Resource Allocation in SVD-assisted Broadband MIMO Systems Using

Polynomial Matrix Factorization

André Sandmann, Andreas Ahrens and Steffen Lochmann

Hochschule Wismar, University of Technology, Business and Design, Philipp-Müller-Straße 14, 23966 Wismar, Germany

Keywords:

Multiple-Input Multiple-Output System, Singular-Value Decomposition, Polynomial Matrix Factorization,

Bit Allocation, Power Allocation, Wireless Transmission.

Abstract:

Removing channel interference in broadband multiple-input multiple-output (MIMO) systems is a task which

can be solved by applying a spatio-temporal vector coding (STVC) channel description and using singular

value decomposition (SVD) in combination with signal pre- and post-processing. In this contribution a poly-

nomial matrix factorization channel description in combination with a speciﬁc SVD algorithm for polynomial

matrices is analyzed and compared to the commonly used STVC SVD. This comparison points out the analo-

gies and differences of both equalization methods. Furthermore, the bit error rate (BER) performance is eval-

uated for two different channel types and is optimized by applying bit-allocation schemes involving a power

loading strategy. Our results, obtained by computer simulation, show that polynomial matrix factorization

such as polynomial matrix SVD could be an alternative signal processing approach compared to conventional

SVD-based MIMO approaches in frequency-selective MIMO channels.

1 INTRODUCTION

The strategy of placing multiple antennas at the trans-

mitter and receiver sides, well-known as multiple-

input multiple-output (MIMO) system, improves the

performance of wireless systems by the use of the

spatial characteristics of the channel. MIMO systems

have become the subject of intensive research over the

past 20 years as MIMO is able to support higher data

rates and shows a higher reliability than single-input

single-output (SISO) systems. Singular-value decom-

position (SVD) is well-established in MIMO signal

processing where the whole MIMO channel is trans-

ferred into a number of weighted SISO channels. The

unequal weighting of the SISO channels has led to

intensive research to reduce the complexity of the re-

quired bit- and power-allocation techniques (Ahrens

and Lange, 2008; Ahrens and Benavente-Peces, 2009;

Kühn, 2006). The polynomial matrix singular-value

decomposition (PMSVD) is a signal processing tech-

nique which decomposes the MIMO channel into

a number of independent frequency-selective SISO

channels so called layers (McWhirter et al., 2007).

The remaining layer-speciﬁc interferences as a result

of the PMSVD-based signal processing can be easily

removed by further signal processing such as zero-

forcing equalization as demonstrated in this work.

The novelty of our contribution is that we demon-

strate the beneﬁts of amalgamating a suitable choice

of MIMO layers activation and number of bits per

layer along with the appropriate allocation of the

transmit power under the constraint of a given ﬁxed

data throughput. Here, bit- and power-loading in

both SVD- and PMSVD-based MIMO transmission

systems are elaborated. Assuming a ﬁxed data rate,

which is required in many applications (e.g., real time

video applications), a two stage optimization process

is proposed. Firstly, the allocation of bits to the num-

ber of SISO channels is optimized and secondly, the

allocation of the available total transmit power is stud-

ied when minimizing the overall bit-error rate (BER)

at a ﬁxed data rate. Our results, obtained by computer

simulation, show that PMSVD could be an alternative

signal processing approach compared to conventional

SVD-based MIMO approaches in frequency-selective

MIMO channels.

The remaining part of this paper is structured as

follows: Section 2 introduces the state of the art SVD-

based MIMO system model. The polynomial ma-

trix singular-value decomposition is analysed in sec-

tion 3. In section 4 the well-know quality criteria is

brieﬂy reviewed and applied to our problem. The pro-

posed power allocation solutions are discussed in sec-

tion 5, while the associated performance results are

317

Sandmann A., Ahrens A. and Lochmann S..

Resource Allocation in SVD-assisted Broadband MIMO Systems Using Polynomial Matrix Factorization.

DOI: 10.5220/0005265403170324

In Proceedings of the 5th International Conference on Pervasive and Embedded Computing and Communication Systems (AMC-2015), pages 317-324

ISBN: 978-989-758-084-0

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

presented and interpreted in section 6. Finally, sec-

tion 7 provides some concluding remarks.

2 STATE OF THE ART

A frequency selective MIMO link, composed of n

transmit and n

receive antennas is given by

u = H ·c + n . (1)

In (1), c is the (N

×1) transmitted data signal vec-

tor containing the complex input symbols transmitted

over n

transmit antennas in K consecutive time slots,

i. e., N

= K n

. The vector u describes the (N

×1)

received signal vector, of the length N

= (K +L

which is extended if compared to the transmitted sig-

nal vector based on the (L

+ 1) non-zero elements

of the resulting symbol rate sampled overall channel

impulse response between the µth transmit and νth re-

ceive antenna. Finally, the (N

×1) vector n in (1) de-

scribes the noise term (Ahrens and Benavente-Peces,

2009).

The (N

× N

) system matrix H of the block-

oriented system model, introduced in (1), results in

H =







.. . H

··· H







, (2)

and consists of n

·n

SISO channel matrices H

νµ

(with ν = 1,.. .,n

and µ = 1,..., n

). The sys-

tem description, called spatio-temporal vector coding

(STVC), was introduced by RALEIGH (Raleigh and

Ciofﬁ, 1998; Raleigh and Jones, 1999). Each of these

matrices H

νµ

with the dimension ((K + L

) ×K) de-

scribes the inﬂuence of the channel from transmit an-

tenna µ to receive antenna ν including transmit and

receive ﬁltering. The channel convolution matrix H

νµ

between the µth transmit and νth receive antenna is

obtained by taking the (L

+ 1) non-zero elements

of resulting symbol rate sampled overall impulse re-

sponse into account and results in

νµ







0 0 ··· 0

0 ···

··· 0

. h

··· h

. h

··· h

0 h

. ··· h

0 0 h

···

0 0 0 ··· h







. (3)

presented and interpreted in section 6. Finally, sec-

ℓ,

Figure 1: Resulting layer-speciﬁc SVD-based broad-

band MIMO system model (with 1 2

and

Figure 1: Resulting layer-speciﬁc SVD-based broad-

band MIMO system model (with ℓ = 1,2, . ..,L and k =

1,2,. .. ,K).

The removal of the interferences between the dif-

ferent antenna’s data streams, which are introduced

by the non-zero off-diagonal elements of the chan-

nel matrix H, requires appropriate signal processing

strategies. Singular-value decomposition (SVD) can

be considered as a promising solution for transfer-

ring the whole MIMO system into a system with non-

interfering channels, so called layers.

Using SVD the system matrix H can be written as

H = S ·V ·D

, where S and D

are unitary matrices

and V is a real-valued diagonal matrix of the positive

square roots of the eigenvalues of the matrix H

sorted in descending order

. For removing the inter-

ferences, the MIMO data vector c is now multiplied

by the matrix D before transmission. In turn, the re-

ceiver multiplies the received vector u by the matrix

. Thereby neither the transmit power nor the noise

power is enhanced given S and D are unitary. The

overall transmission relationship is deﬁned as

y = S

(H ·D ·c + n) = V ·c + w. (4)

As a consequence of the processing in (4), the chan-

nel matrix H is transformed into independent, non-

interfering layers having unequal gains.

With the proposed system structure, the SVD-

based equalization leads to different number of

MIMO layers ℓ (with ℓ = 1, 2,. .. ,L) at the time k

(with k = 1,2, .. ., K). Here it is worth noting that

the number of parallel transmission layers L at the

timeslot k is limited by min(n

). The complex-

value data symbol c

ℓ,k

to be transmitted over the layer

ℓ at the time k is now weighted by the corresponding

positive real-valued singular-value



ℓ,k

and further

disturbed by the additive noise term w

ℓ,k

3 POLYNOMIAL MATRIX

FACTORIZATION

In contrast to the STVC, the polynomial matrix fac-

torization exploits a description of the channel im-

pulse responses in the z-domain. Thus, each fre-

quency selective channel impulse response h

νµ

(k) be-

The transpose and conjugate transpose (Hermitian) of

D are denoted by D

and D

, respectively.

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318

tween the µth transmit and the νth receive antenna of

a (n

×n

) MIMO system is given by

νµ

(z) =

∑

k=0

νµ

[k] z

−k

, (5)

where the underscore denotes a polynomial. Consec-

utively, the broadband MIMO channel is formed by

grouping these impulse responses into the polynomial

channel matrix and thus it can be described as multi-

ple non-polynomial matrices H

multiplied with their

respective delay z

−k

as follows

H(z) =

∑

k=0

−k

H(z) =







(z) h

(z) ··· h

(z)

(z) h

(z) ··· h

(z)

(z) h

(z) ··· h

(z)







(6)

where H(z) ∈C

×n

. Using this polynomial descrip-

tion in the z-domain a MIMO system is described in

analogy to (1) by

u(z) = H(z) c(z) + n(z) , (7)

where c(z) is the (n

×1) transmit signal vector, u(z)

is the (n

×1) receive signal vector and n(z) describes

the additive white Gaussian noise (AWGN) compo-

nent in polynomial notation.

The polynomial channel matrix H(z) can be or-

thogonalized by calculating the polynomial matrix

singular value decomposition (PMSVD) with the help

of the second-order sequential best rotation (SBR2)

algorithm as presented in (McWhirter et al., 2007;

Foster et al., 2010). The decomposition of the polyno-

mial channel matrix results in H(z) = S(z)V(z)



D(z),

where (



·) denotes the para-conjugate operator. The

matrices S(z) ∈C

×n

and



D(z) ∈C

×n

are parau-

nitary matrices and V(z) ∈ C

×n

is assumed to be

a diagonal matrix, because the off-diagonal elements

are negligibly small when the SBR2 algorithm is set

up accordingly. The diagonal matrix has the follow-

ing form (n

= n

)

V(z) =







(z) 0 ··· 0

0 v

(z) ··· 0

0 0 ··· v

(z)







, (8)

where the diagonal polynomial elements are de-

scribed by v

ℓ

(z) =

∑

k=0

ℓ,k

−k

. In contrast to the

singular values



ℓ,k

using SVD, the polynomial co-

efﬁcients of v

ℓ

(z) are complex. In analogy to the SVD

model, the maximal number of activated layers L us-

ing PMSVD is min{n

}. For removing the inter-

ference signal pre-processing at the transmitter and

post-processing at the receiver is applied in analogy

to the classical SVD. Consequently, the transmit data

vector c(z) is multiplied by D(z) so that

u(z) = H(z) D(z)c(z) + n(z)

u(z) = S(z) V(z)



D(z)D(z) c(z) + n(z) ,

(9)

with



D(z)D(z) = I and I describing the identity ma-

trix. The receive vector u(z) is multiplied by



S(z) re-

sulting in

y(z) =



S(z)u(z) =



S(z)



S(z)V(z) c(z) + n(z)





S(z)S(z) V(z)c(z) +



S(z)n(z) .

(10)

where



S(z)S(z) simpliﬁes to the identity matrix I.

Therefore, the orthogonalized system is given by

y(z) = V(z) c(z) + w(z) . (11)

Hereinafter, the resulting system is described by mul-

tiple parallel SISO channels, so called layers. The

layer based discrete-time description is expressed as

ℓ

(k) = v

ℓ

(k) ∗c

ℓ

(k) + w

ℓ

(k) , (12)

where ∗ denotes discrete convolution such that v

ℓ,k

∗

ℓ,k

∑

κ=0

ℓ,κ

·c

ℓ,k−κ

. Herein the parameter L

+ 1

describes the number of non-zero coefﬁcients of the

layer-speciﬁc impulse response. The layer-speciﬁc

model is depicted in Fig. 2. Here in each layer the

singular value decomposition (PMSVD) with the help

of the second-order sequential best rotation (SBR2)

algorithm as presented in (McWhirter et al., 2007;

Foster et al., 2010). The decomposition of the polyno-

denotes the para-conjugate operator. The

are parau-

is assumed to be

a diagonal matrix, because the off-diagonal elements

are negligibly small when the SBR2 algorithm is set

ℓ,

−1

ℓ,

−

ℓ,0

ℓ,1

ℓ,

−1−1

ℓ,

Figure 2: Resulting layer-speciﬁc PMSVD-based broad-

band MIMO system model (with 1 2

and

Figure 2: Resulting layer-speciﬁc PMSVD-based broad-

band MIMO system model (with ℓ = 1,2, . ..,L and k =

1,2,. .. ,K) assuming L

+ 1 non-zero coefﬁcients of the

layer-speciﬁc impulse response.

input symbols c

ℓ

(k) are inﬂuenced by a ﬁnite impulse

response ﬁlter v

ℓ

(k) = (v

ℓ,0

ℓ,1

,. .. , v

ℓ,L

) and hence

inter symbol interference (ISI) occurs on each layer.

In order to remove the ISI a corresponding T-spaced

equalizer f

ℓ

(k) is applied to the received signal y

ℓ

(k)

on each layer so that z

ℓ

(k) = y

ℓ

(k) ∗ f

ℓ

(k) as depicted

in Fig. 3. The equalizer is designed as an FIR ﬁl-

ter with coefﬁcients as described in (Bingham, 2000)

or (Tse and Viswanath, 2005) and therefore comes as

close as possible to the following condition

ℓ

(k) ∗ f

ℓ

(k) = i

ℓ

(k) ,

(13)

ResourceAllocationinSVD-assistedBroadbandMIMOSystemsUsing

PolynomialMatrixFactorization

319

ℓ

( )

ℓ

(

) z

ℓ

( )

ℓ

(

)

ℓ

(

)

ℓ

(

)

Figure 3: Layer-speciﬁc PMSVD-based transmission

model applying a T-spaced equalizer with the coefﬁcients

ℓ

(k) speciﬁcally designed for each layer.

with i

ℓ

(k) = (0 , .. . ,0 ,1 ,0 ,. .. , 0), where the position

of the 1 in i

ℓ

(k) is a degree of freedom in the equalizer

design process. Accordingly, the equalized receive

signal results in

ℓ

(k) = c

ℓ

(k) + w

ℓ

(k) ∗ f

ℓ

(k) . (14)

The corresponding layer-speciﬁc ISI free system

model is shown in Fig. 4, where the transmitted

symbols are received unchanged and the noise w

ℓ

(k)

is weighted by the equalizer coefﬁcients f

ℓ

(k). The

PMSVD-based broadband MIMO system model with

layer-speciﬁc T-spaced equalization is henceforth re-

ferred to as T-PMSVD system model (Sandmann

et al., 2014).

ℓ

( )

ℓ

( )

ℓ

( )

ℓ

( )

Figure 4: ISI free layer-speciﬁc T-PMSVD-based broad-

band MIMO system model.

4 TRANSMISSION QUALITY

CRITERION

In general the quality criterion for transmission sys-

tems can be expressed with using the signal to noise

ratio (SNR) at the detector input as follows

ρ =

(half vertical eye opening)

noise power

)

, (15)

where U

and P

correspond to one quadrature com-

ponent. Considering a layer based MIMO system

with a given SNR ρ

(ℓ,k)

for each layer ℓ and time k and

a M-ary quadrature amplitude modulation (QAM) the

bit error rate (BER) probability is given by (Proakis,

2000)

(ℓ,k)

BER

log

ℓ



1 −

√

ℓ



erfc







(ℓ,k)





(16)

This BER is averaged at each time slot over all ac-

tivated layers taking different constellation sizes at

each layer into account and results in

(k)

BER

∑

ℓ=1

log

ℓ

∑

ℓ=1

log

ℓ

(ℓ,k)

BER

. (17)

In order to obtain the average BER of one data block

consisting of K transmitted symbols the time slot de-

pendent BER has to be averaged as follows

BER

= E



(k)

BER



∀k , (18)

where E{·} denotes the expectation functional. Fi-

nally, when considering time-variant channel condi-

tions, rather than an AWGN channel, the BER can

be derived by considering the different transmission

block SNRs.

For QAM modulated signals the average transmit

power per layer can be expressed as

s,ℓ

ℓ

−1) , (19)

assuming that all M symbols are equally distributed.

Intuitively the total available transmit power P

equally split between the L activated layers and hence

the layer-speciﬁc transmit power is given by: P

s,ℓ

/L. This guarantees that the condition

∑

ℓ=1

s,ℓ

(20)

is complied. With rearranging (19) the half-level

transmit amplitude for each layer results in

s,ℓ



2L (M

ℓ

−1)

. (21)

Considering the SVD layer model the noise power is

unchanged at the receiver. However, the half verti-

cal eye opening U

at each time slot k and layer ℓ

is inﬂuenced by the singular values so that U

(ℓ,k)



ℓ,k

s,ℓ

. Using the T-PMSVD model the equal-

izer fully removes the ISI and thus for each layer

the half vertical eye opening U

A,ℓ

of the receive sig-

nal equals the half-level amplitude of the transmitted

symbol U

s,ℓ

. The drawback of the T-PMSVD is that

the noise and hence the noise power is weighted dif-

ferently on each layer by the equalizer coefﬁcients ex-

pressed by the factor θ

ℓ

so that the noise power on

each layer results in

R,ℓ

= θ

ℓ

, where θ

ℓ

∑

∀k

ℓ,k

. (22)

Taking the inﬂuence of the singular values



ℓ,k

each time slot k in the SVD based layer model into

PECCS2015-5thInternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

320

account and considering the weighing factor of the

noise power θ

ℓ

induced by the T-spaced equalizer co-

efﬁcients in the PMSVD based layer model the corre-

sponding SNR values become

(ℓ,k)

SVD

ℓ,k

s,ℓ

3ξ

ℓ,k

L (M

ℓ

−1)

(23)

and

(ℓ)

T−PMSVD

s,ℓ

ℓ

L (M

ℓ

−1)

, (24)

where E

is the signal energy of the transmit signal.

5 POWER ALLOCATION

The overall bit error rate of a decomposed MIMO sys-

tem is largely determined by the layer with the high-

est BER. In order to balance the bit error rates on

all layers the mean of choice is to equalize the SNR

values ρ

(ℓ,k)

over all layers. This is clearly not the

optimal solution for minimizing the overall BER but

it is is easy to implement and not far away from the

optimum as shown in (Ahrens and Benavente-Peces,

2009; Ahrens and Lange, 2008).

Therefore, the half-level transmit amplitude U

s,ℓ

adjusted on each layer by multiplying it with

√

ℓ,k

as to apply the power allocation (PA) scheme. Conse-

quently the half vertical eye opening of the received

symbols for the SVD-based model becomes

(ℓ,k)

A,PA

√

ℓ,k



ℓ,k

s,ℓ

, (25)

whereas in the T-PMSVD model the factor



ℓ,k

dropped due to the ZF-equaliser. With this adjustment

the SNR values result in

(ℓ,k)

= p

ℓ,k

(ℓ,k)

. (26)

The respective system models for T-PMSVD and

SVD equalization including PA are depicted in Fig. 5

and 6.

ℓ,

√

ℓ,

Figure 5: Resulting layer-speciﬁc SVD-based model with

power allocation by adjusting the half-level amplitude of the

Figure 5: Resulting layer-speciﬁc SVD-based model with

power allocation by adjusting the half-level amplitude of the

transmit symbols c

ℓ,k

with the square root of the PA factor

ℓ,k

Hereinafter, the strategy for choosing the PA fac-

tors p

ℓ,k

is elucidated. As mentioned above, the aim

account and considering the weighing factor of the

induced by the T-spaced equalizer co-

efﬁcients in the PMSVD based layer model the corre-

(23)

ℓ

( )

ℓ

( )

ℓ

( )

ℓ

( )

√

ℓ

Figure 6: Resulting layer-speciﬁc T-PMSVD-based model

with power allocation by adjusting the half-level amplitude

of the transmit symbols c

ℓ,k

with the square root of the PA

factor p

ℓ,k

is to achieve equal SNRs over all activated layers ℓ at

the time k and hence

(ℓ,k)

= constant ∀ℓ (27)

has to be fulﬁlled for all activated MIMO layers. Ad-

ditionally, the overall transmit power after PA needs

to be the same as without PA and thus the second con-

dition

∑

ℓ=1

ℓ,k

s,ℓ

∑

ℓ=1

ℓ,k

∀k , (28)

has to be guaranteed. By combining these two

requirements the PA factor p

ℓ,k

for SVD and T-

PMSVD based systems can be calculated as follows

(Ahrens and Lange, 2008; Ahrens and Benavente-

Peces, 2009)

(SVD)

ℓ,k

ℓ

−1)

ℓ,k

∑

λ=1

−1)

λ,k

(29)

and

(T−PMSVD)

ℓ

= θ

ℓ

−1)

∑

λ=1

−1)

. (30)

6 RESULTS

Hereinafter, the BER quality is studied by using ﬁxed

transmission modes with a spectral efﬁciency of 8

bit/s/Hz. The analyzed QAM constellations, equiv-

alent to how many bits are allocated to each layer, are

shown in Tab. 1.

In order to analyse the T-PMSVD, a two-path

time-invariant (2 ×2) MIMO system is investigated.

The polynomial channel matrix is chosen as

H(z) = H

+ H

−1

(31)

with



1 0.6

0.5 0.8



and H

. (32)

ResourceAllocationinSVD-assistedBroadbandMIMOSystemsUsing

PolynomialMatrixFactorization

321

Table 1: Transmission modes.

throughput layer 1 layer 2 layer 3 layer 4

8 bit/s/Hz 256 0 0 0

8 bit/s/Hz 64 4 0 0

8 bit/s/Hz 16 16 0 0

8 bit/s/Hz 16 4 4 0

8 bit/s/Hz 4 4 4 4

The factor 4/5 is chosen so to guarantee that the

channel is not amplifying in any power allocation sit-

uation. In addition, the number of equalizer coefﬁ-

cients within the T-PMSVD model is chosen to be

10 and thus the factors by which the noise power is

weighted on each layer come out as θ

= 0.9768 and

= 17.7729. Therefore, assuming equal QAM con-

stellations on all layers the modiﬁed noise power af-

fects the SNR of the second layer approximately 18

times more than the SNR of the ﬁrst one. The calcu-

lated BER results as a function of the signal energy

to noise power spectral density E

for both equal-

ization types are depicted separately in Fig. 7 and 8.

15 20 25 30 35

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(256,0) QAM

(64,4) QAM

(16,16) QAM

Figure 7: BER with PA (dotted line) and without PA (solid

line) applying SVD-based equalization when transmitting

over the time-invariant (2×2) MIMO channel given by (31)

and (32) with 8 bit/s/Hz using the transmission modes intro-

duced in Table 1.

Here the (64,4) QAM transmission mode shows

the best results. Furthermore, comparing the SVD and

T-PMSVD results indicate that the quality ranking of

the transmission modes is similar for both equaliza-

tion types. Also, the beneﬁts of using the equal SNR

power allocation method are visible. A direct compar-

ison between the SVD and T-PMSVD results is de-

picted in Fig. 9 and shows that the T-PMSVD quality

outperforms the SVD results.

The previous channel is now extended to a two-

path time-invariant (4 ×4) MIMO system with the

5 is chosen so to guarantee that the

channel is not amplifying in any power allocation sit-

uation. In addition, the number of equalizer coefﬁ-

cients within the T-PMSVD model is chosen to be

10 and thus the factors by which the noise power is

9768 and

15 20 25 30 35

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(256,0) QAM

(64,4) QAM

(16,16) QAM

Figure 8: BER with PA (dotted line) and without PA (solid

line) applying T-PMSVD equalization when transmitting

over the time-invariant (2×2) MIMO channel given by (31)

and (32) with 8 bit/s/Hz using the transmission modes intro-

duced in Table 1.

lated BER results as a function of the signal energy

for both equal-

ization types are depicted separately in Fig. 7 and 8.

15 20 25 30 35

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(256,0) QAM

(64,4) QAM

Figure 9: BER comparison between the SVD-based (dashed

line) and T-PMSVD-based equalization results (solid line)

when transmitting over the time-invariant (2 ×2) MIMO

channel given by (31) with 8 bit/s/Hz using the transmission

modes introduced in Table 1 and applying equal SNR PA.

polynomial channel matrix

H(z) = H

+ H

−1

(33)

with









1 0.6 0.5 0.3

0.5 0.8 0.6 0.4

0.4 0.5 0.7 0.5

0.3 0.4 0.5 0.6







(34)

and

. (35)

The corresponding BER results are shown in Fig.

10 and 11 for SVD as well as T-PMSVD process-

ing. The results show, that the (64,4,0,0) conﬁgura-

PECCS2015-5thInternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

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15 20 25 30 35

−8

−6

−4

−2

BER

→

10 ·log

) (indB) →

(256,0,0,0) QAM

(64,4,0,0) QAM

(16,16,0,0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 10: BER with PA (dotted line) and without PA (solid

line) applying SVD-based equalization when transmitting

over the time-invariant (4×4) MIMO channel given by (33)

and (35) with 8 bit/s/Hz using the transmission modes intro-

duced in Table 1.

15 20 25 30 35

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(256,0,0,0) QAM

(64,4,0,0) QAM

(16,16,0,0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 11: BER with PA (dotted line) and without PA (solid

line) applying T-PMSVD equalization when transmitting

over the time-invariant (4×4) MIMO channel given by (33)

and (35) with 8 bit/s/Hz using the transmission modes intro-

duced in Table 1.

tion shows the best performance. Comparing these re-

sults with the (4,4,4,4) transmission mode, it turns out

that activating all MIMO layers results in a high BER,

based on activating layers with low quality. Directly

comparing both equalization types as shown in Fig.

12 highlights the superior BER performance of the

PMSVD-based equalization.

So far only time-invariant channels were stud-

ied. These investigations are now extended to time-

variant wireless channels. Here, a two path (4 ×4)

MIMO channel without a line-of-sight component is

analyzed (L

= 1), where the amplitudes are modeled

as Rayleigh distributed. The BER results are shown

in Fig. 13 and 14. Here the (16,16,0,0) QAM trans-

mission mode performs best for SVD as well as for

T-PMSVD equalization. Thus, not all layers have to

Figure 10: BER with PA (dotted line) and without PA (solid

15 20 25 30

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(64,4,0,0) QAM

(16,16,0,0) QAM

Figure 12: BER comparison between the SVD-based

(dashed line) and T-PMSVD-based equalization results

(solid line) when transmitting over the time-invariant (4 ×

4) MIMO channel given by (33) with 8 bit/s/Hz using the

transmission modes introduced in Table 1 and applying

equal SNR PA.

Figure 11: BER with PA (dotted line) and without PA (solid

line) applying T-PMSVD equalization when transmitting

MIMO channel given by (33)

10 15 20 25

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(256,0,0,0) QAM

(64,4,0,0) QAM

(16,16,0,0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 13: BER with PA (dotted line) and without PA (solid

line) applying SVD-based equalization when transmitting

over a Rayleigh distributed (4×4) MIMO two path channel

with 8 bit/s/Hz using the transmission modes introduced in

Table 1.

be activated for achieving the best BER performance

results. The transmission mode performance for both

equalization types is also similar. Applying the easy

to implement equal SNR PA results in a signiﬁcant

improvement of the BER. The direct BER perfor-

mance comparison depicted in Fig. 15 shows that the

T-PMSVD BER quality is superior to the SVD BER

quality.

7 CONCLUSION

In this paper broadband MIMO systems have been

described using polynomial matrix factorization. In

order to remove the MIMO channel interference

ResourceAllocationinSVD-assistedBroadbandMIMOSystemsUsing

PolynomialMatrixFactorization

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10 15 20 25

−8

−6

−4

−2

BER

→

10 ·log

) (in dB) →

(256,0,0,0) QAM

(64,4,0,0) QAM

(16,16,0,0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 14: BER with PA (dotted line) and without PA (solid

line) applying T-PMSVD equalization when transmitting

over a Rayleigh distributed (4×4) MIMO two path channel

with 8 bit/s/Hz using the transmission modes introduced in

Table 1.

5 10 15 20

−8

−6

−4

−2

BER

→

10 ·log

) (indB) →

(64,4,0,0) QAM

(16,16,0,0) QAM

Figure 15: BER comparison between the SVD-based

(dashed line) and T-PMSVD-based equalization results

(solid line) when transmitting over a Rayleigh distributed

(4 ×4) MIMO two path channel with 8 bit/s/Hz using the

transmission modes introduced in Table 1 and applying

equal SNR PA.

a particular singular value decomposition algorithm

for polynomial matrices (PMSVD) including layer-

speciﬁc T-spaced equalization for eliminating the re-

maining inter symbol interference has been stud-

ied. This T-PMSVD technique has been compared

in terms of bit error rate performance with the well-

known spatio-temporal vector coding description ap-

plying SVD equalization. The simulation results

demonstrate that using T-PMSVD equalization the

BER performance is superior compared with applying

SVD. For both equalization types bit loading schemes

have been combined with equal SNR power alloca-

tion so as to optimize the BER performance. The

equal SNR criteria for power allocation seems to be

a good sub-optimum solution to improve the channel

performance. Furthermore, the bit and power loading

analogies between both equalization types have been

shown. The two different analyzed channels clarify

that the optimal QAM transmission mode largely de-

pends upon the used channel type and that the activa-

tion of all transmission layers doesn’t always lead to

the best performance.

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