The spatial-domain non-orthogonal multiple access

(NOMA) based on multi-input multi-output (MIMO)

technology is a cornerstone for the ﬁfth-generation (5G)

mobile communication system, to support enhanced mobile

broadband and massive machine-type communications with

the limited frequency-time resources. This naturally leads to

multiuser MIMO communication systems. In the literature,

most existing MIMO system designs adopt the linear MIMO

channel [1]–[10]. The linear MIMO channel assumption

however is only valid when the transmitter high power

ampliﬁer (HPA) operates within its linear dynamic range.

Practical HPAs on the other hand are often nonlinear,

as they exhibit nonlinear saturation and phase distortion

characteristics [11]–[15]. More speciﬁcally, the linear channel

assumption critically depends on the transmitted signal’s

peak-to-average power ratio (PAPR) as well as the average

transmission power. For the modulation constellations with

unity PAPR, such as phase shift keying (PSK), the phase shift

of the HPA’s output is constant for all the symbol points.

Consequently, the HPA does not cause amplitude distortion

in this case, and the MIMO channel is linear. In order to

meet high throughput requirement, however, multiuser MIMO

systems typically utilize the high-throughput quadrature

amplitude modulation (QAM) with multiple bits per symbol

[16]. Since high-throughput QAM constellations have high

PAPR, the nonlinear distortion of the transmitter HPA may

become serious and the linear MIMO channel may no longer

be valid. Note that high-throughput QAM transmission is

achieved by imposing high average transmission power.

Therefore, it is impossible to avoid the nonlinearity of

transmitter HPA by using output back-off (OBO), because the

OBO required would be too severe, which would be unable

to meet the required link power budget.

This paper investigates the challenging single-carrier mul-

tiuser MIMO uplink with high-throughput QAM transmis-

sion, where transmitters are equipped with nonlinear HPAs

(NHPAs). Note that for the single-carrier multiuser nonlinear

MIMO downlink, where the base station (BS) transmits to

multiple mobile users (MUs), effective solution for overcom-

ing nonlinear distortions of NHPAs readily exists. Speciﬁcally,

since the BS possesses sufﬁcient computation capacity, it can

implement digital predistorter [17]–[23] to pre-compensate for

the nonlinear distortions of NHPA in addition to implement

the multiuser transmit precoding to compensate for the MIMO

channel interference. This leads to our recent design using a

B-spline neural network (BSNN) based predistorter for single-

carrier multiuser nonlinear MIMO downlink [24]. In uplink,

by contrast, it is difﬁcult for a mobile handset to implement

the predistorter owning to its limited computation capacity.

As a result, the BS receiver must ﬁrst estimate the multiuser

nonlinear MIMO channel and then performs the nonlinear

multiuser detection (MUD), which is extremely challenging.

In the literature, only few works [25]–[27] attempted to

tackle this difﬁcult task by employing the MIMO Volterra

model to identify the frequency-selective nonlinear MIMO

channel [25]–[27], which not only imposes impossibly heavy

computational burden but also is impractical for implementing

nonlinear MUD for the uplink with high-throughput QAM

transmission. In [28], we proposed a nonlinear MUD scheme

for single-carrier multiuser nonlinear MIMO uplink. However,

our previous work [28] only considers the MIMO systems

with frequency-nonselective or narrowband channels. In prac-

tice, MIMO channels are frequency-selective. Hence, practical

single-carrier multiuser nonlinear MIMO uplink is much more

complex than the case investigated in [28].

In this paper, we develop a BSNN assisted space-time

equalization (STE) scheme for this much more challenging

B-Spline Neural Network Assisted Space-Time Equalization

for Single-Carrier Multiuser Nonlinear Frequency-Selective

MIMO Uplink

SHENG CHEN

School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK

Abstract—This paper designs a nonlinear space-time equalizer based on B-spline neural network (BSNN) for

the singlecarrier high-throughput multiuser frequency-selective multipleinput multiple-output (MIMO)

nonlinear uplink. Specifically, based on a BSNN parametrization of the nonlinear high power amplifiers

(NHPAs) at mobile terminals’ transmitters, a novel nonlinear identification scheme is developed to estimate the

nonlinear dispersive MIMO uplink channel, which includes the BSNN models for the NHPAs at transmitters as

well as the frequency-selective MIMO channel impulse response (CIR) matrix. Furthermore, the BSNN inverse

models of the NHPAs are also estimated in closed-form. This allows the base station to implement nonlinear

multiuser detection effectively using the space-time equalization (STE) based on the estimated

frequencyselective MIMO CIR matrix and followed by compensating for the nonlinear distortion of the

transmitters’ NHPAs based on the estimated BSNN inverse models. Simulation results are utilized to

demonstrate the superior bit error rate performance of our nonlinear STE approach for single-carrier high-

throughput multiuser nonlinear frequency-selective MIMO uplink.

Keywords: Multiple-input multiple-output uplink, frequency-selective channel, nonlinear transmit high power

amplifier, B-spline neural network, nonlinear inversion, nonlinear space-time equalization

Received: August 23, 2021. Revised: March 27, 2022. Accepted: April 26, 2022. Published: June 3, 2022.

1. Introduction

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single-carrier multiuser nonlinear frequency-selective MIMO

uplink with high-throughput QAM transmission and NHPAs.

Our novel contribution is two-fold.

•By extending the work [28] originally derived for mul-

tiuser nonlinear narrowband MIMO channel, we develop

a BSNN assisted identiﬁcation scheme to identify the

multiuser nonlinear frequency-selective MIMO channel.

This includes the BSNN [29]–[36] modeling for the

MUs’ NHPAs and the estimate of the linear frequency-

selective MIMO channel impulse response (CIR) matrix.

By exploiting the results of this nonlinear MIMO channel

identiﬁcation, the BSNN inverse models of the transmit-

ters’ NHPAs are also identiﬁed.

•We implement the minimum mean square error (MMSE)

space-time equalizer [3], [4] to combat the interference

of the frequency-selective MIMO channel using the es-

timated linear frequency-selective MIMO CIR matrix.

Then we compensate for the nonlinear distortions of

the transmitters’ NHPAs with the estimated BSNN in-

verse models for the NHPAs. An extensive simulation

study is carried out to demonstrate the excellent bit

error rate (BER) performance of our proposed nonlinear

STE approach for multiuser nonlinear frequency-selective

MIMO uplink with high-throughput QAM transmission

and NHPAs.

The rest of this paper is structured as follows. The single-

carrier multiuser nonlinear frequency-selective MIMO uplink

system is introduced in Section II. This includes the NHPA

model at each MU’s transmitter and the frequency-selective

MIMO channel model, as well as the nonlinear STE based

MUD at the BS receiver that ﬁrst uses a standard space-

time equalizer to remove both multiuser interference and

self-interference and then removes the nonlinear distortion of

the transmitters’ NHPAs by the nonlinear inversion of the

NHPAs, assuming that both the frequency-selective MIMO

CIR matrix and the inverse mappings of transmitters’ NHPAs

are known at the BS receiver. The proposed BSNN assisted

nonlinear STE scheme is detailed in Section III. By utilizing

a unique parametrization of the frequency-selective MIMO

CIR matrix and the nonlinear transmitters as well as the

effective BSNN modeling of the NHPAs, a new iterative

alternating least squares (ALS) estimator is developed, which

guarantees to attain the unbiased and accurate estimates of

the frequency-selective MIMO CIR matrix and the BSNN

parametrized NHPAs’ models in a few iterations. Based on

the nonlinear frequency-selective MIMO channel identiﬁcation

results, the closed-form BSNN inverse models for the NHPAs

are also obtained. Section IV is devoted to simulation study, to

investigate the effectiveness of our proposed BSNN assisted

nonlinear STE scheme for single-carrier multiuser nonlinear

frequency-selective MIMO uplink with high-throughput QAM

transmission and NHPAs. Our conclusions are offered in

Section V.

Fig. 1 depicts the system diagram of the spatial-domain

NOMA based single-carrier multiuser nonlinear frequency-

...

s (k)

...

......

l,m

Mantenna L

z (k)

n (k)

antenna l

MUM

antenna 1

n (k)

MU 1

HPA

s (k) z (k)

n (k)

HPA

s (k) m

z (k)

x (k)

y (k)

...

s (k−d)

Space−Time Equalizers

Fig. 1. Spatial-domain NOMA based single-carrier multiuser nonlinear

frequency-selective MIMO uplink where the BS is equipped with the L-

element antenna array to receive the data from Msingle-antenna MUs using

the same single resource block.

selective MIMO uplink, where Msingle-antenna MUs trans-

mit to the BS equipped with Lreceive antennas using the same

frequency-time resource block. Note that L > M.

Since we consider the wideband or frequency-selective

channel, the CIR from the mth mobile to the lth antenna of

the BS can be expressed by

hl,m =h0,l,m h1,l,m ···hnH−1,l,mT,(1)

for 1≤l≤Land 1≤m≤M, where for notational

simplicity, all the L·MCIRs are assumed to have the same

CIR length of nH. The kth data symbol transmitted by the

mth MU is denoted by sm(k) = |sm(k)| · exp j∠sm(k),

where j=√−1,|sm(k)|denotes the amplitude of sm(k)

and ∠sm(k)is the phase of sm(k). As we use the U-

QAM constellation with log2(U)bits per symbol, to enhance

the achievable throughput, sm(k)takes the value from the

constellation set:

S=ndS(2l−√U−1) + jdS(2q−√U−1),

1≤l, q ≤√Uo.(2)

The minimum distance between the symbol points of Sis 2dS.

Without loss of generality, the HPAs at all the MUs’

transmitters are assumed to be the same type. Hence under

the same given operation condition, they exhibit the same

nonlinear characteristics. We employ a common and practical

HPA, the solid state power ampliﬁer [14], [15]. For this type

of NHPA, the transmitted signal of the mth MU, 1≤m≤M,

can be expressed as

zm(k) =Ψ (sm(k))

=A|sm(k)|·exp j∠sm(k) + Υ(|sm(k)|),(3)

where Ψ(·)represents the NHPA at a MU’s transmitter, A(·)

is its amplitude response and Υ(·)is its phase response. The

output Ψ(s)of this type of NHPA is speciﬁed by its amplitude

response A(r)and phase response Υ(r), where r=|s|denotes

the amplitude of the input sto the NHPA. Note that the

distortion caused by this type of NHPA depends only on the

2. Nonlinear Frequency-selective

Mimo Uplink

2.1 Channel and transmitter models

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amplitude of the NHPA input. According to [14], [15], the

amplitude response A(r)is given by

A(r) = gar

1 + gar

Asat 2βa1

2βa

,(4)

while the phase response Υ(r)is deﬁned by

Υ(r) = αφrq1

1 + r

βφq2[degree],(5)

where the parameters ga,βaand Asat are known as the

small signal’s gain, the smoothness factor and the saturation

level, respectively, while the parameters αφ,βφ,q1and q2

specify the NHPA’s phase response [14], [15]. If we deﬁne

the maximum output power of the NHPA as Pmax =A2

sat.

Further denote the average output amplitude of the NHPA as

Aave, which means that the the average output power of the

NHPA output signal is Pave =A2

ave. Then the operating status

of the HPA is deﬁned by the ratio of the maximum output

power Pmax of the NHPA to the average output power Pave

of the NHPA output signal, called the OBO, which is given

OBO = 10 ·log10

Pmax

Pave

[dB].(6)

Note that the maximum output power Pmax is the NHPA’s

saturated output power. The smaller the OBO is the deeper

the NHPA is operating into its saturation region and hence

causing more severe nonlinear distortion.

Recall the CIR (1) and denote the mth MU’s transmitted

signal vector by z(h)

m(k) = zm(k)zm(k−1) ···zm(k−nH+

1)Twith zm(k−i) = Ψ(sm(k−i)) for 0≤i≤nH−1.

Then the received signal sample xl(k)at the BS’s lth receiver

antenna can be expressed by

xl(k) =

m=1

l,mz(h)

m(k) + nl(k)

m=1

nH−1

i=0

hi,l,mzm(k−i) + nl(k),(7)

where nl(k)is the complex additive white Gaussian noise

(AWGN) with power Ehnl(k)2i= 2σ2

n. Collect the re-

ceived signals xl(k)for 1≤l≤Las x(h)(k) =

[x1(k)x2(k)···xL(k)]T, which can be expressed as

x(h)(k) = 





1,1hT

1,2··· hT

1,M

2,1hT

2,2··· hT

2,M

.....

L,1hT

L,2··· hT

L,M













z(h)

1(k)

z(h)

2(k)

z(h)

M(k)







+n(h)(k) = Hz(h)(k) + n(h)(k),(8)

XXX

...

x (k)

1∆ ∆ ∆

∆∆∆

∆ ∆ ∆

0,1,m

*w*w

n −1,1,m

w w w

0,2,m 1,2,m n −1,2,m

w w w

0,L,m 1,L,m

* *

y (k)

1,1,m

n −1,L,m

Fig. 2. Space-time equalizer for detecting the mth mobile user’s transmitted

signal.

in which the AWGN vector n(h)(k)=[n1(k)···nL(k)]T.

Assume that the BS knows the multiuser MIMO CIR matrix

H. If all the transmitters’ HPAs are operating in the linear

regions, the MUD for the MUs’ transmitted signals consists

of Mspace-time equalizers [3], [4], one for each mobile, as

illustrated in Fig. 2. Speciﬁcally, each space-time equalizer

has length nF. Further deﬁne the mth space-time equalizer’s

weight vector associated with the BS’s lth receive antenna

as wl,m =w0,l,m w1,l,m ···wnF−1,l,mT, and denote the

corresponding space-time equalizer’s input signal vector by

xl(k) = xl(k)xl(k−1) ···xl(k−nF+ 1)T. Then the

output of the mth space-time equalizer is given by

ym(k) =

l=1

l,mxl(k)

l=1

nF−1

i=0

w∗

i,l,mxl(k−i),1≤m≤M. (9)

Since the HPAs are nonlinear, ym(k)is only a sufﬁcient

statistic for detecting zm(k−d) = Ψ(sm(k−d)), and it is not

a sufﬁcient statistic for detecting the transmitted data symbol

sm(k−d), where dis the decision delay.

Based on linear convolution, xl(k)can be expressed as

xl(k) = PM

m=1 cl,mzm, where the nF×(nF+nH−1)

CIR matrix cl,m associated with the mth MU transmitter

and the lth BS receiver antenna has the structure shown

in (10) at the bottom of the previous page, and zm(k) =

zm(k)zm(k−1) ···zm(k−nF−nH+2)T, for 1≤m≤M.

cl,m =







h0,l,m h1,l,m ··· hnH−1,l,m 0··· 0

0h0,l,m h1,l,m ··· hnH−1,l,m

....

................0

0··· 0h0,l,m h1,l,m ··· hnH−1,l,m







(10)

2.2 Receiver multiuser detection

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By deﬁning the overall system CIR convolution matrix as

C=





c1,1c1,2··· c1,M

c2,1c2,2··· c2,M

.··· .

cL,1cL,2··· cL,M





,(11)

and the MIMO channel input vector as z(k) =

zT

1(k)···zT

M(k)T, the space-time equalizer’s input

vector x(k) = xT

1(k)···xT

L(k)Tcan be expressed as

x(k) =Cz(k) + n(k),(12)

in which the overall noise vector n(k) =

nT

1(k)nT

2(k)···nT

L(k)Twith nl(k) = nl(k)nl(k−

1) ···nl(k−nF+ 1)Tfor 1≤l≤L. Further deﬁne

the overall weight vector of the mth space-time equalizer

by wm=wT

1,m wT

2,m ···wT

L,mT. The mth space-time

equalizer (9) can be expressed concisely as

ym(k) =wH

mx(k).(13)

From [3], we have the following closed-form MMSE solution

for wm:

w(MMSE)m=CCH+2σ2

nI−1

C[ :(m−1)(nF+nH−1)+(d+1)],

(14)

for 1≤m≤M, where Iis the (LnF)×(LnF)identity

matrix and C[ :i]is the ith column of C.

The space-time equalizer (13) provides the estimate bzm(k−

d)for zm(k−d). If the nonlinear inversion Ψ−1(·)of

the complex NHPA’s nonlinear mapping Ψ(·)is known, the

estimate of sm(k−d)is then given by

bsm(k−d) =Ψ−1bzm(k−d).(15)

It can be seen that in order to detect the MUs’ data

sm(k−d),1≤m≤M, the BS needs to acquire the

MIMO channel matrix Hand to invert the unknown complex

nonlinear mapping Ψ(·). This is a very challenging nonlinear

estimation and inversion problem. First, the MIMO channel

input z(h)(k)is unknown to the receiver, and the BS cannot

apply the standard least squares (LS) estimator to identify H.

Second, the model of the MUs’ NHPAs, denoted as z(h), is

multiplicative with the MIMO CIR matrix Has the product

H·z(h). This implies that there are inﬁnitely many equivalent

pairs of the parametrization for the MIMO CIR matrix and the

NHPAs’ model, which causes ambiguity problem and imposes

a signiﬁcant challenge to the task of identifying both he MIMO

CIR matrix and the NHPAs’ model. In order to develop a

meaningful identiﬁcation procedure for both the linear MIMO

CIR matrix and the NHPAs’ model, it is necessary to derive

a unique parametrization of the linear MIMO channel matrix

and the Mnonlinear transmitters.

First, we note that there are inﬁnitely many pairs of the

equivalent parametrization, which can be expressed as HU ·

U∗z(h), where U∈C(MnH)×(M nH)is any unitary matrix.

Second, for any particular model for the MIMO channel matrix

HU , there are also inﬁnitely many pairs of the equivalent

parametrization for the model of the NHPAs U∗z(h). In

order to derive a unique parametrization of the linear MIMO

channel matrix and the Mnonlinear transmitters, therefore, we

need: 1) ﬁrst to determine a particular parametrization of the

MIMO CIR matrix HU , and 2) next to derive a particular

parametrization of the NHPAs’ model U∗z(h). To achieve

these two tasks, we re-express (8) equivalently as

x(h)(k) =







h0,1,1hT

1,11

h0,1,2hT

1,2··· 1

h0,1,M hT

1,M

h0,1,1hT

2,11

h0,1,2hT

2,2··· 1

h0,1,M hT

2,M

.....

h0,1,1hT

L,11

h0,1,2hT

L,2··· 1

h0,1,M hT

L,M













h0,1,1z(h)

1(k)

h0,1,2

h0,1,1h0,1,1z(h)

2(k)

h0,1,M

h0,1,1h0,1,1z(h)

M(k)





+n(H)(k).(16)

From (16), we have a unique parametrized MIMO channel

matrix HU as







h0,1,1hT

1,11

h0,1,2hT

1,2··· 1

h0,1,M hT

1,M

h0,1,1hT

2,11

h0,1,2hT

2,2··· 1

h0,1,M hT

2,M

.....

h0,1,1hT

L,11

h0,1,2hT

L,2··· 1

h0,1,M hT

L,M







.(17)

In (17), we still denote this equivalent linear MIMO channel

matrix HU by Hfor notational simplicity. From (16), we

also note that the Mnonlinear transmitters can be expressed as

zm(k) = h0,1,m

h0,1,1·h0,1,1z(h)

m(k)for 1≤m≤M. Therefore,

we have a unique parametrized NHPAs’ model U∗z(h)as

zm(k) =ζmΨ(sm(k)),1≤m≤M, (18)

with ζ1= 1 and ζm=h0,1,m

h0,1,1∈Cfor 2≤m≤M. Note that

(18) corresponds to absorbing h0,1,1into the NHPA’s response

Ψ(·), and again for notational simplicity, we still denote this

equivalent NHPA’s response h0,1,1Ψ(·)by Ψ(·).

Compared to the nonlinear frequency-nonselective MIMO

channel of [28], the nonlinear frequency-selective MIMO

channel, namely, the nonlinear MIMO Hammerstein system

(17) and (18), is much more complicated. In particular, MIMO

channel matrix H∈CnHLM is nHtimes larger than the

MIMO channel matrix of [28]. Identiﬁcation of such a large-

size MIMO nonlinear system, consisting of the frequency-

selective MIMO CIR multiplicative with the nonlinear model

of the Mnonlinear transmitters, is much more difﬁcult than

the corresponding identiﬁcation task in [28].

As shown in the previous section, implementing the non-

linear space-time equalizer for multiuser nonlinear frequency-

selective MIMO uplink requires the knowledge of the dis-

persive linear MIMO CIR matrix Has well as the inverse

mappings of all the MUs’ nonlinear HPAs. Since the dispersive

linear MIMO CIR matrix His cascaded with the Mnonlinear

transmitters, in order to acquire H, it is necessary to jointly

2.3 Unique parametrization of MIMO uplink

3. Proposed Nonlinear Space-time

Equalization

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estimate both the MIMO CIR matrix Hand the model

of the MNHPAs. Based on the unique parametrization of

Subsection II-C, we develop a BSNN based approach for

estimating both the MIMO CIR matrix Hand the MUs’

nonlinear transmitters ζmΨ(·)for 1≤m≤M. Furthermore,

the results of this nonlinear identiﬁcation enable us to acquire

the inverse mappings ζmΨ−1(·)of ζmΨ(·)for 1≤m≤M.

To jointly estimate the MIMO CIR matrix H(17) and the

MNHPAs (18), we still need to parametrize the complex-

valued NHPA Ψ(·)of (3). We propose to use a complex-valued

BSNN for this parametrization. The reason why we choose

the BSNN rather than other nonlinear models is because

among all the universe approximators for the class of nonlinear

continuous functions in the univariate dimension, the BSNN

has the maximum robustness to estimation error [37]–[39]. In

other words, it is an optimal choice for this task.

First, we establish some physical properties of the NHPA

Ψ(·)and its input s=sR+jsI, which are essential for our

BSNN parametrization. Clearly, the NHPA Ψ(·)is a one-to-

one continuous mapping, and therefore it is invertible. This

establishes the physical base for identifying Ψ(·)as well as

inverting it. The input to the NHPA stakes value from the

QAM constellation Sof (2). Observe from the QAM signaling

(2) that the constellation points are symmetrically distributed,

and they are both upper and lower bounded. In order words,

the distributions of sRand sIare identical and symmetric. In

addition, since −√U+ 1dS< sR, sI<√U−1dS, we

can always specify some known ﬁnite real values, Umin and

Umax, such that Umin < sR, sI< Umax.

1) Univariate BSNN: Consider a generic continuous non-

linear real-valued function y=f(u)deﬁned in the univariate

dimension of u∈R, and its input uis both upper and lower

bounded, i.e., Umin <u<Umax, with the known Umin and

Umax. We use a univariate BSNN with piecewise polynomial

degree of Poand Nrbasis functions to model this nonlinear

function. According to [29], the univariate BSNN is built upon

the so-called knot sequence speciﬁed by (Nr+Po+ 1) knot

values, denoted as {U0, U1,···, UNr+Po}, with the following

relationship

U0< U1<···< UPo−2< UPo−1=Umin < UPo<···

< UNr< UNr+1 =Umax < UNr+2 <···< UNr+Po.(19)

Since the input region is Umin, Umax, we have Nr+ 1 −Po

internal knots inside Umin, Umax, two boundary knots (Umin

and Umax), and 2(Po−1) external knots outside the input

region. Given the set of predetermined knots (19), we can

compute the set of NrB-spline basis functions. Speciﬁcally,

using the well-known De Boor recursion [29], we start from

the zero-order basis functions

B(r,0)

l(u) = 1,if Ul−1≤u < Ul,

0,otherwise,1≤l≤Nr+Po,

(20)

and recursively compute the pth order basis functions with

p= 1,···, Po

B(r,p)

l(u) = u−Ul−1

Up+l−1−Ul−1

B(r,p−1)

l(u)

+Up+l−u

Up+l−Ul

B(r,p−1)

l+1 (u),(21)

for l= 1,···, Nr+Po−p. The BSNN model for y=f(u)

is then produced as

i=1

biB(r,Po)

i(u),(22)

where bifor 1≤i≤Nrare the BSNN model coefﬁcients.

2) Structure determination: We now discuss how to deter-

mine the structure parameters, Poand Nr, for the univariate

BSNN model (22). For modeling the nonlinear functions

commonly encountered in the real world, the polynomial

degree Po= 3 or 4 is sufﬁcient [31]–[36]. Since the input

region Umin, Umaxis a bounded interval, using Nr= 6

to 10 B-spline basis functions is also sufﬁcient for accurately

modeling over the interval Umin, Umax. As regarding how

to determine the knot sequence relationship (19), the two

boundary knots can obviously be set to the known values

Umin and Umax, respectively, and the Nr+ 1 −Pointernal

knots can be uniformly spaced in the interval Umin, Umax.

The 2(Po−1) external knots are used to equip the BSNN

model with extrapolating capability outside the input region

Umin, Umax, and they can be set empirically. Since no data

appears outside Umin, Umax, the choice of these external

knots does not really matter, in terms of modeling accuracy.

Also the physical properties of the system to be modeled can

be taken into account to improve the modeling performance.

For our application with the symmetric QAM signals, the

distribution of u=sRor u=sIis naturally symmetric,

and therefore the knot sequence should be symmetrically

distributed too.

3) Computational complexity: The computational complex-

ity of the univariate BSNN model (22) depends on Po, not Nr.

This is because given any input u∈Umin, Umax, it can be

shown that no more than (Po+ 1) basis functions are nonzero

[40]. In [40], it further demonstrates that the complexity of

the BSNN model (22) is no more than twice of the following

polynomial basis model

i=0

aiui,(23)

with the model coefﬁcients aifor 0≤i≤Po, and the basis

function set

1, u, u2,···, uPo.(24)

4) Maximum robustness to estimation error: The works

[37]–[39] have established the fact that among all the universe

approximators for the class of nonlinear continuous functions

in the univariate dimension, the BSNN model (22) has the

optimal maximum robustness property. This optimal maximum

robustness property of the BSNN is due to the convexity of

its model bases, speciﬁcally, all the B-spline bases are positive

3.1 BSNN based parametrization

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and they sum up to unity. This maximum robustness property

provides the BSNN model with the maximum robustness to

estimation error and enables the BSNN model to attain highly

accurate estimation in noisy-data environments [28], [34],

[36], [40]–[42], outperforming other universe approximators

of similar complexity with non-convex model bases.

For example, the model bases (24) of the polynomial model

(23) do not possess the convexity property, and our previous

applications [28], [34], [36], [40]–[42] have all conﬁrmed that

the BSNN model signiﬁcantly outperforms the polynomial

model, particularly under highly noisy environments. In fact,

the previous analysis given in [40] has explained exactly

why. More speciﬁcally, recall the real-valued true nonlinear

system y=f(u)with y, u ∈R. Assume that this nonlinear

function can be exactly modeled by the polynomial model

(24) of degree Poor by the BSNN model (22) of polynomial

degree Powith Nrbasis functions. Because the training

data are noisy, due to the estimation error, the estimated

model coefﬁcients are perturbed from their true values ai

to bai=ai+εifor the polynomial model, and from their

true values bito b

bi=bi+εifor the BSNN model. Let us

assume that all the estimation errors εiare bounded, namely,

εi< εmax. The modeling error for the polynomial model

satisﬁes the following condition

|y−by|=

i=0

aiui−

i=0 baiui< εmax

i=0

ui.(25)

Observe that the upper bound of the modeling error for the

polynomial model depends not only on the upper bound of

the estimation error but also on the input value uand the

polynomial degree Po. For example, the higher the polynomial

degree Pois, the higher the modeling error of the polynomial

model will be. By contrast, the modeling error y−byfor

the BSNN model meets the following condition owe to the

convexity of its model bases

y−by=

i=1

biB(r,Po)

i(u)−

i=1 b

biB(r,Po)

i(u)

<εmax

i=1

B(r,Po)

i(u)=εmax.(26)

Clearly, the upper bound of the modeling error for the BSNN

model only depends on the upper bound of the estimation

error, and it does not depend on the input value x, the number

of basis functions Nror the polynomial degree Po. Unlike the

polynomial model, given the estimation error, the modeling

error of the BSNN model is not ampliﬁed. This conﬁrms that

the BSNN model has the maximum robustness to estimation

error or noise.

5) Complex-valued BSNN model for NHPA: The input

s=sR+jsIto the NHPA (3) is complex-valued or bivariate

and the output of the NHPA Ψ(s)is also complex-valued.

We now discuss how to construct the complex-valued BSNN

model for the NHPA. First, based on the univariate BSNN

modeling discussed in Subsection III-A1, for the inputs sR

and sI, we can construct the two sets of the univariate B-

spline basis functions, B(R,Po)

r(sR)for 1≤r≤NRand

B(I,Po)

i(sI)for 1≤i≤NI, respectively. Then by applying

the tensor product between these two sets of univariate B-

spline basis functions [30], we obtain the new set of bivariate

B-spline basis functions B(Po)

r,i (s) = B(R,Po)

r(sR)B(I,Po)

i(sI)

for 1≤r≤NRand 1≤i≤NI. This yields the following

complex-valued BSNN model for the NHPA Ψ(·)

bz=b

Ψ(s) =

r=1

i=1

B(Po)

r,i (s)θr,i

r=1

i=1

B(R,Po)

r(sR)B(I,Po)

i(sI)θr,i,(27)

where θr,i ∈C,1≤r≤NRand 1≤i≤NI, are the

complex-valued BSNN model coefﬁcients. By collecting all

the coefﬁcients into a vector form

θ=θ1,1θ1,2···θr,i ···θNR,NIT∈CNB,(28)

where NB=NRNI, the task of identifying the complex-

valued NHPA Ψ(·)becomes one of estimating the complex-

valued parameter vector θ.

From Subsection II-C, the multiuser nonlinear frequency-

selective MIMO uplink is the multiplicative cascade of the M

nonlinear transmitters (18) with the MIMO CIR matrix (17).

Further adopting the BSNN model for the NHPA given in

Subsection III-A, we have the unique parametrization of this

multiuser nonlinear frequency-selective MIMO uplink, which

involves the parameter vectors θand ζ=ζ1ζ2···ζMT,

where ζ1= 1, of the Mcomplex-valued BSNNs as well as

the multiuser MIMO CIR matrix H, where h0,1,m = 1 for

1≤m≤M.

1) Estimation signal representation: We ﬁrst collect a

block of Ktraining data, {s(h)(k),x(h)(k)}K

k=1, in which

the training input s(h)(k) = s(h)

1(k)T···s(h)

MTTwith

s(h)

m(k) = sm(k)sm(k−1) ···sm(k−nH+ 1)T, and

the desired output x(h)(k) = [x1(k)···xL(k)]T. The outputs

bxl(k)of our nonlinear model for modeling the desired outputs

xl(k)for 1≤l≤Lcan be expressed by

bxl(k) =

m=1

nH−1

q=0

hq,l,m bzm(k−q)

m=1

nH−1

q=0

r=1

i=1

B(Po)

r,i (sm(k−q))hq,l,mζmθr,i.(29)

From (29), it can be seen that this nonlinear frequency-

selective MIMO uplink identiﬁcation is a very challenging

nonlinear estimation problem because the parameters to be

estimated enter the model in the nonlinear triple product form

of hq,l,mζmθr,i. To devise an effective iterative estimation

procedure, we need the regression representations that are

‘linear’ in hq,l,m,ζmand θr,i, respectively.

3. Nonlinear frequency-selective

MIMO uplink identification

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1) Linear in Hregression model: Clearly, we can express

the desired output matrix X∈CL×Kas

X=





x1(1) x1(2) ··· x1(K)

x2(1) x2(2) ··· x2(K)

.··· .

xL(1) xL(2) ··· xL(K)





=









.(30)

Recalling the MIMO channel model (8), Xcan be further

expressed as

X=HQ +N,(31)

in which N∈CL×Kdenotes the channel AWGN matrix, and

the ‘regression’ matrix Q∈C(MnH)×Kis given by

Q=



b

z1(1) b

z1(2) ··· b

z1(K)

z2(1) b

z2(2) ··· b

z2(K)

.··· .

zM(1) b

zM(2) ··· b

zM(K)





,(32)

in which b

zm(k) = bzm(k)bzm(k−1) ···bzm(k−nH+ 1)T

with

bzm(k−q) =

r=1

i=1

B(Po)

r,i (sm(k−q))ζmθr,i,

0≤q≤nH−1,(33)

for 1≤m≤M. The regression model (31) is indeed linear

in Hbut its regression matrix Qis nonlinear in the parameter

products ζmθr,i.

2) Linear in θregression model: Next, express the desired

output vectors xl∈CK,1≤l≤L, where xT

lis the lth row

of X, as

xl=Plθ+nl,1≤l≤L, (34)

in which nl∈CKis the corresponding channel AWGN

vector, and the ‘regression’ matrix Pl∈CK×NBis given by

Pl=







φ(l)

1,1(1) φ(l)

1,2(1) ··· φ(l)

NR,NI(1)

φ(l)

1,1(2) φ(l)

1,2(2) ··· φ(l)

NR,NI(2)

.··· .

φ(l)

1,1(K)φ(l)

1,2(K)··· φ(l)

NR,NI(K)







,(35)

with

φ(l)

r,i(k) =

m=1

nH−1

q=0

hq,l,mζmB(Po)

r,i sm(k−q),(36)

for 1≤r≤NRand 1≤i≤NI. Aggregating (34) for

1≤l≤L, we have

l=1

xl=

l=1

Plθ+

l=1

nl⇒x=P θ +n.(37)

This model is linear in θbut its regression matrix Pis

nonlinear in hq,l,mζm.

3) Linear in ζregression model: Similarly, express xl∈

CK,1≤l≤L, as

xl=Slζ+nl,1≤l≤L, (38)

where the ‘regression’ matrix Sl∈CK×Mis given by

Sl=





l,1ψ1(1) hT

l,2ψ2(1) ··· hT

l,M ψM(1)

l,1ψ1(2) hT

l,2ψ2(2) ··· hT

l,M ψM(2)

.··· .

l,1ψ1(K)hT

l,2ψ2(K)··· hT

l,M ψM(K)





,(39)

in which ψm(k) = ψm(k)ψm(k−1) ···ψm(k−nH+ 1)T

and

ψm(k−q) =

r=1

i=1

B(Po)

r,i sm(k−q)θr,i,(40)

for 0≤q≤nH−1and 1≤m≤M. Hence we have

l=1

xl=

l=1

Slζ+

l=1

nl⇒x=Sζ +n.(41)

This model is linear in ζbut its regression matrix Sis

nonlinear in hq,l,mθr,i.

2) Iterative ALS procedure: We extend the estimation al-

gorithm of [28] originally developed for efﬁciently identifying

the multiuser nonlinear narrowband MIMO uplink model to

our current application of the multiuser nonlinear frequency-

selective MIMO uplink, and derive a new iterative ALS

procedure for estimating H,θand ζ. The basic idea is that

if we alternatively ﬁx the two parameters among the triple

parameters H,θand ζ, the third parameter can be obtained

by the least squares (LS) estimator. The estimation procedure

involves two iterative loops with three-stage ALS estimations

of H,θand ζ, respectively, as detailed below.

Algorithm: Two-loop three-stage ALS estimation.

Step 1. Estimation procedure initialization.

1.1. Set the maximum number of the outer loop iterations ςmax

and the maximum number of the inner loop iterations υmax.

1.2. Initialize Hand ζto H[0] and ζ[0]. Speciﬁcally, by

assuming that all the transmitters’ HPAs are linear, we have

the ‘approximate’ regression model X≈SH +N, where

the regression matrix S∈C(nHM)×Kis given by

S=





s1(1) s1(2) ··· s1(K)

s2(1) s2(2) ··· s2(K)

.··· .

sM(1) sM(2) ··· sM(K)





,(42)

with sm(k) = sm(k)sm(k−1) ···sm(k−nH+ 1)Tfor

1≤m≤M. Then we can set H[0] to the following standard

LS estimate

H[0] =XSHSSH−1.(43)

To meet the unique parametrization of the MIMO CIR matrix

as discussed in Subsection II-C, we ‘normalize’ c

H[0] accord-

ing to

h[0]

l,m =1

h[0]

0,1,m b

h[0]

l,m,1≤l≤L, 1≤m≤M. (44)

All the elements of b

ζ[0] can be initialized to 1, i.e., b

ζ[0]

m= 1

for 1≤m≤M.

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Step 2. Start of the outer iterative loop. For 1≤ς≤ςmax,

do:

Step 3. Fix the unknown MIMO CIR Hin the regression

matrices Pand Sto c

H[ς−1], and initialize b

ζ[0 =b

ζ[ς−1].

3.1. Start of the inner iterative loop. For 1≤υ≤υmax,do:

3.2. Fix the unknown ζin the regression matrix Pto b

ζ[υ−1

and denote the resulting Pas P[υ, which is now free from the

unknown Hand ζ. The closed-form regularized LS estimate

of θcan readily be obtained as1

θ[υ=P[υHP[υ+λINB−1P[υHx,(45)

where λis a very small positive regularization parameter, and

INBdenotes the NB×NBidentity matrix.

3.3. Fix the unknown θin the regression matrix Sto b

θ[υ

and denote the resulting Sas S[υ, which is then free from

the unknown Hand θ. The closed-form LS estimate of ζis

readily given by

ζ[υ=S[υHS[υ−1S[υHx.(46)

To meet the unique parametrization of the MNHPAs, we

normalize b

ζ[υwith

ζ[υ

m=b

ζ[υ

mb

ζ[υ

1,1≤m≤M. (47)

3.4. End of the inner iterative loop.

For the ﬁxed MIMO CIR matrix c

H[ς−1], we obtain the

estimated parameter vectors of the MNHPAs as b

θ[ς]=b

θ[υmax

and b

ζ[ς]=b

ζ[υmax .

Step 4. In the regression matrix Q, ﬁx the unknown θto b

θ[ς]

and the unknown ζto b

ζ[ς]. The resultant Qis denoted as

Q[ς], which becomes independent of the unknown θand ζ.

The closed-form LS estimate of His then given by

H[ς]=XQ[ς]HQ[ς]Q[ς]H−1

,(48)

which is followed by the normalization operation

h[ς]

l,m =1

h[ς]

0,1,m b

h[ς]

l,m,1≤l≤L, 1≤m≤M, (49)

to meet the unique parametrization of the MIMO CIR matrix.

Step 5. End of the outer iterative loop.

At the end of Algorithm, we obtain the estimates c

H[ςmax ],b

θ=b

θ[ςmax ]and b

ζ=b

ζ[ςmax ]for the multiuser

nonlinear frequency-selective MIMO uplink.

3) Unbiasedness and efﬁciency analysis: Observe that our

proposed identiﬁcation procedure for the multiuser nonlinear

frequency-selective MIMO uplink model contains the two

iterative loops, namely, the outer iterative loop of Step 1 to

Step 5 and the inner iterative loop of Step 3.1 to Step 3.4,

together with the three stages of ALS estimation, namely, the

closed-form LS estimates (45), (46) (with the normalization

(47)) and (48) (with the normalization (49)). The outer loop

iterates between the two stages of estimating the model of the

MNHPAs and estimating the multiuser MIMO CIR matrix.

Within the ﬁrst stage of the outer iterative loop, the inner loop

1Since the size of θis relatively large, the regularization is applied to avoid

ill-conditioning and enhance estimation accuracy.

iterates between the two stages of estimating θand ζ, which

together forms the model of the Mnonlinear transmitters.

We now analyze why the proposed iterative ALS procedure

converges extremely fast. The initial estimate for the unknown

MIMO CIR matrix is given by c

H[0] of (43), which is an

estimate of Hscaled by the NHPAs’ complex-valued gains.

It can readily be seen that with the normalization operation

(44), c

H[0] is an unbiased unique estimate of Hin (17).

Therefore, give this unbiased estimate c

H[0] of H, the inner

iterative loop converges to the unique estimates of θand ζ

very fast, owing to the unique parametrization of the NHPAs

and the closed-form LS estimates of (45) and (46) (with

the normalization (47)). Given this accurate NHPAs’ model,

Step 4 of the outer iterative loop can further improve the

accuracy of the estimate for the MIMO CIR matrix. Thus,

the outer iterative loop with ςmax = 1 iteration is in fact

sufﬁcient. To further enhance the overall estimation accuracy

of the multiuser nonlinear frequency-selective MIMO uplink

model, we may set ςmax = 2.

It is worth emphasizing that the uniqueness of the solutions

H,θand ζis guaranteed by our unique parametrization of

the multiuser nonlinear MIMO system, (16) to (18). In the

simulation study, we will further investigate empirically the

unbiasedness and efﬁciency property of our proposed iterative

ALS estimation procedure.

To implement the nonlinear STE based MUD for the

multiuser nonlinear frequency-selective MIMO uplink, we also

require the inverse mappings of the MNHPAs. Mathemati-

cally, the complex-valued inverse mappings of the MNHPAs

are deﬁned by

sm(k) =ζmΨ−1(zm(k)) = Φm(zm(k))

=ΦmζmΨsm,1≤m≤M. (50)

It can be seen that the inverse mapping Φm(·)of the mth

NHPA maps the output zmof the NHPA back to the NHPA’s

input sm. This is a challenging complex-valued nonlinear

inversion problem.

Since the BSNN is a universe nonlinear approximator with

the maximum robustness to estimation error, as discussed

in Subsection III-A, it is ideal for this nonlinear inversion

problem. Thus, we employ another complex-valued BSNN to

model Φm(·). By deﬁning the two knots sequences similar to

(19) for the real and imaginary parts of zm=zmR+jzmI,

respectively, the BSNN model for Φm(·)can be constructed

bsm=b

Φm(zm) =

r=1

i=1

B(Po)

r,i (zm)α(m)

r,i

r=1

i=1

B(R,Po)

r(zmR)B(I,Po)

i(zmI)α(m)

r,i ,(51)

for 1≤m≤M, where the two sets of the basis functions,

B(R,Po)

r(zmR)for 1≤r≤NRand B(I,Po)

i(zmI)for 1≤

i≤NI, are similarly calculated according to the De Boor

recursion (20) and (21). It can be seen that inverting the NHPA

3.3 Inverting the NHPAs

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ζmΨ(·)becomes the task of estimating the BSNN’s parameter

vector α(m)∈CNBgiven by

α(m)=α(m)

1,1α(m)

1,2···α(m)

r,i ···α(m)

NR,NIT.(52)

If we can acquire the training input-output data

zm(k), sm(k)K

k=1, then this estimation problem is

easily solved. However, the input zm(k)for this identiﬁcation

task is unobserved and therefore unavailable. To overcome

this problem, we utilize the identiﬁcation results for the

multiuser nonlinear frequency-selective MIMO uplink.

Speciﬁcally, in this identiﬁcation, we have obtained the

MBSNN models b

ζmb

Ψ·;b

θ,1≤m≤M, for the M

NHPAs. Therefore, we can calculate the estimate of zm(k)as

¯zm(k) = b

ζmb

Ψsm(k); b

θ, and use this ‘pseudo’ input b

¯zm(k)

to substitute for the unknown true input zm(k). This enables

us to construct the training data b

¯zm(k), sm(k)K

k=1 for this

inverse modeling. The downside is that b

¯zm(k)is not the true

training input and it is highly noisy, which may potentially

introduce bias in the estimate. Since we employ the BSNN as

the inverse model, we can rely on the maximum robustness

property of BSNN to combat this problem. After constructing

the training data b

¯zm(k), sm(k)K

k=1, we can form the linear

in α(m)regression model from which the LS estimate of

α(m)is readily obtained. Speciﬁcally, by deﬁning the desired

output vector as

sm=sm(1) sm(2) ···sm(K)T,(53)

and the regression matrix e

Bm∈RK×NBas

Bm=







B(Po)

1,1b

¯zm(1)··· B(Po)

NR,NIb

¯zm(1)

B(Po)

1,1b

¯zm(2)··· B(Po)

NR,NIb

¯zm(2)

.··· .

B(Po)

1,1b

¯zm(K)··· B(Po)

NR,NIb

¯zm(K)







,(54)

the closed-form LS estimate α(m)is readily given by

α(m)=e

Bm−1e

msm.(55)

Although the training input b

¯zm(k)is noisy, the optimal

maximum robustness property of the BSNN as discussed in

Subsection III-A4 ensures that the LS estimate (55) is unbiased

and highly accurate.

The simulated multiuser nonlinear frequency-selective

MIMO uplink is speciﬁed in Table I. Since the system has

Lreceiver antennas and Musers, we deﬁne the multiuser

TABLE I

PARAMETERS OF SIMULATED MULTIUSER NONLINEAR

FREQUENCY-SELECTIVE MIMO UPLINK

BS antennas: L= 5; MUs: M= 3; modulation: 64-QAM;

CIR length: nH= 3

NHPA: (4) and (5) with ga= 19,βa= 0.81,Asat = 1.4,

αφ=−48000,βφ= 0.123,q1= 3.8,q2= 3.7

Space-time equalizer length: nF=10,and decision delay: d= 5

TABLE II

STRUCTURE PARAMETERS OF B-SPLINE NEURAL NETWORK.

Polynomial degree: Po= 4, number of basis functions: NR=NI= 8

Knot sequence for sRand sI(modeling of NHPA)

-10.0, -9.0, -1.0, -0.9, -0.05, -0.02, 0.0, 0.02, 0.05, 0.9, 1.0, 9.0, 10.0

Knot sequence for zRand zI(inverse modeling of NHPA)

-20.0, -18.0, -3.0, -1.4, -0.8, -0.4, 0.0, 0.4, 0.8, 1.4, 3.0, 18.0, 20.0

MIMO system’s average signal-to-noise ratio (SNR) as the

ratio of the total transmitted signal power over the total noise

power, given by

Average SNR = PM

m=1 σ2

L·2σ2

,(56)

where σ2

zm=E{|zm(k)|2}is the average power of the

mth MU’s transmitted signal. The BSNNs used for modeling

and inverse modeling of NHPA are speciﬁed in Table II. As

explained in Subsection III-A2, choosing NR=NI= 8 and

Po= 4 is adequate for our application. The knot sequences in

Table II are chosen to cover the NHPA’s operating range and

match the 64-QAM signals. Observe that the knot sequences

for sRand sIare identical, and they are symmetric, since

the distributions of sRand sIare symmetric and identical.

Similarly, the knot sequences for zRand zIare identical, and

they are symmetric.

Since the total number of model parameters for this mul-

tiuser nonlinear frequency-selective MIMO channel is L×M×

nH+NB+M= 112, the number of training samples is

set to K= 1000 for ensuring the estimation accuracy. For

the iterative ALS procedure, we set the number of outer-loop

iterations to ςmax = 2 and the number of inner-loop iterations

to υmax = 2. As explained in Subsection III-B3, this choice

is sufﬁcient for the iterative ALS procedure to converge to the

unique and accurate estimates of H,θand ζ.

We ﬁrst demonstrate that the proposed BSNN based identiﬁ-

cation algorithms presented in Subsections III-B and III-C are

TABLE III

UNIQUE PARAMETRIZED TRUE MULTIUSER NONLINEAR

FREQUENCY-SELECTIVE MIMO CHANNEL.L= 5,M= 3 AND nH= 3.

THE NHPA IS SPECIFIED IN TABLE I.

NHPAs’ true weightings ζT

111

True H=hhT

l,m,1≤m≤3∈C1×9i,1≤l≤5

10.4740 + j1.1054 0.3705 −j0.7751

10.3755 + j0.4018 1.6995 −j0.2905

1−0.1295 −j1.4125 −0.5323 −j0.4941

0.3291 −j0.1268 1.0269 + j0.4665 −0.5798 + j0.8334

−0.5858 −j0.2308 −0.3396 + j0.1845 −0.2193 −j0.3347

1.3517 −j1.3128 −0.6780 + j0.9676 0.8737 −j0.3385

−0.1278 + j0.6590 0.0567 −j0.2107 −0.4374 −j0.5615

−0.5436 −j0.5148 0.7399 + j0.2869 −0.5403 + j0.7881

0.0122 + j0.9869 0.3670 + j0.4122 0.1809 + j0.2305

−1.0084 −j0.4358 −0.0909 −j0.4223 −0.8884 −j0.4641

−0.2137 −j0.2550 −0.1393 −j0.3626 −0.2465 + j0.0176

−1.1740 + j0.7498 −1.7164 + j0.6888 −0.6179 + j0.6992

−0.6067 −j0.7319 0.1464 + j0.5121 −0.4454 + j0.4105

0.0466 −j0.5741 0.8389 −j0.9315 0.1460 −j0.7706

1.0872 + j1.0012 −0.8176 + j1.3148 1.8309 + j0.5452

4. Simulation Evaluation

4.1 Simulation system setup

4.2 Estimation results by our BSNN approach

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TABLE IV

BSNN BASED IDENTIFICATION RESULTS FOR THE MULTIUSER NONLINEAR FREQUENCY-SELECTIVE MIMO CHANNEL OF TABLE III. THE OBO IS 3DB

AND THE AVERAGE SNR IS 20 DB. THE RESULTS ARE OBTAINED OVER 100 INDEPENDENT RUNS,AND ARE PRESENTED AS:AVERAGE ESTIMATE

(±STANDARD DEVIATION). THE BSNN ESTIMATED NHPAS ARE DEPICTED IN FIG. 3.

Estimated weightings of MUs’ HPAs b

ζby BSNN approach

11.0000 + j0.0000 (±0.0023 ±j0.0024)1.0000 −j0.0002 (±0.0023 ±j0.0027)

Estimated MIMO channel matrix c

H=hb

l,m,1≤m≤3∈C1×9i,1≤l≤5, by BSNN approach

10.4735 + j1.1052 (±0.0032 ±j0.0032)0.3704 −j0.7748 (±0.0023 ±j0.0026)

10.3754 + j0.4020 (±0.0020 ±j0.0019)1.6995 −j0.2908 (±0.0036 ±j0.0033)

1−0.1297 −j1.4128 (±0.0032 ±j0.0032)−0.5323 −j0.4944 (±0.0024 ±j0.0023)

0.3291 −j0.1270 (±0.0017 ±j0.0022)1.0270 + j0.4665 (±0.0025 ±j0.0029)−0.5800 + j0.8330 (±0.0027 ±j0.0026)

−0.5859 −j0.2308 (±0.0021 ±j0.0020)−0.3399 + j0.1846 (±0.0019 ±j0.0019)−0.2191 −j0.3343 (±0.0019 ±j0.0017)

1.3521 −j1.3126 (±0.0044 ±j0.0038)−0.6780 + j0.9675 (±0.0026 ±j0.0026)0.8741 −j0.3383 (±0.0022 ±j0.0027)

−0.1277 + j0.6592 (±0.0020 ±j0.0021)0.0565 −j0.2107 (±0.0022 ±j0.0018)−0.4371 −j0.5613 (±0.0022 ±j0.0024)

−0.5436 −j0.5150 (±0.0023 ±j0.0022)0.7398 + j0.2869 (±0.0022 ±j0.0023)−0.5402 + j0.7881 (±0.0025 ±j0.0026)

0.0120 + j0.9866 (±0.0021 ±j0.0025)0.3672 + j0.4126 (±0.0023 ±j0.0020)0.1809 + j0.2304 (±0.0018 ±j0.0019)

−1.0081 −j0.4359 (±0.0025 ±j0.0029)−0.0909 −j0.4222 (±0.0021 ±j0.0022)−0.8883 −j0.4642 (±0.0027 ±j0.0025)

−0.2138 −j0.2547 (±0.0019 ±j0.0019)−0.1395 −j0.3628 (±0.0020 ±j0.0017)−0.2468 + j0.0179 (±0.0019 ±j0.0018)

−1.1740 + j0.7497 (±0.0032 ±j0.0028)−1.7168 + j0.6883 (±0.0037 ±j0.0041)−0.6180 + j0.6993 (±0.0027 ±j0.0026)

−0.6065 −j0.7320 (±0.0023 ±j0.0026)0.1464 + j0.5121 (±0.0021 ±j0.0021)−0.4451 + j0.4104 (±0.0024 ±j0.0023)

0.0466 −j0.5740 (±0.0019 ±j0.0021)0.8388 −j0.9316 (±0.0029 ±j0.0029)0.1457 −j0.7707 (±0.0020 ±j0.0022)

1.0872 + j1.0016 (±0.0035 ±j0.0032)−0.8175 + j1.3143 (±0.0035 ±j0.0032)1.8308 + j0.5451 (±0.0039 ±j0.0037)

0.2

0.4

0.6

0.8

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

Output amplitude

Input amplitude

Mobile 1: true HPA

Average B-spline estimate

0.2

0.4

0.6

0.8

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

Output amplitude

Input amplitude

Mobile 2: true HPA

Average B-spline estimate

0.2

0.4

0.6

0.8

1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

Output amplitude

Input amplitude

Mobile 3: true HPA

Average B-spline estimate

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0 0.02 0.04 0.06 0.08 0.1 0.12

Output phase Shift (rad)

Input amplitude

Mobile 1: true HPA

Average B-spline estimate

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0 0.02 0.04 0.06 0.08 0.1 0.12

Output phase Shift (rad)

Input amplitude

Mobile 2: true HPA

Average B-spline estimate

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0 0.02 0.04 0.06 0.08 0.1 0.12

Output phase Shift (rad)

Input amplitude

Mobile 3: true HPA

Average B-spline estimate

(a) (b) (c)

Fig. 3. The unique parametrized true NHPA’s mapping ζmΨ(·)in comparison with the BSNN estimated NHPA mapping b

ζmb

Ψ(·)averaged over 100

identiﬁcation runs given OBO of 3 dB and average SNR of 20 dB: (a) MU 1, (b) MU 2, and (c) MU 3.

capable of attaining the unbiased and accurate estimates of the

MIMO CIR matrix and the BSNN models of the NHPAs at

the MUs’ transmitters as well as the BSNN inverse models

of the transmitters’ NHPAs. For this purpose, we consider

the true MIMO CIR matrix Hand the true NHPAs’ weights

ζfor a unique parametrized multiuser nonlinear frequency-

selective MIMO channel as given in Table III, where for the

clear representation purpose, each row of His re-arranged

into three subrows:

l,1hT

l,2hT

l,3⇒

l,1

l,2

l,3

,1≤l≤L. (57)

In this set of experiments, we set the NHPAs’ OBO to 3 dB

and the system’s average SNR to 20 dB. The BSNN based

identiﬁcation scheme and the nonlinear STE based MUD

presented in Section III are applied to this multiuser nonlinear

frequency-selective MIMO uplink. The results are obtained

over 100 independent identiﬁcation experiments.

1) Accuracy of MIMO CIR matrix estimate: The MIMO

CIR matrix estimate c

Hobtained by the proposed BSNN based

estimator is tabulated in Table IV, where the estimation results

are presented as average estimate with standard deviation.

Observe from Table IV that the BSNN based estimate c

His a

very accurate unbiased estimate of the true MIMO CIR matrix

Hgiven in Table III, with very small estimation error standard

deviations.

2) Accuracy of BSNN estimates of NHPAs: The estimated

NHPAs’ weighting vector b

ζobtained by the BSNN based

estimator closely matches the true NHPAs’ weighting vector

ζ, as can be clearly seen from Table IV. Fig. 3 compares the

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0.02

0.04

0.06

0.08

0.1

0.12

0 0.02 0.04 0.06 0.08 0.1 0.12

Amplitude response

Input amplitude

Mobile 1:true HPA+true inversion

true HPA+average B-spline inversion

0.02

0.04

0.06

0.08

0.1

0.12

0 0.02 0.04 0.06 0.08 0.1 0.12

Amplitude response

Input amplitude

Mobile 2:true HPA+true inversion

true HPA+average B-spline inversion

0.02

0.04

0.06

0.08

0.1

0.12

0 0.02 0.04 0.06 0.08 0.1 0.12

Amplitude response

Input amplitude

Mobile 3:true HPA+true inversion

true HPA+average B-spline inversion

-0.04

-0.03

-0.02

-0.01

0.01

0.02

0.03

0.04

0 0.02 0.04 0.06 0.08 0.1 0.12

Phase response (rad)

Input amplitude

Mobile 1: true HPA + true inversion

true HPA + average B-spline inversion

-0.04

-0.03

-0.02

-0.01

0.01

0.02

0.03

0.04

0 0.02 0.04 0.06 0.08 0.1 0.12

Phase response (rad)

Input amplitude

Mobile 2: true HPA + true inversion

true HPA + average B-spline inversion

-0.04

-0.03

-0.02

-0.01

0.01

0.02

0.03

0.04

0 0.02 0.04 0.06 0.08 0.1 0.12

Phase response (rad)

Input amplitude

Mobile 3: true HPA + true inversion

true HPA + average B-spline inversion

(a) (b) (c)

Fig. 4. The ideal combined response of the true NHPA ζmΨ(·)and its true inversion Φm(·)in comparison with the combined response of the true HPA

ζmΨ(·)and the estimated BSNN inversion b

Φm(·)averaged over 100 identiﬁcation runs given the OBO of 3 dB and the average SNR of 20 dB: (a) MU 1,

(b) MU 2, and (c) MU 3.

-1.5

-1

-0.5

0.5

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Transmit 1 estimate Im

Transmit 1 estimate Re

-1.5

-1

-0.5

0.5

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Transmit 2 estimate Im

Transmit 2 estimate Re

-1.5

-1

-0.5

0.5

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Transmit 3 estimate Im

Transmit 3 estimate Re

(a) (b) (c)

Fig. 5. Detected MUs’ transmitted signals, zm(k),1≤k≤3, by the MMSE space-time equalizer using the estimated MIMO CIR matrix c

Hobtained

by the BSNN based estimation scheme at a typical identiﬁcation run given OBO of 3 dB and average SNR of 20 dB: (a) bz1(k−5), (b) bz2(k−5), and

-0.15

-0.1

-0.05

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

User 1 estimate Im

User 1 estimate Re

-0.15

-0.1

-0.05

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

User 2 estimate Im

User 2 estimate Re

-0.15

-0.1

-0.05

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

User 3 estimate Im

User 3 estimate Re

(a) (b) (c)

Fig. 6. Detected MUs’ transmitted 64-QAM symbols, sm(k),1≤m≤3, by the nonlinear MMSE space-time equalizer based MUD using the estimated

Hand the BSNN inversions b

Φm(·)obtained by the BSNN based estimation scheme at a typical identiﬁcation run given OBO of 3 dB and average SNR of

20 dB: (a) bs1(k−5), (b) bs2(k−5), and (c) bs3(k−5).

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TABLE V

LINEAR LS ESTIMATE FOR THE TRUE MIMO CHANNEL MATRIX HOF MULTIUSER NONLINEAR FREQUENCY-SELECTIVE MIMO CHANNEL. THE OBO IS

3DBAND THE AVERAGE SNR IS 20 DB. THE RESULTS ARE OBTAINED OVER 100 INDEPENDENT RUNS,AND ARE PRESENTED AS:AVERAGE ESTIMATE

(±STANDARD DEVIATION).

Linear LS estimate (43) c

H[0] =hb

h[0]

l,mT,1≤m≤3∈C1×9i,1≤l≤5

10.5037 −j0.9566 (±0.1368 ±j0.1215)6.0066 + j11.1395 (±0.1139 ±j0.1278)3.1474 −j8.4847 (±0.1183 ±j0.1432)

10.4996 −j0.9583 (±0.1260 ±j0.1184)4.3061 + j3.8817 (±0.1028 ±j0.1349)17.5568 −j4.6649 (±0.1413 ±j0.1154)

10.4797 −j0.9443 (±0.1309 ±j0.1221)−2.7116 −j14.6836 (±0.1099 ±j0.1346)−6.0624 −j4.6978 (±0.1301 ±j0.1274)

3.3110 −j1.6360 (±0.1179 ±j0.1248)11.2270 + j3.9250 (±0.1197 ±j0.1057)−5.2922 + j9.2857 (±0.1101 ±j0.1009)

−6.3678 −j1.8751 (±0.1130 ±j0.1107)−3.3683 + j2.2609 (±0.1089 ±j0.1069)−2.6414 −j3.2991 (±0.1190 ±j0.1083)

12.9403 −j15.0679 (±0.1186 ±j0.1057)−6.1817 + j10.7898 (±0.1185 ±j0.1090)8.8338 −j4.3716 (±0.1151 ±j0.1004)

−0.7102 + j7.0421 (±0.0820 ±j0.0850)0.3957 −j2.2574 (±0.0841 ±j0.0811)−5.1194 −j5.4768 (±0.0831 ±j0.0859)

−6.1958 −j4.8718 (±0.0828 ±j0.0769)8.0256 + j2.3119 (±0.0864 ±j0.0794)−4.9283 + j8.7706 (±0.0720 ±j0.0744)

1.0621 + j10.3361 (±0.0720 ±j0.0977)4.2427 + j3.9866 (±0.0853 ±j0.0825)2.1151 + j2.2470 (±0.0852 ±j0.0763)

−11.0036 −j3.6347 (±0.1145 ±j0.1149)−1.3577 −j4.3382 (±0.1102 ±j0.1123)−9.7641 −j4.0338 (±0.1133 ±j0.1178)

−2.4888 −j2.4631 (±0.1145 ±j0.1159)−1.7963 −j3.6669 (±0.1038 ±j0.1204)−2.5767 + j0.4046 (±0.1148 ±j0.1288)

−11.6016 + j8.9805 (±0.1205 ±j0.1031)−17.3620 + j8.8487 (±0.1294 ±j0.0982)−5.8282 + j7.9291 (±0.1160 ±j0.1159)

−7.0849 −j7.1076 (±0.1373 ±j0.1471)2.0245 + j5.2562 (±0.1213 ±j0.1403)−4.2894 + j4.7420 (±0.1373 ±j0.1430)

−0.0367 −j6.0743 (±0.1442 ±j0.1321)7.9309 −j10.5453 (±0.1296 ±j0.1181)0.7785 −j8.2380 (±0.1362 ±j0.1305)

12.3697 + j9.4738 (±0.1281 ±j0.1386)−7.3169 + j14.5895 (±0.1263 ±j0.1260)19.7142 + j3.9957 (±0.1485 ±j0.1206)

Normalized linear LS estimate (44) c

H[0]

10.4713 + j1.1035 (±0.0197 ±j0.0207)0.3702 −j0.7741 (±0.0151 ±j0.0149)

10.3733 + j0.4038 (±0.0126 ±j0.0145)1.6986 −j0.2892 (±0.0226 ±j0.0219)

1−0.1314 −j1.4130 (±0.0200 ±j0.0196)−0.5338 −j0.4964 (±0.0147 ±j0.0144)

0.3267 −j0.1260 (±0.0114 ±j0.0124)1.0263 + j0.4672 (±0.0139 ±j0.0164)−0.5796 + j0.8312 (±0.0180 ±j0.0150)

−0.5853 −j0.2320 (±0.0125 ±j0.0114)−0.3376 + j0.1845 (±0.0105 ±j0.0101)−0.2210 −j0.3344 (±0.0124 ±j0.0109)

1.3533 −j1.3159 (±0.0222 ±j0.0238)−0.6771 + j0.9686 (±0.0170 ±j0.0162)0.8734 −j0.3384 (±0.0158 ±j0.0157)

−0.1277 + j0.6588 (±0.0090 ±j0.0095)0.0568 −j0.2097 (±0.0090 ±j0.0087)−0.4362 −j0.5612 (±0.0123 ±j0.0129)

−0.5432 −j0.5136 (±0.0112 ±j0.0100)0.7381 + j0.2876 (±0.0110 ±j0.0122)−0.5411 + j0.7859 (±0.0130 ±j0.0135)

0.0124 + j0.9874 (±0.0117 ±j0.0131)0.3676 + j0.4135 (±0.0098 ±j0.0095)0.1810 + j0.2307 (±0.0089 ±j0.0079)

−1.0077 −j0.4378 (±0.0129 ±j0.0110)−0.0909 −j0.4213 (±0.0111 ±j0.0114)−0.8872 −j0.4649 (±0.0140 ±j0.0160)

−0.2139 −j0.2541 (±0.0095 ±j0.0092)−0.1381 −j0.3619 (±0.0101 ±j0.0120)−0.2469 + j0.0160 (±0.0112 ±j0.0123)

−1.1747 + j0.7511 (±0.0186 ±j0.0184)−1.7188 + j0.6895 (±0.0207 ±j0.0224)−0.6193 + j0.7008 (±0.0141 ±j0.0147)

−0.6078 −j0.7320 (±0.0139 ±j0.0154)0.1459 + j0.5137 (±0.0115 ±j0.0122)−0.4458 + j0.4108 (±0.0168 ±j0.0157)

0.0489 −j0.5741 (±0.0140 ±j0.0125)0.8400 −j0.9277 (±0.0208 ±j0.0212)0.1446 −j0.7714 (±0.0160 ±j0.0146)

1.0901 + j1.0022 (±0.0253 ±j0.0238)−0.8170 + j1.3186 (±0.0184 ±j0.0210)1.8319 + j0.5464 (±0.0247 ±j0.0260)

BSNN estimated NHPA mapping b

ζmb

Ψ(·)averaged over 100

independent runs with the true HHPA’s mapping ζmΨ(·). It

can be seen that the amplitude response of b

ζmb

Ψ(·)is almost

identical to the true NHPA’s amplitude response, and the

estimation error of the phase response of b

ζmb

Ψ(·)is no more

than 0.01 radian.

3) Accuracy of BSNN inversions of NHPAs: We now ver-

ify the accuracy of the BSNN inversion estimates b

Φm(·),

1≤m≤3, obtained by the proposed BSNN inverting

scheme. From (50), it can be seen that the ideal combined

response of the true NHPA ζmΨ(·)and its true inversion

Φm(·)satisﬁes sm= ΦmζmΨ(sm). Therefore, we generate

the combined response of the true NHPA ζmΨ(·)and its

BSNN estimated inversion b

Φm(·), and compare this combined

response with the ideal combined response in Fig. 4. Observe

that the combined response b

ΦmζmΨ(·)matches well the

ideal combined response ΦmζmΨ(·), and we have

ΦmζmΨ(sm)≈sm,1≤m≤3.(58)

More speciﬁcally, the combined magnitude response is almost

identical to the ideal combined magnitude response, while

the error between the combined phase response and the ideal

combined phase response is no more than 0.02 radian. This

clearly demonstrates the accuracy of our proposed BSNN

inversion scheme based on the noisy pseudo training data.

4) Overall effectiveness of BSNN based estimation proce-

dure: To further illustrate the overall effectiveness of our

design, the estimated MIMO CIR matrix c

Hand the BSNN

inversions b

Φm(·),1≤m≤3, of the transmitters’ NHPAs,

obtained by the BSNN based estimation procedure in a typical

identiﬁcation run, are used to constructed the nonlinear STE

based MUD. Fig. 5 depicts the detected MUs’ transmitted

signals zm(k),1≤m≤3, by the MMSE space-time

equalizer. The MUs’ transmitted 64-QAM data are detected

by passing the detected transmitted signals bzm(k−5) through

the estimated BSNN inversion b

Φm(·)to compensate for the

distortion of the transmitters’ NHPAs, for 1≤m≤3, which

are shown in Fig. 6.

5) Empirical evidence of unbiasedness and efﬁciency: The

estimation results of Table IV as well as Figs. 3 to 6 clearly

demonstrate the accuracy and efﬁciency of our proposed

BSNN based estimation procedure for the multiuser nonlinear

frequency-selective MIMO uplink. Speciﬁcally, these empiri-

cal results show that the estimated MIMO CIR matrix c

His an

unbiased and accurate estimate for the true MIMO CIR matrix

H, while the identiﬁed BSNN inversions b

Φm(·)are unbiased

and accurate estimates for the true NHPAs’ inversion mappings

Φm(·), for 1≤m≤M. Since we only set the numbers of

iterations in both the outer loop and the inner loop to 2, the

fast convergence of our proposed iterative ALS procedure is

self-evident.

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10-6

10-5

10-4

10-3

10-2

10-1

100

5 10 15 20 25 30 35

User 1 Bit Error Rate

Average SNR (dB)

OBO=3dB, linear

OBO=5dB, linear

OBO=3dB, B-spline

OBO=5dB, B-spline

10-6

10-5

10-4

10-3

10-2

10-1

100

5 10 15 20 25 30 35

User 2 Bit Error Rate

Average SNR (dB)

OBO=3dB, linear

OBO=5dB, linear

OBO=3dB, B-spline

OBO=5dB, B-spline

10-6

10-5

10-4

10-3

10-2

10-1

100

5 10 15 20 25 30 35

User 3 Bit Error Rate

Average SNR (dB)

OBO=3dB, linear

OBO=5dB, linear

OBO=3dB, B-spline

OBO=5dB, B-spline

(a) (b) (c)

Fig. 7. Average bit error rate performance comparison for the proposed BSNN assisted nonlinear space-time equalization based MUD and the standard MMSE

linear space-time equalization scheme over 100 MIMO channel realizations, given the two OBO values of 3 dB and 5 dB: (a) MU 1, (b) MU 2, and (c) MU3.

In Subsection III-B3, we point out that the linear LS

estimate c

H[0] (43) is an estimate of the MIMO CIR matrix

scaled by the MUs’ NHPAs’ complex-valued gains, and its

normalized version (44) is an unbiased estimate of the true

MIMO CIR matrix H, although its estimation accuracy may

be poor. We now supply the empirical evidence to support

this analysis. In Table V, we list the linear LS estimate

H[0] (43) and its normalized version (44). By comparing this

normalized linear LS estimate with the true MIMO CIR matrix

Hgiven in Table III, it is clear that the normalized c

H[0] is an

unbiased estimate of the true MIMO CIR matrix H. Since this

normalized linear LS estimate is used as the initial estimate

of the MIMO channel matrix in our iterative ALS estimator,

it is not surprising that our iterative ALS estimator converges

very fast. Moreover, by comparing this normalized linear LS

estimate c

H[0] with our BSNN approach based estimate c

of Table IV, it can be seen that with only two iterations, the

estimation accuracy of the latter is signiﬁcantly better than that

of the former, since the estimation error standard deviations of

Hare around six times smaller than those of the initial c

H[0].

Hence our estimation results also offer the empirical evidence

to support the analysis of Subsection III-B3.

We now evaluate the ultimate performance metric of our

design, namely, its achievable bit error rate (BER). We

consider the rich scattered wireless environment, where the

entries of the MIMO CIR matrix follow the independent

complex Gaussian distribution CN(0,1). In the simulation,

we randomly generate the multiuser MIMO channel matrix

Hby drawing its coefﬁcients hi,l,m for 0≤i≤nH−1,

1≤l≤Land 1≤m≤Mfrom CN(0,1). A total of

100 channel realizations or MIMO CIR matrices are drawn.

For each MIMO channel realization, joint estimates of H,

θand ζare obtained using the identiﬁcation algorithms of

Section III with K= 1000 training data. Based on the

estimated c

H,b

θand b

ζ, the BSNN assisted nonlinear STE

based MUD is implemented and 10864-QAM data symbols

are transmitted by each MU for the BS to calculate the BER.

The average BER performance over the generated 100 MIMO

channel realizations achieved by our proposed BSNN assisted

nonlinear STE based MUD are depicted in Fig. 7, given the

two OBO values of 3 dB and 5 dB.

It is worth recapping that our proposed scheme is the

ﬁrst effective and practical multiuser nonlinear STE scheme

for the single-carrier multiuser nonlinear frequency-selective

MIMO uplink, and there exists no other effective and practical

nonlinear MUD schemes in the literature to compare with. The

existing STE based MUD schemes for single-carrier multiuser

frequency-selective MIMO uplink typically assume a linear

frequency-selective MIMO channel, which clearly no longer

work for the single-carrier multiuser nonlinear frequency-

selective MIMO uplink. To demonstrate this fact, we also

implement the linear STE for this single-carrier multiuser

nonlinear frequency-selective MIMO uplink. Speciﬁcally, we

ﬁrst estimate the equivalent linear MIMO channel matrix c

H[0]

using the linear LS estimate of (43), and then design the

linear MMSE space-time equalizer based on c

H[0]. The BER

performance achieved by this linear space-time equalizer are

also shown in Fig. 7 for the comparison with our BSNN as-

sisted nonlinear STE based MUD. Not surprisingly, this linear

space-time equalizer exhibits a high BER ﬂoor at the BER

level of 10−2even under the OBO of 5 dB, because it cannot

compensate for the nonlinear distortion of the transmitters’

NHPAs.

A BSNN assisted space-time equalization based MUD

has been proposed for the single-carrier multiuser nonlinear

frequency-selective MIMO uplink employing high-throughput

QAM transmission and with NHPAs at MUs’ transmitters.

First, we have developed a unique parametrization of the mul-

tiuser frequency-selective MIMO CIR matrix and the MUs’

nonlinear transmitters as well as a BSNN parametrization of

the transmitter’s NHPA. Second, we have proposed a highly

efﬁcient and accurate iterative ALS estimation procedure to

jointly estimate the MIMO CIR matrix and the BSNN models

of the MUS’ NHPAs. Third, the BSNN inverse models for

the MUs’ NHPAs have also been estimated. Based on the

estimated MIMO CIR matrix and the constructed BSNN

inversion models of the NHPAs, a BSNN assisted space-

time equalization based MUD has been implemented for the

single-carrier multiuser nonlinear frequency-selective MIMO

4.3 Bit error rate performance 5. Concluding Remarks

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DOI: 10.37394/23204.2022.21.20a

Sheng Chen

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uplink. Simulation results have demonstrated that our proposed

iterative ALS procedure converges very fast to the unbiased

and accurate estimates of both the dispersive MIMO CIR

matrix and the MUs’ NHPAs. Simulation results have also

conﬁrmed the effectiveness of the BSNN assisted space-time

equalization based MUD scheme, in terms of achievable BER

performance.

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