梳狀引導信號輔助之多輸入輸出通道動差預估與分析

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國立交通大學

電子工程學系電子研究所

碩士論文

梳狀引導信號輔助之多輸入輸出通道動差

預估與分析

Channel Moment Estimation and Analysis for

MIMO-OFDM with Comb-type Pilots

研究生：蔡咏廷

指導教授：桑梓賢教授

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梳狀引導信號輔助之多輸入輸出通道動差預估與分析

Channel Moment Estimation and Analysis for

MIMO-OFDM with Comb-type Pilots

研究生：蔡咏廷 Student：Yung-Ting Tsai

指導教授：桑梓賢 Advisor：Tzu-Hsien Sang

國立交通大學

電子工程學系電子研究所

碩士論文

A Thesis

Submitted to Department of Electronics Engineering & Institute of Electronics College of Electrical and Computer Engineering

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master in

Electronics Engineering August 2006

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梳狀引導信號輔助之多輸入輸出通道動差預估與分析

研究生：蔡咏廷

指導教授：桑梓賢

國立交通大學

電子工程學系電子研究所碩士班

摘要

正交多頻分工(OFDM)系統在過去數年間已經成功的應用到許多數位

通訊上。在正交分頻多工系統通道偵測上，根據梳狀引導信號(comb-type

pilots)的輔助，過去已經提出了許多的方法。在這篇論文中，我們提出

了一種創新的引導信號內插法(pilot interpolation)並與其他內插法比

較，例如：線性內插(LI)、木條曲線內插(SCI)…。在這方法中，我們利

用了通道的頻率響應(CFR)的一階與二階動差，並且時域的低通濾波器是

在頻域中完成的，這意味著我們不用多一次的傅立葉轉換，因此可以省下

一個快速傅立葉轉換，節省了系統的成本。另一方面，多輸入多輸出(MIMO)

與正交分頻多工(OFDM)系統的結合，也就是 MIMO-OFDM，近來也是廣泛的

應用在數位通訊上。因此，我們提出了一個 MIMO-OFDM 架構，可以將之前

根據單輸入單輸出正交分頻多工系統(SISO-OFDM)所設計的引導信號內插

演算法套用於上。

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Channel Moment Estimation and Analysis in MIMO-OFDM

with Comb-type Pilots Added

研究生：蔡咏廷 student：Yung-Ting Tsai

指導教授：桑梓賢 Advisors：Tzu-Hsien Sang

Department of Electronics Engineering & Institute of Electronics

National Chiao Tung University

ABSTRACT

Orthogonal frequency division multiplexing (OFDM), has been

successfully applied to a wide variety of digital communications applications

over the past several years. There is many channel estimation strategies are

used in OFDM system with comb-type pilots inserted. This thesis introduces

an innovative pilot interpolation method by using the 1

st

and the 2

nd

moments

of the channel impulse response (CIR) and comparing with other methods, ex.

LI, SCI ... In this innovative method, time-domain low-pass filter is

implemented in frequency-domain; it means that we don’t have to do another

fourier transform. In this way, we can save a FFT block to reduce the system

cost. On the other hand, the combinations of multi-input multi-output (MIMO)

and OFDM, MIMO-OFDM, is widely applied on digital communication

recently. Therefore, we introduce a MIMO-OFDM architecture that can easily

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誌謝

在這將近三年的研究生生涯中，學習了不少研究的方法與做人處世的

道理。首先要感謝指導教授桑梓賢老師，除了專業上的指導與幫助，有時

更也會給我們一點思考上方向的激盪，讓我更懂得以不同的角度來看待事

物。另外，非常有幸的可以到詠發科技，在李宇旼老師的身上學習到了專

業與謙虛的並存，非常有耐心的指導，也對我這篇論文的生出有決定性的

幫助。另外研究室的欣德學長也在有問題疑惑的時候給了我很大的幫助，

同學 B 魔、阿康、Yoyo、晉運、李薇，以及各位學弟的兩年同窗生活，

除了解決的研究上的問題之外，也讓我研究生涯多采多姿。最後要感謝父

母親對我的支持以及栽培，還有女友容容在背後給我很大的支持及鼓勵，

讓我學生生涯畫下個句點。

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LIST OF TABLES

Table 01 Computational Complexity Analysis of Comb-Type Pilot 1-D Interpolation

Methods ... 18

Table 02 The Encoding and Transmittion Sequence for the Two-Branch Transmit Diversity Scheme ... 38

Table 03 Simulations Parameters... 47

LIST OF FIGURES

Fig 1.1 Two types of pilot arrangement ... 2

Fig 1.2 The scattered pilot arrangement... 4

Fig 1.3 A basic digital implementation of a baseband OFDM system... 5

Fig 2.1 Comb-Type Pilot Arrangement ... 11

Fig 2.2 Comb-Type Pilot Arrangement in Single OFDM Symbol... 11

Fig 2.3 The Channle Estimation Block Diagram ... 12

Fig 2.4 Piecewise-Constant Interpolation ... 13

Fig 2.5 Linear Interpolation ... 14

Fig 2.6 The Relationships between ˆ 0H and _n H in Frequency Domain and Time _n Domian……... 18

Fig 2.7 The Low-Pass Filter that Gets the Real CIR... 16

Fig 2.8 The Block Diagram of LPI ... 17

Fig 2.9 TDI Block Diagram ... 17

Fig 2.10 SER Performance versus SNR for the Channel Estimators Based on LS with Comb-Type Pilot Arrangements ... 18

Fig 2.11 The Example of LPI ... 19

Fig 2.12 A More Ideal Filter in Assumption ... 20

Fig 2.13 The Block Diagram of MA-LPITD ... 20

Fig 2.14 The Block Diagram of MA-LPIFD... 21

Fig 2.15 Down Sampling Effects ... 27

Fig 2.16 The Summation Paths of the 1st Moment and its Estimator... 30

Fig 2.17 The Summation Paths of the 2nd Moment and its Estimator... 31

Fig 2.18 The Summation Paths of m1 with Down Sampling Effects ... 32

Fig 2.19 The Summation Paths of m2 with Down Sampling Effects ... 32

Fig 3.1 The Architecture of the MIMO-OFDM Channel Estimation in Current Idea . 38 Fig 3.2 The 2*M MIMO Channel Model... 39

Fig 3.3 The Block Diagram of MIMO-OFDM with Alamouti Scheme... 40

Fig 4.1 Three Types of PDPs... 41

Fig 4.2 The Normalized MSE of m1 Estimators... 43

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Fig 4.4 The m1 Estimator MSE of Theoretical Value and Monte Carlo Method ... 45 Fig 4.5 The m2 Estimator MSE of Theoretical Value and Monte Carlo Method ... 46 Fig 4.6 BER Performance versus SNR for the Equalizer Based On Alamouti with Comb-Type Pilot Arrangements ... 48

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Chapter 1 Introduction

In this chapter, the motivation of the research will be described first. Then, the organization of this thesis will be presented at the end of this chapter.

1.1 Motivation

Orthogonal frequency division multiplexing (OFDM) is a multicarrier transmission technique which uses parallel data transmission and frequency division multiplexing and was drawn firstly in 1960s [1-2]. Because of its’ high channel efficiency, OFDM is wildly applied in the new generation wireless access systems such as digital broadcasting systems [3]and wireless local area network [4]. On the other hand, the concept of multiple-input multiple-output (MIMO) based communications has received wide spread attention from the communication society [5]. The utilization of MIMO techniques in wireless transmission provides a tremendous increase in spectral efficiency. To achieve wideband wireless transmission at high data rate, the combination of these two techniques, MIMO-OFDM [6], is one of the best solutions to high data rate digital transmission.

In wireless communications, the receiver systems have to compensate the channel effects and the channel equalizer and channel estimation techniques are deployed. The channel equalizer recovers the original signal under non-perfect channel environment based on the estimated channel; therefore, the precision of the channel estimation is important. In general, the fading channel of OFDM systems can be viewed as a 2-dimensinal (2D) function (time and frequency) [7]. In the OFDM system with pilot added, the optimal channel estimator in terms of mean-square error is based on 2D Wiener filter interpolation.

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However, the 2D wiener filter is too complex for practical implementation. Therefore, the tradeoff between complexity and performance, usually one-dimensional (1D) interpolation is adopted in OFDM systems. Basically, there are two types of 1D pilot arrangement,one is block-type pilot arrangement, and the other is comb-type pilot arrangement. And the block-type pilot arrangement and comb-type pilot arrangement are illustrated in Fig. 1.1. The block-type pilot arrangement inserts pilots in frequency direction, and its OFDM channel estimation symbols are transmitted periodically, and all subcarriers are used as pilots. On the other hand, the comb-type pilot arrangement inserts pilots in time direction, and every transmitted symbol contain pilot signals. In comb-type pilot arrangement, the pilots are uniformly inserted into the subcarriers of OFDM symbol. The latter type, comb-type pilot channel estimation, is the focus of this thesis.

Fig 1.1 Two Types of Pilot Arrangement

In comb-type pilot arrangement, after exploiting the pilots to acquire the estimated channel frequency response (CFR) on each pilot subcarriers, there are many methods to interpolate the CFR into the data subcarriers, such as linear interpolation (LI), second-order interpolation (SOI), low-pass interpolation (LPI), and time-domain interpolation (TDI). In

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implemented if converts the estimated CFR to time domain by inverse discrete Fourier transform (IDFT). However, if there are 2K/4K/8K subcarriers, the method adding an IDFT block will cost too high to afford, such as the DVB-T/H systems. Therefore, if all the time-domain computations can be operated in frequency domain, then the method, Moments-Assisted LPI in Frequency Domain (MA-LPIFD), will be a practical one.

To do the computations in frequency domain, there are few challenges that we have to solve. First, the computations have to transform into frequency-domain by Fourier transform. Second, the first and second moments of the channel impulse response (CIR) have to be acquired in frequency domain. Third, the accuracy of MA-LPIFD needs to be further improved.

After achieving those tasks, some future works came to my mind. The first one is to adopt the pilot arrangements in DVB-T/H, the scattered pilot, Fig. 1.2 [3], then extend the 1-D channel estimation in frequency-direction to 2-D or two 1-D (2*1D) channel estimations [9]. The second one is to extend the SISO-OFDM system to MIMO-OFDM system to exploit the spatial diversity gain. In scattered pilot arrangement, the channel can be both estimated in time-direction and frequency-direction, and according to [7], 2D channel estimation yields better performance than the 1D scheme, at the expense of higher computational complexity and processing delay. However, in systems that use 2K/4K/8K modulations, if the channel has to be estimated in time-direction, we need adequate symbols to collect information in each computation, which means large amounts of memory is needed and the channel is estimated until receiving enough symbols. Therefore, comparing the first extension with the second extension, the latter one, from SISO-OFDM to MIMO-OFDM, is more interesting.

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1.2 Introductions of Baseband OFDM Model

The main idea of applying OFDM systems is to split a high rate data stream into a several lower rate data streams that are transmitted simultaneously over a number of subcarriers. A basic digital implementation of a Baseband OFDM system is illustrated in Fig. 1.3.

Fig. 1.3 A Basic Digital Implementation of a Baseband OFDM System

First, we have totransform the serial data stream to parallel data stream. After that, the data stream is transmitted in different subcarriers, which are divided in orthogonal frequency. An inverse discrete Fourier transform (IDFT) is used to achieve orthogonal frequency division, and the equation is shown as follows:

2 1 1 [ , ] 1 1 , 0 0 , j kt N kt n t N N N N N n k n k k k x X_⎡ _⎤

e

π X_⎡ _⎤W− ⎣ ⎦ ⎣ ⎦ − − = = =

=

∑

(1.1)

where N is the N-point DFT, and k represents the k-th subcarrier, and WN=e-j 2π/N

. X[n,k] represents the data in k-th subcarrier of the n-th OFDM symbol in frequency domain. x[n,t] represents the data in t-th section of the n-th OFDM symbol in time domain. Define that

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[ ] [ ] [ ] [ ] T ,0 , 1 T ,0 , 1 , , , , n n n N n n n N x x X X − − ⎡ ⎤ = ⎣ ⎡ ⎤ = ⎣ ⎦ x X . ⎦ W (1.2)

Following the IDFT block and parallel-to-serial (P/S) block, the cyclic prefix block (CP) extends the time length TG, which is larger than the expected delay spread. In discrete computation, the extended length is larger than the channel path number.

The digital-to-analog converter (DAC) contains low-pass filter with bandwidth 1/Ts, where Ts is the sampling interval.In the discrete form, the mathematical expression of this block is not shown.

The channel is modeled as a sampled impulse response. Denote and as the -th coefficient of the channel impulse response and complex additive white Gaussian noise (AWGN) at the -th OFDM block respectively. From the discrete Fourier transform, the CFR and noise frequency response are defined as

[ ]n l,

h

n

_{[ ]}_{n l}_, l n 1 , , 0 1 , , 0 L kl N n k n l l L kl N n k n l l n H h N W ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ − = − = = =

∑

(1.3)

where H[n,k] and N[n,k] represent the CFR and noise frequency response at the k-subcarrier of the n-th OFDM block respectively. Denote that L is the known maximum channel length,W_N =e−j2π N and N is the OFDM block size and L<N is assumed [8]. And we defined that [ ] [ ] [ ] [ ] T ,0 , 1 T ,0 , 1 , , , , n n n L n n n N h h H H − − ⎡ ⎤ = ⎣ ⎡ ⎤ = ⎣ ⎦ h H ⎦ (1.4)

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[ ] [ ] [ ] [ ] T ,0 , 1 T ,0 , 1 , , , , n n n L n n n N n n N N − − ⎡ ⎤ = ⎣ ⎡ ⎤ = ⎣ ⎦ n N ⎦ n

h

n

⎦ (1.5)

arenoise impulse response and noise frequency response vectors of n-th OFDM symbol respectively.

And the CIR h_n and CFR

H

_n has the following relationship

n

=

H

F

. (1.6) So as for n_n and

Ν

_n, n

=

N

F

(1.7) where ( )( ) 0 0 0 0 1 1 1 1 0 1 N N N L N N N N L L N N N W W W W W W W W W − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ F _.

At the receiver, after passing through the analog-to-digital converter (ADC) and removing the CP, the S/P block transforms the serial data stream into parallel, and the DFT block transfers the time domain data to frequency domain data. Define that

[ ] [ ] [ ] [ ] T ,0 , 1 T ,0 , 1 , , , , n n n N n n n N y y Y Y − − ⎡ ⎤ = ⎣ ⎡ ⎤ = ⎣ ⎦ y Y (1.8)

where Y[n,k] represents the received data in k-th subcarrier of the n-th OFDM symbol in frequency domain, and y[n,t] represents the received data in t-th section of the n-th OFDM symbol in time domain. And Yn and yn are the vectors of n-th OFDM received

symbol. And as same as transmitted signals, the received signals have following relationship, Y[n,k] =DFT(y[n,t]).

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be expressed as follow n = n n+ Y H X N_n (1.9) Where [ ] [ ] [ ] ,0 ,1 , 1 0 0 0 0 0 0 n n n n N X X X ₋ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ X

In channel estimation, the transmitted signals and received signals are assumed known. Based on (1.9), the channel can be estimated by, for instance, the least square estimator (LS) or the minimum mean square error (MMSE) estimator, and these estimated CFR are expressed as follow

( )

[ ] [ ] [ ] [ ] T 1 ,LS ,0 ,0 , 1 , 1 ˆ _, _, n n n Yn X n Yn N X n N − − − ⎡ = _{= ⎣} H X Y ⎤_⎦ (1.10)

(

)

1 1 2 ,MMSE , , , ,LS ˆ ˆ n n n

σ

n N n n − − ⎡ = _HH_⎣ _HH + H R R X X ⎤_⎦ H_n (1.11) where

(

)(

)

{

}

{ }

H _H , , 2 2 , n n n n n N n

E

σ

=

HH hh

R

h

R

N

F

.

After accessing the estimated CIR (1.10) or (1.11), for comb-type pilot arrangement, the estimated CFR on pilot subcarriers is used to interpolate the unknown CFR on data subcarriers to reconstruct the entire CFR on all subcarriers. Since the CFR is acquired, based on (1.9), the transmitted signal Xnn can be

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1.3 Organization of the Thesis

The organization of this thesis is as follow. In Chapter 2, some comb-type pilot interpolation methods will be introduced, after that, we will discuss our interpolation method in detail. In Chapter 3, the architecture of MIMO-OFDM that adopts the interpolation method we bring out will be introduced. In Chapter 4, we will show the simulations results and analysis of these methods. At last, the conclusion and future works will be discussed in Chapter 5.

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Chapter 2 Comb-Type Pilot Aided Channel Estimation

2.1 The Signal Model of Comb-Type Pilot Channel Estimation in

OFDM System

OFDM is becoming widely applied in digital wireless communication systems due to its high data rate and high bandwidth efficiency. For wideband mobile communication system, the channel between transmitter and receiver is usually frequency-selective and time-variant. However, most channel estimation methods for OFDM system are developed under the assumption that the channel is stationary within one packet, and those methods are not suitable under mobile communication channel. Therefore, a dynamic channel estimation method for frequency-selective and time variant channel is needed. The pilot-based channel estimation has been proven suitable for OFDM systems in mobile communication channels [7].

In comb-type pilot arrangement, as shown in Fig. 2.1, for each transmitted symbol, Np pilots signals are uniformly inserted into N subcarriers in Xn. FS is the distance of two pilots subcarriers, where FS=N/ Np. In Fig. 2.2, a more detail com-type pilot arrangement in single OFDM symbol is illustrated.

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Fig. 2.1 Comb-Type Pilot Arrangement

Fig. 2.2 Comb-type Pilot Arrangement in Single OFDM Symbol

Denote that the subcarrier address of pilots in each OFDM symbol is the same, and can be expressed as follow

( ) ( )

(

)

T

(

)

0 , 1 , , _p 1 , ( ) 1 _s k 1 _p p p p N p k k F ⎡ ⎤ =_⎣ − _⎦ = − p = ∼ N

⎤

⎦

(2.1) and the pilot values are

p T P(N -1) P P(0) P(1)

,

X

⎡

= ⎣

X

_{. (2.2)}

At the receiver, if pilot values and subcarrier address are known, then CFR at the pilot subcarrier can be calculated by some method, for example, LS estimator, equation 1.10, or

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some other algorithms. Denote that the estimated CFR at the pilot subcarriers of n-th OFDM symbol as follow

p T P(N -1) P P(0) P(1)

ˆ

_,

ˆ

_,

ˆ

n

= ⎣

⎡

H

n

H

n

H

n

H

⎤

_⎦

] (2.3) where . Since estimated CFR at the pilot subcarrier are acquired, there are some interpolation methods to interpolate the CFR between two pilot subcarriers. The channel estimation block diagram is shown as Fig. 2.3.

[ P(0) ,P(0)

ˆ

n n

H

=

H

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2.2 Prior Arts in One-Dimensional(1-D) Interpolation Methods

for Comb-type Channel Estimation

One-dimensional interpolation is used to estimate the CFR at the data subcarriers, with reference to the estimated CFR at the pilot subcarriers, and some methods are summarized in the following section [7].

2.2.1 Piecewise-Constant Interpolation

The piecewise interpolation is the simplest interpolation method. With mathematical expression, the data subcarrier CFR between two pilot subcarrier and are estimated as follow, and illustrated in Fig. 2.4.

P( ) ˆ k n H Hˆ_nP(k+1) P( ) P( ) S ˆ k i ˆ k _{,where i=1} _F ₁ n n H + =H ∼ − (2.4)

Fig. 2.4 Piecewise-Constant Interpolation

2.2.2 Linear Interpolation (LI)

The LI interpolation method performs better than piecewise-constant interpolation, and the data subcarrier CFR between two pilot subcarrier and are estimated as follow equation, and illustrated in Fig. 2.5.

P( )

ˆ k n

H HˆnP(k 1)

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[ ] P( 1) P( ) P( ) S ,P( ) ˆ ˆ ˆ ˆ k n k n k _{,where i=1} _F ₁ n n k i S H H H H i F + + − = + × ∼ − (2.5)

Fig. 2.5 Linear Interpolation

2.2.3 Low-pass Interpolation(LPI)

The LPI method is performed by inserting zeros in to , and then applying a low-pass finite-length impulse response (FIR) filter. The data in pilot subcarriers is not changed, and the data in the data subcarriers will be interpolated.

P

ˆ

n

H

This method minimized the mean-square error between real value and interpolated value [7]. The idea of this method is also the inspiration of our algorithms.To explain this method, we need to figure out what padding zeros and low-pass filtering do in both time domain and frequency domain. Assumed that the estimated CFR on pilot subcarrier are equal to the real CFR, that is

p T p T P(N -1) P(N -1) P P(0) P(1) P P(0) P(1)

ˆ

_,

ˆ

_,

ˆ

_,

n

=

⎡

⎣

H

n

H

n

H

n

⎦

⎤

=

n

=

⎣

⎡

H

n

H

n

H

n

⎤

⎦

H

,

assume that the channel path is L, and then do N-points DFT to transform to frequency domain. After padding F -1 zeros into Hˆ P, it becomes

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It is assumed that N/FS>L, which means no aliasing at time domain. And denote

( )

ˆ

₀

_IDFT

ˆ

₀

IDFT

n n n n

=

h

H

h

H

. (2.7)

The relationship between and in frequency domain and time domain is shown

in Fig. 2.6, and the elements of and can be expressed as following equation ˆ 0n H H_n ˆ 0n H H_n [ ] [ ] 2 1 , 0 ,

ˆ 0

S S ki F _j F n k i n k S

H

e

H

F

π − =

=

∑

_(2.8)

We can see that, after padding zeros in , the impulse response of , , is F

P

ˆ

n

H

ˆ 0

_n

h

ˆ 0

_n

S times repetitions of hn and 1/FS of its amplitude. Therefore, a low pass filter is used to pass through is needed to get one copy of the repetition CIR. Theoretically, the width of this filter is 2*N/FS, which is shown in Fig. 2.7. The time domain low-pass filter can be implemented by a finite length impulse (FIR) response filter, and applied after

, Fig. 2.8 shows the block diagram of LPI..

ˆ 0

_n

h

ˆ 0

_n

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Fig. 2.6 The Relationships Between and in Frequency Domain and Time Domain

ˆ 0_n

H H_n

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Fig. 2.8 The Block Diagram of LPI

2.2.4 Time Domain Interpolation (TDI)

The TDI method is a high-resolution interpolation based on DFT/IDFT. It first converts to time domain by IDFT, and then interpolate the time domain data to N-points by piecewise-constant interpolation or linear interpolation method. The block diagram of TDI is shown in Fig. 2.9.

P

ˆ

n

H

Fig. 2.9 TDI Block Diagram

2.2.5 Other Interpolation Methods

Besides the methods we introduced in 2.2.1 to 2.2.4, there are other 1-D interpolation methods, for example, Second-Order Interpolation (SOI) and Spline Cubic Interpolation (SCI) which exploit three adjacent pilot data and continuous polynomial fitting to estimate CFR respectively. The complexity analysis and simulation performance are shown in Table1 and Fig. 2.10 [7].

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Estimation Scheme Complexity Comments PCI Lowest LI Lowest SOI Low

Simple estimation and interpolation methods

SCI LPI TDI

Moderated Interpolation methods are relatively complex, with fitted polynomial, low-pass convolution, and DFT/IDFT calculation, respectively

Table 1: Computational Complexity Analysis of Comb-Type Pilot 1-D Interpolation Methods

Fig. 2.10 SER Performance versus SNR for the Channel Estimators Based on LS with Comb-Type Pilot Arrangements

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2.3 Proposed Moments-Assisted Low Pass Interpolation (MA-LPI)

In those 1-D interpolation methods, with mathematical analysis and simulation results, LPI method is the best performance. In LPI method, the width of low-pass FIR filter is decided by N-points FFT and the distance between two pilot subcarriers. However, there are still someplaces that can be improved. For example, in 2K-points FFT system, the distance of two pilot subcarriers is 4 (FS=4), and the channel length L is 40. In LPI method, the width of low-pass filter will be 2K/4, which is much longer than the channel length L, and a lot noise (AWGN noise, the red part) will be filtered in. This example is shown in Fig. 2.11; we can see that, a lot of undesired noises (the red part) arepassed by the wide width low-pass filter. If the filter width can be tuned to get lesser noise, the signal to noise ratio (SNR) will be raised, and the SER/BER will belowered. In Fig. 2.12, a more ideal filter in assumption is shown. Therefore, the idea is that if the range can be tuned with respect to the CFR or CIR, the new filter will be more ideal [11].

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Fig. 2.12 A more Ideal Filter in Assumption

A bigger question is how to exploit the CIR/CFR information to design the filter. There are two parameters we eager to know, the first one is where the center of the CIR is, and the second one, what’s the length of the CIR is. If the CIR is acquired, these two parameters can be accessed easily. The center of the CIR can be accessed by setting the threshold of the amplitude of CIR, and then find out the center and the width of this band-pass filter. Of course, it needs DFT/IDFT blocks to transform to time domain. This idea, is rather similar to the combination of LPI and TDI, however, this is the first version of this interpolation method.The only drawback of this idea is the high cost of IDFT/DFT block. The block diagram of the current idea, called Parameters-Assisted Low Pass Interpolation in Time Domain (PA-LPITD), is shown in Fig. 2.13.

P

ˆ

n

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access the filter parameters in frequency domain. In this way, all the computation is completed in frequency domain, and the IDFT/DFT blocks are negligible. However, the center and the length of the CIR are not accessible in frequency domain, but the centroid (the first moment) and the second moment of CIR are accessible in frequency domain. Therefore, in the second version algorithm, the center of the CIR is replaced by the first moment of CIR, the length of CIR is replaced by the second centre moment of CIR, and the IDFT/DFT blocks are removed. This method, which is called Moments-Assisted Low Pass Interpolation in Frequency Domain (MA-LPIFD), accesses these parameters in frequency domain, and it is discussed in Chapter 2.4. In Fig. 2.14, the block diagram of MA-LPIFD is shown.

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2.4 The Channel Moment Estimation

In this section, we will focus on the estimation of the 1st moment and 2nd centre moments of CIR. In section 2.4.1, it will be discussed in continuous time, to figure out the method to access moments in frequency domain. In section 2.4.2, the discrete form of the moment estimators will be shown. In section 2.4.3, the down sampling effects will be discussed. In section 2.4.4, the relationships between moments and moments estimators are discussed, and some improvements of the estimators are also introduced in section 2.4.5. Some simulation of this improvement is shown in section 2.4.6.

2.4.1 How to Access the Moments of CIR

Denote that the continuous time CIR is h t , and

( )

h t₀

( )

= h t

( )

2 is the square of absolute of h t

( )

in continuous time.

Because the CIR is complex number, the square of absolute value of CIR is used in following calculation. Of course, there are some differences between h t

( )

and h t

( )

2, but not much differences in the moments calculations. To design the filter, the following information of h t

( )

2 is required: the 1st moment m1, the 2nd centre moment m2,centre and the 2nd moment m2 . They are shown in the following equations:

2 1 2 ( ) ( ) t h t dt m h t dt =

∫

(2.9)

( )

2 2 2 2 1 1 2, 2 1 1 ( ) ( ) ( ) centre t h t m dt t h t dt m h t m dt h t dt − = = −

∫

2 (2.10) 2 2 ( ) t h t dt m =

∫

(2.11)

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and (2.9), (2.10), (2.11) are functions of continuous time. These functions need a bridge to connect frequency domain and time domain, and the Fourier transform is the one to

connect two different domains. The Fourier transforms of h t

( )

2 and h t₁

( )

2 are shown in (2.12a) and (2.12b) respectively. (2.13) is the differential of (2.12) to frequency; (2.14) is the second differential of (2.12) to frequency.

{ }

{

}

2 2 ₂ 2 2 ₂ 1 1 1 ( ) ( ) ( ) (2.12a) ( ) ( ) ( ) j ft HH j ft HH F h t R f h t e dt F h t R f h t e dt π π ∞ ₋ −∞ − = = = =

∫

(2.12b) ∞ −∞

∫

2 ' 2 2 ' 1 2 1 1 ( ) ( ) 2 ( ) (2.13a) ( ) ( ) 2 ( ) j ft HH HH j ft HH HH dR f R f j t h t e dt df dR f R f j t h t e dt df π π π π ∞ ₋ −∞ − = = − = = −

∫

(2.13b) ∞ −∞

∫

2 2 '' 2 2 2 2 2 '' 1 2 2 1 2 1 ( ) ( ) ( 2 ) ( ) (2.14a) ( ) ( ) ( 2 ) ( ) j ft HH HH j ft HH HH d R f R f j t h t e dt df d R f R f j t h t e dt df π π π π ∞ ₋ −∞ − = = − = = −

∫

(2.14b) ∞ −∞

∫

And if f=0, (2.12a), (2.13), (2.14) can be written as follows:

2 (0) ( ) HH R ∞ h t dt −∞ =

_∫

(2.15) ' ' 2 0 ' ' 2 1 1 1 0 ( ) (0) ( ) (2.16a) 2 2 ( ) (0) ( ) 2 2 HH HH f HH HH f R f R t h t dt j j R f R t h t dt j j π π π π ∞ −∞ = = = = − − = = − −

∫

(2.16b) ∞ −∞

∫

(

)

(

)

' '' 2 2 2 2 0 ' '' 2 2 1 1 1 2 2 0 ( ) (0) ( ) (2.17a) 4 2 ( ) (0) ( ) 4 2 HH HH f HH HH f R f R t h t dt j R f R t h t dt j π π π π ∞ −∞ = = = = − − = = − −

∫

(2.17b) ∞ −∞

∫

where _{( )} *( ) ( ) HH f R f ∞ H τ H τ dτ −∞ +

=

_∫

is the autocorrelation of CFR. From (2.15) and (2.16a), the first moment can be expressed in frequency domain:

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1 ' (0) 2 ( HH HH m R j

π

R = − × 0). (2.18)

And the differential functions can express as follows:

1 0 0 ( ) ( ) 2 lim ₂ ₍ HH HH f f _HH m R f f R f f f j

π

R = Δ → = + Δ − − Δ Δ − 0) (2.19)

And the approximate of (2.19) is

1 ( ) ( ) 2 ˆ 2 ( ( ) ( ) 2 2 (0 HH HH HH HH HH f F HH R f R f f m j R R F R F F j R

π

Δ = Δ − −Δ Δ ≈ ₋ − − = ₋ 0) ) (2.20)

where F is division between two subcarrier frequencies. Since are complex conjugate, (2.20) can rewrite as

( ) and ( ) HH HH R f R −f

{

}

{

}

1 2 Im ( ) Im ( ˆ 2 2 (0) 2 (0) HH HH HH HH R F R F m F

π

R

π

FR = = − − ) . (2.21)

From (2.15) and (2.17a), the 2nd moment is written as

'' 2 2 (0) 4 ( HH HH R m R π = − × 0). (2.22) From (2.15) and (2.17b), the 2nd centre moment is written as

'' 1 2, 2 1 (0) 4 ( HH centre HH R m R π = − × 0). (2.23) And the approximation of (2.22) is

2 ' ' 0 2 2 ' ' 2 2 2 ( ) ( ) 2 ˆ 4 ( ( ) ( ) 4 (0) F HH HH f HH f _F _F HH HH HH R f f R f f f m R R R R F π π = Δ = + Δ − − Δ Δ ≈ − − − = ₋ 0). (2.24)

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2 2 ' ' 2 ' 2 ( ) ( ) ( ) 2 ( ) ( ) ( ) 2 ( ) ( ) ( ) 2 F F HH HH HH HH HH F HH f HH HH F HH f R f f R f f R f f R f f R f f R f R f f R f f R f = =− + Δ − − Δ = Δ + Δ − − Δ ∴ = Δ + Δ − − Δ − = Δ ∵

(2.24) can rewrite as follows:

{

}

{

}

2 2 2 2 2 2 2 2 ( ) (0) (0) ( ) ˆ 4 (0) ( ) ( ) 2 (0) 4 (0) 2 Re ( ) 2 (0) 4 (0) Re ( ) 1 (0) 2 HH HH HH HH HH HH HH HH HH HH HH HH HH HH R F R R R F F F m R F R F R F R R F R F R R F R F R F π π π π − − − ⎛ ₋ ⎞ ⎜ ⎟ ⎝ ⎠ = − + − − × = − − × = − − = . (2.25)

And the approximation of (2.23) is similar to the 2nd moment, the only difference is the m1

phase shift, which is 2 1

1( ) ( )

j fm HH

HH

R f =R f ×e π . And the approximation of (2.23) is

{

}

{

}

{

}

{

}

{

}

1 1 2 2 2, 1 2 2 2 Re ( ) 2 (0) ˆ 4 (0) Re ( ) Im ( ) Im ( ) Im ( ) 1 cos sin (0) (0) (0) (0) 2 HH HH centre HH HH HH HH HH HH HH HH HH R F R m R F R F R F R F R F R R R R F π π − × = ₋ ⎛ ⎞ ⎛ ⎞ − × _⎜− _⎟+ _⎜− _⎟ ⎝ ⎠ ⎝ ⎠ = .(2.26)

2.4.2 The Discrete Form of the Moments

Because all simulations are executed in discrete, it’s necessary to transform m1, m2, and m2,centre into discrete form. We can assume that sampling frequency

F

=

1/

N

, where N is the FFT size, and the 1st moment in discrete and the 1st moment estimator in discrete is show as following equations:

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[ ] [ ] 1 ₂ , 0 ,1 1 ₂ , 0 ,1, ,1 ,0 ˆ 2 L n l l n L n l l n i n n h l m h NR m R π − = − = ⎧ ⎪ ⎪ = ⎪ ⎨ ⎪ − ⎪ = ⎪ ⎩

∑

(2.27)

denote that m_n_,1 is the 1st moment, mˆ_n_,1 is the 1st moment estimator,

{

}

, , Re , ( )

n k r n HH

R = R kF ,Rn k i_{, ,} =Im

{

Rn HH_, (kF)

}

,and n is the n-th OFDM symbol. And the equations of the 2nd moment and the 2nd moment estimator in discrete are as follow

[ ] [ ] 1 ₂ 2 , 1 ,2 1 ₂ , 1 ,1, ,0 2 ,2 2 (2.28a) 1 ˆ 2 L n l l n L n l l n r n n h l m h R R m N π − = − = = − = ×

∑

_. (2.28b) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪⎩

And the equations of the 2nd centre moment and the 2nd centre moment estimator in discrete are as follow [ ] [ ] 1 ₂ 2 1 , 0 ,2, 1 ₂ 1 , 0 ,1, ,1, ,0 ,0 2 ,2, (2.29a) 1 cos ˆ L n l l n centre L n l l n r n i n n n centre h l m h R R R R m N − = − = = ⎛ ⎞ − × _⎜_⎜ _⎟_⎟+ ⎝ ⎠ = ×

∑

,1, ,1, ,0 ,0 2 sin (2.29b) 2 n i n i n n R R R R π ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ _⎛ _⎞ ⎪ _⎜_⎜ _⎟_⎟ ⎪ _⎝ _⎠ ⎪⎩

2.4.3 The Effect of Down-Sampling to the Moments Estimators

In chapter 2.4.2, the F is defined as 1/N, which means all subcarrier information is obtained. In fact, a receiver gets information only on pilot subcarriers. The CFR on pilot

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downsampling and

padding zeros autocorrelation P down sampled autocorrelation P

0 (s1)

(s2)

n n n n n n

p

→

H

R

H

R

where _, _{( )}* _, _{( )} 1 1 ( ) p N n n P k n P k m k p Rp m H H N ₌ ⎡⎣ ⎤⎦ ⎡⎣ + =

∑

_⎤⎦ and _{[ ]}* _[ _] , , 1 1 ( ) S N P n n k n k mF k R m H H N = + =

∑

.

In (s1), the pilot signals are acquired at receiver, and then we use those pilots to access the autocorrelations. But all the autocorrelations we mentioned before are defined as steps (s2). It’s important to know what’s different in those two steps. With reference to Fig. 2.15, the down-sampling effects to the CIR in time domain are shown clearly.

P

n

H

Fig. 2.15 Down Sampling Effects

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[ ] [ ] [ ]

downsampling and

padding zeros autocorrelation P 2 , , , down sampled autocorrelation P 2 , , ,

0 (s1)

0

0 (s2)

3

n n n n l n l n l n n n n l n l n l

p

h

→

H

R

H

R

, (2.30) where _{[ ]} _{[ ]} 2 , , 1_{n l} 0_{n l} h = h , _{[ ]} _{[ ]} 2 , , 2_{n l} _{n l}

h = h , h3_{[ ]}_{n l}_, is the FS repetitions of l=1~N/FS. We can see that if the channel length is shorter than N/FS, and the is equivalent to , and the moments estimator is usable. In other words, if the channel length is longer than N/F

[ ],

3_{n l}

h h1_{[ ]}_{n l}_,

S, there are aliasing in , and its autocorrelation will not equal to , and the estimator is not work. With sapling theory, if the number of pilot subcarriers is not enough, the obtained information is unable to recover the original data, which means the N/FS will be shorter than the channel length.

[ ],

0_{n l}

h h1_{[ ]}_{n l}_, h3_{[ ]}_{n l}_,

2.4.4 The Relationships between Moments and Moments Estimator

in Time Domain:

To evaluate the performance of an estimator, there is some background knowledge that needs to know. The relationship of the moments and the moment estimators in time domain can help with the analysis of the estimator performance. Therefore, we have to transform the frequency domain estimator to time domain to compare with the original moments.

The frequency response of channel is (1.3a): 1

L−

∑

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[ ] [ ] [ ] [ ] [ ] [ ] 1 1 1 1 1 1 * 1 , , 1 2 ( ) 2 1 1 * 1 , , 1 0 0 2 2 ( ) 1 1 * 1 , , 1 0 0 ( ) N n N n k n k m k k m l kl N L L _j _j N N N n l n l k l l kl k m l N L L _j N N n l n l k l l R m H H h e h e h h e π π π π + = + − − ₋ = = = − + − − = = = = = =

∑

∑∑∑

1

only when l=l, the elements of R m_n( )≠ , therefore 0 [ ] [ ] [ ] [ ] 2 1 * 1 , , 1 0 2 1 ₂ , 0 2 1 ₂ , 0 ( ) ml N L _j N n N n l n l k l ml L _j N n l l ml N _j N n l l R m h h e h e h e π π π − − = = − − = − − = = = =

∑∑

∑

. (2.31)

Denote that the length of CIR is still L, but the value of from L~N-1 is zero. The reason to use N instead of L is that it’s convenient to combine the effect of down sampling to the estimators. Use (2.31) to the estimators, the autocorrelation, (2.27b) and (2.28b), can be expressed in time domain as follow:

[ ]n l, h

{

}

_{[ ]}

{

}

_{[ ]} 1 ₂ ,1, , 0 1 ₂ ,1, , 0 2 Re ( ) cos 2 Im ( ) sin N S n r n S n l l N S n i n S n l l F l R R F h N F l R R F h N π π − = − = ⎛ ⎞ = = _⎜ _⎟ ⎝ ⎛ ⎞ = = − _⎜ _⎟ ⎝ ⎠

∑

⎠ (2.32)

And the 1st moment estimator of n-th symbol

m

ˆ

_n_,1, (2.27b), is expressed as follows:

[ ] [ ] 1 ₂ , ,1, 0 ,1 1 ₂ ,0 , 0 2 sin ˆ 2 2 N S n l n i l n N n n l l F l N h N R _N m R h π π _π − = − = ⎛ ⎞ ⎜ ⎟ × _⎝ = − =

∑

⎠_(2.33)

And the 2nd moment estimator of n-th symbol

m

ˆ

_n_,2 is,

[ ] [ ] ,1, 2 1 ₂ ,0 2 ,2 2 1 ₂ , 2 0 , 0 1 2 ˆ 1 cos 2 2 n r N n S n N n l l n l l R R N F l m N h N h π π _π − − = = − ⎛ ⎛ ⎛ ⎞⎞ ⎞ = = _⎜ _⎜ − _⎜ _⎟_⎟ ⎝ ⎠ ⎝ ⎠ ⎝

∑

⎠

∑

⎟ (2.34)

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summation path of m1 is linear and the summation path of

m

₁ is a sine wave. Fig 2.16 shows the summation paths of the 1st moment estimator and the1st moment. The left one is the summation path of the estimator, the right one is the summation path of the original moment, and the blue one is the CIR. Denote that the FS equal to F, which means all subcarrier information is obtained, and there is no down sampling effects in Fig. 2.16:

Fig. 2.16 The Summation Paths of the 1st Moment and its Estimator

From Fig. 2.16, (2.27a), and (2.33), we can know that the slope of summation paths is almost equivalent at the beginning paths, which is about N/4 paths. It tells that if the CIR is centered on first few paths, the estimator will be very accurate.

When it comes to the performance of the 2nd moment and the 2nd centre moment, there is a simplification has to do to the estimator, thus we can see the relations more clear. The following equation is the relation between m2 and m2,centre in statistic.

{

}

{

(

( )

)

2

}

{ }

₂

(

{ }

)

2 2,centre

E m =E x−E x =E x − E x ,

where x is the CIR. The simplification is: Assume that m1 is perfectly estimated, which E{x}= m1, and then E{x2}=m2 is the only parameter that affects the performance of the 2nd centre moment estimator. After this simplification, the performance evaluation of the 2nd

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original moment in time domain, and their summation paths are shown in Fig. 2.17. The cosine line is the summation path of the estimator, and the red line is the summation path of the original moment. Denote that FS=F , which is same as the figure of 1st moments.

Fig. 2.17 The Summation Paths of the 2nd Moment and its Estimator

From the upper figure, we can see that the summation path of the estimator is almost equivalent to the one of the original moment before N/3 paths. It means that if the CIR is longer than N/3 paths, then the 2nd moment estimator will be failed.

It is assumed that all subcarrier information is obtained in upper two figures, however, often the FS does not equal to F, and the summation paths also have some changes. These changes are effect from down sampling, which is discussed in Chapter 2.4.3. The

summation path of first moments with down sampling effects is show in Fig. 2.18, and the red line is the summation path of m1,and the black sin waves is the summation path of , we may consider to adjust the amplitude of sine wave to gain a more accurate estimator with comparison to m

1

m

1. With down sampling effects, the m1 estimator equation has a little change, which is as follow

[ ] [ ] / 1 ₂ , ,1, 0 ,1 / 1 ₂ ,0 , 0 2 0 sin ˆ 2 2 0 S S N F S n l n i l n N F n S n l l F l N h N R _N m R F h π π _π − = − = ⎛ ⎞ ⎜ ⎟ × _⎝ = − =

∑

⎠_{. (2.35)}

The summation length is reduced from N-1 to (N-1)/FS because of the FS repetitions in time domain, and the FS in denominator is used to reduce the amplitude of sine wave.

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Fig. 2.18 The Summation Paths of m1 with Down Sampling Effects

On the other hand,m2 estimator has the same down sampling effects as m1 estimator does. These effects are shown in Fig. 2.19. And the m2 estimator has the some changes too, which as follow: [ ] [ ] ,1, / 1 2 ₂ ,0 2 ,2 2 / 1 ₂ , 0 2 2 , 0 1 2 ˆ 1 cos 0 2 2 0 S S n r N F n S n N F n l l S n l l R R N F l m N h N F h π π _π − − = = − ⎛ ⎛ ⎛ ⎞⎞ ⎞ = = _⎜ _⎜ − _⎜ _⎟_⎟ _⎟ ⎝ ⎠ ⎝ ⎠ ⎝

∑

⎠

∑

(2.36)

Fig. 2.19 The Summation Paths of m2 with Down Sampling Effects

And (2.35) and (2.36) are the equations used to evaluate the estimator performance in next section.

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the receiver, we can see that the amplitude of sine wave and cosine wave is the key to the accuracy performance. If we can slightly adjust the amplitude of sine and cosine wave based on the channel power delay profile, it would be a good solution to improve the accuracy of these estimators. To adjust the amplitude, the factor x and y is added into the denominator of m1 and m2 estimators respectively, and the estimators can rewrite as:

[ ] [ ] [ ] [ ] / 1 ₂ , ,1, 0 ,1 / 1 ₂ ,0 , 0 ,1, / 1 2 ₂ ,0 2 ,2 2 / 1 ₂ , 0 2 2 , 0 2 0 sin ˆ 2 2 0 1 2 ˆ 1 cos 0 2 2 0 S S S S N F S n l n i l n N F n S n l l n r N F n S n N F n l l S n l l F l N h N R _N m xR F x h R R N F l m N h y N F y h π π _π π π _π − = − = − − = = ⎛ ⎞ ⎜ ⎟ × _⎝ _⎠ = − = − ⎛ ⎛ ⎛ ⎞⎞ ⎞ = = _⎜ _⎜ − _⎜ _⎟_⎟ _⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

. (2.37)

Use the method of MMSE to find out the optimal x, the procedures are as eq. 2.38, and it is assumed that the total power is constant, which _{[ ]}

/ 1 ₂ , 0 0 = constant S N F n l l h − =

∑

B [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ,1 ,1 / 1 ₂ / 1 ₂ , , 0 0 / 1 ₂ / 1 ₂ , , 0 0 / 1 ₂ / 1 ₂ , , / 1 ₂ 0 0 , 0 2 , ˆ 2 0 sin 0 2 0 0 2 1 0 sin 0 2 0 2 1 0 sin 2 S S S S S S S n n N F N F S n l n l l l N F N F S n l n l l l N F N F S n l n l N F l l S n l l S n l S m m F l N h h l N F x h h F l N h h F x N h F l N h B F x π π π π π π − − = = − − = = − − = = = − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = − ⎛ ⎛ ⎞ ⎞ = _⎜ _⎜ _⎟− _⎟ ⎝ ⎠ ⎝ ⎠ =

∑

l − [ ] / 1 / 1 ₂ , 0 0 0 S S N F N F n l l l h l N − − = = ⎛ ⎛ _{⎞ −} ⎞ ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ ⎝

∑

⎠ . (2.38)

Because the summation range is N/FS, where h0_{[ ]}_{n l}_, = h_{[ ]}_{n l}_, , eq. 2.38 can write as:

[ ] [ ] / 1 ₂ / 1 ,1 ,1 , 0 0 2 1 ˆ sin 2 S S N F N F S n n n l n l l l S F l N m m h h l B F x N π π − − = = ⎛ _⎛ _⎞ ⎞ − = _⎜ _⎜ _⎟− _⎟ ⎝ ⎠ ⎝

∑

⎠ 2 ,

And the MSE and BIAS of the 1st moment estimator are:

{

}

[ ] 2 / 1 ₂ 2 ,1 ,1 2 , 0 2 1 ˆ MSE: sin 2 S N F S n n n l l S F l N E m m E l h B F x N π π − = ⎧ _⎛ _⎞ ⎫ ⎪ ⎛ ⎞ ⎪ − = _⎨ _⎜ − _⎜ _⎟_⎟× _⎬ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩

∑

⎭

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{ } { }

[ ] / 1 ₂ ,1 ,1 , 0 2 1 ˆ BIAS: sin 2 S N F S n n n l l S F l N E m E m E l h B F x N π π − = ⎧ ⎛ ⎞ ⎫ ⎪ ⎛ ⎞ ⎪ − = _⎨ _⎜ − _⎜ _⎟_⎟× _⎬ ⎝ ⎠ ⎪ ⎝ ⎠ ⎪ ⎩

∑

⎭

Assume that h[n,l]~complex normal distribution N(0,σ , then the probability density l2)

function (PDF) of h_{[ ]}_{n l}_, is Rayleigh distribution R(σ_l π₂, 2σ_l2), where R(a, b): a=E{x}, b= E{x2}, and E x{ } 124 = σ4, and the MSE and BIAS will be:

/ 1 2 2 2 2 2 2 2 0 2 2 1

MSE : sin sin 12

4 S N F S S l l S S F l F l N N l l B F x N F x N π π σ 4 π π − = ⎛ _⎛ _⎞ _⎛ _⎞⎞ − + ⎜ ⎜ ⎟ ⎜ ⎟⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

× / 1 2 0 2 1 BIAS: sin 2 2 S N F S l l S F l N l B F x N π _σ π − = ⎛ _⎛ _⎞⎞ − × ⎜ ⎜ ⎟⎟ ⎝ ⎠ ⎝ ⎠

∑

We can see that MSE function is also a function of factor x:

/ 1 2 2 2 2 2 2 0 2 2 ( ) sin sin 12 4 S N F S S l l S S F l F l N N f x l l F x N F x N

π

σ

4

π

− = ⎛ _⎛ _⎞ _⎛ _⎞⎞ = _⎜ − _⎜ _⎟+ _⎜ _⎟_⎟× ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

Then the differential to x of f(x) is

/ 1 2 2 4 2 2 2 3 0 2 2 ( ) sin sin 12 0 2 S N F S S l l S S F l F l df x N N l dx F x N F x N

π

_σ

π

− = ⎛ _⎛ _⎞ _⎛ _⎞⎞ = _⎜ _⎜ _⎟− _⎜ _⎟_⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

× =

The f(x) will be minimized when x equals to / 1 4 2 1 / 1 4 1 2 sin 2 2 sin S S N F S l l MMSE N F S S l l F l N N x F F l N

π

σ

π

_σ

π

− = − = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

∑

l

And the m1 estimator with factor x is

,1, ,1, ,0 ˆ 2 n i n MMSE MMSE n NR m x R π − =

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2 / 1 4 / 1 0 4 2 / 1 2 0 4 2 0 2 sin 12 MMSE : 2 sin S S S N F S l N F l l N F l S l l F l l N l B F N π σ σ π σ − − = − = = ⎡ _⎛ _⎛ _⎞_⎞ ⎤ ⎢ _⎜ _⎜ _⎟_⎟ ⎥ ⎝ ⎠ ⎢ ₋⎝ ⎠ ⎥ ⎢ _⎛ _⎞ ⎥ ⎢ _⎜ _⎟ ⎥ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦

∑

l / 1 / 1 2 4 / 1 0 1 2 / 1 0 4 2 1 2 2 sin sin 2 BIAS with x: 2 sin S S S S N F N F S S l l N F l l l N F l S l l F l F l l N N l B F N π π σ σ σ π σ − − − = = − = = ⎡ ⎛ _⎛ _⎞⎞⎛ _⎛ _⎞ ⎤ ⎢ ⎜ ⎜ ⎟⎟⎜ ⎜ ⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎢ ₋ ⎥ ⎢ _⎛ _⎞ ⎥ ⎢ ⎜ ⎟ ⎥ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦

∑

l ⎞ ⎟ ⎠

It’s obviously that m1 is a biased estimator. After evaluated m1 estimator, we use the same method to find out the y factor of the 2nd moment, the MSE and BIAS of m2 estimator are:

{

}

[ ] 2 2 / 1 ₂ 2 ₂ ,2 ,2 2 2 , 0 2 1 cos 1 ˆ MSE: 2 S S N F n n n l l S F l N N E m m E l h B F y π π − = ⎧ _⎛ _⎛ _⎛ _⎞_⎞ _⎞ ⎫ ⎪ _⎜ _⎜ − _⎜ _⎟_⎟ _⎟ ⎪ ⎪ _⎜ ⎝ ⎝ ⎠⎠ _⎟ ⎪ − = _⎨ − × _⎬ ⎜ ⎟ ⎪ _⎜ _⎟ ⎪ ⎪ _⎝ _⎠ ⎪ ⎩ ⎭

∑

{ } { }

[ ] 2 / 1 ₂ 2 ,2 ,2 2 , 0 2 1 cos 1 ˆ BIAS: 2 S S N F n n n l l S F l N N E m E m E l h B F y π π − = ⎧ ⎛ ⎛ ₋ ⎛ ⎞⎞ ⎞ ⎫ ⎪ ⎜ ⎜ ⎜ ⎟⎟ ⎟ ⎪ ⎪ _⎜ ⎝ ⎝ ⎠⎠ _⎟ ⎪ − = _⎨ − × _⎬ ⎜ ⎟ ⎪ _⎜ _⎟ ⎪ ⎪ _⎝ _⎠ ⎪ ⎩ ⎭

∑

The definition of channel is as same as m1, h_{[ ]}_{n l}_, is Rayleigh distribution R( l ₂, 2 l2

π ₎

σ σ .

And MSE and BIAS become:

2 2 / 1 2 4 2 2 0 2 1 cos 1 MSE: 12 2 S S N F l S F l N N l B F y π σ π − = ⎛ ⎛ ₋ ⎛ ⎞⎞ ⎞ ⎜ ⎜ ⎜_⎝ ⎟_⎠⎟ ⎟ ⎝ ⎠ ⎜ ₋ ⎟ _× ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

∑

{ } { }

2 / 1 2 2 ,2 ,2 2 0 2 1 cos 1 ˆ BIAS: 2 2 S S N F n n l S F l N N E m E m l B F y π σ π − = ⎛ ⎛ ₋ ⎛ ⎞⎞ ⎞ ⎜ ⎜ ⎜_⎝ ⎟_⎠⎟ ⎟ ⎝ ⎠ ⎜ ⎟ − = − × ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

∑

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2 / 1 4 2 1 / 1 2 2 4 2 1 2 1 cos 2 2 1 cos S S N F S l l MMSE N F S S l l F l N N y F F l l N

π

σ

π

_σ

π

− = − = ⎛ ₋ ⎛ ⎞⎞ ⎜ ⎟ ⎜ _⎝ _⎠⎟ ⎝ ⎠ = ⎛ ₋ ⎛ ⎞⎞ ⎜ ⎟ ⎜ _⎝ _⎠⎟ ⎝ ⎠

∑

And mn,2,centre and mn,2 estimators with factor y can write as:

,1, ,1, ,1, ,1, ,0 ,0 ,0 ,0 2 ,2, 2 2 1 cos sin ˆ 2 n r n i n i n i n n n n centre S MMSE R R R R R R R R m N F y π ⎛ ⎞ ⎛ ⎞ − × ⎜_⎜ ⎟_⎟+ ⎜_⎜ ⎟_⎟ ⎝ ⎠ ⎝ ⎠ = × n ,1, ,0 2 ,2 2 2 1 ˆ 2 n r n n S MMSE R R m N F y π − =

And MMSE of m2 and BIAS of m2 with factor y are:

2 / 1 2 4 / 1 0 4 4 2 2 _/ ₁ 0 ₄ 0 2 1 cos 12 MMSE: 2 1 cos S S S N F S l N F l l _{N F} l S l l F l l N l B _F _l N π _σ σ π _σ − − = − = = ⎛ _⎛ _⎛ _⎛ _⎞_⎞ _⎞ ⎞ ⎜ _⎜ _⎜ − _⎜ _⎟_⎟ _⎟ ⎟ ⎝ ⎠ ⎜ ₋⎝ ⎝ ⎠ ⎠ ⎟ ⎜ _⎛ _⎞ ⎟ ⎛ ⎞ ⎜ _⎜ ₋ _⎜ _⎟_⎟ ⎟ ⎜ _⎝ _⎝ _⎠_⎠ ⎟ ⎝ ⎠

∑

2 / 1 2 2 2 0 2 1 cos 2 BIAS: 2 2 S S N F l S F l N N l B F y π σ π − = ⎛ ⎛ ₋ ⎛ ⎞⎞ ⎞ ⎜ ⎜ ⎜_⎝ ⎟_⎠⎟ ⎟ ⎝ ⎠ ⎜ ₋ ⎟_× ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

∑

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Chapter 3 Pilot-Aided Channel Estimation for

MIMO-OFDM

3.1 Reasons to extend SISO-OFDM to MIMO-OFDM

Extending SISO-OFDM to MIMO-OFDM aims to provide spatial diversity gains by multi-antenna. The multi-antenna technique is a popular method, and there are many new algorithms designed to fully utilize its tremendous potential. Channel estimation for MIMO systems is important; the algorithm proposed in Chapter2 is designed based on SISO-OFDM, and if we want to use the MA-LPIFA method in MIMO-OFDM system without changing the algorithm, the architecture of

MIMO-OFDM must be considered. In this chapter the Alamouti scheme and MA-LPIFA are combined to provide an effective channel estimation method for MIMO-OFDM.

3.2 The Architecture of the MIMO-OFDM

To adapt those pilot interpolation methods in MIMO without changing the interpolation algorithms, the pilot interpolation must be completed in single channel information, which means there is a space decoder block needed to separate the multi-input for each receiver. Fig. 3.1 shows the idea of the architecture of MIMO-OFDM with comb-type pilot inserted, noted that the architecture after the spatial decoder is as same as the single channel estimation.

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Fig. 3.1 The Architecture of the MIMO-OFDM Channel Estimation in Current Idea

3.3 The Space-Time CODEC for MIMO-OFDM

The Space-Time codes are exploited to extract the relations between different antennas; one of the famous S-T code schemes is Alamouti scheme [12]. In this these, the Alamouti scheme is the only method we use to extract the multi-antenna equations. In Alamouti scheme, the antennas are design as 2 transmitters and M receiver, which is 2M diversity. The encoding and transmission sequence for the two-branch transmit diversity scheme is shown in table2.

Antenna0 Antenna1

Time t (n-th symbol) S0 S1

Time t+T (n+1-th symbol) -S1* S0*

Table 2 The Encoding and Transmission Sequence for the Two-Branch Transmit Diversity Scheme

In Table2, S0 and S1 are denoted as the n-th transmitted symbol by antenna0 and antenna1 respectively, and * is the complex conjugate notation. In Fig. 3.2 is the channel model with 2*M diversity.

梳狀引導信號輔助之多輸入輸出通道動差預估與分析

國立交通大學

電子工程學系 電子研究所

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