CHAPTER 3

CHANNEL ESTIMATION

SCHEMES

59

CHAPTER 3

CHANNEL ESTIMATION SCHEMES

hannel uncertainty focuses the attention on the way to model the unreliable channel and its

influence on the communication systems. The most common unreliable channel is the

channel with partial channel state information. The unreliability includes the measurement errors, channel

variations and the inaccuracy in the channel modeling. This Chapter deals with the techniques which have

been utilized for estimating the partial channel correctly without the loss of reliability of the transmitted

signal and the requirement of the power by the MIMO system should be as low as possible.

3.1. INTRODUCTION There are so many types of channel estimation schemes in the wireless communication systems.

The main focus is to identify the proper channel estimation scheme which can easily accommodate

different types of the available method of the modulation schemes. There are various estimation methods

which help the wireless communication system to make it robust and decrease the probability of error like

different types of training of the modulating signal and their estimation schemes, which is going to be

discussed in this Chapter. A new method is also proposed with less mean square error (MSE).

3.2. PILOT ASSISTED CHANNEL ESTIMATION SCHEMES In MIMO system, the transmitter modulates the message bit sequence into PSK/QAM and

different modulation schemes, performs Inverse Fractional Fourier Transform (IFFT) on the symbols to

convert them into time-domain signals, and sends them out through a (wireless) channel. The received

signal is usually distorted by the channel characteristics. In order to recover the transmitted bits, the

channel effect must be estimated and compensated in the receiver [Jiang et al. (2003)]. Each subcarrier

can be regarded as an independent channel, as long as no ICI (Inter-Carrier Interference) occurs, and thus

preserving the orthogonality among subcarriers. The orthogonality allows each subcarrier component of

the received signal to be expressed as the product of the transmitted signal and channel frequency

response at the subcarrier. Thus, the transmitted signal can be recovered by estimating the channel

response just at each subcarrier. In general, the channel can be estimated by using a preamble or pilot

symbols known to both transmitter and receiver, which employ various interpolation techniques to

estimate the channel response of the subcarriers between pilot tones. In general, data signal as well as

C

60

training signal, or both, can be used for channel estimation. In order to choose the channel estimation

technique for the MIMO system under consideration, many different aspects of implementations,

including the required performance, computational complexity and time-variation of the channel must be

taken into account. Various pilot assisted schemes are available for the review purpose but all of them are

based on some basic techniques which are explained with following sections. [Cover and Thomas (2006) ;

Kay (1998) ; Oestges and Clerckx (2007)]

3.2.1. Block Type Pilot Scheme

In this type of pilot schemes, the transmitted symbols with pilots are transmitted from the

transmitter using all subcarriers. A time domain interpolation is performed to estimate the channel along

the time axis. If suppose TJ denotes the period of pilot symbols in time as shown in Figure 3.1 and each

block is showing one transmitted symbol, now to keep track of the time varying channel characteristic, the

pilot symbols must be placed as frequently as the coherence time. Since, the coherence time is shown in

an inverse form of the Doppler frequency df in the channel, the pilot symbol period must satisfy the

following inequality:

1

Td

Jf

� (3.2.1)

Since, pilot tones are inserted into all subcarriers of pilot symbols with a period in time, the block type

pilot arrangement is suitable for frequency selective channels. For the fast fading channels, it might give

too much overhead to track the channel variations by reducing the pilot symbol [Coleri et al. (2002) ;

Zijian and Leus (2007)]. �����������������������������������������������������

Figure 3.1. Block type pilot arrangement for channel estimation

61

3.2.2. Comb Type Pilot Scheme�

In this type of pilot arrangement, which has been shown in Figure 3.2, each transmitted symbol

has pilot tone at the periodically repeated subcarrier which can be utilized as to estimate the channel along

the frequency axis using frequency domain interpolation. To main the track of the frequency selective

channel characteristics, the pilot symbols must be placed as frequently as coherent bandwidth. As the

coherence bandwidth range is determined by inversion of the maximum delay spread max� , the pilot

symbol period must satisfy the following inequality:

max

1fJ

�� (3.2.2)

Since, this arrangement is completely inverse in terms of the basic domain, the comb-type pilot

arrangement is suitable for fast fading channels but not for frequency selective channels.

Figure 3.2. Comb type pilot arrangement for channel estimation

3.2.3. Lattice Type Pilot Scheme�

In this type of pilot scheme, both time and frequency axes are utilized by inserting the pilot

symbols along both time and frequency axes with the provided symbol period as shown in Figure 3.3.

Pilot symbols are distributed in both time and frequency axes which helps the user to detect the channel

using both time and frequency domain interpolation. To keep tracking of the time varying and frequency

selective channel characteristic, the pilot symbol arrangement should satisfy both (3.2.1) and (3.2.2). This

pilot arrangement scheme has been the prime contributor in the proposed pilot arrangement scheme which

will be shown ahead.

62

�

Figure 3.3. Lattice pilot symbol arrangement for channel estimation

SNR for the lattice type pilot scheme

The per symbol SNR can be denoted as -

� � � �

� �2 2

, ,

2,

k l k l

k l

E x E hSNR

E n� (3.2.3)

and for the relative number of pilots J , reduced by -

� �1010log 1lossSNR J dB� (3.2.4)

It can be observed that data rate is '(1 ) / sN J T symbol/sec, i.e., it depends on J . If denser pilot pattern

is chosen then the per symbol SNR will also get loosened up, for which only those pilot will be included

who are carrying data with them. The design of pilot pattern is the trade-off between good channel

estimation (closely spaced pilots) and high SNR (sparsely spaced pilots).

3.3. TRAINING SYMBOL BASED CHANNEL ESTIMATION

Training symbols can be used for channel estimation which provides good performance but the

transmission efficiencies are reduced due to the required overhead of training symbols such as preamble or

63

pilot symbols which have been transmitted with the data symbols. The Least square (LS) and MMSE

techniques are generally used for channel estimation when training symbols are transmitted [Coleri et al.

(2002)].

The mathematical design for the pilot training symbols for 'N subcarriers which are orthogonal to each

other can be represented in the following matrix form -

(0) 0 00 (1)

00 0 ( ' 1)

xx

x N

�� � �� � � �

X

�� �

� � ��

(3.3.1)

here, ( )x k denotes pilot symbols at the thk subcarrier, with � �( ) 0E X k � and � � 2var ( ) xX k �� with

0,1,2,....., ' 1k N� . Further, the received MIMO signal with pilot symbols can be expressed as -

(0) (0) 0 0 (0) (0)(1) 0 (1) (1) (1)

0( ' 1) 0 0 ( ' 1) ( ' 1) ( ' 1)

y x H ny x H n

y N x N n N y N

� � � �� � � � � � � � � � �� � � � � � � � � � � � � � � �

y

�� �

� � � � � ��

(3.3.2)

where, H is a channel vector given as � �ˆ (0), (1),...., ( ' 1) TH H H N� H and n is a noise vector given as

� �(0), (1),...., ( ' 1) Tn n n N� n with � �( ) 0E X k � and � � 2var ( ) xX k �� with 0,1,2,....., ' 1k N� and H be

the estimate of the channel H .

3.3.1. Least Square (LS) Channel Estimation

The least square (LS) channel estimation method helps to determine the channel estimate H in

such a way to minimize the cost function as -

� � � �

2

. .ˆ ˆ( )

ˆ ˆ

ˆ ˆ ˆ

c f

H

H H H H H H

J �

�

� �

H Y XH

Y XH Y XH

Y Y Y XH H X Y H X XY

(3.3.3)

By assuming the derivative of the function to make the H to zero,

. . * *ˆ( ) ˆ2( ) 2( ) 0ˆ

c f H HJ�� � �

�

HX Y X XH

H (3.3.4)

It is observed that ˆH H�X XH X Y , which gives the solution to the LS channel matrix as -

64

� � 1 1ˆ H HLS

� �H X X X Y X Y (3.3.5)

LS channel estimate ˆLSH can be denoted for each subcarrier by considering � �LSH k for the given range

which has been shown below:

� � � �� �

ˆ , 0,1,2,.... ' 1LS

Y kk for k N

X k� � H (3.3.6)

The mean square error (MSE) of this LS channel estimate is given as -

� � � �� �� � � �� �� � � �� �

� �

1 1

1 1

21

2

ˆ ˆ

( ) ��

�

�

�

� �

H H H H

H X Y H X Y

X Z X Z

Z XX Z

H

LS LS LS

H

H

H H z

x

MSE E

E

E

E

(3.3.7)

MSE in (3.3.7) is inversely proportional to the SNR 2 2/z x� � , which shows that this may contribute in

noise enhancement specially when the channel is very rich scattered but still it is widely used estimation

scheme due to its simplicity [Bottomley (2012) ; Jo (2008) ; Song (2008)].

3.3.2. Minimum Mean Square Error (MMSE) Channel Estimation

Again considering the solution for the LS channel estimation in (3.3.5), the MMSE estimator will

utilize the weight matrix � which can be defined as H �H�� . The MSE channel estimator has been shown

in Figure 3.4, where, MSE of the channel estimate has been given as -

� � � � � �22ˆ ˆMSE E e E� � H H H (3.3.8)

Figure 3.4. MMSE channel estimator

65

This method shows that the it can estimate the channel in a better way by utilizing the weight

vectors or matrix for updating the received symbols at the receiver side so as to minimize the mean square

error as shown in (3.3.8). It has already been told that the pilot symbols are orthogonal with each other and

shows that the estimation error vector ˆe � H H is orthogonal to H� ,

� � � �� �� �� �

� � � �

ˆ

0

�

�

�

� �HH HH

H H H H

H �H H

HH � HH

r �r� �

� �

� �

� � �

H H

H

H H

E e E

E E

E E (3.3.9)

here, HHr � shows the cross correlation matrix of N N� matrices with the LS channel estimate as -

1 1 � � �H X Y H X Z� (3.3.10)

Now on solving (3.3.9) for the weight vectors,

1� HH HH� r r� � (3.3.11)

and further the HHr � � which is the autocorrelation matrix of H� , and has been formed by calculating which

gave -

� � �2

2

��

� �HHr HH I� �H z

x

E � (3.3.12)�

using (3.3.12), the MMSE channel estimate will be formed as -

12

12

ˆ ��

� �� � � �� �

� �HHHH HH HHH �H H I H� � � �

� � �z

x

r r r r (3.3.13)

Hereafter, to estimate the channel for data symbols, the pilot symbols available with the subcarriers must

be interpolated which are of many types including linear interpolation, second order interpolation and

cubic spline interpolation.

As discussed earlier that this type of estimator will be used in this thesis for the estimation of the

semiblind channel by detecting the pilot symbol series out from the received signals and combining them

with the another estimation scheme so as to make more robust system for semiblind channel estimation.

3.3.3. Decision Directed Channel Estimation (DDCE)

The next channel estimator which is going to be explored in this thesis and will be combined with

the MMSE scheme to estimate the semiblind channel with the help of adaptive pilot symbol based

modulation scheme is the Decision Directed Channel Estimation (DDCE). Once initial channel would be

estimated with the help of pilot symbols, the coefficients of channel will be feedback to compare with the

66

original signal but delayed version. If this symbol is detected properly, then it is analogous to a known

pilot symbol and the channel coefficients corresponding to this received symbol can be estimated in the

same way which further can be updated with DDCE method which doesn’t require any pilots or training

symbols. This technique uses the detected signal as a feedback system to track the time varying channel

whereas at the same time it will be utilizing the channel estimates to detect the signal by making it as

reference data for the next coming data.[Blogh and Hanzo (2002) ; Du and Swamy (2010) ; Hanzo et al.

(2008) ; Parsons (2000) ; Prasad et al. (2009)]

3.3.4. Channel Estimation Using Superimposed Channel

In this type of channel estimation, the pilot symbols are merged with the data symbol with low

power which then used at the receiver for channel estimation without losing data rate. In this technique,

the power allocated to the pilot symbols or training sequence is wasted [Hoeher and Tufvesson (1999)].

For the thl transmitted symbol at the thk subcarrier, the superimposed signal can be expressed as

( ) ( )l lx k J k� , where, ( )lx k and ( )lJ k denote the data and pilot signals. The received signal can be

denoted as -

� � � � � � ( )l l l lk k k n k� �Y H X (3.3.14)

where, � � � � � �l l lX k x k J k� � and � �l kY and � �ln k denote the received noise at the thk subcarrier in the

thl symbol period.�

Here, the response has been assumed constant over the time and frequency axis of the channel and the

pilot symbols are also set to be constant for both time and frequency interval of length LM . Hence the

average received signal over time or frequency interval LM with zero mean and i.i.d. distribution will

become as -

� �� � � � � � � �� �� �1

0

:1

� � ��

� � � ��Y H HJL

L L L

L

M

l l m l m l mmL

Timedomain

E k x k J k n kM

(3.3.15)

� �� � � � � � � �� �� �1

0

:1

�

� � � � � � ��Y H HJL

L

M

l l L l L l LmL

Frequency domain

E k x k m J k m n k mM

(3.3.16)

These two relations given in (3.3.15) and (3.3.16) assumed that the LM is large enough to make the

average of data signal nearly zero as shown below-

67

� �1

0

1 0L

L

M

l LmL

x k mM

�

� �� (3.3.17)

then the channel estimate will be formed as -

� �� �ˆ lE k

H �YJ

(3.3.18)

Here, it is observed that no pilot symbols are required, this channel estimation scheme is good in respect

of data rate even though it requires an additional power for transmitting pilot signals and a long time

interval to make the average of data signal zero.

3.3.5. Channel Estimation Using Expectation Maximization (EM Based) Algorithm

This scheme has been widely used in large number of areas that deal with unknown factors

affecting the receiver such as signal processing and it’s an iterative technique for determining the

maximum likelihood (ML) estimates of a channel [Aldana et al. (2003) ; Moon (1996)]. This is the only

technique which has been termed as semiblind method since it can be implemented with the system where

transmits symbols are not arriving at the receiver in good manner or they are being missed due to fading

effect.

Considering x as one of the m-ary symbols of constellation of size A , such that the � �1 2, ,..., Ax x x x�

where, ix denotes the thi symbol in the constellation. The conditional probability density function of the

received signal can be denoted as -

� � 222

1 1| , exp | . |22

f���

� �� � �� �

Y H X Y H X (3.3.19)

It has already been discussed that for the MIMO system, equal power is divided into subchannels, then it

is assumed that � � 1

Ai i

x�

are transmitted with the same probability of 1 / A , the conditional p.d.f. of the

received signal can be denoted as -

� � 222

1

1 1 1| , exp | . |22

A

ii

f xA ����

� �� � �� �

�Y H X Y H (3.3.20)

According to the theory of EM algorithm, the received data is called “incomplete” or can be termed as

missed received data if only ix has been considered as the received data which is having some missed

data. Only the received data and the observed data can be considered as the “complete” data. It is very

difficult to estimate the channel with “incomplete” data, hence the p.d.f. of this will be converted to the

68

p.d.f. of “complete” data. The p.d.f. of “complete” data can be formed using the log-likelihood function of

the “incomplete” data as -

� � � �1

1log | , log | ,D

d d

df f

A�

!� " #$ %

�Y H X Y H X (3.3.21)

where, d denotes the incomplete data variable. In most likely estimation algorithm, H is estimated by

maximizing the likelihood function � �|f Y H in (3.3.20). Here, channel response is identified with the

help of increasing the iterations in (3.3.21).

Basically the EM algorithm consists of two separate iteration phases, first is Expectation and the second is

maximization step. In first step, the expected value of the log-likelihood function of H is computed by

taking the expectation over the range of X , depending on Y and using the latest estimate of H as shown

ahead -

� � � � � �� �

� �� �� �

( )

( )

( )1 1

| , | | ,

| ,1log | ,|

JJx

d jA Di

d i d ji d

E f

ff

A Af� �

!� " #$ %

��

Q H H Y X H Y H

Y H XY H X

Y H

�

(3.3.22)

where, � �JH denotes the latest estimate of the channel with the help of pilot symbols. It means the log-

likelihood functions of “complete” data in (3.3.21) are averaged for the D received symbols. In the next

iterations step, i.e., maximization step, � �1J �H is determined my maximizing (3.3.22) over all the possible

values of H . In more general way, the maximized value will be obtained by differentiating (3.3.22) w.r.t.

H and making its derivative to zero to formulate the following result:

� � � �� �� �� �

� �� �

1

1( ) ( )2 * 2

( ) ( )1 1 1 1

arg max |

| , | ,| | | |

| |

J J

H

d j d jA D A Di id

i id j d ji d i d

f fx Y x

Af Af

�

� � � �

�

�� � �� � ��� ��

H Q H H

Y H X Y H X

Y H Y H

(3.3.23)

This relation can be viewed as a weighted least square solution where an estimate of cross correlation

function is divided by an estimate of auto correlation function, each being weighted by the corresponding

Probability Distribution Function (p.d.f.).

The technique is very useful for the channel estimation when the data or training signal is

incomplete or unavailable. It is known that the focus has to be given on the partial channel state

information condition for which this technique is of prime importance and the partial CSI conditions

required for the coherent decoding between the transmit and receive antenna. Since, the other

conventional schemes cannot be utilized for the conditions when the received signal is superimposed of

the signals transmitted from different antennas for each subcarrier, EM based technique can be used to

69

convert the superimposed or multiple channel estimation into a single input channel estimation problem.

The use of this technique for semiblind scheme with partial CSI condition and APASBCE scheme has

been included in Chapter 6. Although the computation complexity increases exponentially using this EM

based estimation when the number of transmitted signal or constellation increases. This computational

complexity has been tried to reduce by making the combination of APASBCE scheme and EM based

scheme after making some modifications in precoder and decoder as shown in Chapter 4.

3.3.6. Blind Channel Estimation

This type of channel estimation scheme utilizes the statistical properties of received signals and

the channel can be estimated without the use of the pilot signals. Obviously, such a blind channel

estimation technique has an advantage of not incurring an overhead with training signals. However, it

generally needs a large number of received symbols to extract statistical properties. Furthermore, their

performance is usually worse than that of other conventional channel estimation techniques that employ

the training signal.

The subspace-based channel estimation technique is the blind channel estimation techniques

developed for OFDM systems [Dinh-Thuan and Dinh-Thanh (2010) ; Wei-Chieh et al. (2010)]. It is

derived by using the second-order statistical properties and orthogonal properties of a received signal.

Since, the received signal space can be divided into signal subspace and noise subspace, the channel can

be estimated by using the property of the noise subspace which is orthogonal to the signal subspace. The

subspace-based channel estimation technique needs a high computational complexity to separate the

signal subspace from noise subspace; this requires a computation of correlation from the received signal

and then, Eigen-decomposition. Also, a large number of received signals are required to estimate the

statistical properties of received signals. Different approaches, such as increasing the number of equations

by oversampling or employing a precoder matrix with full rank, have been investigated for the subspace-

based channel estimation.

In this estimator, the system is purely dependent on the received symbols which correspond to the

unknown data symbols without any pilot symbols, available at the receiver.

3.4. SYTEM MODEL AND PROBLEM STATEMENT A MIMO system has been considered with TM transmit antennas and RM receive antennas,

which communicates over a flat fading channels, and is abbreviated as T RM M� receive MIMO channel

matrix H. The system is described by ( ) ( ) ( )y k x k n k� �H , where, x is 1 2( ),. ( ),.... ( )T

T

Mx k x k x k �� � which is

70

the transmitted symbol vector of TM transmitter with the symbol energy given by 2 2| ( ) |mE x k � � �� � for

1 Tm M& & and covariance matrix ( )HE�Q XX , y denotes the received vector

1 2( ) ( ), ( ),..... ( )R

T

My k y k y k y k �� � � and 1 2( ) ( ), ( ),.... ( )R

T

Mn k n k n k n k �� � � is the complex valued gaussian

white noise vector at the receiving end for MIMO channels with energy 2( ) ( ) 2R

Hn ME n k n k � � �� � I

distributed according to 2(0, )RC n MI�N assumed to be zero mean, spatially and temporally white and

independent of both channel and data fades. The channel model considered here denoted by 1/2 1/2R T'�H R H R [Da-Shan et al. (2000)] with &T RR R representing the normalized transmit and receive

correlation matrices with identity matrix. The entries of 'H are independent and identically distributed

(i.i.d.) Nc(0,1).

Here, the CSIR is described by -

1/2 1/2 ' 1/2 ' 1/2ˆ ˆ, ,R T n R TE' '� � � �H H E' H R H R R E R (3.4.1)

where, H is the estimate of H and 'E is the overall channel estimation error matrix, 'ˆ &' 'H E are

white matrices spatially uncorrelated with i.i.d. entries distributed according to � �2'0,1C � EN and

� �2'0,C � EN with variance 2

'E� of channel estimation error [Musavian et al. (2007)].

It is assumed that the system is having lossless feedback, i.e., CSIT and CSIR both are same. Thus 2 2

'ˆ , , , &R T n� �EH R R represents that the CSI is known to both the ends. With the partial CSI model, the

channel output can be considered as ˆ 'y x x n� � �H E with the total noise given by ' x n�E with mean

zero and covariance matrix given by -

� �� �' ''t

H

n x n x n �� � �� � �R E E E (3.4.2)

1/2 ' 1/2 1/2 ' 1/2 2

2 2'

' ( )( ) ( )

( )R

R

H H H HR T T R n M

T R n M

XX

tr' ' �

� �

�� �� �� �E

E R E R R E R I

R Q R I (3.4.3)

where, the expectation is w.r.t. the distribution of x, n and ''E . It is known that Tn is not gaussian and it is

not easy to obtain the exact capacity relation. Thus tight upper and lower bounds can be taken in

consideration for system design. The mutual information with partial CSI for unpredictable capacity with

Gaussian distribution can be denoted as -

1

ˆ( , | )ˆ ˆ, log 2 |

� �

� �

H

I HQH RR t

L U

HL M n

I I x y I

where I (3.4.4)

71

� �2 22 2 'log | | ' log | ( ) |

t R

HU L n E T R n MI I x x� �� � �R E R R I (3.4.5)

&L UI I denotes the lower and upper bounds on the maximum achievable mutual information with the

error 'E considered over the distribution of x .

Since, the information is passing through the parallel subcarriers for the system with covariance matrix Q

is given, the achievable rates are given as -

'' 1 '( ,.... )nn n kC C C�

where, k denotes the number of subcarrier and with users,

' ' 2'

log det'R

H

Kkn N M

k N

C BN B

� �� �� �� �� �� �� �

��

HQHI (3.4.6)

where, 'NB is subcarrier bandwidth, Q is the covariance matrix and 'N is total number of subcarriers.

The capacity of uplink region with all subcarriers and users will then become -

1( ,...... )kC C C�

then (3.4.6) states that -

' 0, 1,..... , ' 1,..... ';KNQ k K n N( � � (3.4.7)

'

'' 1

& ( ) , 1,......N

kn kn

tr Q P k K�

� �� (3.4.8)

where, kP is the power constraint for k subcarrier or users and this (3.4.8) shows the inequality of the

uplink individual power constraint. It is known that (3.4.7) and (3.4.8) can be formulated with the

following total power constraint which will provide us the downlink channel capacity using the uplink

capacity region as given above [Vishwanath et al. (2003)] -

'

'1 ' 1

( )K N

knk n

tr Q P� �

��� (3.4.9)

where, 1

K

kk

P P�

�� is the downlink power constraint. Here, capacity lower bound (3.4.4) has been

considered for design criteria as also has been considered in [Musavian et al. (2007) ; Taesang et al.

(2004) ; Yoo and Goldsmith (2006)]. To obtain the highest data rate using the capacity lower bound

[Hassibi and Hochwald (2003)] , i.e., to get the best estimates out of all received data estimates, the

following problem is required to be solved with reference to the (2.4.57) as given in Chapter 2, [Musavian

et al. (2007) ; Taesang et al. (2004)],

72

2 2 20, { }

ˆ ˆmax log

( )RT

R

H

L MQ tr Q PE T R n M

Itr� �( �

� ��

HQHIR Q R I

(3.4.10)

where, the lower bound on the ergodic capacity is [ ]L LC E I� with the expectation w.r.t. the fading

channel distribution.

Alternatively, a common detection strategy for frequency selective MIMO channels is to achieve temporal

equalization first and then spatial equalization [Papadias (2004)]. Temporal equalization can be easily

found using blind equalization methods based on second order statistics (SOS) [Han-Fu et al. (2005) ;

Junqiang and Zhi (2001)]. These SOS based blind equalization schemes achieve temporal equalization

while leaving the ambiguity of an instantaneous combination which is equivalent to a narrowband MIMO

system [Inouye and Ruey-Wen (2002)]. The narrowband MIMO channel matrix can be denoted by

,[ ] 1 1� � � � �HR TM M R R T Th for M N and M N , where, ,R TM Mh denotes the non-dispersive channel

coefficient linking the TM transmitter to the RM receiver. In this scenario, a slow fading environment has

been considered for remaining all the entries unchanged in the MIMO channel matrix H at the time of

transmitting block or frame. The channel impulse response ,T RM Mh assumed to be i.i.d. complex valued

gaussian variable with zero mean and , 1. � � �T R

T

M ME h The modulation scheme considered is M-QAM

and the values from the defined M-QAM symbol set as -

� �, , 1 ,i q i qS S u ju i q M� � � �� (3.4.11)

with the real part symbol , 2 1i q iR S u i M � � � � � and the imaginary part symbol

, 2 1i q qI S u q M � � � � � . The average SNR of the system is defined as -

2 2/ / 2T s nS N N � �� � (3.4.12)

An spatial filter bank or equalizer are used to detect the transmitted symbols MT

S for 1 T TM N� � -

( ) ( ), 1T T

HM M T Ty k K M N)� � � � (3.4.13)

where, TM) is the 1RN � complex valued weight vector of the mth spatial equalizer. The MMSE solution

for the NT spatial equalizer considering the channel to be perfectly known can be shown by-

12

, 2

2 , 1��

� �

� � � �� �� �

HH IT R T

H nMMSE M N M T T

s

V h M N (3.4.14)

73

where, TMh denotes the mth coloumn of the channel matrix H. It is known that in spatial domain, the short

term power is applicable, i.e., No temporal power allocation can be considered. On the other hand it is

know that the power constraint is applicable across each antenna at each fading state for a given H . The

expectation of mutual information over fading channel distributed can be maximized by maximizing the

mutual information [Caire and Shamai (1999)] , i.e.,

0, { } 0, { },max [ ] max

T TL LQ tr Q P Q tr Q P

E I E I( � (

�� � � � (3.4.15)

Here, the right hand side is the maximum value which can be easily achieved with short term power

constraints and with the partial channel state information for which the author is trying to formulate more

convenient and robust pilot assisted scheme which can work for both time and frequency axis as discussed

in next section.

3.5. REQUIREMENT OF PROPOSED METHOD

Figure 3.5. Model diagram for the blind estimation of transmitted symbols using the MIMO systems

The block diagram in Figure 3.5 shows the model for the proposed method which will employ the

proposed training method and the channel estimation applicable for the same. Before analyzing the

proposed pilot assistance scheme, it is required to know that what basically the system wants.

When the transmitter is considered for both spatial pre-filtering matrix and power allocation, the mutual

information of the MIMO system corresponding to nth subcarrier is given by [Paulraj et al. (2003)] -

2

( ) ( ) ( ) ( ) ( )( ( ); ( )) log detR

H H

MT

H n n P n n H nI y n x n IM

' '*

!� �" #

$ % (3.5.1)

74

where, ( ) & ( )n P n' is the spatial pre filter matrix and power allocation. Assuming the pilot symbols be

J which can be denoted as � �(1) (2)..... ( )JZ Z Z Z J� and � �' '(1) '(2)...... '( )JX x x x J� as the available

training data. The channel estimation of the MIMO channel H using least square method relies upon

� �( ), '( )Z J X J which gave -

� � 1' ' 'ˆ H HJ J J JH z x x x

� (3.5.2)

and noise variance is given by this expression is

22 '1 ˆˆ2

.n J JR

Z HxJ M

� � (3.5.3)

It is required to have training pilots as low as possible, for which � �( ), '( )Z J x J should have full rank. To

achieve this, choosing � �max ,T R RJ M M M� � , i.e., assuming RJ M� which can be treated as the

lowest number of pilot symbols.

The MMSE solution gave the spatial equalizer weight vectors using roughly estimated value H of

channel-

12

2

ˆ2 ˆˆ , 1T R T

H nM M M T T

s

h M N�)

�

� �

� � � �� �� �

HH I (3.5.4)

where, ˆTMh denotes the mth coloumn of the estimated channel matrix H . The weight vectors (3.5.4) are

not sufficient to estimate correctly since because the pilot symbols are insufficient. It is not easy to use

direct decision adaptation, hence a new adaptive method has been proposed. However the covariance

matrix adaptation (CMA) and Soft decision directed (SDD) blind estimation [Chen and Chng (2004) ;

Chen et al. (2004)] can be utilized to adapt the spatial filters (3.4.13) with (3.5.4) as their initial weight

vectors.

3.6. PROPOSED PILOT ASSISTANCE SYSTEM

It is known that the pilot symbols should be placed in such a way that they appear repeatedly like

a periodic signal with the data signal. In this condition only, the channel will able to be identified at the

receiver, otherwise it is not going to steady state condition in the time varying channel with mobile

receiver. It is desirous to make the system in such a way that, it can track the data in the time varying

environment with less number of tracking symbols with the help of the non periodic pilot symbols.

75

As it is known that, the IEEE 802.16e standard for wimax has total 1024 subcarriers with 60

clusters in an OFDM symbol. If the system considers both the time and frequency axis as shown in Figure

3.6, which shows that the pilot is being inserted in the comb pilot fashion in each subcarrier of the cluster

which corresponds to the number of the MIMO antennas available at the transmitter. In next k processing

step, the pilot symbols are moved by one step ahead by one position of the symbol period and the same

process continues repeatedly for forming the block time pilot assistance into the lattice type pilot assisted

structure of combination of three processing steps, which responds to both the frequency and the time axis

and is able to be utilized in the fast fading environment too. If the pilot symbol is inserted on the first k

processing step of one antenna, another processing step is not going to transmit the pilot symbol on the

same subcarrier, which will be shifted by one position in each processing step. This will avoid the inter

antenna interference. Since, the system will be known about the periodic behavior of the pilot symbols, it

will be very easy to assume that the pilot symbols are periodic in nature whereas in real the next pilot

symbol will be arriving in one subcarrier after completion of the whole data segment. When the receiver

starts to detect this training symbol and send feedback to the transmitter to make it semiblind system, the

transmitter then will start to reduce to send the pilot symbols from different number of antennas at the

same time for the same data frame and this will be depending upon the tracking ability of the adaptive

receiver. This adaptive receiver repeatedly tracks to identify the most appropriate channel with the help of

the transmitted pilot symbols strength and its power level. When the desired channel has been locked with

the receiver for identifying the received signals, the feedback stage following the tracking by the receiver

will direct the transmitter to reduce the transmission of pilot symbols so that the data symbols may be sent

in lieu of the undesired transmitting pilot symbols as shown in Figure 3.7.

3.6.1. Pilot Symbol Assignment With Data Symbols

Considering the initial time period of arranging the pilot symbol in the data signal arranged with

pilots as JT which will be equivalent to the length of the smallest block over which the pilot symbol will

show its periodicity. The data signal with pilot symbols can be denoted as 1 2, ,.......... nj j jX x x x �� � � and the

pilot symbol can be denoted as � �1 2, ,..... nJ j j j� comprising of total time period of � �1n i i

J i JT X J�� + � .

Now assuming the indexing position to be defined as JI which will be positioned at the beginning of the

first subcarrier within one period and further it will move by one position but the JT will remain same for

all the conditions.

76

At the receiver end, the channel estimate H from the MMSE channel estimator will be sent to the

most likely (ML) detector since it is the global optimum sequence detector with the received signal vector

� �y k . For a particular indexing, the received symbol corresponding to the pilot symbol is JlT ky � , where,

� �0,1,....Jk I for l� � . The MMSE channel estimator will identify the entire available pilot symbol and

their corresponding data signal for both the current and the past conditions. The expression for the MMSE

channel estimator at the instant � �JlT k� for the index JI will become -

� �ˆ ( ) | ,J J JlT k J lT k lT k JI E y for j k j I� � �� � �H H (3.6.1)

Since, the MMSE estimator has been used, it requires to store all the past and present pilot symbols

sequences at the receiver as well as computing a matrix inversion at every time instant to estimate the

channel for the next coming data. This whole process increases the problem of memory and computation

both, for which adaptive filters or estimation schemes (which has been discussed in Chapter 6) can be

utilized for overcoming the problem created by the excessive burden by this. The estimation techniques

shown in Chapter 6 will consider these received signals with some data missing at certain forced positions

and will detect to find the estimates using expectation maximization technique as shown in section 3.2.5.

If the channel estimation is known h at processing step k , the optimal detection is given by ML detector

for the pilot symbol with equi-probable data. If the position of the pilot symbol is fixed at some particular

position PJ , the MMSE of the channel estimate at processing step k can be denoted as depending on any

received data with ZMCSCG,

� � � � � �� �2ˆ; P k P k PMMSE k J E h J h J� (3.6.2)

The ML estimation rules will be changed depending upon the type of the constellation like phase shift

keying constellation for which 2 2k xx �� and which shows that the ML detector used for the known

channel estimation can be used by replacing the channel estimate values as shown ahead -

� �� � � �

� �

� �

1* *

* *

2

ˆˆ arg max , |

ˆ ˆarg min

ˆarg max Re

ˆarg min

k

k

k

k

k k k kx

Tk k P k k Pk

x

k k P kx

k k P kx

x E y h x

y h J y h J

x h J y

y h J x

�

� �� � � � �

�� � �

�

� (3.6.3)

77

3.6.2. Pilot Symbol Optimization Criteria With Data Symbols

It is known that the means square error and bit error rate of the pilot based modulation system is

never stable. Therefore, it is required to send the pilot symbols till the channel is estimated and the data

starts to be received accurately at the receiver. The estimation at the receiver end will be done by the

adaptive method using the Gauss-Markov model which generally produces the inaccurate estimates at the

initial conditions but the adaptive nature of the proposed model starts to refine the accuracy of the system

in some good manner which will be discussed ahead. The updated and repeated channel MMSE

estimation after receiving the training block for � � ,P Jk J I�

� �� � � �� �� � � �� �

2 2 2 2

2 2 2 2 2

1 1;

1 1f J f h

J Pf J f h J

MMSE lT kMMSE lT k J

MMSE lT k

� �

� � �

� � � �

� � �

r r

r r (3.6.4)

The adaptive expectation maximization technique predicts the partial channel state information at the

transmitter end with the help of the estimated signal comprises of pilot symbols index’s position at the end

of the whole transmitted signal of duration JT . The MMSE in such case is given by -

� � � � � �2 2 21 1,2....k kJ J J J J h J JMMSE lT I k r MMSE lT I r for k T�� � � � � � (3.6.5)

It can be seen with this relation that if l ,- , then the system will receive the steady state periodic pilot

symbols positions which is not considered in partial channel state conditions and also the MMSE increases

with the increase in the processing steps k and depends upon the fading correlation coefficient Jr .

For making it feasible, it is assumed to have the index position, i.e., J J JY I X�� of the pilot symbol at the

last symbol in the thi data stream. Then the maximum MMSE in that particular block will be considered

equal to JX J� for which the energy will then considered as the -

� � � � � �: 1max max

JP

J k J Y Jk K J i nE T MMSE I MMSE I

. � ��� (3.6.6)

The optimized position of the pilot symbol with optimized MSE which minimizes the maximum stabilized

channel MMSE can be obtained as -

� �1

arg min ( ) arg min maxJ

J J

OptJ MMSE J Y Ji nI I

I E I MMSE I� �

� � (3.6.7)

here, it can be observed that the bit error rate is directly proportional to the channel MMSE, and the thesis

motive is to find the optimized positioning of the pilot symbol with less numbers which also can minimize

the maximum stabilized BER. Considering the Bit error rate of this stabilized MMSE at the first thk

position of the placement of the optimized pilot symbol, the following optimization is required -

78

� �:

arg min max :JJ

optJk K II

BER BER k I.

� (3.6.8)

where, opt optJ MMSEI BER� , when BPSK and QPSK modulation technique is considered under the

assumption of the Gauss-Markov channel model, if the channel estimator is used with ML detector.

3.6.3. Implementation Of The Optimization Technique In MIMO System

Assuming the data segment of size M, i.e., estimated data vector size is M+N-1, where, N is the

length of the Inter symbol interference (ISI) channel. It is required to calculate ( 1)2 M N� branch metric

which then be calculated by following relation of unknown estimates sequence / to avoid sacrifice of

tracking ability of channel, i.e.,

ˆ ˆ( ) ( )Tr rh h h h/ � (3.6.9)

where, 1ˆ ˆ .h a y� and rh is the reference channel which is an ideal ISI free channel of the same standards

and span length as the blindly estimated channel.

Consider (L+1) bits transmitted through an ISI channel of length 2 (using Jake’s model) in which

L samples which contains the desired block of information. Now it is required to process M samples from

these L samples of information by moving one sample forward in each step for getting the M size segment

vector. By doing so, k processing steps which equal L-M+1 will be formed which has been shown in

Fig.3.6.

Figure 3.6. Proposed adaptive pilot assistance training before iterations.

79

Figure 3.7. Proposed adaptive pilot assistance training with reduced pilot symbols and high power levels after

number of iterations which resembles like the lattice type pilot schemes as shown in section 3.1.3.

Now calculating the branch metric � �( 1)' 2 M NR � � for k steps followed by calculating the path

metric for all possible paths from the values in matrix H Now choosing the path with minimum path

metric or gain ,i L0 and track the bits through the path which will be considered as detected sequence.

Now take a short time average value of the detected sequence, i.e.,

.K K KJ1 /� (3.6.10)

Further, calculating the branch metric K1 for all possible estimated vector gives us the selected

estimates with minimum branch metric K1 . This unique set of selected estimates with minimum K1 can

be processed further to track surviving states with minimum value from the matrix H and eventually the

possible block of transmitted sequence which has been shown in Figure 3.7. and the algorithm for the

same has been shown in table 3.1.

80

Table 3.1. Algorithm for the proposed adaptive pilot assisted estimation

Proposed Algorithm

1 . Assuming data segment of size M, i.e., estimated data vector size is M+N-1. 2 . Calculate branch metric 2(M+N-1) using unknown estimate sequence � relation:

ˆ ˆ( ) ( )Tr rh h h h/ �

3 . Calculate tracking ability of the channel, if found suitable, then terminate.

4 . If not found suitable, process M samples from the L samples of information block.

5 . Move one sample forward in each step for getting M size segment vector.

6 . Process the forwarding of sample for k times.

7 . Again calculate the branch metric using step 2 for all possible paths.

8 . If found suitable, calculate path metric for all possible paths from the values in H .

9 . Choose path with minimum path metric or gain ,i L0 .

10. Track the bits through the chosen path (optimum path).

11 . If channel gets changed, restart step3.

12 . Take short time average value of the detected sequence found in step 10, i.e.,

.K K KJ1 /�

13 . Calculate the minimum path metric K1 for all possible estimated vectors.

14 . Repeat again if surviving state with minimum value not found, otherwise terminate.

To implement this proposed method with (3.5.4), the weight vector of mth spatial equalizer has

been assumed with output at step k as ' ( ) ( ) ( )TMy k k Z k)� . As discussed above,

1 2( ), ( ),...... ( )T T T TM M M M mk k k) ) ) )� , where, ( )

TM mk) denotes the weights for the kth step which

corresponds to the M samples from the L samples of information. Now 1( )TM k) will be processed one

sample forward resulting ( )TM m ik) � , where, i denotes the one step forwarding of each weight vector.

Calculating the branch metric R for k steps followed by calculating the path metric for all possible paths -

( ) ( )T TM m M k kk k Z) ) 1� � (3.6.11)

Finally, the weight vector TM) is updated at each step using number of iterations which gives the

minimum required pilot symbols to detect the signal in a semiblind manner as shown in Figure 3.7. This

updating process of weight vectors has been utilized here from [Chen et al. (1995)]. Since, M-QAM

81

modulation has been considered, the complex values will be divided in M/4 regions, each containing four

symbol points as -

� �, ,' , 2 1, 2 , 2 1, 2i l r sS x r i i s l l� � � (3.6.12)

where, 1 , / 2i l M� � . If the spatial equalizers output ,( )TM i ly k S� , marginal probability density

function approximation has been given by [Chen and Chng (2004) ; Chen et al. (2004)] -

2,( )

2 22

2 1 2 1

1ˆ( , ( )8

M r sT

T T

y k xl i

M Ms l r i

P y k e 2)�2

� �

� � � (3.6.13)

where, 2 denotes the segment size associated with each segment of each region ,i lS .

Now, considering the forwarded step of weight ( 1)TM k) � , which has been updated as -

( 1)

( ( 1), ( 1)( 1) ( 1)

( 1)T T

T T

T

pil M MM M k

M

J k y kk k

k)

) ) 1)�

� � �� � � �

� � (3.6.14)

where, � �ˆ( , ) log ( ( ), ( ))T T T Tpil M M M MJ y P k y k) 2 ) �� � � , in which the log of the marginal probability

distribution function (p.d.f.) is to be maximized using a stochastic gradient optimization [Chen and Chng

(2004) ; Chen et al. (2004)] and

� �2

,( 1)2 2

2,

2 2 1

{ ( 1) ( 1)} 1 ( 1) . ( 1)( 1)

M r sT

T T

T

T

y k xi l

pil M Mr s M

r i l s lM

J k y ke x y k Z k

k2)

) *

�

� �

� � �� � �

� � � � (3.6.15)

with the normalization factor

2,( 1)

2 2 2

2 1 2 1

M r sTy k xi l

r i s le

2*

�

� �

� � �

It is required to maintain the value of 2 less than 1 since because minimum distance between two

neighboring constellation point is 2, which further will ensure the power separation of the four clusters of

,i lS . If the value of 2 is chosen less, then the algorithm may tightly control the segment size and may

create problem in identifying the proper estimates. On the other hand, on increasing the value of 2 ,

degree of separation may not be achieved desirably. For higher SNR conditions, small value of 2 is

required whereas for lower SNR conditions, larger 2 is required since the value of 2 is related to the

variance which is 22T T

Hn M M� ) ) . After receiving the information received by the use of pilot symbol in the

form of initial weight vector (3.5.4), if it is compared with the pure blind adaptation case as in [Chen and

Chng (2004) ; Chen et al. (2004)], smaller value of 2 can be used which also gives us the study

82

performance. Alternative estimation are also considered in region (3.6.12) that includes [ ( )]TMq y k , i.e.,

single hard estimation, where, q denotes the quantization operator and each arriving estimate is weighted

by an exponential term, which is a function of the distance between the equalizer’s calculated output

( )TMy k and the arriving estimate ,r sx . This interactive calculation by equalizer will substantially reduce

the risk of propagation error and also gives fast convergence, compared with [Chen and Chng (2004) ;

Chen et al. (2004)].

3.7. PERFORMANCE ANALYSIS

Simulations for the pilot symbols have been performed and the Constellation analysis, pilot length

analysis, and MSE analysis have been compared with [Chen et al. (2004)] which has simulated the blind

space time equalizer for Single Input Multiple Output (SIMO) systems using constant modulus algorithm

and soft decision directed algorithms. The proposed scheme in this Chapter has shown the better results in

case of the semiblind environment which is moreover very near to the perfect CSI conditions. The

received constellations for the adaptive pilot schemes have been shown in Figure 3.8 – 3.14 for different

conditions of correlation coefficients and higher SNR levels. Similarly, MSE comparison with [Feng, Zhu,

et al. (2008)] has also been shown in Figure 3.15 which shows that the said has proposed the training

based least square criterion along with a blind constraint for MIMO-OFDM following the weighing in the

semiblind cost function, whereas, the proposed scheme in this Chapter has been utilized in Chapter 6 for

implementing with the MIMO-OFDM scheme which has shown better results due to the adaptive nature

of the proposed method. Here, the adaptive nature of the APASBCE scheme changes accordingly to the

requirement of the pilots for robust estimation in the system till the channel is remained acquired for the

estimation e.g. if the initial pilot requirement of pilots is 32 in case of 512 subcarriers or users, for

acquiring the channel initially. After acquisition of channel, the APASBCE scheme will start reducing the

pilots and reaches to the level of two minimum till the channel is firm. If the channel dropped due to fast

fading environment, then the process will be restarted to acquire the channel using 32 pilots again. The

MSE comparison of the proposed scheme has been compared with the propositions in [Feng, Zhu, et al.

(2008)], i.e., training based Least Square (LS) approach with blind constraints, nulling based algorithm in

[Feng, Wei-Ping, et al. (2008)] and whitening rotation based semiblind channel estimation in

[Jagannatham and Rao (2006)] and found that the proposed scheme in this Chapter is performing better

than the above said methods and finally found the results as shown in Figure 3.15. Further, MSE for Blind

channel estimation found in [Cui and Tellambura (2007)] has been compared in Chapter 6 with the

application of proposed scheme with OFDM.

83

Figure 3.8 to Figure 3.14 have been shown for different training sequences that have been deployed with

the transmitting signal sequence. Here, in Figure 3.8, constellation for training sequence in transmitting

signal sequence has been shown for the MIMO channel with partial CSI with the correlation coefficient of

0.5 between the transmitting antenna elements, whereas in Figure 3.9, Gold training sequence has been

taken in consideration with correlation coefficient of 0.5. The gold sequence is denoted as � �2 1

0

nn n n

Z Z

��

with length 2n , where, 1 , 01 , 1

nn

n

ZZ

Z� �

" �$. This generation is repeated for m times depending on the

number of sub-channels. The resultant modulated signal for mth will be 2exp

2mm m

n n nS U C U j mnM�� �� � � �

� �

which have a good periodic autocorrelation characteristic. It can be seen that the comparison of both

Figure 3.8 and 3.9 shows that the gold sequence constellation is tightly bounded under the exploring

region and does not interferes the other signals.

Fig.3.8. Training sequence constellation in channel with known channel state information with correlation

coefficient of 0.5.

84

Figure. 3.9. Training sequence constellation in channel with known channel state information with correlation

coefficient of 0.5 for gold training sequence.

Figure 3.10. Training sequence constellation of the training sequence in channel with partial channel state

information with correlation coefficient of 0.1 for proposed scheme with less SNR values.

85

Figure 3.11. Training sequence constellation for channel with partial CSI and proposed estimation scheme for

correlation coefficient of 0.5.

Figure 3.12. Training sequence constellation for SNR 35dB with correlation coefficient of 0.1 for channel with

partial CSI.

86

Figure 3.13. Constellation of the training sequence in channel with partial CSI using APASBCE scheme and

correlation coefficient of 0, i.e., no correlation at transmitter end.

Figure 3.14. Training sequence analysis for SNR 35dB with correlation coefficient of 0.5 channel and partial CSI

with proposed estimation scheme.

87

In Figure 3.10 and Figure 3.11, the training sequence with partial CSI channel and proposed

estimation based channel has been shown, in which Figure 3.10 significantly shows that the constellation

is moving away or reaching at the edges of its exploring boundaries when the correlation coefficient of 0.1

has been considered, whereas in Figure 3.11, where, the proposed scheme has been implemented with the

correlation coefficient of 0.5, can be easily seen exploring inside its region.

Similarly Figure 3.12 and Figure 3.13 has been found for the higher SNR with correlation

coefficient of 0.1 (very less correlation) and 0 (i.e., no correlation between antenna elements), in which it

can be seen that for the partial CSI channel conditions, the constellation of the training sequence are

moving out of the exploring region (which is higher in Figure 3.13) and may not be detectable at the

receiver end, which leads the system to estimate the transmitted signal incorrectly. Whereas in Figure

3.14, it can be seen that a few training symbols are diverged with the proposed APASBCE scheme but are

confined under the exploring boundaries even at higher SNR level of 35dB with correlation coefficient of

0.5, which leads the MIMO system to estimate the data at the receiver end. To make the symbols more

confined, number of iterations can be raised which will lead to more accurate symbol confinement.

Figure 3.15 shows the comparative analysis of the Mean Square Error found with different

existing schemes with the proposed pilot estimation method in APASBCE scheme. The Figure shows the

improvement in the APASBCE scheme as compared with the LS scheme proposed by [Feng, Zhu, et al.

(2008)] by at least 1dB and by 1.2dB when compared with the nulling based LS estimation in [Feng, Wei-

Ping, et al. (2008)] at the SNR level of 30 dB whereas it is quit much more than the MSE found with the

whitening rotation based algorithm in [Jagannatham and Rao (2006)].

Figure 3.16 shows the power amplification of the pilot symbols contributing to the channel

estimation purpose. The Figure shows the improvement of the APASBCE scheme pilot symbol power

amplification as compared with the closed form semiblind channel estimation (CFSB) scheme proposed in

[Murthy et al. (2006)]. The result shows that the APASBCE scheme is giving better output with 0.1 dB as

compared with the CFSB due to the use of the MMSE estimation used in the proposed scheme, but still

less than the perfect channel and closed form semiblind perfect channel estimation (CSFB perfect). This

can be increased with the increase in number of iterations at the cost of time and complexity of the system.

88

Figure 3.15. Comparison of MSE proposed with the existing MSE.

Figure 3.16. Average power amplification associated with the pilot symbols for the channel gain with

2T RM M� � MIMO antenna system.

89

Figure 3.17. BER comparison of the training symbols proposed in APASBCE scheme with the existing semiblind

and blind schemes proposed by authors shown in caption.

The BER comparative analysis of the training symbols proposed in APASBCE scheme with existing

semiblind and blind channel estimation schemes proposed by [Cui and Tellambura (2007) ; Feng, Zhu, et

al. (2008) ; Murthy et al. (2006)] has been shown in Figure 3.17.

3.8. SUMMARY

In this Chapter, different methods for the insertion of pilot symbols into the transmitting signal of

MIMO antenna system have been presented. The pilot insertion methods have been combined with

different algorithms for the estimation of the channel variations in frequency and time domain. Further on,

the pilot insertion methods have been investigated in terms of their respective influence on constellation of

the received pilots in the signal. For the introduced pilot assisted channel estimation algorithms, the pilot

symbol parameters as MMSE, SNR and BER and the MSE performance, has been presented and the

results have been analysed and compared in the previous section.

The sampling theorem in frequency and time domain has to be fulfilled in order to estimate the

channel variations in frequency and time domain by pilot assisted channel estimation. That means, at least

one subcarrier per coherence bandwidth of the channel and one Interleaved Frequency division multiple

Access (IFDMA) symbol per coherence time of the channel has to be used for pilot transmission. For

90

IFDMA, the fulfillment of the sampling theorem in frequency and time domain entails the usage of each

of the K allocated subcarriers for pilot transmission in frequency domain if the number of channel delay

taps is larger than K . That means, for IFDMA, interpolation in frequency domain is solely feasible if K

is large, i.e., the transmitted data rate is high. The application of subcarrierwise pilot insertion with Wiener

interpolation in frequency and time domain leads to the best MSE performance in comparison to the other

presented algorithms. The changes made according to the requirement of the MIMO system with our

motive has been shown in Figure 3.18 by marking the particular block in yellow colour.

Figure 3.18. Changes made according to the requirement of the motive of this Chapter.