Generalized Receiver Designs for OFDM Systems with Alamouti

Decoding in Fast Fading Channels

VYACHESLAV TUZLUKOV

Department of Technical Maintenance of Aviation and Radio Electronic Equipment

Belarusian State Aviation Academy

77, Uborevicha Str., 220096 Minsk

BELARUS

Abstract: - In the present paper, the generalized receiver designs for orthogonal frequency-division multiplex-

ing (OFDM) systems that exploit the Alamouti transmit diversity technique are addressed. In Alamouti space-

time coded OFDM systems, the simple Alamouti decoding at the generalized receiver relies on the assumption

that the channels do not change over an Alamouti codeword period (two consecutive OFDM symbol periods).

Unfortunately, when the channel is fast fading, the assumption is not met, resulting in severe performance deg-

radation. In the present paper, a sequential decision feedback sequence estimation (SDFSE) scheme based on

the generalized receiver with an adaptive threshold, a traditionally single-carrier equalization technique, is used

to mitigate the performance degradation. A new method to set the threshold value is proposed. For small signal

constellations like binary phase-shift keying (BPSK) and quadrature phase shift-keying (QPSK), the SDFSE

generalized receiver with the adaptive threshold requires much lower complexity than a previous minimum me-

an square error (MMSE) approach based on the generalized receiver at the cost of small performance degradati-

on. Furthermore, we show that the performance difference becomes smaller when the channel estimation error

is included. Adaptive effort sequence estimation (AESE) scheme based on the generalized receiver is also pro-

posed to further reduce the average complexity of the SDFSE generalized receiver scheme with the adaptive

threshold. The AESE generalized receiver scheme is based on the observation that a high Doppler frequency

does not necessarily mean significant instantaneous channel variations. Simulations demonstrate the efficacy of

the proposed SDFSE generalized receiver with the adaptive threshold and AESE generalized receiver schemes.

Key-Words: - Generalized receiver, equalization, orthogonal frequency-division multiplexing (OFDM), sequen-

ce estimation, transmit diversity.

Received: August 7, 2021. Revised: March 14, 2022. Accepted: April 17, 2022. Published: May 11, 2022.

1 Introduction

In recent years, transmit diversity techniques have

received attention because they increase transmissi-

on reliability over wireless fading channels without

penalty in bandwidth efficiency [1], [2]. Space-time

coding at the transmitter does not require channel

state information, thus no feedback from the receiv-

er to the transmitter is necessary [3]-[5]. One popu-

lar and practical transmit diversity technique is the

Alamouti scheme [6], in which the maximum-likeli-

hood decoding naturally decouples the signals trans-

mitted from different antennas. The simple Alamou-

ti decoding scheme works well when channels are

flat fading and time-invariant over the Alamouti co-

deword period.

Unfortunately, high data rate applications neces-

sitate data transmission over broadband frequency

selective channels, which cause severe intersymbol

interference. However, a frequency selective chann-

el can be divided into a set of parallel flat fading

channels by combining the Alamouti decoding tech-

nique with an orthogonal frequency-division multip-

lexing (OFDM) modulation method. In this Alamo-

uti coded OFDM system, the simple Alamouti deco-

ding at each subchannel requires that channels have

to be constant over two OFDM symbol periods.

When the quasistatic channel condition is met and

an appropriate cyclic prefix is used, the simple Ala-

mouti decoding works well.

The combination of Alamouti technique and

OFDM modulation, however, makes the degrading

time varying channel effects more severe. Since

OFDM systems have much longer symbol duration

than single-carrier systems, a channel that is quasi-

static for single-carrier systems may not be quasista-

tic for OFDM systems. Consequently, rapidly chan-

ging channels cause more severe performance degr-

adation in the Alamouti coded OFDM systems than

in Alamouti coded single-carrier systems. When the

channel is fast fading, a channel variation within an

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

OFDM symbol gives rise to the interchannel interfe-

rence or coupling between the symbols in different

codewords (the intercodeword coupling). In additi-

on, the channel variation between two consecutive

OFDM symbols causes coupling between symbols

in a codeword at each subchannel (the intracode-

word coupling). With two coupling effects lowering

the effective signal-to-noise ratio (SNR) at the recei-

ver, as will be shown in Section 4.3, the Alamouti

decoding performance degradation motivates the ne-

ed for a decoding scheme that improves the perfor-

mance at moderate complexity.

The effect of a fast fading channel on the bit-err-

or rate (BER) performance of OFDM systems was

analyzed in [10], however, no transmit diversity tec-

hnique was considered in [10]. Performance degrad-

ation due to fast fading channels in systems with tra-

nsmit diversity using the Alamouti code was consi-

dered in [11]. Since a single-carrier system was con-

sidered in [11], however, the interchannel interfere-

nce was not included as a performance degrading fa-

ctor. In [12], a new model of signal-to-interference

plus noise ratio (SINR) in a multiple-input multiple-

output (MIMO) OFDM system, including the imp-

act of time-varying channels was proposed. In the

present paper, we separate the impact of the time-

varying channel into the intercodeword and intraco-

deword couplings (both are defined in Section 4.2)

and use them to analyze the effect of channel varia-

tion within an OFDM symbol period.

Various decoding schemes for space-time coded

systems have been proposed. In [13], it was reported

that transmit diversity exploiting the Alamouti code

and its simple decoding can be used for a single-car-

rier system even when channels are not quasistatic.

This is possible due to in part to a relatively short

symbol period of single-carrier systems when com-

pared with OFDM systems. In [14] a simplified ma-

ximum likelihood (ML) decoder for a space-time

block-coded single-carrier system was proposed

when channels are time selective. A decoding mat-

rix was proposed in an effort to make the resultant

matrix (channel matrix multiplied by the decoding

matrix) diagonal, eliminating interantenna interfere-

nce. An adaptive frequency domain equalization

scheme was also proposed for single-carrier systems

in [15] to track channel variations within a transmis-

sion block. The previous approaches [13]-[15] con-

sider single-carrier systems, where the interchannel

interference is not applicable, thus these approaches

do not apply to OFDM systems.

Decoding schemes in OFDM systems with trans-

mit diversity were reported in [12], [16], and [17]. A

time-domain filtering approach was proposed in

[12] for MIMO OFDM systems in the fast fading

channels. We compare our proposed approaches ma-

inly with this previous approach. In [12], a time-va-

riant filter has been designed in the time domain so

that SINR including a channel variation effect is ma-

ximized. As will be shown in detail later, however,

the design process as well as the filtering process is

computationally expensive for some system parame-

ters. In [16], a space-frequency encoding/decoding

scheme for wideband OFDM system was proposed

to improve performance, concatenating space-time

block coding with trellis coded modulation (thereby

increasing complexity). However, a slow fading

channel is assumed in [16], which is not the case we

consider in the present paper. OFDM systems with

the interchannel interference and intersymbol inter-

ference were considered in [17] and a decision feed-

back equalization structured equalizer was designed.

However, the interchannel interference considered

in [17] is due to insufficient cyclic prefix length rat-

her than fast fading channels.

A differential space-time block coding scheme

was developed based on the Alamouti scheme in

[18], eliminating the need for channel estimation.

The differential scheme becomes relatively more ba-

ndwidth efficient, when compared with a coherent

scheme, in the fast fading channels because no train-

ing symbols is required to estimate the channels.

The differential scheme, however, assumes that the

channels do not change over two Alamouti code-

word periods, which is not true under the fast fading

channels under consideration. Therefore, the fast ti-

me variation of the channels is likely to have a more

severe impact on performance of differentially space

-time block coded systems than on coherent systems

We compare the performance of the proposed gene-

ralized receivers with the differentially coded syst-

em through simulations at a high Doppler frequency

and demonstrate the advantages of coherent approa-

ches in Section 6. Although a more thorough comp-

arison could be made considering the different Dop-

pler frequencies at the different channel state infor-

mation estimation schemes, we consider only the fi-

xed high Doppler frequency and one channel state

information scheme.

In the present paper, we analyze the intercode-

word and intracodeword coupling effects and demo-

nstrate their severity via simulation. Then, we use

sequence estimation schemes [19]-[25] that are tra-

ditionally single-carrier equalization techniques to

alleviate the performance degradation due to the co-

upling effects. Firstly, we formulate the maximum

likelihood sequence estimation (MLSE) scheme bas-

ed on the generalized receiver in the frequency dom-

ain. In [26], it was argued that small normalized Do-

ppler frequency, the product of the Doppler frequen-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

cy and symbol period, implies that the interchannel

interference is from only a few nearest subchannels.

In the MLSE formulation, we take an advantage of

an observation that even when the normalized Dop-

pler frequency is somewhat large, we need to comp-

ensate for the interchannel interference from only a

few nearest subchannels because of the channel esti-

mation error as well as less significance of the inter-

channel interference from far away subchannels. Se-

condly, a sequential decision feedback sequence es-

timation (SDFSE) scheme based on the generalized

receiver with the adaptive threshold is described as a

suboptimal scheme that reduces the high computati-

onal complexity of the maximum likelihood sequen-

ce estimation. The complexity of the SDFSE gene-

ralized receiver scheme is again reduced by using

the adaptive threshold. Twice the intercodeword co-

upling, which will be defined later, is used as a thre-

shold value. The SDFSE generalized receiver sche-

me with the adaptive threshold is composed of can-

didate selection step and sequence estimation step.

The applicability of the adaptive threshold idea is

based on the observation that the intercodeword co-

upling is much weaker than the intracodeword coup-

ling. The relatively small intercodeword coupling

keeps the number of candidates small to make effi-

cient the SDFSE generalized receiver scheme with

the adaptive threshold.

To further reduce the average complexity of the

SDFSE generalized receiver scheme with the adapti-

ve threshold, we propose an adaptive effort symbol

estimation (AESE) scheme based on the generalized

receiver. Basically, the simple Alamouti decoding

scheme is selected when the instantaneous channel

variation is negligible, and the SDFSE generalized

receiver scheme with the adaptive threshold is used

when the channel variation is significant. The deg-

ree of the channel variation is measured in terms of

the intracodeword coupling, which is defined later.

When the intracodeword coupling is larger than a

certain threshold, instantaneous channel parameter

variation is considered as significant, and vice versa.

The threshold value in the AESE generalized receiv-

er scheme is set from simulation experiments. The

AESE generalized receiver scheme is motivated by

observation that the high Doppler frequency does

not necessarily mean the instantaneous significant

channel variation. Therefore, even when the Dopp-

ler frequency is very high, the transmitted symbols

are estimated via the Alamouti decoding when the

instantaneous channel variation is negligible. The

Alamouti decoding, the SDFSE scheme with the ad-

aptive threshold, AESE scheme, and the time-doma-

in MMSE approach (all of them are constructed bas-

ed on the generalized receiver) are compared in

terms of complexity and performance via simulation

Since each signal estimation scheme may react diff-

erently to channel estimation error, we consider both

cases with and without the ideal channel state infor-

mation. We use the channel estimation technique in-

volving pilot tone and interpolation in [12] to estim-

ate the channel state information. All proposed sche-

mes, namely, SDFSE, AESE, MMSE, MLSE, discu-

ssed in the present paper are based on the generaliz-

ed approach to signal processing in noise (see Secti-

on 3) [27]-[29].

The remainder of this paper is organized as foll-

ows. In Section 2, the Alamouti coded OFDM syst-

em with two transmit antennas and one receive ante-

nna is described. In Section 3, the main functioning

principles of the generalized receiver constructed

based on the generalized approach to signal process-

ing in noise are discussed. In Section 4, the Alamou-

ti decoding scheme is investigated under both quasi-

static and fast fading channel environments. The re-

lative significance of the two coupling effects and

consequent performance degradation are both analy-

zed and demonstrated via simulation. In Section 5,

symbol estimation schemes in the fast fading chann-

els are described. In Section 6, computer simulation

experiments are conducted to compare the perform-

ance of the schemes characterized by different levels

of complexity. Conclusions are presented in Section

In the present paper, a boldface letter denotes the

vector or matrix, as will be clear from the context;

denotes the

MM 

identity matrix;



)(

denotes

the complex conjugate;

)(

denotes the transpose;

)(

denotes the Hermitian transpose;

||

denotes the

absolute value;

|||| 

denotes the

norm of matrix or

vector; in general, a lowercase letter stands for the

time-domain signal while an uppercase letter deno-

tes frequency domain signal. If

),,,( zba 



is a

sequence,



\ a

),,,( zcb 

. The notation

mm |][(X

)



, where

is the set, denotes a sequence whose

elements indexes are increasingly ordered.

2 System Model

In the present paper, we consider an OFDM system

with transmit diversity as illustrated in Fig.1. The

bandwidth

TB 1

,where

is the sampling interv-

al, is divided into N equally spaced subcarriers at

frequencies

1,,1,0,  Nkfk 

with

NBf 

. At

the transmitter, information bits are grouped and

mapped into complex symbols. In the present paper,

quadrature phase shift-keying (QPSK) with constel-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

lation

QPSK

is assumed for the symbol mapping. Ac-

cording to the Alamouti code,

]}[ ][{ 21 kXkX

are

transmitted by two antennas simultaneously during

the first symbol period

)1( l

for each

k

. During

the second symbol period

]}[ ][{),2( 12 kXkXl 



are transmitted by two antennas for each

k

. The

set

}1)2(,),2{(  cc NNNN 

(1)

is the set of data carrying subcarrier indexes, and

is the number of subcarriers carrying data; N is the

fast Fourier transform (FFT) size; consequently, the

number of virtual carriers is

NN 

.We assume half

Fig.1. OFDM system with transmit diversity: 1- QPSK modulator; 2 – STBC encoder;

3,4 – inverse fast Fourier transform; 5 – fast Fourier transform; 6 – channel estimation;

7 – space-time decoder; 8 – QPSK demodulator

of the virtual carriers are on both ends of the spectr-

al band. The inverse FFT (IFFT) converts each

1N

complex vector into a time-domain signal and the

copy of the last D samples is appended as a cyclic

prefix. Thus, the length of an OFDM symbol is

TD)

. The time-domain signals transmitted by the

antenna i during the lth symbol period

 nnx li 0],[

}2,1{ },2,1{ ,1  liDN

are expressed as







klili NDnkjkSnx }/)(2exp{][][ ,,



, (2)

where

][

,kS li

denotes the complex symbol transmit-

ted by the i-th antenna during the l-th symbol period

in the Alamouti codeword over the k-th subchannel.

The index for the Alamouti codeword is omitted to

keep the notation simple.

The signals from the two transmitting antennas

go through independent channels. The wireless cha-

nnel can be described as L resolved multipath com-

ponents

}1,,1,0{  Lp 

, each characterized by the

amplitude

],[

,pnh li

and delay

, where

],[

,pnh li

stands for the p-th resolved multipath component

amplitude between the i-th transmit antenna and the

receive antenna at the time n (the sample index) du-

ring the l-th symbol period. The maximum delay

spread of the two channels is assumed to be the sa-

me and equals to

TL )1( 

The received signals during the Alamouti code-

word period take the following form

}2,1{ , ][][],[][ 2

0,,  







lnwpnxpnhny l

plilil

(3)

where

][nwl

is the circularly symmetric zero-mean

white complex Gaussian random process. It can be

observed that the received signals are the superposi-

tion of signals generated by two transmitting anten-

nas. If the cyclic prefix length D is longer than

1L

the received signals given by (3) after removing the

prefix can be considered as the circular convolution

result of the transmitted signal given by (2) and the

channel. Consequently, the demodulated signals in

the frequency domain via the FFT are expressed as

}2,1{ , ][],[][][ 2

1,,  

 

lkWmkamSkY l

i m lilil

(4)

where







 1

0,,, }/2exp{][],[ L

pplili NmpjmkHmka



; (5)







 1

0,,, }/)(2exp{],[

][ N

nlipli Nnmkjpnh

mkH



(6)

The notation

][

,, mkH pli 

represents the FFT of

the p-th multipath component between the i-th trans-

mitting antenna and the receive antenna during the l-

th symbol period. Note that

kmmka li ],,[

denotes

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

the interchannel interference from the m-th subcha-

nnel to the k-th subchannel for each transmit anten-

na index

}2,1{i

and symbol period

}2,1{l

. An ad-

ditional interpretation of

][

,, mkH pli 

and

],[

,mka li

provided in [26].

3 Generalized Receiver: Main

Functioning Principles

The generalized receiver is constructed in accordan-

ce with the generalized approach to signal process-

ing in noise [27]-[29]. The generalized approach to

signal processing in noise introduces an additional

noise source that does not carry any information ab-

out the parameters of desired transmitted signal with

the purpose to improve the signal processing system

performance. This additional noise can be consider-

ed as the reference noise without any information

about the parameters of the signal to be detected.

The jointly sufficient statistics of the mean and

variance of the likelihood function is obtained under

the generalized approach to signal processing in noi-

se employment, while the classical and modern sig-

nal processing theories can deliver only the suffici-

ent statistics of the mean or variance of the likeliho-

od function. Thus, the generalized approach to sig-

nal processing in noise implementation allows us to

obtain more information about the parameters of the

desired transmitted signal incoming at the generaliz-

ed receiver input. Owing to this fact, the detectors

constructed based on the generalized approach to si-

gnal processing in noise technology are able to imp-

rove the signal detection performance of signal pro-

cessing systems in comparison with employment of

other conventional detectors.

The generalized receiver (GR) consists of three

channels (see Fig. 2): the GR correlation detector

channel (GR CD) – the preliminary filter (PF), the

multipliers 1 and 2, the model signal generator

(MSG); the GR energy detector channel (GR ED) –

the PF, the additional filter (AF), the multipliers 3

and 4, the summator 1; and the GR compensation

channel (GR CC) – the summators 2 and 3, the acc-

umulator 1. The threshold apparatus (THRA) device

defines the GR threshold.

As we can see from Fig.2, there are two bandpass

filters, i.e., the linear systems, at the GR input, nam-

ely, the PF and AF. We assume for simplicity that

these two filters or linear systems have the same am-

plitude-frequency characteristics or impulse respon-

ses. The AF central frequency is detuned relative to

the PF central frequency.

There is a need to note the PF bandwidth is mat-

ched with the transmitted signal bandwidth. If the

detuning value between the PF and AF central freq-

uencies is more than 4 or 5 times the transmitted si-

gnal bandwidth to be detected, i.e.,

f54

, where

f

is the transmitted signal bandwidth, we can beli-

eve that the processes at the PF and AF outputs are

uncorrelated because the coefficient of correlation

between them is negligible (not more than 0.05).

This fact was confirmed experimentally in [30] and

[31] independently. Thus, the transmitted signal plus

noise can be appeared at the GR PF output and the

noise only is appeared at the GR AF output. The sto-

chastic processes at the GR AF and GR PF outputs

present the input stochastic samples from two inde-

pendent frequency-time regions. If the discrete-time

noise

][kwi

at the GR PF and GR AF inputs is Gaus-

sian, the discrete-time noise

][k



at the GR PF out-

put is Gaussian too, and the reference discrete-time

noise

][k



at the GR AF output is Gaussian owing

to the fact that the GR PF and GR AF are the linear

systems and we believe that these linear systems do

not change the statistical parameters of the input

process. Thus, the GR AF can be considered as a ge-

nerator of the reference noise with a priori informa-

tion a “no” transmitted signal (the reference noise

sample) [28, Chapter 5]. The noise at the GR PF

and GR AF outputs can be presented as























miAFi

miPFi

mkwmgk

, ][][][

; ][][][





(7)

where

][mgPF

and

][mgAF

are the impulse responses

of the GR PF and GR AF, respectively.

In a general, under practical implementation of

any detector in wireless communication system with

sensor array, the bandwidth of the spectrum to be

sensed is defined. Thus, the GR AF bandwidth and

central frequency can be assigned, too (this band-

width cannot be used by the transmitted signal beca-

use it is out of its spectrum). The case when there

are interfering signals within the GR AF bandwidth,

the action of this interference on the GR detection

performance, and the case of non-ideal condition

when the noise at the GR PF and GR AF outputs is

not the same by statistical parameters are discussed

in [32] and [33].

Under the hypothesis

(“a yes” transmitted sig-

nal), the GR CD generates the signal component

][][ ksks i

caused by interaction between the model

signal

][ksm

, forming at the MSG output, and the in-

coming signal

][ksi

, and the noise component

][ksm

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

][k





caused by interaction between the model sig-

nal

][ksm

and the noise

][k



at the PF output. GR

ED generates the transmitted signal energy

][

2ksi

and

the random component

][][ kks ii



caused by interac-

tion between the transmitted signal

][ksi

and the noi-

][k



at the PF output. The main purpose of the

GR CC is to cancel completely in the statistical sen-

se the GR CD noise component

][][ kks i



and the

GR ED random component

][][ kks ii



based on the

same nature of the noise

][k



. The relation between

the transmitted signal to be detected

][ksi

and the

model signal

][ksm

is defined as:

, ][ ][ ksks i





(8)

where



is the coefficient of proportionality.

Fig. 2. Generalized receiver.

The main functioning condition under the GR

employment in any signal processing system includ-

ing the communication one with radar sensors is the

equality between the parameters of the model signal

][ksm

and the incoming signal

][ksi

, for example, by

amplitude. Under this condition it is possible to can-

cel completely in the statistical sense the noise com-

ponent

][][ kks i



of the GR CD and the random co-

mponent

][][ kks ii



of the GR ED. Satisfying the GR

main functioning condition given by (8),

][ksm

][ksi

1



, we are able to detect the transmitted si-

gnal with the high probability of detection at the low

SNR and define the transmitted signal parameters

with the required high accuracy.

Practical realization of the condition (8) at





requires increasing in the complexity of GR structu-

re and, consequently, leads us to increasing in com-

putation cost. For example, there is a need to emp-

loy the amplitude tracking system or to use the off-

line data samples processing. Under the hypothesis

(“a no” transmitted signal), satisfying the main

GR functioning condition (8) at

1



we obtain on-

ly the background noise

][][ 22 kk ii





at the GR out-

put.

Under practical implementation, the real structu-

re of GR depends on specificity of signal processing

systems and their applications, for example, the rad-

ar sensor systems, adaptive wireless communication

systems, cognitive radio systems, satellite communi-

cation systems, mobile communication systems and

so on. In the present paper, the GR circuitry (Fig.2)

is demonstrated with the purpose to explain the ma-

in functioning principles. Because of this, the GR

flowchart presented in the paper should be consider-

ed under this viewpoint. Satisfying the GR main fu-

nctioning condition (8) at

1



, the ideal case, for

the wireless communication systems with radar sen-

sor applications we are able to detect the transmitted

signal with very high probability of detection and

define accurately its parameters.

In the present paper, we discuss the GR implem-

entation in the broadband space-time spreading MC

DS-CDMA wireless communication system. Since

the presented GR test statistics is defined by the sig-

nal energy and noise power, the equality between

the parameters of the model signal

][ksm

and trans-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

mitted signal to be detected

][ksi

, in particular by

amplitude, is required that leads us to high circuitry

complexity in practice.

For example, there is a need to employ the ampli-

tude tracking system or off-line data sample proces-

sing. Detailed discussion about the main GR functi-

oning principles if there is no a priori information

and there is an uncertainty about the parameters of

transmitted signal, i.e., the transmitted signal para-

meters are random, can be found in [27], [28, Chap-

ter 6, pp.611–621 and Chapter 7, pp. 631–695].

The complete matching between the model signal

][ksm

and the incoming signal

][ksi

, for example by

amplitude, is a very hard problem in practice becau-

se the incoming signal

][ksi

depends on both the fad-

ing and the transmitted signal parameters and it is

impractical to estimate the fading gain at the low

SNR. This matching is possible in the ideal case on-

ly. The GD detection performance will be deteriora-

ted under mismatching in parameters between the

model signal

][ksm

and the transmitted signal

][ksi

and the impact of this problem is discussed in [34]-

[37], where a complete analysis about the violation

of the main GR functioning requirements is presen-

ted. The GR decision statistics requires an estimati-

on of the noise variance





using the reference noi-

][k



at the AF output.

Under the hypothesis

, the signal at the GR PF

output, see Fig. 2, can be defined as

][][][ kkskx iii





, (9)

where

][k



is the noise at the PF output and

][][][ kskhks ii 

, (10)

where

][khi

are the channel coefficients. Under the

hypothesis

and for all i and k, the process

][kxi

][k



at the PF output is subjected to the complex

Gaussian distribution law and can be considered as

the i.i.d. process.

In the ideal case, we can think that the signal at

the GR AF output is the reference noise

][k



with

the same statistical parameters as the noise

][k



. In

practice, there is a difference between the statistical

parameters of the noise

][k



and

][k



. How this di-

fference impacts on the GR detection performance is

discussed in detail in [28, Chapter 7, pp. 631-695]

and in [34]-[37],

The decision statistics at the GR output present-

ed in [30] and [31, Chapter 3] is extended for the ca-

se of antenna array when an adoption of multiple an-

tennas and antenna arrays is effective to mitigate the

negative attenuation and fading effects. The GR de-

cision statistics can be presented in the following

form:





 

1

0 1

][][2)( N

iiGR kskxT X

][][

0 1

2GR

iiTHRkkx 



 







 





, (11)

where

 

)1(),...,0(  NxxX

(12)

is the vector of the random process at the GR PF

output and

THR

is the GR detection threshold.

Under the hypotheses

and

when the amplitu-

de of the transmitted signal is equal to the amplitude

of the model signal,

][][ ksks i

i

1



, the GR de-

cision statistics

)(X

takes the following form in

the statistical sense, respectively:



















 



 

0 1

]}[][{)(:

]}[][][{)(:

iiGD

kkT

kkksT



. (13)

In (13) the term

iiEks 

 



 

0 1 2][

corresponds to

the average transmitted signal energy, and the term

   

 



  1

0 1 2

0 1 2][][ N

iikk



is the background

noise at the GR output. The GR output background

noise is the difference between the noise power at

the GR PF and GR AF outputs. Practical implemen-

tation of the GR decision statistics requires an esti-

mation of the noise variance





using the reference

noise

][k



at the AF output.

4 Alamouti Decoding Scheme

In this section, the Alamouti decoding scheme is bri-

efly reviewed under assumption of quasistatic chan-

nel. Then, the performance degradation of the sche-

me in a fast fading channel is both analyzed and de-

monstrated via computer simulations.

4.1 Slow fading channel

If the channel is slow fading, the interchannel inter-

ference terms are not significant as described in [8]

and

kmmka li  0],[

. (14)

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022

As a result, the received signal given by (4) is expre-

ssed as a set of simultaneous equations

 kkkkkk ],[][],[][ WXAY

(15)

where

















 mk

mkamka

mk ,for

],[],[

],[ 2,12,2

1,21,1

;

(16)

 













.][][][

;][][][

kWkWk

kXkXk

kYkYk

(17)

Via the assumption that the channels do not cha-

nge over an Alamouti codeword period, i.e.,













],[],[],[

22,21,2

12,11,1

kkkakka



(18)

space-time decoding is performed by multiplying

both sides of (15) with

],[ kk

to estimate the tra-

nsmitted symbols

. ][],[][)|][||][(|][

1kkkkkkk HWAXX 



(19)

Note that two symbols in

][

~kX

are decoupled from

each other. The final decisions are made independe-

ntly

}2,1{ , ||][][][

||minarg][

ˆ ikXkkXkX iCXi QPSK



(20)

with

1|][||][|][ kkk





. (21)

4.2 Fast fading channel

When the channel is fast fading, however, approxi-

mation (14) is not more valid and the received sig-

nal (4) is split into

equations as follows

 





kkmmkkkkk

],[][],[][],[][ WXAXAY

(22)

If we define















][][

][][

][



, (23)

where

],[][ and ],[][ 1,221,11 kkakkkak 



, (24)

then

][][][

~IAA kkk





, (25)

where

is the unit matrix with the size

22

The following equation can be derived from (22)

using the above identity

][][][][][

~kkkkk INTRA XCXY 



 





kkmmk

mINTER , ][

][],[ WXC

(26)

where

















].[

],[][

];[][

][

];,[][

],[

];[][

][

];[][

][

kkkk

kkk

mkkmk

kkk

INTER

INTRA

AAA

WAW

AAC

YAY

(27)

It can be observed that the second and third terms

on the right side of (26) show the effect of the time-

variant channel. The second term shows the coupl-

ing effect between symbols in the codeword (the in-

tracodeword coupling) and the third term shows the

coupling effect between symbols in the different co-

dewords (the intercodeword coupling) or the inter-

channel interference. These two coupling effects

create the interference that is lower than the effecti-

ve SNR at each subchannel, thereby degrading the

performance [38].

To show the relative significance of the two cou-

pling effects, the following two statistics and the co-

upling function are defined as







kINTRA

INTRA kE

kC ; ]||][[||

][ 2

(28)







kINTER

INTER kkkE

kC ; ]||],[[||

][ 2

00 C

(29)











otherwise, ],,[

0 if ],[

],[

0kkk

INTER

INTRA

(30)

where

][E

is the mathematical expectation. From

(28)-(30), the following statistic is obtained







kkE

k]||],[[||

][ 2

00 ΨΨ

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

Volume 21, 2022











. otherwise ],[

0 if ],[

kkC

INTER

INTRA

(31)

The statistic

]0[Ψ

shows the average intracodeword

coupling degree. When

][,0 00 kk Ψ

is the average

intercodeword coupling amount from a subchannel

that is

times the subcarrier spacing away from an

observed subchannel.

4.3 Numerical examples

In this section, the two coupling effects and perfor-

mance degradation due to the coupling effects are

demonstrated via simulation. A two transmitting an-

tenna and one receive antenna OFDM system is si-

mulated. Exact channel estimation at the receiver is

assumed. The bandwidth is

400B

kHz, the FFT si-

128N

, the cyclic prefix length

32D

, and the

number of data carrying subchannels

120

; con-

sequently, the number of virtual carriers is

 c

, and the OFDM symbol period

400)(  s

TDN

μs. Four subchannels on both ends of the spectrum

are not used for data transmission. Each subcarrier is

modulated by QPSK symbols. The performance cri-

terion is the BER versus SNR at the receiver input.

The total signal power from two transmitting anten-

nas is used for the calculation of the SNR. The mobi-

le channel used for simulation is a two-path channel

with equal power and delays of zero and

, respe-

ctively, with each path experiencing independent

Rayleigh fading. Jakes’ model was used for the Ray-

leigh fading channel simulation [39]. Doppler frequ-

ency considered is 297 Hz, which results in more se-

vere channel variation than the scenario in [12]. For

the statistic (31) and the BER measurement, 1000

OFDM symbols (500 Alamouti codewords) are tran-

smitted and estimated.

Figure 3 demonstrates an empirical

],0[][ 0ΨΨ k

}10,,1,0{

0 k

when

297

Hz. The simulati-

on result suggests that the intracodeword coupling is

much stronger than the intercodeword coupling. Fig-

ure 4 displays the BER as a function of SNR at the

receiver input for Doppler frequencies of 50 and 297

Hz and demonstrates the performance degradation

in the fast fading channels when the standard Alam-

outi decoding scheme is used. As the Doppler frequ-

ency increases from 50 to 297 Hz, the error perfor-

mance is degraded significantly, especially at the

high SNR, i.e., SNR >15 dB. Given the relative sig-

nificance of the two coupling effects in Fig. 3, it can

be said that the performance degradation is mainly

due to the intracodeword coupling effect rather then

the intercodeword coupling effect. The performance

degradation motivates a novel symbol estimation

scheme, which compensates for the coupling effects

at a moderate complexity. In the next section, symb-

ol estimation schemes are described that improve

the performance under the fast fading channel envir-

onment.

Fig. 3. Relative significance of the couplings caused by

the time-variant channel

)]0[][lg(10 0ΨΨ k

versus

297

Fig. 4. BER performance versus SNR of the Alamouti

decoding scheme: 1-

297

Hz; 2 -

50

Hz.

5 Symbol Estimation under Fast

Fading Channel Environment

In this section, symbol estimation schemes are des-

cribed in the presence of the coupling discussed in

Section 4.2. In Section 5.1, the MLSE generalized

receiver approach is formulated for the system under

consideration. In Section 5.2, the SDFSE generaliz-

ed receiver scheme with the adaptive threshold is

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

100

Volume 21, 2022

described as the suboptimal scheme reducing the co-

mplexity of the MLSE generalized receiver appro-

ach. Section 5.3 describes the AESE generalized re-

ceiver scheme. Section 5.4 considers the required

computational complexity, especially compared

with the MMSE generalized receiver approach.

5.1 MLSE formulation

In this section, the MLSE generalized receiver sche-

me is formulated. In the present paper, the Alamouti

coded OFDM system is considered, while a trellis-

based space-time code was considered in [7] and

[9]. When the channel is the fast fading, the two co-

upling effects described in Section 4.2 need to be

compensated. It was argued that if the normalized

Doppler frequency is small, we can assume that in-

terchannel interference is from only a few nearest

subchannels [26]. We argue that even when the nor-

malized Doppler frequency is pretty large, we have

only to consider the interchannel interference from a

few nearest subchannels due to channel estimation

error as well as less significance of interchannel in-

terference from far away subchannels. By this assu-

mption, the received signal (22) is simplified into

][],[][ kkkk XAY 

],[][],[ ]12)(,min[

],2)(,max[

kmmk

NNqk

kmNNqkm

WXA  



(32)

where 2q is the number of subchannels considered

as the causing intercodeword couplings.

It can be observed in (32) that the received signal

is composed of the attenuated desired signal, inter-

channel interference from 2q other subchannel, and

the additive noise.

Let

})2,1{,

|}2,1{,(  ilP iil XXY

(33)

is the conditional probability that

}2,1{, l

are re-

ceived under assumption that the

}2,1{,

ˆi

are tra-

nsmitted. In the MLSE generalized receiver scheme,

we estimate the transmitted sequence to be the sequ-

ence that maximizes the likelihood in (33). Since

][kW

in (32) is the white complex Gaussian random

process, we can show that the MLSE generalized re-

ceiver scheme amounts to computing [19]







kMLSE

mm kmm ][minarg)|][

()|][

(



, (34)

where







]12)(,min[

],2)(,max[

][],[)(][ c

NNqk

kmNNqkm

MLSE mmkkk XAY



. (35)

After a simple modification of the coupling func-

tion (30), the following function is defined

















else. 0,

},2,1,{ if ],,[

0 if ],,[][

],[ 00

002

0qkkk

kkkk

imp Ψ

ΨI



(36)

If the above function is used, an equivalent metric to

(35) can be written as

.][],[)(

][ ]12)(,min[

],2)(,max[ 



 c

NNqk

kmNNqkm impMLSE mkmkkk XΨY



(37)

The function

],[ mk

imp

can be considered as the sort

of non-causal time-variant impulse response at time

k with the channel memory of

12 q

, while

mm |][X



is the symbol sequence we need to detect. The

state at k is defined as

),2)(,{max(|][( ck NNqkmm  X



)})1)2)(,min(  c

NNqk

. (38)

Dynamic programming based on the principle of

optimality, such as the Viterbi algorithm, can simp-

lify the minimization problem (34). The minimizati-

on problem under consideration, however, requires

)2(2 q

MLSE generalized receiver states, where Q is

the constellation size. Even for small Q, the minimi-

zation problem might be computationally prohibiti-

ve. In addition, merging is the random phenomenon

and it is possible that no decision is made until the

end of the entire sequence. Given that the length of

the sequence

mm |][X

, this may result in

codeword period delay [19].

The next section describes the SDFSE generaliz-

ed receiver scheme with the adaptive threshold,

which mitigates both the delay and the very high co-

mplexity problem of the MLSE generalized receiver

scheme.

5.2 SDFSE scheme with adaptive threshold

Among suboptimal but computationally feasible se-

quence estimation techniques is sequential sequence

estimation. Sequential sequence estimation, which

relieves the delay problem of the MLSE generalized

receiver scheme, can also be combined with decisi-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

101

Volume 21, 2022

on feedback scheme to further reduce complexity

[20]-[23]. By assuming that we have correctly reco-

vered the sequence

})1,),(5.0,{max(|][

(

kNNqkmm ck X



(39)

by the time we try to recover

][

ˆkX

an SDFSE sche-

me can be formulated as: for

 NNNk c(:2)(

12) 

















discarded is ][

and adopted is ][

,][minarg

SDFSEk







, (40)

where

)})1)2/)(,min(,,{  c

NNqkk 

; (41)



mCCm QPSKQPSKk |][(X



)})1)2/)(,min(,,{  c

NNqkk 

; (41)







]12)(,min[ ][],[)(][ c

NNqk

SDFSE mmkkk XAY





; (42)







 1

]2)(,max[

][],[][)( k

NNqkm c

mmkkk XAYY



. (43)

Now the required number of metric calculations is

)1(2 q

for the estimation of symbols in a codeword.

Further complexity reduction via the adaptive

threshold. Now further complexity reduction is acc-

omplished by using the adaptive threshold. The idea

of thresholding that is referred as the T-algorithm,

was introduced to reduce the decoding complexity

of the convolutional codes in [22], but no formula

was proposed for selecting the threshold value. A si-

milar idea was used in [25], in which a posteriori

probabilities associated with the one-step previous

states are calculated and the state is removed when

the corresponding a posteriori probability is less

than a threshold.

Another method to set a threshold value was pro-

posed in [24]. In [24], the threshold value is set so

that the removal probability of the correct state is

less than the target error probability. In this scheme,

the instantaneous SNR is necessary to calculate the

instantaneous threshold value though. The maximal

possible threshold value can be used to avoid the in-

stantaneous threshold value calculation, which, in

turn, decreases the efficiency of the threshold idea.

On the other hand, in the present paper, the thresh-

old value is decided based on the time-variation de-

gree of the channel without requiring the instantane-

ous SNR or the a posteriori probability [40].

Under the assumption that a sequence

k



was re-

covered correctly, a simple metric is defined

][],[)(][ kkkkk

simple XAY  



. (44)

A comparison between (42) and (44) shows that

the metric (42) involves a sequence

k



, while the

simple metric (44) considers only

][kX

. In other

words, the metric (44) considers only the intracode-

word coupling effects, while the metric (42) takes

into account both coupling effects. From (42) and

(44), the difference between the two metrics is bou-

nded as

QPSKQPSK CCk  ][X

and

k



with the

][kX

we obtain



k

kk SDFSE

simple





][][ ][X







]12)(,min[

][

],[

NNqk

mmk XA

][],[2 ]12)(,min[

kBmk

NNqk

 



. (45)

Note that the norm

QPSKQPSK CCk  ][,2 XX

by assuming a constellation of unit amplitude symb-

ols. We can observe that the bound

][kB

is a functi-

on of the intercodeword coupling.

Let

][k

subopt

be the minimizer of the metric (44),

then

][][ ][][ k

simple

simple kk subopt XX





])[][](,[ kkkk subopt XXA 

][][][],[ kCkkkk subopt  XXA

. (46)

The bound

][kC

is the function of the intracodeword

coupling effect.

The following relationship between the above

two bounds can be induced from Fig. 3

]}.[{2}][{

]}[{ kBEk,kEkCE 



A

(47)

The above inequality again implies

QPSKQPSKk CCk  ][{XS

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

102

Volume 21, 2022

}][][][ ][][ kCkk k

simple

simple subopt  XX



, (48)

where

QPSKQPSK CCk  ][{X

}][2][][ ][][ kBkk k

simple

simple subopt  XX



. (49)

Meanwhile, if we let

][k

opt

be the minimizer of the

metric (42), the relationship between the two mini-

mizers is derived from (33)

}][2][][ ][][ kBkk k

simple

simple optsubopt  XX



, (50)

which implies that

S][k

opt

A small set of probable minimizers of the metric

(42) can be chosen via the simple metric (44). This

is where the idea of the adaptive threshold method is

used. The smaller the bound

][kB

compared to

][kC

the smaller

becomes, where

is defined as the

number of elements in

. The set

can be regard-

ed as the candidate set of

][kX

. After the selection

of the candidates of

][kX

, which requires

metric

calculations of

],[Qk

simple 



metric calculations

are necessary to find the minimizer of

][k

SDFSE



. In

other words, the intracodeword coupling effects are

first considered for the estimation of the transmitted

symbols and the intercodeword coupling effects are

considered only when there are more than one con-

tender. Note that when the large q needs to be consi-

dered, the idea of the threshold can be further explo-

ited by defining another simple metric including a

few the intercodeword coupling effects. We also no-

te that the adaptive threshold decreases not only the

number of metric calculations but also the complexi-

ty of each metric calculation via (44).

5.3 AESE scheme

To further reduce the average complexity of the

SDFSE generalized scheme with the adaptive thre-

shold, the AESE generalized receiver scheme is pro-

posed in this section. The basic idea is that when the

instantaneous channel variation is small,

][k

INTRA

for each

k

is smaller than the threshold

AESE

the simple Alamouti decoding generalized receiver

scheme is used. On the other hand, the SDFSE gene-

ralized receiver scheme with the adaptive threshold

is used to mitigate the performance degradation

when the instantaneous channel variation is large.

Consequently,

][k

INTRA

is larger than the thre-

shold. The block diagram of the proposed adaptive

effort with the generalized receiver is presented in

Fig. 5. This scheme is based on the observation that

the high Doppler frequency implies the statistical

fast fading channel but it does not necessarily mean

significant instantaneous channel variation. To qua-

ntify the effectiveness of the proposed AESE gene-

ralized receiver scheme, we define the following

probability that an instantaneous channel variation is

significant and, consequently, an Alamouti code-

word is estimated via the SDFSE generalized receiv-

er scheme with the adaptive threshold [41]

)],[( AESEINTRAAESE TkkPP  C

. (51)

The trade off between complexity and performa-

nce can be made via

AESE

in the AESE generalized

receiver scheme. Larger

AESE

means that more sym-

bols are estimated via the simple Alamouti decod-

ing generalized receiver. Thus, a complexity can be

reduced by using larger

AESE

but the performance

will be degraded at the same time.

5.4 Computational complexity

In this section, the computational complexity of va-

rious schemes is compared. The required number of

metric calculations for the proposed schemes is su-

mmarized in Table I. Since we are considering

1q

the number of metric calculations required in

SDFSE generalized receiver with the adaptive thre-

shold is

)1( Q

S

, where







)(

ikk iiP SS

. (52)

In the process of decoding

Alamouti codewords,

)1(4 QN kc S

multiplications are required per Ala-

mouti codeword period. The average complexity can

be further reduced via the adaptive effort scheme

based on the generalized receiver with only negligi-

ble performance degradation.

To assess the complexity of the time-domain

MMSE approach, the MMSE generalized receiver

scheme design procedure as well as the MMSE filte-

ring procedure needs to be included. In the filter de-

sign procedure, the correlation matrix

in [12, Se-

ction IVB] is constructed first with the size

NN 22 

for the two-transmit and one-received antenna syst-

ems, requiring

multiplications. Using the sparse

structure of the corresponding matrices, the comple-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

103

Volume 21, 2022

xity is lowered to

LN 24 2

, where L is the number

of multipaths for each channel. Then, the inverse of

the correlation matrix is calculated that requires

multiplications. Finally, a time-variant filter is desi-

gned as described in Section 5.2 (

multiplicati-

ons). In the filtering process,

multiplications are

required since the length of the filter is 2N. Therefo-

re, roughly

83 N

4N

multiplications are requi-

red per Alamouti codeword period. The low rank

approximation of the correlation matrix that was us-

ed in [42] can be adopted to reduce complexity in

the filter design process. Unlike in [42], however, a

low rank approximation is necessary per codeword

period. It seems that the computationally expensive

singular value decomposition in the approximation

process does not reduce the complexity dramatical-

ly.

Since we are considering the parameters

128N

and

4Q

, the complexity of the proposed SDFSE

generalized receiver with the adaptive threshold is

much lower in comparison with the MMSE genera-

lized receiver approach. Since the MMSE generali-

zed receiver scheme complexity does not depend on

the constellation size Q, the relative complexity of

the proposed schemes grows as larger constellations

are used. But under the harsh channel environment

we are considering here, the small signal constellati-

on may be used to achieve a proper error performan-

ce.

Fig. 5. Block diagram of the proposed adaptive effort generalized receiver. When the instantaneous

channel time variation is significant the SDPSE with the adaptive threshold is used, and when

it is not significant the simple Alamouti decoding is used: 1- fast Fourier transform; 2 – genera-

lized receiver; 3 – channel estimation; 4 – Alamouti decoding; 5 – SDFSE with the adaptive

threshold; 6 – QPSK demodulator.

Table I. Required number of metric calculations for an Alamouti codeword

estimation in various symbol estimation schemes based on the

generalized receiver.

Decoding scheme

Required number of metric calculations

Alamouti decoding scheme

2Q

SDFSE

)1(2 q

SDFSE with the adaptive

threshold

kQSQ 22 ||

AESE

QPQSQP AESE

kAESE 2)1()( 22 || 

6 Simulation Results

In this section, the error performance and complexi-

ty of the following schemes are compared via simu-

lation: the Alamouti decoding generalized receiver

scheme, the SDFSE generalized receiver scheme

with the adaptive threshold, the AESE generalized

receiver scheme, the STBC generalized receiver

scheme, and the MMSE generalized receiver appro-

ach. The simulation scenario is the same as in Secti-

on 4.3. We consider only the higher Doppler freque-

ncy of 297 Hz resulting in the normalized Doppler

frequency of

12.0)(  sD TDNf

. Unlike in Section

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

104

Volume 21, 2022

4.3, we consider both cases with and without ideal

channel state information at the receiver. Consider-

ed values for

AESE

in the AESE generalized receiver

scheme are 0.3 and 0.4. Exact noise power is assum-

ed to be available for the MMSE generalized receiv-

er approach.

The BER performance as a function of the SNR

for various decoding schemes is presented in Fig. 6

when the ideal channel state information is assumed.

The time-domain MMSE generalized receiver appr-

oach shows the best performance. The performance

of the proposed SDFSE generalized receiver with

the adaptive threshold falls between the Alamouti

decoding generalized receiver performance and that

of the MMSE generalized receiver approach. It can

be observed that as the parameter q increases in the

proposed SDFSE generalized receiver with the ada-

ptive threshold, the performance approaches that of

the MMSE generalized receiver. For the SNR range

from 5 to 25 dB, the proposed SDFSE generalized

receiver with the adaptive threshold

)2( q

shows

almost the same performance as that of the MMSE

generalized receiver. When the SNR is as high as 30

dB, there exists a performance difference though.

The performance gap seems to be due to the fact that

the MMSE generalized approach considers the inter-

codeword coupling from all subchannels while the

SDFSE generalized receiver with the adaptive thre-

shold considers intercodeword coupling from only

2q adjacent subchannels. The performance gap sug-

gests that, as the SNR gets higher, more subchannels

need to be considered from which the intercodeword

coupling is caused.

Fig. 6. BER performance (perfect channel state in-

formation) of the Alamouti decoding (1), proposed

SDFSE with the adaptive threshold (2-

,0q

q

)2q

, and time-domain MMSE schemes (5)

based on the generalized receiver.

The BER performance when the channel state in-

formation is estimated via the channel estimation te-

chnique discussed in [12] is presented in Fig. 7. The

performance degradation due to channel estimation

error can be observed. The decoding schemes react

differently to the channel estimation error. The error

performance of the Alamouti decoding generalized

receiver and the SDFSE generalized receiver with

the adaptive threshold

)0( q

is almost identical as

when the ideal channel state information is assumed.

With channel estimation error, however, the SDFSE

generalized receiver with the adaptive threshold

)2

shows almost the same performance as that of

the SDFSE generalized receiver with the adaptive

threshold

)1( q

, which suggests that the intercode-

word coupling

]2,[ kkA

is more susceptible to cha-

nnel state information estimation error than

],[ kkA

and

]1,[ kkA

are. The MMSE generalized receiver

approach shows the most significant performance

gap between the ideal and estimated channel state

information.

Fig. 7. BER performance (perfect channel state in-

formation) of the Alamouti decoding (1), proposed

SDFSE with the adaptive threshold (2-

,0q

q

)2q

, time-domain MMSE (5) schemes based

on the generalized receiver, and differential STBC

scheme (no channel state information).

The BER performance shown in Fig. 7 suggests

that with the channel state information estimation

error, large q does not have to be adopted in the

SDFSE generalized receiver with the adaptive thre-

shold. It shows also that with the channel state infor-

mation estimated the proposed SDFSE generalized

receiver with the adaptive threshold

)1( q

demonst-

rates a negligible performance degradation compar-

ed with the MMSE generalized receiver approach.

Figure 7 also presents the performance of the dif-

ferential scheme [18]. Although the differential

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

105

Volume 21, 2022

scheme eliminates the need for the channel state in-

formation, which is more beneficial when the chan-

nel parameters change very fast, its performance

loss due to the non-quasistatic channels is much mo-

re severe than the performance less of other coher-

ent schemes due to the channel state information es-

timation error. It seems that the performance loss is

caused by the strict assumption of the differential

scheme that the channels do not change over two

Alamouti codeword periods, i.e., four OFDM symb-

ol periods. Note that the Alamouti decoding genera-

lized receiver scheme assumes that the channel pa-

rameters do not change over only two OFDM sym-

bol periods.

Fig. 8. Number of candidates in the proposed

SDFSE generalized receiver with the adaptive

threshold

)1( q

for various SNR values:

1 - SNR = 5 dB; 2- SNR = 10 dB; 3- SNR = 20

dB; 4- SNR = 30 dB. Estimated channel state in-

formation is used.

The probability of the number of candidates in

the SDFSE generalized receiver with the adaptive

threshold scheme

)1( q

is presented in Fig. 8. It can

be observed that more candidates are selected when

the SNR is low. This is because that more constella-

tion points satisfy the constraint (49) due to domin-

ant background noise. As the SNR increases, the nu-

mber of candidates significantly decreases. When

the SNR is 20 dB, only one candidate is selected

with the probability of 0.9. The corresponding



2.0

; hence, the complexity of the SDFSE generali-

zed receiver with the adaptive threshold is

 2.1

2Q

multiplications per the Alamouti codeword

period. The complexity of the proposed approach is

much lower than

23 482 NN 

of the MMSE gene-

ralized receiver approach when a moderate Q is as-

sumed. The complexity ratio of the proposed recei-

ver to the MMSE generalized receiver approach is

approximately

22 245NQ

The performance of the SDFSE generalized rece-

iver with the adaptive threshold

)1( q

, AESE gene-

ralized receiver with

4.0;3.0

AESE

, and the MMSE

generalized receiver approaches are presented in

Fig. 9. As can been seen from the Fig. 9, the AESE

generalized receiver scheme shows negligible perfo-

rmance degradation compared with SDFSE genera-

lized receiver with the adaptive threshold

)1( q

scheme when the threshold value

3.0

AESE

. There-

fore, about 42% of transmitted signals are estimated

via the Alamouti decoding generalized receiver

scheme even when the Doppler frequency is as high

297 Hz, if

3.0

AESE

. Although the higher

AESE

can

be used to further decrease the complexity, when

AESE

= 0.4 the probability goes down to 35%, there

exist significant performance gaps. Therefore, it can

be concluded that the proposed AESE generalized

receiver scheme with appropriate

AESE

is the attract-

ive receiver for the Alamouti coded OFDM systems

in the fast fading channels.

Fig. 9. BER performance (estimated channel state

information) of: 1 - SDFSE generalized receiver

with the adaptive threshold

)1( q

; 2 - AESE gene-

ralized receiver with

3.0

AESE

; 3 - AESE with

generalized receiver

4.0

AESE

; 4 - time-domain

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

106

Volume 21, 2022

MMSE generalized receiver scheme.

7 Conclusions

In Alamouti coded OFDM systems, the time variati-

on of channel causes both the intercodeword coupl-

ings, which significantly degrade the performance

of the Alamouti decoding generalized receiver per-

formance. We showed that the performance degra-

dation can be mitigated by the SDFSE generalized

receiver scheme with the adaptive threshold at a

much lower complexity when compared with the

previous MMSE generalized receiver approach and

a small constellation is assumed, exploiting the rela-

tive significance of the two couplings. It was also

shown that the performance difference between the

MMSE generalized receiver and the SDFSE genera-

lized receiver with the adaptive threshold schemes

becomes smaller when the channel state information

estimation error is taken into account. When the ve-

ry large constellation and small FFT size are adopt-

ed, the SDFSE generalized receiver with the adapti-

ve threshold scheme may require higher complexity

was achieved based on the observation that the high

Doppler frequency does not necessarily mean signi-

ficant instantaneous channel variation all the time,

which motivated the development of the adaptive

effort receiver. The simulation demonstrated the eff-

icacy of the proposed SDFSE generalized receiver

with the adaptive threshold and AESE generalized

receiver schemes.

References

[1] V. Tarokh, N. Seshadri, and A.R. Calderbank,

Space-time codes for high data rate wireless co-

mmunication: performance criterion and code

construction//IEEE Transactions on Informati-

on Theory.1998, Vol. 44, No.2, pp. 744-765.

[2] A.F. Naguib, N. Seshadri, and A.R. Calderbank,

Increasing data rate over wireless channel//IEEE

Signal Processing Magazine. 2000, Vol. 17, No.

3, pp.76-92.

[3] V. Tarokh, H. Jafarkhani, and A.R. Caldebank,

Space-time block codes from orthogonal des-

igns//IEEE Transactions on Information Theo-

ry. 1999, Vol.45, No. 5, pp. 1456-1467.

[4] V. Tarokh, H. Jafarkhani, and A.R. Caldebank,

Space-time block coding for wireless communi-

cations: Performance results//IEEE Journal on

Selected Areas on Communications. 1999, Vol.

17, No. 3, pp. 451-460.

[5] X. Li, N. Luo, G. Yue, and C.Yin, A squariung

method to simplify the decoding of orthogonal

space-time block codes//IEEE Transactions on

Communications. 2001, Vol. 49, No. 10, pp.

1700-1703.

[6] S.M. Alamouti, A simple transmit diversity tec-

hnique for wireless communications//IEEE Jo-

urnal on Selected Areas on Communications.

1998, Vol. 16, No. 10, pp. 1451-1458.

[7] M. Hoseinzade, K. Mohamedpour, S. Medhi

Hosseini Andargoli, Decision feedback channel

estimation for Alamouti coded OFDM systems//

International Journal of Information & Comm-

unication Technology Research. 2012, Vol. 4,

Issue 3, pp.1-11.

[8] R. Bhadada, R. Taparia, Performance analysis

of Alamouti code MIMO OFDM systems for

error control and IQ impairments//Communica-

tions on Applied Electronics. 2015, Vol. 1, No.

2, pp.14-17.

[9] M.A. Youssefi, N. Bounouader, Z. Guennoun,

J.E. Abdadi, Adaptive switching between spa-

ce-time and space-frequency block coded

OFDM systems in Rayleigh fading channel//In-

ternational Journal of Communications, Net-

work and System Sciences. 2013, Vol. 6, No. 6,

Article ID:3230,8 pages;doi:10.4236/ijcns.2013

.66034

[10] B.-S. Kim, K.H. Choi, Over-sampling effect in

distributed Alamouti coded OFDM with freque-

ncy offset//IET Communications. 2016, Vol. 10,

Issue 17, pp. 2344-2351.

[11] H.A. Bakir, F. Debhat, and F.T. Bendimerad,

Performance enhancement of OSTBC applied

OFDM modulation for wireless communication

systems//Journal of Applied Sciences. 2016,

Vol.16, pp. 419-428; doi:10.3923/jas.2016.419.

428

[12] A. Stamoulis, S.N. Diggavi, N. Al-Dhahir, Inte-

rcarrier interference in MIMO OFDM//IEEE

Transactions on Signal Processing. 2002, Vol.

50, No. 10, pp. 2451-2464.

[13] Z. Liu, X. Ma, G.B. Giannakis, Space-time co-

ding and Kalman filtering for time-selective fa-

ding channels//IEEE Transactions on Commu-

nications.2002, Vol. 50, No. 2, pp. 183-186.

[14] B.S. Kim, D. Na, K.H. Choi, Partial ML detec-

tion for frequency-asynchronous distributed

Alamouti-coded (FADAC) OFDM//Wireless

Communications and Mobile Computing. Spe-

cial Issue: Antenna Design Techniques for 5G

Mobile Communications. 2019, Volume 2019,

Article ID:4319802;doi:10/1155/2019/4319802

[15] B.S. Kim, K.H.Choi, Iterative detection for fre-

quency-asynchronous distributed Alamouti-co-

ded (FADAC) OFDM//EURASIP Journal on

Wireless Communications and Networking.

2017,39(2017);doi:10.1186/S13638-017-0819-1

[16] I.B. Djordjevich, L. Xu, T. Wang, Alamouti-ty-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

107

Volume 21, 2022

pe polarization-time coding in coded-modulati-

on schemes with coherent detection//Optic Ex-

press. 2008, Vol. 16, Issue 18,pp.14163-14172.

[17] K. Choi, Intercarrier interference free Alamouti

coded OFDM for cooperative systems with fre-

quency offsets in non-selective fading environ-

ments//IET Communications. 2011, Vol. 5, Iss-

ue 15, pp.2125-2129.

[18] S.N. Figgavi, N. Al-Dhahir, A Stamoulis, A.R.

Calderbank, Diferential space-time coding for

frequency-selective channels//IEEE Communi-

cations Letters. 2002, Vol.6, No.6, pp.253-255.

[19] J.F. Hayes, The Viterbi algorithm applied to di-

gital data transmission//IEEE Communications

Magazine. 2002, Vol. 40, No. 5, pp. 26-32

[20] A. Duel-Hallen, C. Heegard, Delayed decision-

feedback sequence estimation//IEEE Transacti-

ons on Communications. 1989, Vol. 37, No. 5,

pp. 428-436.

[21] W. Younis, N. Al-Dhahir, Joint prefiltering and

MLSE equalization of space-time-coded trans-

missions over frequency-selective channels//

IEEE Transactions on Vehicular Technology.

2002, Vol. 51, No. 1, pp. 144-154.

[22] S.J. Simmons, Breadth-first trellis decoding

with adaptive effort////IEEE Transactions on

Communications. 1989, Vol. 37, No. 5, pp. 428

-436.

[23] W. Gerstacker, R. Schober, Equalization conc-

epts for EDGE//IEEE Transactions on Wire-

less Communications. 2002, Vol.1, No. 1, pp.

190-199.

[24] H. Zamiri-Jafarian, S. Pasupathy, Complexity

reduction of the MLSD/MLSDE receiver using

the adaptive state allocation algorithm// IEEE

Transactions on Wireless Communications.

2002, Vol.1, No. 1, pp. 190-199.

[25] J.P. Seymour, M.P. Fitz, Near-optimal symbol-

by-symbol detection schemes for flat Rayleigh

fading//IEEE Transactions on Communications

2002, Vol.1, No. 2/3/4, pp. 1525-1533.

[26] W. Jeon, K. Chang, Y. Cho, An equalization te-

chnique for orthogonal frequency division mul-

tiplexing systems in time-variant multipath

channels// IEEE Transactions on Communica-

tions. 1999, Vol.47, No. 1, pp. 27-32.

[27] V. Tuzlukov, A new approach to signal detecti-

on theory. Digital Signal Processing: Review

Journal. 1998, Vol.8, No. 3, pp.166–184.

[28] V. Tuzlukov, Signal Detection Theory. Spring-

er-Verlag, New York, USA, 2001, 746 pp.

[29] V. Tuzlukov, Signal Processing Noise. CRC

Press, Boca Raton, London, New York, Washi-

ngton, D.C., USA, 2002, 692 pp.

[30] M. Maximov, Joint correlation of fluctuative

noise at outputs of frequency filters. Radio En-

gineering. 1956, No. 9, pp. 28–38.

[31] Y. Chernyak, Joint correlation of noise voltage

at outputs of amplifiers with no overlapping re-

sponses. Radio Physics and Electronics. 1960,

No. 4, pp. 551–561.

[32] Shbat, M., Tuzlukov, V.P. Definition of adapti-

ve detection threshold under employment of

the generalized detector in radar sensor syst-

ems. IET Signal Processing. 2014, Vol. 8, Iss-

ue 6, pp. 622–632.

[33] Tuzlukov, V.P. DS-CDMA downlink systems

with fading channel employing the generaliz-

ed. Digital Signal Processing. 2011. Vol. 21,

No. 6, pp. 725–733.

[34] Communication Systems: New Research, Edit-

or: Vyacheslav Tuzlukov, NOVA Science Pub-

lishers, Inc., New York, USA, 2013, 423 pp.

[35] Shbat, M., Tuzlukov, V.P. Primary signal dete-

ction algorithms for spectrum sensing at low

SNR over fading channels in cognitive radio.

Digital Signal Processing (2019). https://doi.

org/10.1016/j.dsp. 2019.07.16. Digital Signal

Processing. 2019, Vol. 93, No. 5, pp. 187- 207.

[36] V. Tuzlukov, Signal processing by generalized

receiver in wireless communications systems

over fading channels,” Chapter 2 in Advances

in Signal Processing. IFSA Publishing Corp.

Barcelona, Spain. 2021. pp. 55-111.

[37] V. Tuzlukov, Interference Cancellation for

MIMO Systems Employing the Generalized

Receiver with High Spectral Efficiency”.

WSEAS Transactions on Signal Processing.

2021, Vol. 17, Article #1, pp. 1-15.

[38] M. Shbat and V. Tuzlukov, SNR wall effect al-

leviation by generalized detector employment

in cognitive radio networks. Sensors, 2015, 15

(7), pp.16105-16135;doi:10.3390/s150716105.

[39] V. Tuzlukov, Signal processing by generalized

detector in DS-CDMA wireless communication

systems with frequency-selective channels. Ci-

rcuits, Systems, and Signal Processing. Publi-

shed on-line on February 2, 2011, doi:10.1007/

s00034-011-9273-1; 2011, Vol.30, No.6, pp.

1197–1230.

[40] M. Shbat, V. Tuzlukov, Generalized detector

as a spectrum sensor in cognitive radio net-

works, ”Radioengineering , 2015, Vol. 24, No.

2, pp. 558–571.

[41] W.C. Jakes, Microwave Mobile Communicati-

ons. New York: Wiley, 1974.

[42] Y. Choi, P.J. Voltz, F.A. Cassara, On channel

estimation and detection for multicarrier sig-

nals in fast and selective Rayleigh fading chan-

nels//IEEE Transactions on Communications.

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

108

Volume 21, 2022

2001, Vol.49, No. 8, pp. 1375-1387.

Dr. Vyacheslav Tuzlukov

received the MSc and PhD degrees in radio physics from

the Belarusian State University, Minsk, Belarus in 1976

and 1990, respectively, and DSc degree in radio physics

from the Kotelnikov Institute of Radioengineering and

Electronics of Russian Academy of Sciences in 1995.

Starting from 1995 and till 1998 Dr. Tuzlukov was a Vi-

siting Professor at the University of San-Diego, San-Die-

go, California, USA. In 1998 Dr. Tuzlukov relocated to

Adelaide, South Australia, where he served as a Visiting

Professor at the University of Adelaide till 2000. From

2000 to 2002 he was a Visiting Professor at the Universi-

ty of Aizu, Aizu-Wakamatsu City, Fukushima, Japan and

from 2003 to 2007 served as an Invited Professor at the

Ajou University, Suwon, South Korea, within the Depart-

ment of Electrical and Computer Engineering. Starting

from March 2008 to February 2009 he joined as a Full

Professor at the Yeungnam University, Gyeonsang, South

Korea within the School of Electronic Engineering, Com-

munication Engineering, and Computer Science. Starting

from March 1, 2009 Dr. Tuzlukov served as Full Profes-

sor and Director of Signal Processing Lab at the Depart-

ment of Communication and Information Technologies,

School of Electronics Engineering, College of IT Engine-

ering, Kyungpook National University, Daegu, South Ko-

rea. Currently, Dr. Tuzlukov is the Head of Department

of Technical Exploitation of Aviation and Radio Engine-

ering Equipment, Belarusian State Academy of Aviation,

Minsk, Belarus. His research emphasis is on signal pro-

cessing in radar, wireless communications, wireless sen-

sor networks, remote sensing, sonar, satellite communi-

cations, mobile communications, and other signal proce-

ssing systems. He is the author over 280 journal and con-

ference papers, seventeenth books in signal processing

published by Springer-Verlag and CRC Press. Some of

them are Signal Detection Theory (2001), Signal Proces-

sing Noise (2002), Signal and Image Processing in Navi-

gational Systems (2005), Signal Processing in Radar Sys-

tems (2012), Editor of the book Communication Systems:

New Research (2013), Nova Science Publishers, Inc,

USA, and has also contributed Chapters “Underwater

Acoustical Signal Processing” and “Satellite Communi-

cations Systems: Applications” to Electrical Engineering

Handbook: 3rd Edition, 2005, CRC Press; “Generalized

Approach to Signal Processing in Wireless Communicati-

ons: The Main Aspects and Some Examples” to Wireless

Communications and Networks: Recent Advances,

InTech, 2012; “Radar Sensor Detectors for Vehicle Safe-

ty Systems” to Electrical and Hybrid Vehicles: Advanced

Systems, Automotive Technologies, and Environmental

and Social Implications, Nova Science Publishers, Inc.,

USA, 2014; “Wireless Communications: Generalized Ap-

proach to Signal Processing” and “Radio Resource Mana-

gement and Femtocell Employment in LTE Networks”,

to Communication Systems: New Research, Nova Science

Publishers, Inc., USA, 2013; “Radar Sensor Detectors for

Vehicle Safety Systems” to Autonomous Vehicles: Intelli-

gent Transport Systems and Automotive Technologies,

Publishing House, University of Pitesti, Romania, 2013;

“Radar Sensor Detectors for Vehicle Safety Systems,” to

Autonomous Vehicles: Intelligent Transport Systems and

Smart Technologies, Nova Science Publishers, Inc., New

York, USA, 2014; “Signal Processing by Generalized Re-

ceiver in DS-CDMA Wireless Communication Systems,”

to Contemporary Issues in Wireless Communications.

INTECH, CROATIA, 2014; “Detection of Spatially Dist-

ributed Signals by Generalized Receiver Using Radar Se-

nsor Array in Wireless Communications,” to Advances in

Communications and Media Research. NOVA Science

Publishers, Inc., New York, USA, 2015; “Signal Process-

ing by Generalized Receiver in Wireless Communication

Systems over Fading Channels” to Advances in Signal

Processing. IFSA Publishing Corp. Barcelona, Spain.

2021; “Generalized Receiver: Signal Processing in DS-

CDMA Wireless Communication Systems over Fading

Channels” to Book Title: Human Assisted Intelligent Co-

mputing: Modelling, Simulations and Its Applications.

IOP Publishing, Bristol, United Kingdom, 2022. He par-

ticipates as the General Chair, Keynote Speaker, Plenary

Lecturer, Chair of Sessions, Tutorial Instructor and orga-

nizes Special Sections at the major International Confere-

nces and Symposia on signal processing.

Dr. Tuzlukov was highly recommended by U.S. experts

of Defence Research and Engineering (DDR& E) of the

United States Department of Defence as a recognized ex-

pert in the field of humanitarian demining and minefield

sensing technologies and had been awarded by Special

Prize of the United States Department of Defence in 1999

Dr. Tuzlukov is distinguished as one of the leading achie-

vers from around the world by Marquis Who’s Who and

his name and biography have been included in the Who’s

Who in the World, 2006-2013; Who’s Who in World,

25th Silver Anniversary Edition, 2008, Marquis Publisher,

NJ, USA; Who’s Who in Science and Engineering,

2006-2012 and Who’s Who in Science and Engineering,

10th Anniversary Edition, 2008-2009, Marquis Publisher,

NJ, USA; 2009-2010 Princeton Premier Business Leaders

and Professionals Honours Edition, Princeton Premier

Publisher, NY, USA; 2009 Strathmore’s Who’s Who

Edition, Strathmore’s Who’s Who Publisher, NY, USA;

2009 Presidental Who’s Who Edition, Presidental Who’s

Who Publisher, NY, USA; Who’s Who among Executi-

ves and Professionals, 2010 Edition, Marquis Publisher,

NJ, USA; Who’s Who in Asia 2012, 2nd Edition, Marqu-

is Publisher, NJ, USA; Top 100 Executives of 2013 Mag-

azine, Super Network Publisher, New York, USA, 2013;

2013/2014 Edition of the Global Professional Network,

Business Network Publisher, New York, USA, 2013;

2013/2014 Edition of the Who’s Who Network Online,

Business Network Publisher, New York, USA, 2014; On-

line Professional Gateway, 2014 Edition, Business Netw-

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

109

Volume 21, 2022

ork Publisher, New York, USA, 2014; 2014 Worldwide

Who's Who", Marquis Publisher, NJ, USA; 2015 Strath-

more Professional Biographies, Strathmore’s Who’s Who

Publisher, NY, USA; Who’s Who in World, 2015, Marq-

uis Publisher, NJ, USA; 2015-2016 Membership in Excl-

usive Top 100 network of professionals in the world, NY,

USA, 2015; 2015 Who’s Who of Executives and Profes-

sionals Honors Edition, Marquis Publisher, NJ, USA;

Worldwide Who’s Who – Top 100 Business Networking,

San Diego, CA, USA, 2015.

Phone: +375173453283; Email: slava.tuzlukov@mail.ru

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the Creative

Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

WSEAS TRANSACTIONS on COMMUNICATIONS

DOI: 10.37394/23204.2022.21.14

Vyacheslav Tuzlukov

E-ISSN: 2224-2864

110

Volume 21, 2022