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Chapter 2 OFDM System

2.5 The Pros and Cons of OFDM System

Since the OFDM system transmits several narrowband signals instead of a wideband

signal, a frequency selective fading channel can be transformed into a flat fading channel

over each OFDM sub-channel if the sub-channel is sufficiently narrow-banded. As a result,

an one-tap equalizer can be applied for OFDM systems, and it is much simpler than the

equalizer used for conventional single-carrier modulation systems.

Besides, the longer symbol duration with the insertion of the guard interval makes

OFDM systems less sensitive to timing synchronization errors as well as more robust

against the ISI effect, as compared with the single-carrier modulation systems.

In addition, the subcarrier frequencies are chosen so that all subcarriers are orthogonal

to each other and therefore the OFDM system can achieve higher spectral efficiency. This

orthogonality also permits the use of the FFT operation for the efficient implementation of

the modulator and the demodulator in OFDM systems.

However, there are some disadvantages in the OFDM systems and they are discussed in

the following. First of all, because of the orthogonality of subcarriers, the OFDM system is

very sensitive to carrier frequency synchronization, and the imperfect frequency

synchronization will cause ICI among subcarriers.

Moreover, the OFDM systems usually suffer from the problem of large peak-to-average

power ratio, abbreviated as PAPR. This is due to that fact that when P independent data

symbols modulated onto subcarriers within an OFDM symbol are added coherently (with

the same phase) in time domain after the IDFT operation, the peak power of the time

domain signal is P times larger than the average power (with the assumption of constant

power modulation scheme, e.g. QPSK). For non-constant power modulation scheme, e.g.

16QAM, the PAPR will become much larger. A large PAPR will increase the cost of

analog-to-digital and digital-to-analog converters and decrease the power efficiency of

power amplifier due to the effect of non-linear distortion.

Chapter 3

DFT-Based Channel Estimation

Blind channel estimation, which merely relies on the received signals, is very attractive

due to its bandwidth saving advantage. However, it requires a long data record, involves

high computational complexity and only applies to slowly time-varying channels. On the

contrary, pilot-aided (PA) channel estimation, which uses pilot tones known to the receiver,

is widely applied in mobile wireless communication, despite of the fact that the use of pilot

tones ends up with lower data rates.

A wide variety of PA channel estimation methods have been proposed. Among these

methods, DFT-based channel estimation, derived from either ML criterion or MMSE

criterion, for OFDM systems with pilot preambles was intensively investigated. The

advantage of the former is simpler to implement, as no information on the channel statistics

or the operating signal-to-noise ratio (SNR) is needed in this scheme [1] [2] [5]. On the

other hand, DFT-based channel estimation with the MMSE criterion is expected to have

better performance as it exploits prior information about the channel statistics and the

operating SNR. Furthermore, it has been shown that for DFT-based channel estimation and

at intermediate or high SNR values, the performance of an ML estimator is comparable to

that of an MMSE estimator when the number of pilot tones is sufficiently larger than the

maximal channel length (in samples) [2].

The DFT-based channel estimation method implemented through four components: a

least-square (LS) estimator, an inverse DFT (IDFT) matrix, a weighting matrix, and a DFT

matrix, as shown in Figure 3.1.

Figure 3.1 The block diagram of the DFT-based channel estimation method

The LS estimate is a noisy observation of channel frequency response. After taking the

IDFT operation to transform the estimate to time domain, we can improve this estimate by

using a weighting matrix which depends on the performance criterion used. Finally, the

enhanced estimate is transformed back to frequency domain to obtain a new estimate of

channel frequency response.

3.1 Signal Model

We first describe the signal model for the received OFDM symbol as

( ) ( ) ( ) ( ),

y n =h nx n +z n (3.1)

where n=0, ,N G+ − is the time index, N is the total number of subcarriers, G is 1

the length of guard time, ( )y n is the received OFDM signal, ( )x n is the transmitted pilot

signal (including the guard interval), ( )h n is channel impulse response with finite length

G, and ( )z n is additive white Gaussian noise (AWGN) and * denotes the convolution.

After removing the guard interval and transforming Eq.(3.1) into frequency domain, we

get

subcarriers (or FFT size) and N is a FFT normalization factor.

The system model can be rewritten as a vector form:

,

diagonal matrix containing the transmitted pilot symbols, { (0)X X N( −1)} ,

and is modeled as independent and identically distributed (i.i.d.) complex Gaussian random

vector with zero mean and variance σn N2I .

3.2 Maximum Likelihood Estimator

The LS estimate for the channel frequency response can be obtained by

2

and p(HˆLS) is a constant value irrelevant to h . With the assumption of ( )p h is the

As a result, we obtain the channel estimation corresponding to h as follows:

ˆ ˆ

response is composed of two components. One is the true channel impulse response and the

other is the noise term.

Chapter 4

Conventional Path Selection Methods

In order to obtain more accurate channel estimation, we can use path selection methods

to suppress the noise in the ML estimate of Eq.(3.11). That is, we pick and reserve

significant channel paths by setting the remaining elements in ˆ

hML as zero.

Figure 4.1 depicts the block diagram of channel estimation with path selection. The LS

estimate of channel frequency response is first transformed into time domain to obtain the

corresponding channel impulse response. Afterward, the estimate of channel impulse

response is passed through a path selection unit to get a refined estimate. This refined

estimate is then transformed back into frequency domain to obtain the estimated channel

frequency response.

Figure 4.1 The block diagram of the DFT-based channel estimation method with path selection.

This chapter introduces two conventional path selection methods in common use:

threshold setting method and number of path setting method. The main strategy of these two

conventional path selection methods is to select those elements with larger amplitude (or

energy) in ˆ

hML and to suppress noise by setting the remaining elements as zero.

4.1 Threshold Setting Method

In the threshold setting method, we first define a threshold and the maximum energy of

the ML estimate of channel impulse response hˆML as TdB and max

{

hˆML( )n 2

}

,

respectively. In order to select main paths, we reserve those elements in ˆ

hML whose energy (in dB scale) is larger than the value of max

{

ˆML( )2

}

dB- TdB

h n , where max

{

ˆML( )2

}

dB

h n is

in dB scale, and then discard all the remaining elements by setting their values of hˆ ( )ML n as zero. The algorithm of the threshold setting method is presented as follows.

Denote TL as the linear scale of TdB, i.e.,

TdB

10 .10

TL = (4.1) Then we can express the estimated channel impulse response with path selection as

2 2

For example, TdB can be set to 20 dB. The elements of the ML estimate of channel transformed back to frequency domain to get the estimated channel frequency response, i.e.,

ˆth =FFT{ }ˆth

H h . (4.3)

4.2 Number of Path Setting Method

To improve the ML estimate of channel impulse response, the number of path setting

method first defines a parameter Np, which represents the desired number of paths. Only

Np elements with larger amplitudes in hˆML are preserved and the other paths are discarded. In consequence, the algorithm of the number of path setting method is presented

in the following: [7]

For example, let Np be 10. Then, only 10 elements with larger amplitude are said to be

the valid elements while the remaining paths are considered as noise and set as zero.

Similar to the threshold setting method, the refined estimate, ˆh , is finally transformed p

back to frequency domain to get the estimated channel frequency response, that is,

ˆ p =FFT{ }ˆp

H h . (4.5)

Chapter 5

Proposed Path Selection Method

Although the conventional path selection methods in the previous chapter, including the

threshold setting method and the number of path setting method, are widely used for

improving channel estimation, the drawback of these two methods is that they are heuristic

approach to the problem and sensitive to channel power delay profiles as well as the

operating SNR.

In general, how to set the threshold, TdB, in the threshold setting method, is relevant to

the structure of multipath power delay profiles. If the threshold is set too large, the path

selection method might pick noise. On the other hand, if the threshold is set too small, the

path selection method might lose true channel paths. Therefore, it is difficult to set a proper

threshold for all kinds of channel environments, and improper setting of the threshold will

decrease the performance of channel estimation significantly.

As to the number of path setting method, it is also difficult to know how many paths

exist in wireless channel environments. Thus, it is potentially required to estimate channel length to choose a proper parameter Np that represents the desired number of paths, or the

system may suffer from performance degradation. Similar to the threshold setting method,

Np with a too large value makes noise included in the estimated channel impulse response, and Np with a too small value excludes the true channel paths in the estimated channel

impulse response.

As shown in Figure 6.45 and 6.46 in the next chapter, the simulation results show that

inaccurate path selection raises the average SE of channel estimation, especially for the case

of losing channel paths. Furthermore, inaccurate path selection influences not only the BER

performance of the system, but also the complexity of channel tracking since the channel

paths are usually tracked path-by-path in the tracking stage. Selecting more paths than the

true channel paths existing in practical environments increases the complexity of the

channel tracking. For example, in a two-path channel, the complexity of picking 10 paths in

path selection methods is fivefold than the complexity of picking 2 paths.

According to the aforementioned discussion, we find that the conventional path

selection methods are sensitive to the setting of parameters, channel conditions and the

operating SNR. Thus, we would like to develop a novel path selection method which can

improve the BER performance, reduce the average SE of channel estimation, and increase

the probability of picking correct paths. The cost function for the proposed path selection

method is then presented in the next section.

5.1 Cost Function

Eq.(5.1) and Eq.(5.2) represent the ML estimate for channel impulse response by using

the signals on even and odd subcarriers, respectively:

2 vectors (or matrices) which involve even and odd subcarriers, respectively.

Afterward, a variable A which is a diagonal matrix of size ( , )G G is further

introduced to indicate which elements in ˆ

heven and ˆ

hodd are desired paths. Thus, the cost function for the proposed path selection method can be formulated as follows:

2 2

denoted as ˆ( )A n . If ˆ( ) 1A n = , the nth element of ˆh is considered as a valid path;

could be deemed as the estimated channel impulse response (corresponding to channel h ),

corrupted by noise. Note that the elements of z and 1 z are denoted as 2 z n1( ) and

2( )

z n , respectively, for n=0,1, ,G− , each of which is with zero-mean and variance 1

2 2

2σ σn /( XN).

The meaning of the cost function can be analyzed by substituting Eq.(5.4) and Eq.(5.5)

into Eq.(5.3). We discuss the problem with two cases: a path does or does not exist at the

nth element. When there is indeed a path at the nth element of h , if ( ) 1A n = , the cost

function with respect to the nth element of Eq(5.3) becomes as

z n1( )−z n2( )2+ z n2( )−z n1( )2 (5.6)

and if ( ) 0A n = , the cost function with respect to the nth element of Eq(5.3) becomes as

h n( )+z n1( )2+ h n( )+z n2( )2. (5.7)

As we can see, value of Eq(5.7) tends to be larger than value of Eq(5.6) when the

amplitude of a channel path is sufficiently larger than noise, and this result sides with the

solution of ( ) 1A n = . On the other hand, when there is no path at the nth element of h ,

if ( ) 1A n = , the cost function with respect to the nth element of Eq(5.3) becomes as

z n1( )−z n2( )2+ z n2( )−z n1( )2 (5.8)

and if ( ) 0A n = , the cost function with respect to the nth element of Eq(5.3) becomes as

z n1( )2+ z n2( )2. (5.9)

In this case, the variance of Eq(5.8) is twice the variance of Eq(5.9), and this result sides

with the solution of ( ) 0A n = .

5.2 Optimum Solution

To solve the minimization problem in Eq(5.3), we expand it as follows:

( ) ( ) ( ) ( )

even odd even odd odd even odd even

A

even even even odd odd even odd odd

A

Obviously, the minimization problem of Eq.(5.10) can be viewed as an ILP problem or a 0-1

programming problem; that is, we have [10]:

{ }

min f(0) (0)A + f(1) (1)A + + f L( −1) (A L−1) ,

A (5.12)

where A n( ) and f n( ) , for n=0, , -1G , are diagonal elements of A and f ,

respectively.

Since the optimum solution of a linear programming (LP) problem can only be vertices

of the feasible set [10], the ILP problem of Eq.(5.14) can be equivalently transformed into a

simple LP problem. To do this, we use 0≤A n( ) 1≤ , for n=0, , -1G instead of the

integer constraint of ( )A n , i.e., ( ) 0 or 1A n = . The process of this constraint release is

reasonable because for this new constraint, each element of the vertices in the feasible set is

either 0 or 1. Consequently, the ILP problem of Eq(5.12) can be transformed into a simple

LP. Even though the LP problem can be solved by the simplex method, the computation still

requires considerable effort to achieve the optimal solution. In order to reduce the

computation complexity, we then develop a more simple method to find the optimum

solution. In fact, it can be easily observed that the optimum solution of Eq(5.12) is to make

( )

A n corresponding to the negative value of ( )f n be one; otherwise, ( )A n is set to zero.

As a result, we have the optimum solution of Eq(5.14):

1, if ( ) 0

suppress noise and reduce the probability of false alarm by finding a proper threshold which

is insensitive to channel power delay profiles.

5.3.1 Analysis of False Alarm

By substituting Eq.(5.4) and Eq.(5.5) into Eq.(5.11), we can obtain

ˆ ˆ ˆ ˆ ˆ ˆ

According to Eq.(5.13), the analysis of false alarm can be done point-by-point due to the

fact that the vaule of ˆ( )A n merely depends on the value of ( )f n and the elements of z 1

the position n. Instead of deriving a closed-form probability density function (PDF) for the

variable ( )f n in Eq.(5.15), we use bootstrap (or resampling) techniques in Monte Carlo

methods to simulate the PDF. The basic idea of the bootstrap is to evaluate the PDF through

the empirical samples [11], and the PDF of ( )f n in Eq.(5.15) is then simulated in Figure

5.1. As shown in Figure 5.1, even if there is no path at the position ,n there is still a

certain probability for the negative value of ( )f n , resulting in the event of false alarm.

Although the range of the values for the variable ( )f n in Eq.(5.15) varies with the

variance of ( )Z k , the shape and proportion of the PDF is invariant to the value of

2/( 2 )

n XN

σ σ . To reduce the probability of false alarm, we propose a refined path selection

method in the next subsection.

Figure 5.1 The PDF of ( )f n evaluated through the empirical samples with 1,000,000 samples. generated from the random variable ( )f n . Therefore, we propose the refined path selection

method to suppress noise and to reduce the probability of false alarm in the following:

1, if ( )

5.3.3 Determination of the Parameter U

The determination of the value of the parameter U will depend on the probability of

false alarm and miss detection. Moreover, the probability of miss detection is related to the

strength of a channel path. In this subsection, we first analyze the influence of the miss

this path will introduce channel estimation error ∆H k( ) in the estimated channel

frequency response, i.e., we have the estimated channel frequency response:

ˆ ( ) ( ) ( )

for k =0,...,N− . From Eq.(3.2) and Eq.(5.18), the received signal after channel matching 1

can be written as

2 2

By using the assumption of Eq.(5.22), SNER of Eq.(5.21) can be approximated as

2 2

significant influence on the SNER. That is, when N µ, we have

2 2

In this thesis, N is set as 256. Hence, we will concern the probability of miss detection of

a path with energy h n( )2 =25σn2/

(

σX2N

)

as a design reference, i.e., set µ =25.

Figure 5.2 shows the CDF of the false alarm (µ = ) and the miss detection (0 µ=25)

of the variable ( )f n . According to the solid line, we have Pr(f n( ) 23.2≥ σn2)=1/64. In

other words, among f n( ) , for n G= ,...,

(

N/ 2 1

)

, the occurrence of the event of

{

f n( ) 23.2 σn2

}

is

( (

N/ 2

)

G

)

/ 64 on average. For example, when N and G are set as 256 and 64, respectively, we can acquire one point of ( )f n whose value is larger than

23.2σn2, i.e., we have max f n{ ( )} 23.2≥ σn2 and therefore Rth ≤ − ×U 23.2σn2 (in average

sense). Accordingly, Table 5.1 lists the probability of false alarm (µ = ) and miss detection 0

(µ=25) with U as a parameter. We denote RNth and f nN( ) as normalized terms, i.e.,

( )

(

2 2

)

/ /

Nth th n X

R =R σ σ N and f nN( )= f n( ) /

(

σn2/

(

σX2N

) )

. From this table, we can determine the value of U which can achieve a desired probability of false alarm and miss

detection. In this thesis, we can choose the value of U as 0.5 such that the probability of

false alarm is less than 7.55 10× 4 and the probability of miss detection is larger than

102.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Figure 5.2 The CDF of ( )f n evaluated through the empirical samples with 1,000,000 samples.

Table 5.1 The probability of false alarm and miss detection with U as a parameter.

U RNth ≤ False Alarm(µ= ): 0

The algorithm of Eq.(5.16) can be implemented by sorting the values of ( )f n , for than the two conventional path selection methods, aforementioned in the previous chapter.

{ ( )},

Figure 5.3 The proposed algorithm can be implemented by sorting the values of ( )f n , for 0,1, , 1

n= G− , in ascending order

Chapter 6 Simulation Results

In this chapter, we simulate the BER, average SE and probability of picking wrong

paths to demonstrate the performance of our proposed path selection method for channel

estimation in OFDM systems. Besides, we also compare the performance with the two

conventional path selection methods, including the number of path setting method and the

threshold setting method.

Table 6.2 Power delay profiles of channel environments ITU-Veh. A channel 0, -1, -9, -10, -15, -20 (dB) ITU-Veh. B channel -2.5, 0, -12.8, -10, -25.2, -16 (dB)

Two-path channel 0, -1 (dB)

Thirty-path exponentially decayed channel 0, -1.3029, -2.6058, -3.9087, -5.2116, -6.5144, -7.8173, -9.1202, -10.4231,

The simulation parameters are listed in Table 6.1. Throughout the simulations, carrier

frequency synchronization and symbol timing synchronization are assumed to be perfect.

Moreover, the simulations are conducted at baseband using the complex low-pass

equivalent representation. The ratio of energy between the pilot signal and the data signal

(on a subcarrier) is set to 1.

Only the small-scale fading is considered in our simulations. Besides, we use four

typical channel power delay profiles, including International Telecommunication Union

(ITU)- Vehicular A and Vehicular B fading channels, a two-path equal power fading

channel, and a thirty-path exponentially decayed fading channel, to demonstrate the

performance. The power delay profiles defined by the recommendations of the ITU are

well-established channel models for research of mobile communication systems. They

specify channel conditions for various operating environments encountered in

third-generation wireless systems, e.g the Universal Mobile Telecommunication Systems

(UMTS) Terrestrial Radio Access System (UTRA) standardised by 3GPP[12]. Both the Veh.

A and Veh. B channels are six-path channels with power delay profiles: 0, -1, -9, -10, -15,

-20 (dB) and -2.5, 0, -12.8, -10, -25.2, -16 (dB), respectively. For the two-path equal power

fading channel, the power delay profile is 0, 0 (dB). For the thirty-path exponentially

decayed fading channel, the power delay profile (linear scale) is given by [13]:

( )

delay profile of the thirty-path channel is listed in Table 6.2.

6.1 Threshold for Refined Path Selection Method

In Section 5.3, we set the value of U as 0.5 for the threshold Rth = − ×U max f n{ ( )}

in the refined path selection method. The algorithm of the refined path selection method is

that if ( )f n is smaller than the threshold Rth = −0.5×max f n{ ( )}, we say that there is a

10dB and 40dB, respectively, with U as a parameter. Figure 6.5 and Figure 6.7 show the

BER performance for the proposed path selection method in the thirty-path channel at

0

E N =10dB and 40dB, respectively, with U as a parameter. Figure 6.6 and Figure 6.8 b

show the average SE performance for the proposed path selection method in the thirty-path

channel at E N = 10dB and 40dB, respectively, with U as a parameter. Figure 6.9 and b 0

Figure 6.11 show the BER performance for the proposed path selection method in the

two-path channel at E N =10dB and 40dB, respectively, with U as a parameter. Figure b 0

6.10 and Figure 6.12 show the average SE performance for the proposed path selection

method in the two-path channel at E N = 10dB and 40dB, respectively, with U as a b 0

parameter.

We can observe that for the BER performance shown in Figure 6.1, Figure 6.3, Figure

6.5, Figure 6.7, Figure 6.9, and Figure 6.11, the threshold R which ranges from 0 to th

6 max f n{ ( )}

− × has no significant influence on BER performance of our proposed method.

As shown in Fig. 6.2 and Fig. 6.4, we can find that for the Veh. A channel, the minimum

average SE is achieved at a threshold between 0.4− ×max f n{ ( )} and 0.6− ×max f n{ ( )}.

Moreover, as shown in Figure 6.6 and Figure 6.8, we can find that for the thirty-path

Moreover, as shown in Figure 6.6 and Figure 6.8, we can find that for the thirty-path

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