∎

11institutetext: M. Lari 22institutetext: Electrical and Computer Engineering
Faculty, Semnan University, Semnan, Iran
22email<EMAIL_ADDRESS>
# Matched-Filter Design to Improve Self-Interference Cancellation in Full-
Duplex Communication Systems

Mohammad Lari

###### Abstract

A new method for capacity and spectral efficiency increases is a full-duplex
(FD) communication, where sending and receiving are done simultaneously.
Hence, severe interference leaked from the transmitter to the receiver, which
can disrupt the system’s operation completely. For interference reduction, the
transceiver tries to estimate the interfering symbols to remove their effects.
A typical method is to use the Hammerstein model. In this method, nonlinear
power amplifier (PA) and multipath channel are modeled with a successive
nonlinear system and a finite impulse response (FIR) filter. Then, the model
parameters are adjusted, and interference symbols are estimated from the
transmitted symbols. In the Hammerstein method, the interference symbols are
estimated directly from the transmitted symbols. But practically, the
transmitted symbols first pass through the pulse-shaping filter and become a
signal. Then, this signal passes through the nonlinear PA and communication
channel. Finally, the received signal is filtered by the matched filter (MF)
at the receiver and converted to the symbols again. In this procedure, the
amplifier and the communication channel affect the transmitted signal directly
and distort transmitted symbols indirectly. Therefore, in the practical
situation, when we consider the transmitter’s pulse-shaping filter and the
receiver’s MF, the estimated symbols with the Hammerstein method are
erroneous. To solve this problem, a new MF at the receiver is proposed and
adjusted according to the interfering signal. We have shown that this method
is far better than the Hammerstein method.

###### Keywords:

Full-duplex Hammerstein Matched-filter Pulse-shaping filter Self-interference
cancellation

††journal: Wireless Networks

Received: date / Accepted: date

##  1 Introduction

With the increasing number of telecommunication subscribers and their need for
higher data rates, service providers use new and various methods to respond to
these requests. One promising method is the use of full-duplex (FD) instead of
the familiar half-duplex (HD) communication systems ref:01_Lari_Asaeian . In
most communication systems, such as mobile cellular networks, the number of
nodes (transceivers) is more than one. These nodes sometimes send data in the
role of transmitter and sometimes receive data in the role of receiver. These
transceivers with HD function send and receive at different time slots or
different frequency slots. Since send and receive are separate in time or
frequency, the transmitted signal does not interfere with the received signal.
Therefore, each transceiver needs two time slots or two frequency slots for
its operation, which is not an efficient use of network resources. If this
number of time or frequency slots reduce to one, the spectral efficiency of
each user will increase. At the same time, it is possible to give the released
time or frequency slot to a new one and increase the number of users. If only
one time-frequency slot assigns to nodes, each transceiver has an FD function.
Then, due to sending and receiving at the same time and at the same frequency,
severe interference is introduced to the receiver from the transmitter. If
there is not enough attention paid to this self-interference (SI), the
receiver will be completely disturbed by its transmitter, then, an outage
occurs. So, the FD node can’t receive the signal of interest (SoI) from the
other side. For this reason, despite their great advantage, FD systems have
not been widely implemented yet ref:02 . FD systems have other advantages
ref:03_Lari , such as reduction of end-to-end delay in delay-sensitive
applications ref:04_Lari_Keshavarz .

To achieve the advantages of the FD system, it is necessary to reduce the
input interference impressively. However, the transmitted power is much higher
than the typical received power. Therefore, the power of SI is much stronger
than the received SoI power. This interference power may even be 100 dB higher
than the noise floor ref:05 . Therefore, its reduction is made in several
stages and in three different points of receiver ref:06 . To prevent the
saturation of radio frequency (RF) circuits, at the first stage, proper design
and arrangement of the transmitter and receiver antennas reduce the input
interference to some extent. In the next step and to use the maximum dynamic
range of the analog to digital converter (ADC), the second part of the
interference cancellation is done in the analog part of the receiver. Here,
part of the transmitted signal is injected into the receiver and subtracted
from the received signal. With this method, the main component (direct path
component) of the interference is removed. The residual interference after
these two steps is digitally removed in the baseband section of the receiver.
Due to the high processing capability of digital processors, cancellation in
this section is of particular importance.

The RF transmitter and receiver chains are not ideal and cause distortions in
the transmitted and received signal. One of the severe problem is the
nonlinearity of the transmitter power amplifier (PA). The output signal of the
amplifier contains linear and nonlinear components of the input signal. In
addition, there are usually obstacles around the transmitter that cause the
transmitted signal enters the receiver from different angles and paths.
Accordingly, the interference signal includes the linear and nonlinear
components of the transmitted signal that enter the receiver through the
multipath communication channel. The most significant linear component of the
interference signal is often removed in the analog domain, and the rest of the
components are eliminated as much as possible in the baseband ref:07 . For
this purpose, first, the nonlinear behavior of the PA along with multipath
channel coefficients is modeled. Then, the model parameters are estimated
using pilot data. Using this model, a symbol similar to the residual
interference is created and subtracted from it in the digital domain. If the
appropriate model is used and its parameters are accurately estimated, the
residual interference will be significantly reduced ref:07 .

In practical systems, the input of the PA is not symbols. The PA captures and
amplifies the output signal from the pulse-shaping filter. So, the nonlinear
PA distorts the pulse-shaping output signal instead of the transmitted symbols
(please pay attention to the difference between symbol and signal well). This
issue causes the interference symbol not to be reconstructed correctly from
the transmitted symbol.

When the PA is linear, the pulse-shaping filter effect is completely reversed
by a similar filter in the receiver called matched-filter (MF) ref:08_Proakis
. In addition, since we have downsampling at the MF output, the data rate at
the output of MF is much less than the data rate in the input. Therefore, it
is more convenient to ignore both pulse-shaping filter and MF in simulation
and analysis and investigate the simple baseband transceiver in the symbol
domain. However, with the nonlinear PA, the effect of the pulse-shaping filter
is not entirely removed by MF in the receiver. This issue, which has been
ignored in most previous SI cancellation research, will be considered here.
First, we show that the SI is not entirely removed, even in the perfect
situation, by considering the interference in the symbol space. This problem
arises because the MF does not match the pulse-shaping filter in the presence
of the nonlinear PA, leading to different errors. Then, we propose a method to
reduce this error with a new MF, which differs from the pulse-shaping filter.
We will see that the performance of this new filter is better than the
standard method for SI cancellation.

In Section 2, we will have a brief review of previous research on SI
cancellation in the digital domain. Then the model of the discussed system
will examine in Section 3 and the cancellation of SI in Section 4. The
weakness of the common method that does not consider the effect of the pulse-
shaping filter and the MF will also explain in Section 4. In Section 5, our
proposed method is presented. In Section 6, various simulations are given to
show the better performance of the proposed method. Finally, in Section 7, the
conclusions of the research are stated.

Since the number of parameters used is relatively large, for convenience, the
most important parameters are given in Table 1 along with their brief
explanation. In addition, for standard writing, numerical variables are
written with small italic letters, vectors are written with lower case and
bold letters, and matrices are written with bold capital letters.

Table 1: The most important parameters with a short explanation Parameter | Explanation
---|---
$s[n]$ | Transmitted symbol from the first node (node 1)
$x[k]$ | Transmitted signal from the first node (node 1)
$M$ | Upsampling factor in the pulse-shaping filter or downsampling factor in the MF
$g_{T}[k]$ | Pulse-shaping filter coefficients
$g_{R}[k]$ | MF coefficients in the regular method
$N$ | Total number of transmitted symbols
$L_{g}$ | Pulse-shaping filter and MF span in the terms of symbols
$h[k]$ | Multipath channel impulse response
$y[k]$ | Received interference signal at the first node
$r[n]$ | Received interference symbol at the first node
$w[k]$ | Received noise signal at the first node
$\omega[n]$ | Received noise symbol at the first node
$u[n]$ | Transmitted symbol from the second node to the first node
$z[k]$ | Received signal from the second node at the first node
$v[n]$ | Received symbol from the second node at the first node
$\eta[k]$ | Total received signal at the first node
$\lambda[n]$ | Total received symbol at the first node
$\mathbf{q}$ | Model’s parameters, (8)
$L_{q}$ | Hammerstein FIR filter length in the terms of symbols
$P$ | Hammerstein polynomial maximum degree
$g_{1}[k]$ | MF coefficients in the proposed method

Please note the difference between symbol and signal. The output of the
digital modulator before the pulse-shaping filter at the transmitter is a
symbol. In addition, the output of the MF before the digital demodulator at
the receiver is a symbol. But, the output of the pulse-shaping filter and the
input of the MF is a signal. We also have a signal in the communication
channel. Therefore, here $n$ shows the discrete-time of the symbols, and $k$
represents the discrete-time of the signal.

##  2 Related Work

FD communication systems have been receiving specific attention for several
years, and various research studies have been conducted ref:09 . The proper
operation of the FD system depends on eliminating SI from the receiver. To
remove the interference sufficiently, this work is usually done in several
sections, including the antenna, the RF and analog, and the digital section of
the receiver ref:06 . Due to the availability of more processing capability in
the digital domain, many researchers have considered the issue of eliminating
SI in the digital section. Some of these references, which are more closely
related to the current article, have been briefly reviewed in this section.

The received SI has passed through the nonlinear PA and the multipath channel.
So, to remove it better, we need to model the amplifier along with the
multipath channel. The Hammerstein model is a relatively simple and
appropriate method to do this, and this model has used in many studies. For
example, in ref:10 , the nonlinear model for the PA is derived using the
generalized Fourier expansion with orthogonal Laguerre bases. Then the
interference symbols are estimated using the Hammerstein model and subtracted
from the received symbols in the receiver. In ref:11 , all transmitter and
receiver RF impairments have modeled with the help of the Hammerstein method.
Then, the interference has removed in the receiver. In the article ref:12 ,
the time-varying communication channel and the changes that occur in the PA
over time have investigated. The Hammerstein with the state space model, along
with the Kalman filter, are used to detect these changes. The Hammerstein
model develops in ref:13  using several nonlinear operators. Therefore, the
nonlinear effect of the low noise amplifier (LNA) in the receiver is also
considered. According to the author’s claim, LNA impairment cannot model in
the usual Hammerstein method. In order to reduce the complexity and the
possibility of implementing the model on a field programming gate array
(FPGA), the interpolation technique is used along with the Hammerstein model
in ref:14 . Finally, the FD system has been implemented and tested in real
conditions. Several articles, such as ref:15 ; ref:16 ; ref:17  have made
modeling and interference cancellation using neural networks. But for data
generation and network training, they have used the Hammerstein method to
model the adverse effects of the transmitter and receiver RF chain.

Some research studies such as ref:18_Majidi ; ref:Eriksson  have considered
the effect of the pulse-shaping filter and the MF along with the nonlinear PA.
For example, in ref:18_Majidi , the pulse-shaping filter is assumed, and an
exact analytical expression of the spectral regrowth at the output of a
nonlinear PA is derived. In ref:Eriksson , the authors analytically provide
statistical results on how a nonlinear amplifier affects the quality of
transmitted signals. Then, describe the nonlinear distortion as a function of
the input signal’s average and instantaneous power and captures the pulse-
shaping and MF effects. Such research is not numerous, and their review has
been done in general HD communication systems. To the best of our knowledge,
the effects of the pulse-shaping filter and the MF in the FD systems do not
take into account yet.

##  3 System Model

The main parts of the transmitter and receiver of a communication node are
drawn in Fig. 1. We consider this node as the first node (node 1), which
intends to send and receive data as FD to other nodes. Since our discussion
here is the cancellation of SI, the other node and its details are not shown
in Fig. 1. However, its main blocks are completely similar to Fig. 1.

In the baseband part of the transmitter, all necessary processes such as
coding, digital modulation, or orthogonal frequency division multiplexing
(OFDM) modulation are performed, and the symbol $s[n]$ leaves from it. We
consider $T_{\mathrm{sym}}$ as the duration of each symbol. The symbol $s[n]$
is upsampled and filtered in the pulse-shaping filter. If the upsampling rate
is $M$ and the filter coefficients are $g_{T}[k]$, the output signal of the
filter is

$x[k]=\sum\limits_{n=0}^{N-1}{s[n]{g_{T}}[k-nM]}$ (1)

where $N$ is the total number of transmitted symbols,
$T_{g}=L_{g}T_{\mathrm{sym}}$ is the pulse-shaping filter span, which is equal
to the $L_{g}$ symbols duration. According to the upsampling rate, the number
of coefficients of this filter will be $ML_{g}$. The discrete time signal
$x[k]$ passes through the digital to analog converter (DAC) and produces the
continuous time signal $x(t)$. Then, this signal is transferred to the
passband and passes through the PA. The amplifier output $x_{\mathrm{RF}}(t)$
is a nonlinear function of the input and can be written as

${x_{{\rm{RF}}}}(t)={\cal F}\left({x(t){{\mathop{\rm
e}\nolimits}^{j2\pi{f_{c}}t}}}\right).$ (2)

In this equation, $\cal{F}(.)$ is the nonlinear function between the output
and input of the PA. Note that all symbols and signals are complex. However,
this is not shown explicitly in Fig. 1. Due to the nonlinearity of
$\mathcal{F}(.)$, a part of the PA output signal is placed in-band and around
the carrier frequency $f_{c}$. Some parts are also located in out-of-band
region around the frequency $\alpha f_{c},~{}\alpha=2,3,...$ ref:18_Majidi ;
ref:19_Majidi . To show this, we can rewrite (2) as

${x_{{\rm{RF}}}}(t)={{\cal F}_{1}}\left({x(t)}\right){{\mathop{\rm
e}\nolimits}^{j2\pi{f_{c}}t}}+\sum\limits_{\alpha=2}^{\infty}{{{\cal
F}_{\alpha}}\left({x(t)}\right){{\mathop{\rm
e}\nolimits}^{j2\pi\alpha{f_{c}}t}}}$ (3)

where $\mathcal{F}_{1}(.)$ is a nonlinear function of the PA at the baseband,
located around $f_{c}$. In the same way, $\mathcal{F}_{\alpha}(.)$ presents a
nonlinear function of the PA at the baseband whose output components are
placed around the frequency $\alpha f_{c}$.

Figure 1: Block diagram of transmitter and receiver of an FD communication
node.

The amplified signal $x_{\mathrm{RF}}(t)$ is transmitted from the first node.
But at the same time, it leaks to its receiver through the multipath channel.
The interference signal at the receiver is $y_{\mathrm{RF}}(t)$. For
simplicity, the signal received from the other node is not considered here.
The signal is amplified in the LNA of the receiver. We assume the LNA has a
linear function and completely removes the out-of-band interference.
Therefore, the interference signal after passing through the mixer is

$y(t)={{\cal F}_{1}}\left({x(t)}\right)*h(t)$ (4)

where $h(t)$ represents the impulse response of the interference channel at
the baseband and $*$ is the convolution operator. The continuous time
interference signal $y(t)$ is sampled at the analog to digital converter (ADC)
with rate $F_{s}=\frac{1}{MT_{\mathrm{sym}}}$ and converted into the discrete
time signal $y[k]=\mathcal{F}_{1}(x[k])*h[k]$. Finally, the discrete signal is
filtered and downsampled at the MF with $g_{R}[k]$ coefficients. The output of
MF is

$\begin{array}[]{@{}lclll}r[n]&=&\sum\limits_{m=0}^{MN-1}{y[m]{g_{R}}[nM-m]}\\\
&=&\sum\limits_{m=0}^{MN-1}{\left({{{\cal
F}_{1}}\left({x[m]}\right)*h[m]}\right){g_{R}}[nM-m]}.\end{array}$ (5)

In addition, the input noise to the receiver is $w_{\mathrm{RF}}(t)$, and it
is converted to $w(t)$ after the mixer, $w[k]$ after the ADC and $\omega[n]$
after the MF.

On the other hand, node 2 wants to communicate with node 1. Therefore, the
transmitter of node 2 first converts the $u[n]$ symbol into a discrete-time
signal and then into a continuous-time signal. Then the signal is transferred
to the passband around the carrier frequency $f_{c}$ and amplified. Finally,
this signal is sent to node 1 through the multipath channel. The received
signal passes through the RF chain of node 1 and presents after the mixer, the
ADC, and the MF as $z(t)$, $z[k]$, and $v[n]$. According to the FD function of
node 1, interference, desired signal, and noise enter the receiver at the same
time. Therefore, the input signal to the MF is $\eta[k]=y[k]+z[k]+w[k]$, and
the output symbol will be $\lambda[n]=r[n]+v[n]+\omega[n]$.

##  4 Self-Interference Cancellation

###  4.1 Conventional Method Based on Hammerstein Model

In the conventional method, to eliminate the interference, the interference
regenerator tries to estimate the SI symbol $r[n]$ with minimum error and in
the form of $\hat{r}[n]$. This block, shown with a dashed line in Fig. 1,
usually uses the Hammerstein method to model the nonlinear PA and the
multipath channel simultaneously ref:20 . According to Fig. 2, the Hammerstein
model is made of a nonlinear system and a finite impulse response (FIR) filter
sequentially ref:20 . The nonlinear part is often modeled with a polynomial of
degree $P$. So, the output of this part is equal to

$\rho[n]=\sum\limits_{p=1,{\rm{}}p=2\ell+1}^{P}{{a_{p}}s[n]{{\left|{s[n]}\right|}^{p-1}}}.$
(6)

$P$ is an odd number and $\left\\{{{a_{1}},{a_{3}},...,{a_{P}}}\right\\}$
represents the polynomial coefficients. Note that the PA is often modeled with
odd degree polynomials. Therefore,
$\left\\{{{a_{0}},{a_{2}},...,{a_{P}-1}}\right\\}$ coefficients are not seen
in (6). The coefficients of the FIR filter
$\left\\{{{q_{0}},{q_{1}},...,{q_{L_{q}-1}}}\right\\}$ with $L_{q}$ unknown
parameters are also considered as parameters of the model.

Figure 2: Block diagram of Hammerstein model.

To adjust the parameters of the Hammerstein model, the transmitter of node 2
does not send a signal for a certain duration. Instead, the first node
transmits $N^{(\mathrm{p})}$ pilot symbols
${s^{({\rm{p}})}}[n],~{}n=0,1,...,{N^{({\rm{p}})}}-1$ and these symbols are
received as
${\lambda^{({\rm{p}})}}[n]={r^{({\rm{p}})}}[n]+{\omega^{({\rm{p}})}}[n],~{}n=0,1,...,{N^{({\rm{p}})}}-1$,
including interference symbols and noise ref:16 . Therefore, the relationship
between the input symbols, model parameters, and the output of the
interference regenerator will be in the form

${{\bf{\lambda}}^{({\rm{p}})}}={{\bf{S}}^{({\rm{p}})}}\times{\bf{q}}$ (7)

where model parameters and received symbols are given in (8). The input
symbols matrix is also written in (9).

${\bf{q}}=\left({\begin{array}[]{*{20}{c}}{{q_{0}}{a_{1}}}\\\ \vdots\\\
{{q_{0}}{a_{P}}}\\\ {{q_{1}}{a_{1}}}\\\ \vdots\\\ {{q_{1}}{a_{P}}}\\\
\vdots\\\ {{q_{{L_{q}}-1}}{a_{1}}}\\\ {\begin{array}[]{*{20}{c}}\vdots\\\
{{q_{{L_{q}}-1}}{a_{P}}}\end{array}}\end{array}}\right),\quad{{\bf{\lambda}}^{({\rm{p}})}}=\left({\begin{array}[]{*{20}{c}}{{\lambda^{({\rm{p}})}}[0]}\\\
{{\lambda^{({\rm{p}})}}[1]}\\\ \vdots\\\
{{\lambda^{({\rm{p}})}}[{N^{({\rm{p}})}}-1]}\end{array}}\right)$ (8)

$\begin{array}[]{l}{{\bf{S}}^{({\rm{p}})}}=\\\
\left({\begin{array}[]{*{20}{c}}{{s^{({\rm{p}})}}[0]}&{{s^{({\rm{p}})}}[0]{{\left|{{s^{({\rm{p}})}}[0]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[0]{{\left|{{s^{({\rm{p}})}}[0]}\right|}^{P-1}}}&0&0&{...}&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots&\vdots\\\
{{s^{({\rm{p}})}}[{L_{q}}-1]}&{{s^{({\rm{p}})}}[{L_{q}}-1]{{\left|{{s^{({\rm{p}})}}[{L_{q}}-1]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[{L_{q}}-1]{{\left|{{s^{({\rm{p}})}}[{L_{q}}-1]}\right|}^{P-1}}}&{...}&{{s^{({\rm{p}})}}[0]}&{{s^{({\rm{p}})}}[0]{{\left|{{s^{({\rm{p}})}}[0]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[0]{{\left|{{s^{({\rm{p}})}}[0]}\right|}^{P-1}}}\\\
{{s^{({\rm{p}})}}[{L_{\rm{q}}}]}&{{s^{({\rm{p}})}}[{L_{q}}]{{\left|{{s^{({\rm{p}})}}[{L_{q}}]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[{L_{q}}]{{\left|{{s^{({\rm{p}})}}[{L_{q}}]}\right|}^{P-1}}}&{...}&{{s^{({\rm{p}})}}[1]}&{{s^{({\rm{p}})}}[1]{{\left|{{s^{({\rm{p}})}}[1]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[1]{{\left|{{s^{({\rm{p}})}}[1]}\right|}^{P-1}}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-1]}&{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-1]{{\left|{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-1]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-1]{{\left|{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-1]}\right|}^{P-1}}}&{...}&{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-{L_{q}}+1]}&{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-{L_{q}}+1]{{\left|{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-{L_{q}}+1]}\right|}^{2}}}&{...}&{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-{L_{q}}+1]{{\left|{{s^{({\rm{p}})}}[{N^{({\rm{p}})}}-{L_{q}}+1]}\right|}^{P-1}}}\end{array}}\right)\end{array}$

(9)

According to the fact that the transmitted and received pilot symbols are
known, the parameters of the Hammerstein model are estimated with least square
(LS) estimation as

${\bf{q}}={\left({{{\bf{S}}^{({\rm{p}}){\dagger}}}{{\bf{S}}^{({\rm{p}})}}}\right)^{-1}}{{\bf{S}}^{({\rm{p}})}}\times{{\bf{\lambda}}^{({\rm{p}})}}$
(10)

After this, transmitters of node 1 and node 2 start sending their data as FD.
In addition to receiving the symbols from node 2, the first node also receives
its transmitted symbols along with noise. Therefore, the symbols received at
the receiver of node 1 are $\lambda[n]=r[n]+v[n]+\omega[n],~{}n=0,1,...,N-1$.
The SI regenerator also estimates the interference symbol as
$\hat{r}[n],~{}n=0,1,...,N-1$. Then, $\lambda[n]$ and $\hat{r}[n]$ enter the
baseband section to eliminate interference. If the estimated symbol is similar
to the interference symbol, $\epsilon[n]=r[n]-\hat{r}[n]$ will have a small
value and the detection and demodulation of the SoI sent by node 2 will be
done well using $\lambda[n]=v[n]+\left(\epsilon{[n]+\omega[n]}\right)$.

###  4.2 Weakness of Hammerstein Model

According to Fig. 1, we can rewrite (5) as

$r[n]={{\cal D}_{M}}\left\\{{{g_{R}}[k]*h[k]*{{\cal
F}_{1}}\left({{g_{T}}[k]*{{\cal
U}_{M}}\left\\{{s[n]}\right\\}}\right)}\right\\}$ (11)

where ${{\cal U}_{M}}\\{.\\}$ and ${{\cal D}_{M}}\\{\\}$ represent the
upsampling and downsampling operations with order $M$. Therefore, ${{\cal
U}_{M}}\\{s[n]\\}$ is equal to ${{\cal
U}_{M}}\left\\{{s[n]}\right\\}=s\left[{{\raise
3.01385pt\hbox{$n$}\\!\mathord{\left/{\vphantom{nM}}\right.\kern-1.2pt}\\!\lower
3.01385pt\hbox{$M$}}}\right]=s[k]$ and ${D_{M}}\left\\{{r[k]}\right\\}$ is
equal to ${D_{M}}\left\\{{r[k]}\right\\}=r[kM]=r[n]$. Note that $n$ represents
discrete time with sampling frequency $\frac{1}{T_{\mathrm{sym}}}$ and $k$
represents discrete time with the sampling frequency
$\frac{M}{T_{\mathrm{sym}}}$. Because the convolution operator has commutative
property, we can write (11) as

$\begin{array}[]{@{}lclll}r[n]&=&{{\cal D}_{M}}\left\\{{h[k]*{g_{R}}[k]*{{\cal
F}_{1}}\left({{g_{T}}[k]*{{\cal
U}_{M}}\left\\{{s[n]}\right\\}}\right)}\right\\}\\\ &=&{{\cal
D}_{M}}\left\\{{h[k]*{g_{R}}[k]*{{\cal
F}_{1}}\left({{g_{T}}[k]*s[k]}\right)}\right\\}.\end{array}$ (12)

Now, assuming the nonlinear PA, we can simplify (12) in the form of ref:21

$\begin{array}[]{@{}lclll}r[n]&=&{{\cal D}_{M}}\left\\{{h[k]*{g_{R}}[k]*{{\cal
F}_{1}}\left({{g_{T}}[k]*s[k]}\right)}\right\\}\\\
&=&\sum\limits_{m=0}^{M-1}{\left({{{\cal
D}_{M}}\left\\{{h[k-m]}\right\\}}\right.}\left.{{\rm{}}*{{\cal
D}_{M}}\left\\{{{g_{R}}[k+m]*{{\cal
F}_{1}}\left({{g_{T}}[k+m]*s[k+m]}\right)}\right\\}}\right).\end{array}$ (13)

Because the length of the pulse-shaping filter is longer than one symbol,
$L_{g}>1$, the term ${g_{T}}[k+m]*s[k+m]$ in (13) is a linear combination of a
number of symbols sent by the transmitter. When this expression passes through
the nonlinear function $\mathcal{F}_{1}(.)$, some symbols are generated which
cannot be regenerated using the Hammerstein model in Fig. 2. Therefore, the
use of Hammerstein’s model in typical communication systems is inaccurate and
is only approximate. Although this approximation is relatively suitable, we
will show in the following that our proposed method can make this
approximation better and more accurate.

For example, suppose the linear combination of transmitted symbols is as
simple as $s[0]+s[1]$. We also consider the PA nonlinear function as
$\mathcal{F}_{1}(x)=x^{3}$. In this way, ${{\cal
F}_{1}}\left({s[0]+s[1]}\right)$ becomes equal to

$\begin{array}[]{@{}lclll}{{\cal
F}_{1}}\left({s[0]+s[1]}\right)={s^{3}}[0]+{s^{3}}[1]+\underbrace{3{s^{2}}[0]s[1]+3s[0]{s^{2}}[1]}_{{\rm{multiplicative~{}terms}}}.\end{array}$
(14)

If the relation (14) is placed in (13) and simplified, the interference symbol
$r[n]$ will also depend on the multiplicative symbols such as ${s^{2}}[0]s[1]$
and $s[0]s^{2}[1]$. Regarding Fig. 2 and equation (6), it is clear that the
Hammerstein model is incapable of generating these multiplicative terms.
Therefore, it is not able to estimate the SI symbol without error. This
problem is not solved by increasing the degree of Hammerstein’s nonlinear
model. In the following, we will consider this issue differently and try to
solve the problem as much as possible.

##  5 Proposed Method for Self-Interference Cancellation

###  5.1 New Matched Filter

The proposed method is simple but efficient. In this method, the MF
coefficients are changed and the new coefficients do not match the pulse-
shaping filter $g_{T}[k]$. That’s why we call it $g_{1}[k]$. The length of
this filter is similar to the length of $g_{R}[k]$ and equals $L_{g}$ symbols.
The coefficients of the new filter are determined so that for the interfering
input signal $y[k]$, the output of the filter is equal to the SI symbol
$s[n]$. Because the SI symbol $s[n]$ is known to the receiver, the baseband
part of the receiver easily removes the interfering symbol. In this method,
the Hammerstein model or its similar structure is not needed between the
baseband part of the transmitter and receiver. So, the dashed line block in
Fig. 1 can remove. In the new method, the MF acts like a fractionally spaced
equalizer (FSE) and tries to convert the transmitted symbols, which have
passed through the nonlinear PA and multipath channel, back into the same
transmitted symbol again ref:22 . If this is done correctly, the SI will be
canceled quickly. In ref:17 , without mentioning the reason, a structure
similar to FSE is used with a neural network for SI cancellation. However, the
effects and performance improvement of FSE have not been investigated.

In order to adjust the coefficients $g_{1}[k]$, the transmitter of node 2 does
not send a signal in a certain period of time, but, the transmitter of node 1
sends $N^{(\mathrm{p})}$ pilot symbols
${s^{({\rm{p}})}}[n],~{}n=0,1,...,{N^{({\rm{p}})}}-1$. These symbols pass the
transmitter RF chain, the multipath channel and the receiver RF chain and
receive by the MF as
${\eta^{({\rm{p}})}}[k]={y^{({\rm{p}})}}[k]+{w^{({\rm{p}})}}[k],~{}k=0,1,...,M{N^{({\rm{p}})}}-1$.
Like the equalizer, the MF tries to fix the SI symbols
${\eta^{({\rm{p}})}}[k],~{}k=0,1,...,M{N^{({\rm{p}})}}-1$ and regenerate
${s^{({\rm{p}})}}[n],~{}n=0,1,...,{N^{({\rm{p}})}}-1$. Because the MF has
downsampling, the relation between the input signal and the output symbol of
the MF will be as

${{\bf{s}}^{({\rm{p}})}}={{\bf{E}}^{({\rm{p}})}}\times{{\bf{g}}_{1}}.$ (15)

According to (15), the vector ${{\bf{s}}^{({\rm{p}})}}$ includes
${N^{({\rm{p}})}}$ pilot symbols, ${{\bf{g}}_{1}}$ is the proposed MF
coefficients and ${{\bf{E}}^{({\rm{p}})}}$ is given as

${{\bf{E}}^{({\rm{p}})}}=\left({\begin{array}[]{*{20}{c}}{{\eta^{({\rm{p}})}}[0]}&{{\eta^{({\rm{p}})}}[1]}&{...}&{{\eta^{({\rm{p}})}}[M{L_{g}}-1]}\\\
{{\eta^{({\rm{p}})}}[M]}&{{\eta^{({\rm{p}})}}[M+1]}&{...}&{{\eta^{({\rm{p}})}}[M+M{L_{g}}-1]}\\\
\vdots&\vdots&\ddots&\vdots\\\
{{\eta^{({\rm{p}})}}[({N^{({\rm{p}})}}-1)M]}&{{\eta^{({\rm{p}})}}[({N^{({\rm{p}})}}-1)M+1]}&{...}&{{\eta^{({\rm{p}})}}[({N^{({\rm{p}})}}-1)M+M{L_{g}}-1]}\end{array}}\right).$
(16)

It should note that the downsampling action of the MF is also considered in
relation (15). In this way, the coefficients of the proposed MF can be
estimated with LS as

${{\bf{g}}_{1}}={\left({{{\bf{E}}^{({\rm{p}}){\dagger}}}{{\bf{E}}^{({\rm{p}})}}}\right)^{-1}}{{\bf{E}}^{({\rm{p}}){\dagger}}}\times{{\bf{s}}^{({\rm{p}})}}.$
(17)

After this, node 1 informs node 2 and sends the new coefficients
$g_{1}[k],~{}k=0,1,...,ML_{g}-1$ for the second node and node 2 uses these
coefficients as a new pulse-shaping filter. More details are explained in the
Use Case subsection.

When two nodes start transmission, the first transmitter sends
$s[n],~{}n=0,1,...,N-1$ and the second transmitter sends
$u[n],~{}n=0,1,...,N-1$ symbols to the other party. In addition to receiving
the SoI sent by node 2, the first node also receives the SI symbols along with
noise. Therefore, the received symbols of the first node are
$\lambda[n]=r[n]+v[n]+\omega[n],~{}n=0,1,...,N-1$. In the new proposed method,
the estimation of received SI symbols is as
$r[n]=\hat{s}[n],~{}n=0,1,...,N-1$. Then $\lambda[n]$ enters the baseband
section and $s[n]$ is subtracted from it. If $r[n]$ and $s[n]$ are similar
with a good accuracy,$\epsilon[n]=r[n]-s[n]$ will have a small value and the
detection and demodulation of the SoI will be done well using the symbol
$\lambda[n]=v[n]+\left({\epsilon[n]+\omega[n]}\right)$.

###  5.2 Use Case

Figure 3: An FD base station with two HD users in uplink and downlink in TDD
mode.

Since in the new proposed method, the receiver’s MF is changed, so for the
correct functionality, the transmitter’s pulse-shaping filter must also be
changed. Therefore, one suitable structure for implementing this new technique
is a cellular communication system with an FD base station and two HD users.
This can be seen in Fig. 3 where the base station or node 1 transmits in the
downlink channel while receives from node 2 in the uplink channel
simultaneously. For simplicity, two users are HD and transmit or receive in
time division duplexing (TDD) mode. In the new proposed method, node 1 uses
the conventional pulse-shaping filter such as root-raised cosine with
coefficients $g_{T}[k]$ and transmits in the downlink channel for node 3. This
user is matched with the transmitter and uses $g_{T}[k]$ in the receiver’s MF.
However, this node is not important in our new proposed scenario. On the other
hand, node 1 is FD and receives severe SI from its transmitter. So, node 1
calculates the new MF coefficients $g_{1}[k]$ and uses this filter in its
receiver. As we explained, this new MF can reduce the interference more
efficiently. In addition, node 1 informs node 2 about the new coefficients.
Therefore, node 2 uses the new coefficients $g_{1}[k]$ in its transmitter’s
pulse-shaping filter. Since node 2 is in transmit mode, it does not require
the MF in this mode. Pulse-shaping and MF coefficients of all transmitters and
receivers are also depicted in Fig. 3. After a while, the roles of node 2 and
node 3 are changed and node 2 receives signal on the downlink channel and node
3 transmits in the uplink channel. Then, the same process will start and the
pulse-shaping and MF of all nodes are determined again.

###  5.3 Computational Complexity

The comparison of the computational complexity of the newly proposed method
with the Hammerstein method can check in two parts, during training and
estimation of model parameters, as well as after training and when executing
the algorithm. When training the model and estimating the parameters, the
computational complexity of the Hammerstein model and the proposed method are
almost the same, though the computational complexity of the Hammerstein may be
lower. However, the parameter setting is usually done once, and then several
tens or hundreds of data packets are sent. Therefore, the computational
complexity of the algorithms during the training is negligible compared to the
computational complexity during the algorithm’s execution. So, the
computational complexity of the two methods during the training has been
calculated and compared in the appendix. Here, we compare the complexity
during the regular system operation. The number of real multiplications
indicates the complexity of each method.

In the Hammerstein model, the MF at the receiver first performs a matrix-
vector multiplication. The matrix has $N\times ML_{g}$ rows and $N\times
ML_{g}$ columns and the vector has $N\times ML_{g}$ rows and its computational
complexity is $2N(ML_{g}+1)$. The coefficient of 2 is required; because the
input signal to the MF is complex, but the filter coefficients are real.
Therefore, a real complex multiplication is equivalent to two real
multiplications. Then, to remove the interference, the SI regenerator must
create a matrix of (9) and, after multiplying it by $\mathbf{q}$, regenerate
the SI symbols. According to the number of multiplications to build the matrix
$\mathbf{S}$, its computational complexity is $2N\frac{{P+1}}{2}{L_{q}}$.
Also, the complexity of ${\bf{S}}\times{\bf{q}}$ is
$4N\frac{{P+1}}{2}{L_{q}}$. Finally, the computational complexity of the
Hammerstein method is equal to

${{\cal
O}_{{\rm{Hammerstein}}}}=2N\left({M{L_{g}}+1}\right)+6N\frac{{P+1}}{2}{L_{q}}.$
(18)

But, with the new proposed MF, we only need to do a $N\times ML_{g}$ square
matrix multiplication to a $ML_{g}\times 1$ vector. Therefore, the
computational complexity of this new method is equal to

${{\cal O}_{\rm{1}}}=2N\left({M{L_{g}}+1}\right).$ (19)

##  6 Simulation Results

To simulate and numerically check the proposed method, the system parameters
are set according to Table 2. If any parameter changes, its new value is
mentioned in the appropriate place.

Table 2: Simulation parameters Parameter | Value
---|---
$N={N^{({\rm{p}})}}$ | 128
$M$ | 8
$L_{g}$ | 4
$L_{q}$ | 4
$P$ | 3

To better excite the nonlinear PA, symbols similar to OFDM are produced and
used as transmitted symbols. The peak to average power ratio (PAPR) is about
13dB. The pulse-shaping filter is a root-raised cosine filter with a roll-off
factor of 0.35. In the Hammerstein model, the MF is matched to the pulse-
shaping filter. But in the proposed method, the MF is calculated with
appropriate coefficients. For a fair comparison, $L_{g}$ for the proposed
method is adjusted for the equal computational complexity with the
Hammerstein. The communication channel is multipath and has a length of 4
symbols. Since our purpose here is to compare the amount of SI cancellation
and calculate residual SI power, therefore, the transmitted symbols of node 2
is only considered in Fig. 10. The signal to noise power ratio (SNR) also
shows the amount of SI power to noise power before the receiver MF. The
nonlinear PA is modeled as Rapp with a smoothness factor of 2 ref:10 . The
input back-off (IBO) is 5dB which means that the PA input power has retreated
5dB from 3dB saturation point. To make a better comparison, the number of
10,000 packets including ${N^{{\rm{(p)}}}}+N$ symbols passed through 10,000
independent channel instances, then the average residual power in the
Hammerstein method and the proposed method is compared. Each transmitted
packet contains ${N^{{\rm{(p)}}}}$ pilot symbols followed by $N$ data symbols.
When sending pilot symbols, the receiver estimates the parameters of the
desired model to eliminate interference. Then, when it receives the leakage
data symbols from its own transmitter, it removes the interference as much as
possible and compares the remaining SI power of different methods.

The performance of the Hammerstein method and the proposed method are compared
in Fig. 4. SNR value is 0dB. It is clear that in some iterations, the
performance of the Hammerstein method is better, and its residual interference
is less than the proposed method. However, it is often not the case, and the
performance of the proposed method is better than the Hammerstein. It is also
clear that, on average, the performance of the proposed method is much better
than the Hammerstein method, and the residual SI power variation (variance)
are also less in this method.

Figure 4: Residual SI power in different iterations in the Hammerstein and
proposed methods with new MF.

After removing the interference by the Hammerstein method and proposed
methods, the average residual SI power compares in terms of SNR in Fig. 5. We
can see that the performance of the proposed method is better than the
performance of the Hammerstein in all SNRs. Also, with the increased SNR,
naturally, the model parameters are estimated with less error, and the
interference is reduced with more accuracy. Therefore, the remaining
interference power in both graphs of Fig. 5 decreases with the increase of
SNR.

Figure 5: The average residual SI power in terms of SNR, in the Hammerstein
and proposed method with new MF.

For a better comparison, we assume the ratio of the residual SI power in the
Hammerstein method to the residual SI power in the proposed method as the
cancellation gain and plot it in Fig. 6. The curve is convex, and the benefit
of SI cancellation is high at first and decreases with the increase of SNR.
When the gain becomes lower, the performance of Hammerstein and new methods
approach each other. The lowest gain is obtained around SNR of 3dB, where the
gain reaches about 16dB. With a further increase of SNR, the gain increases
again, and the curve ascends. At high SNR, SI is strong. Therefore, the
weakness of the Hammerstein method, which we mentioned in section 4.2, shows
itself more, and the better performance of the presented method will be more
specific. But even at low SNR, the proposed method has a high benefit. The
reason for this good performance is the change of MF in the proposed way. In
fact, the filter is more compatible with the SI signal and can reduce the
noise better.

Figure 6: The SI cancellation gain, in the Hammerstein and proposed method
with new MF.

In Fig. 7, the average residual SI power at SNR value of 0dB is plotted versus
$L_{g}$. This parameter shows the length of the pulse-shaping filter and the
length of the MF in the Hammerstein model in terms of the number of symbols.
The performance of the Hammerstein method is almost not dependent on $L_{g}$.

As we explained, when the pulse-shaping filter and the MF consider, the
Hammerstein method is not a powerful method. So, the amount of interference
power removed by this method is reduced. On the other hand, the proposed
method performs better than the Hammerstein, and the residual SI power of this
method does not change much in the shared values between 1 to 8. But in larger
values, the new MF suffers from over-fitting, and the performance of this
method also decreases. It means that the too-long length MF memorizes the
pilot symbols instead of learning them. Therefore, it can not generalize, and
after the training phase, can not remove the interference properly. The
Hammerstein MF is the root-raised cosine and is the same as the pulse-shaping
filter. The coefficients of this filter move quickly to zero. So, the increase
of $L_{g}$ does not have much effect on the performance of this method.
Therefore, it should note that increasing the length of MF in the presented
way does not improve the performance.

Figure 7: The average residual SI power in terms of $L_{g}$, in the
Hammerstein and proposed method with new MF.

Another comparison between the average residual SI power is made in Fig. 8
versus $M$, in SNR of 0dB. As expected, the performance of the Hammerstein
method decreases with the increase of $M$; Because when $M$ is increased, the
effect of the pulse-shaping filter increases, and the weakness of the
Hammerstein method becomes more apparent. But because the presented new method
considers this effect of filters, it has almost the same performance in all
values of $M$. It should be noted that too much increasing of $M$ is not
appropriate, and as shown in Fig. 7, we may suffer from over-fitting. But the
more interesting point is the same performance of both methods in $M=1$. In
this case, there is practically no pulse-shaping filter and MF, and sending
and receiving are done in the form of consecutive symbols. Therefore, the
Hammerstein method does not suffer from the problem we examined in Section
4.2. Consequently, the performance of the Hammerstein and the proposed method
will be the same. In summary, this brief explanation shows and proves the main
idea of this article.

Figure 8: The average residual SI power in terms of $M$, in the Hammerstein
and proposed method with new MF.

Again, the average residual interference power is plotted versus IBO in Fig. 9
and compared for two methods. The larger the IBO, the small part of the input
signal enters the nonlinear region of the PA. Therefore, the nonlinear
distortion of the amplifier is less. As you can see, the proposed method has
almost constant performance in all IBO values. But, the dependency of the
Hammerstein method on the IBO and the amount of nonlinear distortion is much
higher. At high IBO values, the amplifier is nearly linear. But the
Hammerstein method still tries to regenerate interference symbols nonlinearly.
This issue causes the performance of this method to drop. In other words,
similar to the emphasis made in ref:10 , in the Hammerstein method,

The PA degree of nonlinearity should know, and the model’s parameters should
adjust accordingly. On the other hand, the presented new method is relatively
resistant to this issue. Therefore, in addition to the better performance, the
proposed method is also suggested when previous information about the
amplifier nonlinearity is insufficient.

Figure 9: The average residual SI power in terms of IBO, in the Hammerstein
and proposed method with new MF.

Finally, to evaluate the overall performance of the system, the bit error rate
(BER) curve is plotted in Fig. 10. According to Fig. 3, the simulated system
includes an FD base station and two HD users. In addition to receiving the
desired signal from node 2, node 1 also receives interference from its own
transmitter. The SoI and the SI power at the input of node 1 are assumed equal
to one. At node 1, the receiver first removes the interference using the
Hammerstein method and detects the desired signal. Once again, the receiver
removes the interference using the new MF and detects the SoI. The BER curves
for these two modes and BPSK modulation is plotted in Fig. 10 versus SNR.
Here, SNR shows the ratio of the desired signal power to the noise power at
the input of node 1 receiver. As it is known, with the increase of SNR, the
estimation of the Hammerstein model parameters and the new MF becomes better.
Therefore, more SI is removed and the BER is reduced. Meanwhile, the lower BER
in the proposed method indicates the better performance of this method
compared to the conventional Hasmmerstein method.

Figure 10: BER Comparison with the Hammerstein and proposed new MF.

##  7 Conclusion

The problem of digital SI cancellation in the FD communication system has been
studied here. The transmitter PA is nonlinear. Then, to eliminate the
interference, we need to estimate the interference by considering the
nonlinear effect of the amplifier. The Hammerstein model is an excellent way
to do this. However, we showed that in practical systems, when the pulse-
shaping filter is considered in the transmitter and the MF in the receiver,
the Hammerstein method can not estimate the SI symbols accurately. To solve
this weakness, we did not consider the regular MF, which matches the
transmitter pulse-shaping filter. Instead, we presented a new filter that acts
like an equalizer and tries to regenerate the interference symbol sent by the
transmitter from the distorted signal received from the communication channel.
The effect of the pulse-shaping filter considers in our new method. Therefore,
the weakness of the Hammerstein method can be solved using this new method.
Various simulations have been done to show this performance improvement. In
addition, there is no need for previous information about the amount of
nonlinearity of the PA.

Since the interference has nonlinear distortion, it will probably be more
efficient to use a filter which takes this nonlinearity into account.
Therefore, considering the optimal nonlinear MF will be regarded as our future
research study.

##  Appendix A Computational Complexity of the Hammerstein and Proposed Method
During the Training

The computational complexity of the two methods during the training calculate
here. In the Hammerstein method, first in the MR, there is a $N\times ML_{g}$
rows and $N\times ML_{g}$ columns matrix multiplication into a $ML_{g}\times
1$ vector. So, its computational complexity is $2N\left(ML_{g}+1\right)$.
Then, to construct the matrix ${{\bf{S}}^{({\rm{p}})}}$ in (9),
$2N\tilde{P}L_{g}$ real multiplication is required. $\tilde{P}$ is equal to
$\tilde{P}=\frac{P+1}{2}$. Matrix multiplication
${{\bf{S}}^{({\rm{p}})}}^{\dagger}{{\bf{S}}^{({\rm{p}})}}$ has
$4N\left(\tilde{P}L_{g}\right)^{2}$ real multiplications. The matrix inversion
${\left({{{\bf{S}}^{({\rm{p}}){\dagger}}}{{\bf{S}}^{({\rm{p}})}}}\right)^{-1}}$
requires $4{(\tilde{P}{L_{g}})^{3}}$ real multiplications in the simple
inversion method. In the same way, the matrix operations
${\left({{{\bf{S}}^{({\rm{p}}){\dagger}}}{{\bf{S}}^{({\rm{p}})}}}\right)^{-1}}{{\bf{S}}^{({\rm{p}}){\dagger}}}$
and
${\left({{{\bf{S}}^{({\rm{p}}){\dagger}}}{{\bf{S}}^{({\rm{p}})}}}\right)^{-1}}{{\bf{S}}^{({\rm{p}})}}\times{{\bf{\lambda}}^{({\rm{p}})}}$
require $4N{(\tilde{P}{L_{g}})^{2}}$ and $4N\tilde{P}{L_{g}}$ real
multiplications respectively. If we assume ${N^{({\rm{p}})}}=N$, the total
number of real multiplications and the computational complexity associated
with it in the Hammerstein method is equal to

${{\cal
O}_{{\rm{Hammerstein}}}}=2N\left({M{L_{g}}+1}\right)+2N\tilde{P}{L_{g}}\left({4\tilde{P}{L_{g}}+3}\right)+4{\left({\tilde{P}{L_{g}}}\right)^{3}}.$
(A-1)

In the proposed new method according to equation (17), matrix multiplication
by matrix, matrix inversion, inverse matrix multiplication by matrix and
finally matrix to vector multiplication have $4N{(M{L_{g}})^{2}}$,
$4{(M{L_{g}})^{3}}$, $4N{(M{L_{g}})^{2}}$ and $4N(M{L_{g}})$ real
multiplications, respectively. Therefore, the total number of real
multiplications and related computational complexity in the presented method
is

${{\cal
O}_{\rm{1}}}=4N\left({2{{\left({M{L_{g}}}\right)}^{2}}+M{L_{g}}}\right)+4{\left({M{L_{g}}}\right)^{3}}.$
(A-2)

If we assume that $M$, $L_{g}$ and $\tilde{P}$ have an equal order or almost
similar to $M$ and $N\gg M$, then the computational complexity of both methods
is of the same order of $NM^{2}$. However, the number of multiplications in
the Hammerstein method may be less than the proposed method.

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