIMO (?) channels included for comparison.

SNR vs. average SER curves for each spatial mode k in either

−20−15−10−5

signal to noise ratio (dB)

05 101520

10

−2

10

−1

10

0

symbol error rate

(3−digit; k = 1,2)

(5−digit; k = 2)

(5−digit; k = 1)

(6−digit; k = 2)

(6−digit; k = 1)

(7−, 9− digit;

k = 2)

(7−, 9− digit; k = 1)

Fig. 5.

Node-2 direction of {4,2}2↔{4,2}2MIMO SDD configuration, assuming:

3-, 5-, 6-, 7-, and 9- digit computational precision.

SNR vs. average SER curves for each spatial mode k in Node-1 to

Rayleigh fading (just as corresponding Hi does). The loss

of diversity gains is implicit. Since only 5 spatial modes

are facilitated with 11 antennas at Node-1, and 8 antennas

at Node-2, a loss of 3 spatial DoFs is also apparent. These

losses represent the cost of SDD; the benefit is, obviously, the

duplexing capability.

From a mathematical point of view, the SDD techniques we

have examined suppress the self-interference perfectly. That

is not so in practice, when finite computational precision (in

transmitter- and receiver- signal processing) and/ or quantiza-

tion errors (at analog-to-digital conversion) are in effect.

Fig. 5 depicts approximately4, how the number of significant

digits of computation affects the average SER, using the

{4,2}2↔{4,2}2MIMO SDD configuration. 105-point Monte-

Carlo simulation has been used; other assumptions are as

same as before. The SER floors hint the presence of un-

4Approximate, because the internal precision of MATLAB’s ‘svd’ routine

was not restricted. Inputs and outputs of the routine were nevertheless

truncated to have the desired number of significant digits.

−10010203040

10

−4

10

−3

10

−2

10

−1

10

0

signal to noise ratio (dB)

symbol error rate

10 bit, k = 2

10 bit, k = 1

12 bit, k = 2

14 bit, k = 2

16 bit, k = 2

12 bit, k = 1

14 bit, k = 1

16 bit, k = 1

Fig. 6.

Node-2 direction of {4,2}2↔{4,2}2 MIMO SDD configuration, assuming

10-, 12-, 14-, and 16- bit ADCs.

SNR vs. average SER curves for each spatial mode k in Node-1 to

mitigated interference. Apparently, self-interference does not

get suppressed for precisions less than 6-digits. The effect of

truncation errors is evident even at 6-digit precision. However,

the error performance improves rapidly as the number of

significant digits of computation increases beyond a threshold,

that depends on the ratio of transmit and receive signal

strengths (note: log10

Low resolution of the ADC is another concern. It gives rise

to quantization errors, and hence, to increased average SERs.

?√100dB

?

=1

2log10

?1010?= 5).

Example 2: If each element of Gis is zero mean complex

Gaussian with 2σ2variance, each of the real and imag-

inary components of the self-interference may lie in the

(−8σ,8σ) range, at erf?8/√2?

for that range would be able to resolve only up to Δ =

16σ

2n =

be insignificant with respect to both the desired signal and

the self-interference.

= 0.9999999999999988

(i.e. practically 1) probability. An n-bit linear quantizer

σ

2n−4. Corresponding quantization errorΔ

2needs to

The effect of quantization is severe than that of finite com-

putational precision, because linear quantization at a wide

dynamic range is required for SDD to function properly. Self-

interference dominates the received signal; hence, the dynamic

range depends on the Gis. Linear quantization is required

since the self-interference is additive.

Fig. 6 illustrates the effect the quantization errors the 10-,

12-, 14- and 16-bit ADCs introduce have on the average SER.

Assumptions: Elements of His have unit variance, while

those of Gis have 40dB variance5. Midtread quantization

at a dynamic range of 16σ is considered, where σ =

?104/2. 10 data symbols, per spatial mode, per channel

transmit antenna, per channel realization. Least square

method is used for channel estimation.

realization are assumed; along with 10 pilot symbols, per

5An order of separation above 40dB is not achievable with the ADC

resolutions considered. Additional K dB separation would approximately

require extra1

2log2

?100.1K?bit precision at the ADC.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Page 5

−10010 20 30 40

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

signal to noise ratio (dB)

symbol error rate

k = 2

k = 1

Ch Est + Qnt D&P

Ch Est + Qnt D

Perf CSI + Qnt D

Ch Est + No Qnt

Perf CSI + No Qnt

Fig. 7.

1 to Node 2 direction of {4,2}2↔{4,2}2 MIMO SDD configuration,

assuming 14-bit ADC. Five cases reflecting realistic to idealistic assumptions

on quantization and channel estimation errors are compared.

SNR vs. average SER curves for each spatial mode k in Node

106-point Monte-Carlo simulation has been used; and other

assumptions are as with Fig. 3. Error rates improve with

the number of bits the ADCs output per sample. An abrupt

degradation of error performance can be seen in the first spatial

mode (i.e. k = 1) as the precision reduces from 12-bits to 10-

bit. A likely reason for it is having log2(16σ) = 10.1439.

Moreover, an error floor, which is common with systems

affected by SNR invariant errors, can be seen in the average

SER curves.

Quantization of the pilot symbols gives rise to channel

estimation errors, which significantly influence the error rates.

Fig. 7 confirms the fact for a 14-bit ADC, and the same

MIMO SDD configuration and assumptions as with Fig. 6,

by comparing the error performance for the following cases.

i) Ch Est + Qnt D&P: both the data and pilots (used for

channel estimation) quantized;

ii) Ch Est + Qnt D: data quantized, but not the pilots;

iii) Perf CSI + Qnt D: data quantized, perfect CSI assumed;

iv) Ch Est + No Qnt: neither data nor the pilots quantized;

v) Perf CSI + No Qnt: with perfect CSI and no quantization.

106-point Monte-Carlo simulation has been used. The case i)

is realistic; while the cases ii) through v) depict increasingly

idealistic scenarios.

The curves corresponding to cases i) and ii) highlight the

degradation of the performance quantization of pilots induces.

Quantization induced channel estimation errors set an error

floor in both the spatial modes. Quantization of data appears

to have a less significant effect; an error floor is apparent

only with k = 1. The cases iv) and v) let the effect of

channel estimation errors be assessed in isolation. An error

floor does not appear, evidently because the least square

method of estimation improves with the SNR. To sum, the non-

availability of perfect CSI appears to be the main contributing

factor for errors when ADC resolution is coarse.

IV. CONCLUSION

Beamforming for eigenmode transmission over MIMO

space division duplexing was examined. Associated loss of

diversity and multiplexing benefits was highlighted. Further

insights were obtained on the adverse effects of finite compu-

tational precision and quantization errors on the error rate.

General purpose ADCs operating above 107samples per

second do not currently have resolutions beyond 16-bits [13],

[14]. Improving both the sampling rate and the resolution

appears to be challenging due to high data rates, and other fac-

tors such as synchronization and jitter. This, along with non-

linearities in the amplifiers makes suppressing self-interference

greater than 40dB challenging at present. Amount of self-

interference suppression required could be manageable for

short-range links; it may be reduced further by using direc-

tional antennas. But SDD, as was discussed here, may not be

feasible in general until the hardware limitations are overcome.

ACKNOWLEDGMENT

This work is supported in part by the Alberta Ingenuity

Fund through the iCORE ICT Graduate Student Award.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings