正交分頻多工系統中用於通道估計之新型路徑選取方法

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A Novel Path Selection Method for

Channel Estimation in OFDM Systems

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A Novel Path Selection Method for

Channel Estimation in OFDM Systems

Student

Hsin-Yi Tu

Advisor

Dr. Chia-Chi Huang

A Thesis

Submitted to Department of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Communication Engineering

July 2007

Hsinchu, Taiwan, Republic of China

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A Novel Path Selection Method for

Channel Estimation in OFDM Systems

Student

Hsin-Yi Tu Advisor

Dr. Chia-Chi Huang

Department of Communication Engineering

National Chiao Tung University

ABSTRACT

In this thesis, we propose a novel path selection method for channel estimation in OFDM systems. The formulated cost function of the proposed method can first be looked upon as an integer linear programming (ILP) problem. We then find that the ILP problem can be further simplified into a sorting problem. In order to refine the proposed algorithm, we further set a new threshold for the algorithm by analyzing the event of false alarm and miss detection and signal to noise and estimation error ratio. Comparing to the two simple conventional methods, our simulation results show that the proposed method can improve bit error rate (BER) performance, reduce square error (SE) of channel estimation, and increase the probability of correctly selecting channel paths. Finally, our simulation results also show that the proposed method is insensitive to the multipath power delay profile as well as the operating signal-to-noise ratio (SNR).

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List of Tables

Table 5.1 The probability of false alarm and miss detection with U as a parameter...39

Table 6.1 Simulation parameters ...41

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List of Figures

Figure 2.1 (a) Transmitter of a multi-carrier system and (b) receiver of a multi-carrier

system...6

Figure 2.2 An arrangement for subcarriers that avoids the ICI but occupies too large...8

Figure 2.3 An arrangement for subcarriers that avoids the ICI and save a lot of bandwidth...8

Figure 2.4 FFT implementation for an OFDM system...9

Figure 2.5 Spectrum of mutual orthogonal subcarriers. ...12

Figure 2.6 Effect of multipath with silent guard interval in an OFDM system; the delayed sucarrier 2 causes ICI on subcarrier 1, and vice versa...13

Figure 2.7 Complete OFDM symbol; Ng is the length of CP and N is the length of useful symbol...14

Figure 3.1 The block diagram of the DFT-based channel estimation method...17

Figure 4.1 The block diagram of the DFT-based channel estimation method with path...22

Figure 6.1 The BER performance for the proposed path selection method in the Veh. A channel at E Nb 0= 10dB with threshold as a parameter. ...46

Figure 6.2 The average SE for the proposed path selection method in the Veh. A channel at 0 b E N = 10dB with threshold as a parameter. ...46

Figure 6.3 The BER performance for the proposed path selection method in the Veh. A channel at E Nb 0= 40dB with threshold as a parameter. ...47

Figure 6.4 The average SE for the proposed path selection method in the Veh. A channel at 0 b E N = 40dB with threshold as a parameter. ...47

Figure 6.5 The BER performance for the proposed path selection method in the thirty-path channel at E Nb 0= 10dB with threshold as a parameter. ...48

Figure 6.6 The average SE for the proposed path selection method in the thirty-path channel at E Nb 0 = 10dB with threshold as a parameter. ...48

Figure 6.7 The BER performance for the proposed path selection method in the thirty-path channel at E Nb 0= 40dB with threshold as a parameter. ...49

Figure 6.8 The average SE for the proposed path selection method in the thirty-path channel at E Nb 0 = 40dB with threshold as a parameter. ...49

Figure 6.9 The BER performance for the proposed path selection method in the two-path channel at E Nb 0= 10dB with threshold as a parameter. ...50

Figure 6.10 The average SE for the proposed path selection method in the two-path channel at E Nb 0 = 10dB with threshold as a parameter. ...50

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Figure 6.11 The BER performance for the proposed path selection method in the two-path channel at E Nb ₀= 40dB with threshold as a parameter. ...51

Figure 6.12 The average SE for the proposed path selection method in the two-path channel at E Nb ₀ = 40dB with threshold as a parameter. ...51

Figure 6.13 The BER performance for the three path selection methods in the Veh. A

channel...54 Figure 6.14 The average SE for the three path selection methods in the Veh. A channel. ...54 Figure 6.15 The CDF of false alarm for the three path selection methods at E Nb ₀= 10dB

in the Veh. A channel. ...55 Figure 6.16 The CDF of miss detection for the three path selection methods at

0

b

E N =10dB in the Veh. A channel. ...55 Figure 6.17 The CDF of false alarm for the three path selection methods at E Nb ₀= 25dB

0

b

E N =25dB in the Veh. A channel. ...56 Figure 6.19 The CDF of false alarm for the three path selection methods at E Nb ₀= 40dB

0

b

E N =40dB in the Veh. A channel. ...57 Figure 6.21 The BER performance for the three path selection methods in the Veh. B

channel...59 Figure 6.22 The average SE for the three path selection methods in the Veh. B channel. ...59 Figure 6.23 The CDF of false alarm for the three path selection methods at E Nb ₀= 10dB

in the Veh. Bchannel. ...60 Figure 6.24 The CDF of miss detection for the three path selection methods at

0

b

E N =10dB in the Veh. B channel. ...60 Figure 6.25 The CDF of false alarm for the three path selection methods at E Nb ₀= 25dB

in the Veh. B channel. ...61 Figure 6.26 The CDF of miss detection for the three path selection methods at

0

b

E N =25dB in the Veh. B channel. ...61 Figure 6.27 The CDF of false alarm for the three path selection methods at E Nb ₀= 40dB

in the Veh. Bchannel. ...62 Figure 6.28 The CDF of miss detection for the three path selection methods at

0

b

E N =40dB in the Veh. B channel. ...62 Figure 6.29 The BER performance for the three path selection methods in the two-path

channel...65 Figure 6.30 The average SE for the three path selection methods in the two-path channel...65

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Figure 6.31 The CDF of the false alarm for the three path selection methods at E Nb ₀ =

10dB in the two-path channel...66 Figure 6.32 The CDF of miss detection for the three path selection methods at E Nb ₀=

10dB in the two-path channel...66 Figure 6.33 The CDF of the false alarm for the three path selection methods at E Nb ₀ =

25dB in the two-path channel...67 Figure 6.35 The CDF of the false alarm for the three path selection methods at E Nb ₀ =

40dB in the two-path channel...68 Figure 6.37 The BER performance for the three path selection methods in the thirty-path

channel...71 Figure 6.38 The average SE performance for the three path selection methods in the

thirty-path channel...71 Figure 6.39 The CDF of false alarm for the three path selection methods at E Nb ₀= 10dB

in the thirty-path channel...72 Figure 6.40 The CDF of miss detection for the three path selection methods at E Nb ₀=

10dB in the thirty-path channel. ...72 Figure 6.41 The CDF of false alarm for the three path selection methods at E Nb ₀= 25dB

25dB in the thirty-path channel. ...73 Figure 6.43 The CDF of false alarm for the three path selection methods at E Nb ₀= 40dB

40dB in the thirty-path channel. ...74 Figure 6.45 The BER performance for the number of path setting method with Np as a

parameter in the thirty-path channel...76 Figure 6.46 The average SE for the number of path setting method with Np as a parameter

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Chapter 1 Introduction

Orthogonal frequency division multiplexing (OFDM) has received considerable interest

in recent years. Due to its advantages in high-data-rate transmissions over

frequency-selective fading channels and ability to provide a substantial reduction in

equalization complexity compared to classical modulation techniques [1] [2], OFDM is

used for high-data-rate wireless local area network (WLAN) standards, such as ETSI

Hiperlan II and IEEE 802.11a, providing data rates up to 54Mbits/s, and considered for the

fourth-generation (4G) mobile wireless systems and beyond [3].

Although some differential modulation schemes such as differential

phase-shift-keying (DPSK) can be used without channel state information [4], these

differential modulation schemes degrade system performance in signal-to-noise ratio

(SNR) as compared with non-differential modulation schemes such as quadrature

phase-shift-keying (QPSK). In general, non-differential modulation schemes require

channel state information for coherent demodulation at the receiver side. In spite of the

fact that some blind techniques that exploit statistical or deterministic properties of the

transmitted and received signals are developed, their applications are only limited in

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demodulation that needs to estimate and track parameters of fading channel is still needed

[1]. Therefore, channel estimation is an essential issue for implementing a successful

OFDM system.

A number of algorithms have been presented for channel estimation in OFDM

systems. One of them uses Discrete Fourier transform (DFT) to perform channel

estimation and this method is called DFT-based channel estimation. The DFT-based

channel estimation method, derived from either maximum likelihood (ML) criterion or

minimum mean square error (MMSE), was originally proposed for single-input

single-output (SISO)/OFDM systems with pilot preambles. This method is composed of a

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

DFT matrix [1] [2] [5]. The LS estimator exploits pilot symbols to produce an LS

estimate, which is a noisy observation of channel frequency response. After taking the

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

weighting matrix which depends on the performance criterion used (MMSE or ML

criterion). Note that when equally spaced pilot symbols are used, the weighting matrix

degenerates into an identity matrix. Finally, the enhanced estimate is transformed back to

frequency domain to obtain a new estimate of channel frequency response. To further

improve the DFT-based channel estimation method, we can utilize a path selection

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The robustness of path selection has great influence on the system performance and the complexity of channel tracking. There are two conventional path selection methods: number of path setting method and threshold setting method [6] [7]. For these two methods, if the value of the desired number of total paths (a parameter in the number of path setting method) or the value of the threshold (a parameter in the threshold setting method) are set too large, the ability to suppress noise in the path selection method is reduced, thereby lowering the improvement of system performance and increasing the complexity of channel tracking. On the other hand, if these two values are set too small, true channel paths are excluded and the system performance degrades. As a result, it is difficult to set proper values of these two parameters for the two conventional path selection methods, since the setting of the parameters are sensitive to channel environments.

In this thesis, we propose a novel path selection method for channel estimation in

OFDM systems. The proposed method makes use of the linear programming method to find

an optimum solution which selects channel paths more precisely and less sensitively to the

parameters. We first form a cost function for the proposed path selection method and then

this cost function can be represented as an integer linear programming problem (ILP).

Moreover, since the variables involved are binary, we can finally simplify this problem as a

sorting problem. In addition, a threshold which is insensitive to channel conditions (e.g.,

power delay profiles) is introduced to refine the proposed path selection method.

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method derived from ML criterion. Chapter 4 presents two conventional time domain path

selection methods: the number of path setting method and the threshold setting method.

Chapter 5 proposes a novel path selection method which isless sensitive to parameters and

channel conditions than the two conventional methods. The performance of the proposed

method is then evaluated in Chapter 6. Finally, some conclusions and future works are

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Chapter 2 OFDM System

OFDM, which was brought up in the mid 60’s, is a digital multi-carrier modulation

scheme. In recent years, it has been adopted for many applications in wireless

communication systems, such as WLAN and digital video broadcasting (DVB) [2], as a

result of its capability of high-rate transmission and low-complexity implementation over

frequency-selective fading channels.

The basic idea of OFDM is that it divides the available spectrum into several orthogonal

subcarriers. Because these subcarriers are narrow-band, they experience flat fading channel

and equalization method of the system becomes very simple. Furthermore, it possesses high

spectral efficiency by overlapping these orthogonal subcarriers [8]. Moreover, the insertion

of cyclic prefix (CP), which preserves the periodic extensions of the transmitted signal, can

eliminate intersymbol and intercarrier interference caused by multipath environments.

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2.1 System Model

An OFDM system is a kind of multi-carrier modulation schemes. It splits a high-rate

bit stream into a number of low-rate bit streams. These low-rate bit streams are

modulated onto different subcarriers and transmitted simultaneously [9].The increase in

symbol duration due to the narrowband subcarriers makes the amount of multipath delays

relatively small and lets the OFDM system easily implemented in frequency-selective

fading channels. 0 2 j f t e π 1 2 j f t e π s f (0) C (1) C ( 1) C K− 1 2 K j f t e π − 1( ) K d − t s f K 0( ) d t 1( ) d t (a) 1 2 K j f t e− π − 0 2 j f t e− π 1 2 j f t e− π (b)

Figure 2.1 (a) Transmitter of a multi-carrier system and (b) receiver of a multi-carrier system

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Figure 2.1 (a) and (b) are the transmitter and the receiver of a multi-carrier system,

respectively. After serial-to-parallel (S/P) converter, all data streams are modulated onto

subcarriers simultaneously. Data rate of these parallel streams becomes f Ks/ , where the

data rate of the original data stream is fs. Thus the system is more resistant to the effect of

intersymbol interference (ISI).

The signal on the kth subcarrier can be represented by

2

( ) ( ) j f tk

k

d t =c k e π (2.1)

where ( )c k is the modulated data symbol, fk is the carrier frequency of the kth

subcarrier, and t is the time index.

The output of the transmitter is a juxtapositionof individual signal of each subcarrier:

1 2 0 ( ) ( ) k K j f t k d t − c k e π = = (2.2) It needs to choose K proper frequencies, i.e. f f₀, , ,₁ fK−₁, to avoid intercarrier interference (ICI) among subcarriers. Figure 2.2 provides a kind of arrangement for subcarriers. Although this arrangment meets the purpose of no ICI among the subcarriers, it occupies too large bandwidth. On the other hand, the arrangement of the orthogonal subcarriers shown in Figure 2.3 can not only avoid the ICI among subcarrers, but also save a lot of bandwidth.

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Figure 2.2 An arrangement for subcarriers that avoids the ICI but occupies too large bandwidth.

Figure 2.3 An arrangement for subcarriers that avoids the ICI and save a lot of bandwidth.

2.2 DFT Implementation for OFDM

Systems

A system shown in Figure 2.1 needs K oscillators to generate K orthogonal subcarriers. We can introduce a DFT operation as an alternative to make the implementation much easier.

The OFDM signals by using the operation of DFT are generated as follows. By sampling Eq.(2.2) with sampling rate 1/ ,Ts we can obtain

1 2 0 ( ) ( ) k s. K j f nT s k d nT − c k e π = = (2.3)

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1 ₂ 0 [ ] ( ) . kn K _j K k d n − c k e π = = (2.4)

According to the formula of the IDFT

1 ₂ 0 [ ] [ ] , kn K _j K k b n − B k e π = = (2.5) we get [ ] { ( )}. d n =IDFT c k (2.6) Consequently, the DFT operation can be applied to an OFDM system. In addition, when

K is equal to 2m₍_m_{is a positive integer), the fast Fourier transform (FFT) operation}

can be used for faster implementation in an OFDM system, as shown in Figure 2.4.

(0) C (1) C ( 1) C K−

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In Figure 2.4, we consider an OFDM system employing K subcarriers for transmission. After the S/P converter, a serial high-rate bit stream is partitioned into K parallel low-rate streams which are then mapped into symbols by using a modulation constellation scheme, such as QAM and PSK. Each of the K modulated data symbols

( )

c k , for k=0,...,K− , is mapped from several bits by using the corresponding 1 modulation scheme. These symbols are fed to a K -point IDFT unit to generate time-domain samples. The parallel time-domain samples are then converted into serial ones by using the parallel-to-serial (P/S) converter. Afterwards, a cyclic prefix is inserted in front of these time-domain samples as a guard interval to eliminate ISI between adjacent OFDM symbols. The length of a guard interval is chosen larger than the delay spread. More details about the guard interval will be discussed in Section 2.4. Finally, the OFDM symbols is passed through a digital-to-analog converter (DAC) and transmitted in air. The receiver in Figure 2.4 behaves as inverse operation of the transmitter.

2.3 Orthogonality

Unlike the conventional frequency division multiplexing (FDM) systems, orthogonality among subcarriers greatly simplifies the design of both the transmitter and the receiver. Due to the orthogonality, a simplified one-tap equalizer can be used for each subcarrier. The

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orthogonality allows high spectral efficiency as well, and almost the whole available frequency band can be utilized.

For an OFDM system, we can use the DFT operation to produce the subcarriers which are orthogonal to each other. We show this feature in the following. Assume that subcarriers are of the form:

2

( ) j f tk , 0,1, , 1

k t e π k K

ϕ = = − (2.7) where fk =k KT/ s and Ts is a sampling period. For the k th1 and the k th2 subcarriers,

we have 1 2 1 2 1 2 1 2 2 2 * ( ) 2 ( ) 2 ( ) 2 ( ) 1 2 1 2 1 2 ( )( ) ( ) [1 ] 1 2 ( ) ( ) ( - ) 0 s s s s s k k j t j t b _KT _KT a k k j t b _KT a b a b j k k j k k KT KT s s e e dt e dt e e j k k KT for k k b a and b a KT for k k π π π π π

π

− − − − = − = − = − = = ≠ (2.8)

As can be seen in Eq. (2.7) and Eq(2.8), when the subcarrier frequencies fk are chosen

as integer multiples of 1/KTs, they will be orthogonal to each other for an integration

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Figure 2.5 Spectrum of mutual orthogonal subcarriers.

Moreover, as shown in Figure 2.5, if we sample at the peak of each subcarrier, there will be no ICI among these subcarriers.

2.4 Guard Interval

One of the most important feature for an OFDM system is its efficient way to handle multipath interference. Since the system bandwidth is divided into K subcarriers, the symbol duration is increased and the ISI caused by a time-dispersive fading environment is mitigated. To eliminate ISI completely, a guard interval is inserted at the beginning of each OFDM symbol. The guard interval is chosen longer than the maximum channel delay spread, in order to avoid multipath components from one OFDM symbol interfering with the next adjacent OFDM symbol.

If a silent guard interval is adopted in an OFDM system, rather than the cyclic prefix, the effect of ICI would arise among subcarriers due to the fact that the orthogonality of

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subcarriers is no longer preserved [9]. The effect of ICI is illustrated in Figure 2.6.

Figure 2.6 Effect of multipath with silent guard interval in an OFDM system; the delayed sucarrier 2 causes ICI on subcarrier 1, and vice versa.

As shown in Figure 2.6, there are two time-domain sinusoids, corresponding to two orthogonal subcarriers in frequency domain, called subcarrer 1 and subcarrier 2, respectively. Due to the multipath delay, the orthogonality betwen the subcarrier 1 and the subcarrier 2 is destroyed within the DFT interval. When an OFDM receiver tries to demodulate the subcarrier 1, the delayed subcarrier 2 will induce interference to the subcarrier 1.

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adopted in most OFDM-based wireless communication systems. For this method, the cyclic extension of an OFDM symbol is inserted in front of the OFDM symbol, as show in Figure 2.7. Therefore, as long as the channel delay is smaller than the guard time, there will be no ICI among subcarriers when the FFT operation is performed in the OFDM demodulator, due to the fact that integration over a period of a sinusoid is achievable.

Figure 2.7 Complete OFDM symbol; Ng is the length of CP and N is the length of

useful symbol.

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

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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.

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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

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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.

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3.1 Signal Model

We first describe the signal model for the received OFDM symbol as ( ) ( ) ( ) ( ),

y n =h n ∗x 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

( ) ( ) ( ) ( ),

Y k = N X k H k +Z k (3.2) where k is the subcarrier index, ( )Y k is the received OFDM signal in frequency domain,

( )

X k is the transmitted pilot signal, H k( ) is channel frequency response, ( )Z k is frequency domain AWGN with zero mean and variance 2

n

σ

, N is the total number of subcarriers (or FFT size) and N is a FFT normalization factor.

The system model can be rewritten as a vector form:

,

N

N _h _h

Y = XH + Z

= XF h + F z (3.3)

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diagonal matrix containing the transmitted pilot symbols, { (0)X X N( −1)} , [ (0), , (_H _{H N} 1)]T

= −

H is the channel impulse response, _h₌[h(0), h(_G₋1)]T_{is the}

channel impulse response with finite length G , F is a truncated DFT matrix of size _h

N G× , and Z and z represent AWGN vector in frequency and time domain respectively and is modeled as independent and identically distributed (i.i.d.) complex Gaussian random vector with zero mean and variance 2

n N

σ

I .

3.2 Maximum Likelihood Estimator

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

2 1 ˆ _, x N

σ

= H = LS h H X Y F h + Z (3.4) where ₂1 x N

σ

=

Z X . By using the LS estimate, we can derive the ML estimate as follows. According to the deterministic model of Eq.(3.4), we would like to find an h

which maximizes the posteriori probability of p( |h Hˆ_LS), i.e., ˆ_ML ₌_{arg max ( |}_p ˆ _). LS h h h H (3.5) In details, we have ˆ ( | ) ( ) ˆ ( | ) _ˆ , ( ) p p p p = LS LS LS H h h h H H (3.6)

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

same for every h , Eq.(3.5) can be rewritten as

2 2 ˆ 2 2 2 ˆ _{arg max (}ˆ _{| )} 1 arg max ( 2 ) ˆ arg min . n ML N n p e σ

πσ

− − = = = − LS h LS h H F h h LS h h h H h H F h (3.7)

Let J( )h = Hˆ_LS−F h_h 2, ( )J h can be expanded as a quadratic form

ˆ ˆ ˆ ( ) 2 ˆ ˆ ˆ 2 . J = − + = − + H H H LS LS LS h h h H H H H LS LS LS h h h h H H H F h (F h) F h H H H F h h F F h (3.8)

Making the derivative of ( )J h with respect to h be zero, we have

( ) ₂ H _ˆ ₂ H _0.

h h

J

∂ _{= −} ₊ ₌

∂_hh F HLS F F hh (3.9)

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

ˆ ˆ ˆ , ML = = H -1 H h h h LS H h LS h (F F ) F H F H (3.10) where H = . h h G

F F I Substitute Eq.(3.4) into Eq.(3.10), we get

h 2 h 2 1 ˆ 1 , H H H x x N N

σ

= = = ML H h F X Y h + F X Fz h + z (3.11) where ₂ h 1 H _H x N

σ

=

z F X Fz and the elements of the vector hˆML are denoted as ˆh ( )ML n ,

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

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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 hˆ_ML 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.

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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 hˆ_ML 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

{

}

2

ˆ

max hML( )n , respectively. In order to select main paths, we reserve those elements in hˆML whose energy

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

{

ˆML( )2

}

- TdB

dB

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.,

dB T 10 10 . L T = (4.1) Then we can express the estimated channel impulse response with path selection as

2 2 ˆ ˆ ˆ _{( )}_{, if} _{( )} _max{ _{( ) }/} ˆ ( ) , , 0 _otherwise ML ML L ML th h n h n T h n h n = ≥ (4.2)

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For example, TdB can be set to 20 dB. The elements of the ML estimate of channel

impulse response, hˆ ( )ML n , are selected if their energy,

2

ˆ ( )ML

h n , in dB scale is larger than the value of max

{

ˆML( )2

}

20

dB

h n − dB (or equivalently, in linear scale, we have

2 2

ˆ _{( )} _max{ ˆ _{( ) }/}

ML ML L

h n ≥ h n T ). On the other hand, the elements whose energy (in dB scale) is smaller than the value of max

{

ˆML( )2

}

20

dB

h n − dB are discarded and set as zero.

Finally, the refined estimate after the threshold setting path selection method, hˆth, is

transformed back to frequency domain to get the estimated channel frequency response, i.e.,

ˆ_th ₌_FFT_{{ }}ˆ_th

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

p

N 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]

ˆ

ˆ _{( )}_, _if _{( ) is one of the} _{larger values} ˆ ( ) , 0 _otherwise ML p ML p h n N h n h n = (4.4) for n=0, , -1G .

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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_{{ }}ˆ_p

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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

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p

N 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.

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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 2 ˆ X N

σ

= H H

even even even even

h F X Y / (5.1) and 2 2 ˆ _. X N

σ

= H H

odd odd odd odd

h F X Y / (5.2)

where hˆ_even and hˆ_odd are both of size ( , 1)G , F and _even F are the truncated FFT _odd

matrices of size ( / 2, )N G , Y and _even Y are the received signal vectors of size _odd

( / 2, 1)N , X_even and X are the diagonal matrices of size ( / 2, / 2)_odd N N and their diagonal elements are the transmitted pilot signals X k( ), and 2

x

σ

is the energy of the transmitted pilot signal, i.e., 2 _{( )}2

x X k

σ = . The subindices “even” and “odd” indicate 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 hˆ_even and hˆ_odd are desired paths. Thus, the cost function for the proposed path selection method can be formulated as follows:

2 2

ˆ ˆ ˆ ˆ

min _even− _odd + _odd− _even ,

A h Ah h Ah (5.3)

where the diagonal elements of A are

{

A(0), , (A G−1)

}

, each of which is either 0 or 1. We would like to find A that minimizes the SE of Eq.(5.3). The estimate of ( )A n is

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denoted as ˆ( )A n . If ˆ( ) 1A n = , the nth element of ˆh is considered as a valid path; otherwise, we treat the element as noise and let h nˆ( ) 0= , where

(

)

ˆ ˆ

ˆ ₌ ˆ ₊ˆ _{/ 2 [ (0), , (}₌ _h _{h G}₋_1)]T

odd even

h h h .

To examine the meaning of our cost function, we have the following equations by substituting Eq.(3.3) into Eq.(5.2):

2 2 2 2 2 2 ˆ ₍ _{) /} 2 ( / ) 2 ( / ) 2 1 ( / ) 2 , H H X H H X H H H H X H H X X N N N N N

σ

= + = + = + = + = +

odd odd odd odd odd odd

odd odd odd odd odd

odd odd odd odd odd odd odd

odd odd odd

1 h F X X H Z F X X F h Z F X X F h F X Z h F X Z h z (5.4) where _{(2 /} 2₎ H H _/ _. X N

σ

=

1 odd odd odd

z F X Z Similarly, we have ˆ _{= +} _, even 2 h h z (5.5) where _{(2 /} 2₎ H H _/ _. X N

σ

=

2 even even even

z F X Z Eq.(5.4) and Eq.(5.5) show that hˆ_odd and hˆ_even

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

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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 = .

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5.2 Optimum Solution

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

(

) (

)

2 2 ˆ ˆ ˆ ˆ min ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ min ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ min ˆ ˆ ˆ ˆ ˆ H H H H H H H H H − + − = − − + − − = − − + + − −

even odd odd even

A

even odd even odd odd even odd even

A

even even even odd odd even odd odd

A

odd odd odd even even

h Ah h Ah h Ah h Ah h Ah h Ah h h h Ah h Ah h Ah h h h Ah h

(

)

(

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ min ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ min ˆ H H H H H H H H H H tr tr + = − − + − − + = − − + −

odd even even

even odd odd even odd odd

A

odd even even odd even even

odd even even odd odd odd

A even Ah h Ah h Ah h Ah h Ah h Ah h Ah h Ah h h A h h A h h A h

)

(

)

(

)

(

)

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ min 4 H H H H H H tr e − + = + − ℜ

odd odd even even even

odd odd even even odd even

A

h A h h A h h A

h h h h h h A

(5.10)

Let us define a matrix f as follows:

ˆ ˆ ˆ ˆ ˆ ˆ

( H ) 4 _e( H ).

= H − ℜ

f h h + h h h h (5.11) 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.

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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 ˆ( ) 0, otherwise f n A n = < (5.13) forn=0, , -1G .

5.3 Refined Path Selection Method

In the following subsections, we further propose a refined path selection method to suppress noise and reduce the probability of false alarm by finding a proper threshold which is insensitive to channel power delay profiles.

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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 ( ) 4 ( ) ( )( ) ( )( ) 4 (( )( ) ) 4 ( ) 2 H H H H H H H H H H H H H H H H H H H e e e = − ℜ = + + − ℜ = + + + + + + + − ℜ + + + = + + 1 2 H

1 1 2 2 1 2 1 1 1 2 2 2 1 2 2 1 1 1 f h h + h h h h h z h + z h + z h + z h + z h + z hh z z z h hz hh z z z h hz hh z z hz z h hh z z z H _{− ℜ}4 _e( H) 2_{− ℜ}_e( H) 2_{− ℜ}_e( H) 4_{− ℜ}_e( H) 2 2z hh z h1 z h2 z z1 2 (5.14)

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

and z are i.i.d. Assuming that ( ) 0₂ h n = , we can therefore calculate ( )f n as

1 1 2 2 1 2

( ) ( ) ( )H ( ) ( ) 4H ( ( ) ( )),H

f n =z n z n +z n z n − ℜe z n z n (5.15)

It is noted that when the value of ( )f n in Eq.(5.15) is negative, false alarm occurs at 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

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2_/( 2 ₎

n XN

σ σ

. To reduce the probability of false alarm, we propose a refined path selection method in the next subsection.

-20 -10 0 10 20 30 40 50 0 2000 4000 6000 8000 10000 12000 14000 f(n) ( Unit: σ_n2/(σ_x2N) ) N um be r o f s am pl es h(n)=0

Figure 5.1 The PDF of ( )f n evaluated through the empirical samples with 1,000,000 samples.

5.3.2 Algorithm

Since the channel length is less than G , the values of ( )f n , for those positions

(

)

,..., / 2 1

n G= N − corresponding to ( ) 0h n = , can be looked upon as samples which are 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:

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1, if ( ) ˆ( ) 0, otherwise th f n R A n = < (5.16)

for n=0,1, ,G− and 1 Rth = − ×U max f n{ ( )}, for n G= ,...,

(

N/ 2 1

)

− , where the value of U is to be determined.

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 detection of a path on the signal-to-noise and estimation error power ratio (SNER), and develop a rule of thumb to suggest the strength of a path whose probability of miss detection should be concerned. We then develop a method to determine the value of the parameter U which can achieve a desired probability of false alarm and miss detection. Recall that hˆ =

(

hˆ_odd+hˆ_even

)

/ 2, hˆ_odd = h + z₂ in Eq.(5.4), and hˆ_even = h + z₁ in Eq. (5.5). Note that each element of ˆh is denoted as ˆh n . Hence, we have

( )

ˆ = +

h h z (5.17)

where z=

(

z z₁+ ₂

)

/ 2, and each element of z is denoted as z n

( )

with zero mean and variance 2_/

(

2

)

n XN

σ

. Assume that there is a channel path at position n with path energy

(

)

2 ₂ ₂

( ) n / X

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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 ( ) ( ) ( ) H k =H k + ∆H k (5.18) where ( ) 1 ₀1 ( )

(

)

2 j kn N _N n H k h n n n e N π

δ

− − = ∆ = − . Thus, we have 2 2 2 1 ( ) n X H k N N

σ

µ

σ

∆ = × (5.19) for k =0,...,N− . From Eq.(3.2) and Eq.(5.18), the received signal after channel matching 1 can be written as 2 ˆ ( ) ( ) ˆ ( ) H k Y k H k ∗

(

)

(

)

2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) H k H k N X k H k Z k H k H k ∗ + ∆ = + + ∆

(

)

₍

₎

(

)

2 2 ( ) ( ) ( ) ( ) ( )( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H k H k H k H k N X k H k H k Z k N X k H k H k H k H k H k ∗ ∗ + ∆ + ∆ = + ∆ + − ∆ + ∆ + ∆

(

)

(

)

2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H k H k H k H k N X k Z k N X k H k H k H k H k H k ∗ ∗ + ∆ + ∆ = + − ∆ + ∆ + ∆ (5.20) Hence, the SNER can be calculated as

2 2 2 2 2 2 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) X n X N SNER H k H k H k H k N H k H k H k H k H k

σ

= + ∆ + ∆ + ∆ + ∆ + ∆ 2 2 2 2 2 ( ) ( ) ( ) X n X N H k H k N H k

σ

+ ∆ = + ∆

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2 2 2 2 2 2 ( ) ( ) 1 X n n X X N H k H k N N N

σ

µ

_σ

+ ∆ = + × × (5.21) 2 2 2 2 ( ) ( ) X n n N H k H k N

σ

µ

+ ∆ = + Moreover, for high SNR, we have 2 _{( ) /}2 2 ₁

X n

N

σ

H k

σ

. If N≥ , we then have

µ

2 ₂ ₂ 2

( ) n /( X) ( )

H k

σ

N

σ

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

2 2 2 2 ( ) X n n N H k SNER N

σ

µ

≈ + (5.23) As we can observe in Eq.(5.23), when N

µ

, the miss detection of the path will not have significant influence on the SNER. That is, when N

µ

, we have

2 2 2 2 2 2 2 ( ) ( ) X X n n n N H k N H k SNER SNR N

σ

µ

≈ ≈ = + (5.24) In this thesis, N is set as 256. Hence, we will concern the probability of miss detection of a path with energy _{( )}2 ₂₅ 2_/

(

2

)

n X

h n =

σ

N 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( _{( ) 23.2} 2

n

f n ≥

σ

)=1/64. In other words, among f n( ) , for n G= ,...,

(

N/ 2 1

)

− , the occurrence of the event of

正交分頻多工系統中用於通道估計之新型路徑選取方法

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A Novel Path Selection Method for

Channel Estimation in OFDM Systems

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A Novel Path Selection Method for

Channel Estimation in OFDM Systems

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Student

Hsin-Yi Tu

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Advisor

Dr. Chia-Chi Huang

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A Thesis

Submitted to Department of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Communication Engineering

July 2007

Hsinchu, Taiwan, Republic of China

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