正交時空區塊碼之頻域等化及其基於訓練機制之多輸入多輸出的擇頻通道估測

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國

立

交

通

大

學

電機與控制工程學系

碩

士

論

文

正交時空區塊碼之頻域等化及其基於訓練機制

之多輸入多輸出的擇頻通道估測

On Frequency-Domain Equalization with Training-Based Channel

Estimation for Orthogonal Space-Time Block Coded System via

MIMO Frequency-Selective Fading Channels

研究生：楊傑智

楊傑智

指導教授：林清安教授

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正交時空區塊碼之頻域等化及其基於訓練機制之多輸入多輸出的擇頻通道估測

On Frequency-Domain Equalization with Training-Based Channel Estimation for Orthogonal Space-Time Block Coded System via MIMO Frequency-Selective Fading Channels

研究生：楊傑智

Student：Chieh-Chih Yang

指導教授：林清安 Advisor：Ching-An Lin

國立交通大學

電機與控制工程學系

碩士論文

A Thesis

Submitted to Department of Electrical and Control Engineering College of Electrical Engineering and Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Electrical and Control Engineering

August 2007

Hsinchu, Taiwan, Republic of China

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論文題目：正交時空區塊碼之頻域等化及其基於訓練機制之多輸入多

輸出的擇頻通道估測

指導教授：林清安

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論文題目：正交時空區塊碼之頻域等化及其基於訓練機制之多輸入多

輸出的擇頻通道估測

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輸出的擇頻通道估測

指導教授：林清安

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正交時空區塊碼之頻域等化及其基於訓練機制之多輸入多輸出的

擇頻通道估測

學生：楊傑智指導教授：林清安

國立交通大學電機與控制工程學系（研究所）碩士班

摘

要

針對正交時空區塊碼應用於多輸入多輸出之擇頻通道上的問題，吾人於本文中提出了一個嶄新且完整的推導方式；另著墨於此架構下所能達到的最大之多重增益－包括來自於天線以及多重路徑所貢獻者。藉由多次的模擬實驗結果發現，吾人可以得到一個於實際考量下為合理的假設，進而由此推導錯誤率分析。錯誤率分析的推導結果證實了吾人所提出之架構確實是有能力達到完整的多重增益。另一方面，藉由接收端的頻域等化機制以及所提出之傳輸架構，吾人可以進一步找到基於訓練機制時最佳的通道估測所需之相對應的訊號設計模式。本論文之結果為線性等化器能否達到完整的多重增益提供了一個正面的佐證。

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On Frequency-Domain Equalization with Training-Based Channel

Estimation for Orthogonal Space-Time Block Coded System via MIMO

Frequency-Selective Fading Channels

Student：Chieh-Chih Yang Advisor：Ching-An Lin

Department（Institute）of Electrical and Control Engineering

National Chiao Tung University

Abstract

We propose an instructive derivation for the generalized block-level orthogonal space-time block encoder, capable of achieving full spatial diversity via frequency- selective fading environment provided that channel order is known. Instead of dealing with special case and then extending the results intuitively, we provide an alternative by starting with the general signal model with multiple transmit and multiple receive antennas, from which a general form of block-level orthogonality is established. In particular, transmit diversity with more than two transmit antennas can be achieved without compromise by means of frequency-domain equalization, in contrast to the QO-STBC-based approach. Pairwise error probability analysis is derived, under certain assumption which is numerically supported by simulation results, for analytical verifications of our claim on full diversity, inclusive of transmit-receive diversity and the multipath one. Moreover, the encoder structure enables us to generalize a training-based channel estimation technique, originally proposed for flat-fading scenario, to the frequency-selective fading scenario. Surprisingly we even obtain similar optimality criteria for optimal training block design which in our case, the signal block are fixed as OSTBC-based and the design derivation reduces to derive optimal power constraint over the training blocks. The optimality criteria for the training blocks are easy to satisfy when randomness of signal constellation is not a concern. Simulation results validate our discussion of the behaviors of the least-squares and linear MMSE channel estimates.

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Acknowledgement

I would like thank my instructor, Prof. Ching-An Lin, for his teaching, enlightenment and support over the past two years. The thesis would never be complete without his dedication to the revisions and his provision of ingenuity. The valuable joint advice from Prof. Yuan-Pei Lin, Prof. Wing-Kin Ma and Dr. Jwo-Yu Wu serves no less than the above efforts. Especially, issues regarding simulation accuracy and rigorousness of proofs are greatly improved with their assistances. Moreover, the generous helps from the seniors of our laboratory and my friends see me through many difficulties, and I am cordially grateful to all of them. Above all, I would like thank my family for their encouragement.

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List of Figures

Fig. 1 Transmission scheme based on BGOSTBC encoder and the associated FDE

based on training-based channel estimation. ... 48

Fig. 2 A BGOSTBC encoder employing K=4, Mt=3 configuration... 48

Fig. 3-1 Condition numbers generated for 106_{independent simulation runs under}

(N,Mr,Mt,L,K)=(10,1,3,1,4)... 49

(N,Mr,Mt,L,K)=(16,1,3,1,4)... 49

(N,Mr,Mt,L,K)=(16,1,3,2,4)... 50

(N,Mr,Mt,L,K)=(10,2,6,1,8)... 50

(N,Mr,Mt,L,K)=(10,2,6,3,8)... 51

Fig. 3 Compare NMSE of LS and linear MMSE estimates, with arbitrary training. 51

Fig. 4 Compare NMSE of arbitrary training and optimal/orthogonal training... 52

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Chapter 1 Introduction

1.1 Motivation and Background

Orthogonal space-time block code (OSTBC) was introduced in [1] for combating channel fading by exploiting diversities in multiple-input-multiple-output (MIMO) antenna configuration. By virtue of orthogonal design, the symbol decoding relies on only linear processing with relatively lower computational complexity than its trellis counterpart, provided that the channel state information is available. Hence both blind channel estimation and joint symbol detection using OSTBC architecture are well studied in flat-fading scenario.

Various transmission schemes and algorithms based on suitably manipulated orthogonality conditions have been proposed for flat-fading environment[19],[20]. In [19], joint signal detection for a flat-fading MIMO communication under OSTBC scheme was studied, where the exponent, embedded with OSTBC structure, of the closed-form log-likelihood was to be minimized in the maximum likelihood (ML) sense. A non-polynomial time (NP-hard) ML exhaustive search was cleverly reformulated and relaxed as a convex optimization problem which had been well studied theoretically and numerically. The relaxation is known as semidefinite relaxation (SDR) with corresponding numerical algorithm tailored in [21]. The resultant SDR-ML proposed by Ma et al. performs substantially better than the cyclic ML method[22]. It is noteworthy that the reformulation itself exploited the orthogonality of OSTBC, by which a near optimum signal detection with relatively lower computational cost than the optimal ML/sphere decoding was achieved. In [20], a closed-form channel estimation technique based on a variant of the generic orthogonality of OSTBC was developed. This variant was first derived by Alex B. Gershman et al. in [23], where real parts and imaginary parts of the received signal model were separated with discretion. The closed-form channel estimation, unlike the other approaches such as subspace method, suffers from sign ambiguity only due to its special real-valued formulation of signal models. Several

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sufficient antenna configurations for unique channel estimate were studied numerically. The uniqueness of the channel estimate was determined by the discrepancy between the eigenvectors of a certain objective which was formed by exploiting the orthogonality. For those configurations where uniqueness failed, i.e., algebraic multiplicity of eigenvectors greater than 1, the authors proposed a diagonal precoder to alienate the eigenvectors. Benefited from the orthogonality, this can be surprisingly easily done by assigning the power weighting coefficients with sufficient discrepancy in an ad-hoc manner.

However, it remains challenging for the above mentioned methods to be effectively extended to frequency-selective fading environment which is a more practical consideration. By effective extension we mean the algorithm on which the estimation or decoding is based should enjoy either full or partial diversity due to orthogonal nature of the coding itself. To apply the OSTBC structure to frequency- selective case without compromising the code orthogonality, E. Lindskog and A. Paulraj [7] cleverly combine the cyclic-prefix (CP) mechanism with time-reversal operation on symbol blocks, bringing the space-time concept to “block-level”. It was then incorporated with Alamouti scheme[8] and known as time-reversed Alamouti-like (TR-Alamouti) scheme[9]. The TR-Alamouti scheme is unique in that it enjoys full 2-fold transmit diversity and nearly full transmission rate, when neglecting the CP overheads, at the same time. A general block-level orthogonal space-time block code was first proposed by Z. Liu et al.[14] for consideration regarding preserving orthogonality over frequency- selective channels with more than two transmit antennas, where the generalized complex orthogonal design (GOSTBC) [1] was adopted with ZP assistance for mitigating channel distortion. On the other hand, there have been extensive studies based on the 2-fold diversity scheme with frequency-domain equalization (FDE), such as [11],[12] and [13]. As for a general FDE scheme having more than 2 transmit antennas, a compromising method based on quasi-orthogonal STBC (QO-STBC) [10] was proposed in [2], where the nearly full rate was preserved at the cost of achieving only partial spatial diversity. Achieving full spatial diversity, particularly based on FDE and cast into the general structure in [14], was reported in [26]. However, the derivation in [26] is based on the 2-fold structure as a start and then generalized to multiple-antenna scenario intuitively by introducing the block-level concept in [14]. It is as instructive as

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Aside from the derivation issue, we will counterbalance the deduction in [26] that the “CP-only” scheme cannot exploit full multipath diversity, by giving a PEP analysis. Although a premise which is numerically supported by simulations must be met for justification of our PEP analysis, it does shade some light on extending the discussions in [17] to a more general scheme.

Also, optimal training design for MIMO communications in either flat-fading or frequency-selective fading environments is an important topic in practice. A pilot symbol-aided linear MMSE-based training scheme with optimal/orthogonal training is considered in [15]. In [16], the discussion in [15] was generalized to take advantage of space-time diversity. Nonetheless, none of which particularly considered the OSTBC class, and hence the optimal designs were quite involved. It will be shown that with the structure of training blocks fixed as OSTBC, the optimal training design can be simplified to optimal power allocation design.

1.2 Thesis Overview

In this thesis, we introduce a generalized FDE technique based on structure of [14] and by extending the training-based channel estimation approach in [3] we arrive at similar design criteria for optimal/orthogonal training as those obtained in [3] which considers flat-fading scenario. We propose an extended block-level OSTBC scheme capable of achieving full spatial and multipath diversities over frequency-selective fading channels when more than two transmit antennas involved by using FDE. What differs from [26] is that instead of starting from the 2-fold special case, we provide a new and instructive derivation based on general multiple-antenna signal model. The proposed scheme has nearly 1/2 symbol rate when discarding CP overheads. Pairwise error probability (PEP) analysis for demonstrating full spatial and full multipath diversities will be given. It will also be shown that since the signal model resembles that in flat-fading scenario, the optimal training designs such as those developed by M. Biguesh et al.[3] can be generalized to the frequency-selective fading scenario in a straightforward manner. The extended training-based channel estimation is optimal in least-squares (LS) sense in frequency-domain under a given power constraint, provided that a power criterion is satisfied. Also a time-domain linear MMSE channel estimation technique can be developed in a similar fashion as [3]. Adopting the Alamouti scheme for the construction

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of the proposed encoder with proper changes in dimension and scalar factor makes the 2-fold diversity scheme a special case.

In the next chapter, we will propose a transmission scheme and outline that entire system model. Post-processing including the subsequent FDE for data transmission and the training-based channel estimation for training mode will be introduced in chapter 3 and 5, respectively. Also, the optimality conditions for achieving lowest possible NMSE will be derived accordingly. In chapter 4, we will derive our PEP analysis. Simulation results of NMSE vs. SNR for training-based channel estimation are discussed in chapter 6.

1.3 Notations

The following notations are adopted throughout the thesis: P*_{, P}T_{and P}H_denote conjugate, transpose and conjugate-transpose of matrix P, respectively. A⊗B stands for the kronecker product of matrix A and B. Let Re{P} and Im{P} stand for the real and imaginary parts of matrix P, respectively. Let Tr{P} and vec(P) denote the trace and the vectorization of matrix P, respectively. For _y_∈_{, Diag(y)}N _∈N N× _{stands for an N} by N diagonal matrix with y on its main diagonal. For _A_∈N N× _{, Diag(A)}_∈N N× stands for the vector whose ith

entry is the ith

diagonal entry of A. For a matrix N M×

∈

A _,

_{[ ]}

ij

A denotes the entry at the ith

row-and-jth

column position of A . P(i:j,m:n) denotes an extracted submatrix consists of from ith

to jth

rows and from mth to nth

columns of matrix P. P(:,m:n) indicates that all rows ranging from mth to nth

columns are referred. Similarly define P(i:j,:). The symbol F is preserved for N by N normalized discrete-time Fourier transform (DFT) matrix, with the (m,n)th

entry of F being [ ] 2 ( 1)( 1) 1 m n j N mn N e π − − − = F , 1≤m n, ≤N.

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Chapter 2 System Model

In this chapter, we will introduce a transmission scheme based on a block-level extension of OSTBC. As a preliminary, we will also review some basic properties of orthogonal design of space-time block codes, which are to be employed for introducing block-level generalized orthogonal space-time block code (BGOSTBC) encoder as depicted in Fig. 2. The mechanism of CP insertion and CP removal for combating inter-block interference (IBI) will be reviewed at the bottom of this chapter.

2.1 System Configuration

As depicted in Fig. 1, the overall system consists of a BGOSTBC encoder followed by CP insertions at the transmitting end while the receiving end comprises CP removal, DFT and the subsequent channel estimation plus FDE. Let Mt and Mr denote the number of transmit antennas and receive antennas, respectively. Assume that the channel order is L (L+1 taps) for all the subchannels and known a priori. We assume the CP length is exactly L. Let N be the block length and N ≥ + . The information L 1 symbol blocks to be transmitted are accumulated over K blocks, each one of which will be sent by a certain transmit antenna during a specific time epoch with CP insertion. The block ordering is set up according to the proposed BGOSTBC encoder which occupies 2K time epochs for transmitting KN symbols. Each time epoch lasts for N+L symbol periods, where the redundant L symbol duration accounts for the CP insertion. At the receiving end, the received signal blocks over the entire 2K time epochs are buffered after CP removals. Then the equivalent IBI-free received blocks are Fourier transformed. With the outputs at the DFT system block available, we can either acquire channel estimation in training mode or perform FDE on the transmitted information symbols.

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In this subsection, the referred time epoch lasts for N symbol periods only since no CP insertion involved. Now suppose we have collected K symbol blocks, c₁, , c₂ , cK. Let C = c1 c2 cK, where

N k ∈

c denotes the kth

information symbol block to be transmitted, whose nth

symbol is denoted by c n_k( ), 0≤ ≤n N −1, 1≤ ≤k K. The transmission scheme of the BGOSTBC encoder can be regarded as a forward transmission mode followed by a reversed transmission mode, each of them occupying exactly K epochs. In the latter mode each incoming signal block is time-reversed and conjugated prior to transmission, while in the former mode transmission is carried out with blocks unmodified.

Now, let us define the encoder output. The encoder has single serial input and parallel Mt outputs. Let

( )p N m ∈

s denote the signal block transmitted from the mth

encoder output path during the pth

time epoch. Collecting encoder output over 2K time epochs and across Mt encoder output paths yields a matrix of dimension 2NK by Mt, whose ((p-1)N+1:pN,m)th

block-entry is ( )p m

s by definition. The so obtained matrix is essentially a block-level extension of generalized complex orthogonal design[1]. See also [14]. Let us define G as the overall output at the BGOSTBC encoder stacking across B 2K epochs and herein list some properties of it:

* 2NK Mt 1 , k k K B A k B k k × =  _⊗ ₊ _⊗ _∈    

∑

G ΧΧΧΧ c ΧΧΧΧ d (2.1) t 2K M where (0) (1) ( 1) . ( ) ((- ) ), 0 1. , , 1 . , 1 _. , 1 , 1 . , 1 , k k t k l l k t k l l k k l t t T k k k k k k N A B M T A A T A A M T B B T B B T A B M M d d d N d n c n n N k K k l K k l K k l K k l K × ×   = __ − __ = ≤ ≤ − ∈ ≤ ≤  ≤ = ≤  = _− ≤ ≠ ≤   ≤ = ≤  = _− ≤ ≠ ≤  = d I I 0 Χ Χ ΧΧ ΧΧ Χ Χ Χ Χ Χ ΧΧ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ ΧΧ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ ΧΧ Χ Χ Χ ₁ k l_, K_.                 _≤ _≤  Specifically

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and k k k k t K K K A A B A K K K K M × × ×     _ _   _  = _ _ =_ __         0 I X X I 0 0 Χ ΧΧ Χ G , (2.2) where Mt K k A × ∈

G are generic constituent matrices of certain OSTBC design of dimensions depending of choices of K and Mt. See also Appendix A3 and [1] for a detailed description about the construction of constituent matrices of OSTBC. The time-reversal, d nk( )=ck((- ) )n N , is carried out via the modulo-N operation defined as (- )n N =N− for 1n ≤ ≤n N− and (- )1 n N = for n = 0. Note that 0 Χ ΧΧ ΧΧ ΧΧ ΧAk, Bk ∈

t

2K M×

_{, 1}_{≤ ≤}_k _K _{, are constituent matrices of a certain GOSTBC determined by} choice of (K,Mt). ΧΧΧΧA_k and ΧΧΧΧ are non-overlapping matrices consisting of only ones B_k and zeros up to sign changes, and are responsible for designating space-time ordering of the forward transmission mode and the reversed one, respectively, i.e., ( )p

m s takes ck if k A _pm    

ΧΧΧΧ  is one, for 1≤ ≤p K . Similarly, s( )mp takes * k

d if ΧΧΧΧB_k_pm is one, for

1 2

K + ≤ ≤p K .

2.1.2 BGOSTBC Encoder (with CP Insertions)

Note that in Fig. 2, (K,Mt) = (4,3) and CP insertions following the encoding have been taken into account. The mth

encoder output path is followed by a CP insertion whose output is connected to the mth

transmit antenna, for all values of m. CP is inserted prior to the transmission of each symbol block for mitigating the channel distortion. From here on each time epoch lasts for N+L symbol periods. Since the constituent matrices simply serve as designating the space-time ordering for signal blocks, CP insertions directly apply to ( )p_{, 1} ₂

m ≤ ≤p K

s , 1≤m≤Mt. Let M=N+L. Then the signal blocks collected at the output of the CP insertions over 2K time epochs and across Mt transmit antennas can be represented in matrix from as

(

)

(

)

(

)

2 CP * CP CP 1 = _k _k B _K _B K A k B k k= ⊗  _⊗ ₊ _⊗     

∑

G I I G I c I d Χ Χ Χ Χ Χ Χ Χ Χ * 2MK Mt 1 = _k _k , K A k B k k × =  _⊗ ₊ _⊗ _∈    

∑

ΧΧΧΧ c ΧΧΧΧ d (2.3) where

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M CP * * M CP ( ) _{M N} CP ; ;

, the CP insertion matrix.

k k k k L N L L N × − _× ∈ ∈    _{ ∈}     c I c d I d 0 I I I

As a building block, next we review the transmission of a single signal block with CP through a frequency-selective fading channel.

2.2 Block Transmission and IBI-free Model upon CP Removal

Let the sequence sm(pM), sm(pM +1), , sm(pM +M −1)

_{denotes the symbols}

transmitted from mth

antenna during pth

time epoch. Let ( )p

[

₍ ₎ ₍ ₁₎ m sm pM sm pM + s M ( 1)T m s pM +M − _ ∈ _{. By noticing} ( ) ( ) CP p p m = _m s I s , we know that ( )p m s takes ck if k A _pm     Χ  ΧΧ

Χ _{is one, for 1}≤ ≤p K . Similarly, s( )mp takes * k d if ΧΧΧΧB_k_pm is one, for 1 K + ≤ p≤2K.

2.2.1 MIMO Frequency-selective Fading Channel and Noise Model

We consider a frequency-selective channel where the channel impulse response of L+1 taps between mth_{transmit antenna and j}th_{receive antenna is defined as}

jm h L+1 (0) (1) ( )T jm jm jm h h h L  _{ ∈}  

  . Throughout the thesis, we assume independence between channel taps for all hjm, 1≤ ≤j Mr, 1≤m≤Mt . Assume that hjm is circular symmetric Gaussian distributed, i.e., 2

1

( , )

jm σh L+

h ∼ CN 0 I _{. Hence the real and} imaginary parts of each of the entries of hjm are i.i.d. zero-mean Gaussian with variance _0.5 2

h

σ each, i.e., _{(0, 0.5 )}2 h σ

N _{. Assume that the channel remains fixed during} 2K time epochs. Let the sequence ηj(pM), (ηj pM +1),, (ηj pM +M −1) denotes the additive noise samples, circular symmetric Gaussian distributed, at the jth_receiver during the pth

time epoch. Hence, the noise vector defined as ( )p ₍ ₎ j ηj pM η M ( 1) ( 1)T j pM j pM M η + η + − _ ∈_assumes _{( ,} 2 ₎ w M σ 0 I CN _.

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Let the sequence v pMj( ), (v pMj +1),, (v pMj +M −1) denote the received symbols at the jth_{receiver during the p}th_{time epoch. Define} ( )p ₍ _{) (} ₁₎

j =v pM v pMj j + v M ( 1)T j v pM +M − _ ∈ _{. For 0}_{≤ ≤}_n _M _{− , 1}₁ _{≤ ≤}_j _M_r_, 1 0 ( ) ( ) ( ) ( ). t M L m j jm j m l v pM n h l s pM n l η pM n = = +

∑∑

+ − + + We can write ( )p j v as ( ) ( ) ( 1) ( ) 1 t M p tr p IBI p p m m j jm jm j m − =   =

∑

_ + _+ v H s H s η , where (0) 0 0 0 (1) (0) 0 0 (1) ; ( ) 0 (0) 0 0 0 ( ) (1) (0) 0 ( ) (2) (1) 0 0 ( ) (2) 0 0 0 0 0 ( ) 0 0 0 jm jm jm jm tr jm jm jm jm jm jm jm jm jm jm jm IBI jm jm h h h h h L h h L h h h L h h h L h h L  _  _  _  _        _  _  _  _  _  _    _  _  _  _  _  _    _   H H . 0 0 0  _  _  _  _  _      _  _  _  _  _      _  _  _  _  _  _      

After CP removal, we have the IBI-free received signal model, ( )p ( )p j _ N L× N_ j y 0 I v ₌ N ( ) ( 1) ( 1)T j j j y pN y pN y pN N  ₊ ₊ ₋  _∈     , where y ij( ), pN ≤ ≤i pN +N − , 1

are the received symbols at the jth_{receiver during the p}th_{time epoch after CP removals.}

(

)

( ) ( ) ( ) ( 1) 1 ( ) ( 1) ( 1) t p j j j j M tr p IBI p p m m N L N jm N L N jm N L N j m v pM L v pM L v pM M − × × × =   =__ + + + + − __       =

∑

__ __ +__ __ +__ __ y 0 I H s 0 I H s 0 I η

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( ) _{( )} _{( )} 1 ( ) ( ) 1 ( ) (0) 0 0 0 0 0 0 ( ) 0 0 0 0 0 0 ( ) (0) = , t t jm jm M _jm L N L L _p _p m j m N jm jm M p p jm m j m h L h h L h L h × − = =            _{ } _  _{ } _ = _ _{ } _ +                 +

∑

0 I s w I H s w where ( )p ( )p j _ N L× N_ j w 0 I _η _and (0) 0 0 0 ( ) (1) (1) (0) 0 (0) 0 ( ) . ( ) ( 1) (0) 0 ( ) 0 0 0 0 ( ) ( 1) (0) jm jm jm jm jm jm jm jm jm jm jm jm jm jm jm h h L h h h h h L h L h L h h L h L h L h                ₋           ₋      H Let ₍ ₎ N 1 1 = T T jm  _jm _{× − −}_{N L}  ∈  

h h 0 . We note that Hjm is an N by N circulant matrix with its first column being hjm. It follows that, after CP removal, the signal received at the jth

receive antenna on the pth

time epoch is IBI-free and given by, for 1≤ ≤p 2K ,

( ) ( ) ( ) 1 . t M p p p j jm m j m= =

∑

+ y H s w (2.4)

Let us define the following terms which will be used through out the following chapters: t t r t M (L+1) 1 2 M N 1 2 M M (L+1) 1 2 t t r T T T T j j j jM T T T T j j j jM T T T T all M   _{ ∈}  _ _      _ _  ∈  _ _   _ _  ∈  _ _  h h h h h h h h h h h h . (2.5)

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Chapter 3 Frequency-Domain Equalization

In this chapter, we propose a generalized FDE scheme based on block-level extension of generalized complex orthogonal designs (GOSTBC). The extended block-level OSTBC scheme with more than two transmit antenna is capable of achieving full transmit- receive diversity using FDE over frequency-selective channels. This shows an alternative to the block-level extension resorting to QO-STBC[2], which can indeed achieve perfect FDE in more-than-two transmit antenna scenario but at the cost of additional hardware complexity accounting for adders. Aside from this complexity drawback, its spatial diversity is halved.

3.1 Overall Frequency-Domain Received Signal Model

In the subsequent discussion, we will only derive the received signal model at the jth receive antenna as all receivers have the same signal model except for different channel impulse responses. First recall that ( )p

m

s in (2.4) takes ck if ΧΧΧΧA_k_pm is one, for 1≤ ≤p K and takes *

k

d if ΧΧΧΧB_k_pm is one, for K+ ≤ ≤1 p 2K . With the above observations, we have the following received signal model after collecting the IBI-free received signal blocks over 2K time epochs (Without loss of generality, we have assumed the transmission started from time epoch index 1 and collect the received blocks all the way up to index 2K)

( ) ( )

( ) (

)

(

)

(

)

( ) ( )

( ) (

)

(

)

' (1) (2) ( ) ( 1) (2 ) (1) (2) ( ) ( 1) (2 ) ' 2 1 t T T T _K T _K T _K T j j j j j j M _T _T _T _T _T T K K K K jm m m m m m j m + + =         = ⊗ _ _ +  

∑

y y y y y y I H s s s s s w

(

)

' 2 1 (:, ) , t M K jm B j m m = =

∑

I ⊗H G +w (3.1) where '

( ) ( )

(1) T (2) T

(

(2 )K

)

T T j = _ j j j _

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time epochs.

3.1.1 Space-Time Combining Using Signal DFT’s

Define ( )p ( )p_{, 1} ₂ j yj ≤ ≤p K Y F , '

(

)

' 2 j IK ⊗ yj Y F and '

(

)

' 2 j IK ⊗ wj W F . So

performing block-wise DFT on y in (3.1) leads to 'j

( ) ( )

(

) (

)

(

)

(

)

₍

₎

(

)

' (1) (2) ( ) ( 1) (2 ) ' 2 2 2 1 (:, ) t T T T _K T _K T _K T j j j j j j M K K jm B K j m m + =   = _ _ =

∑

I ⊗ I ⊗H G + I ⊗ w Y Y Y Y Y Y F F

(

)

(

)

' 2 1 (:, ) . t M K jm B j m m = =

∑

I ⊗ FH G +W (3.2) See the tailored version for 2-fold diversity in [9],[26]. Since Hjm is circulant, we can

decompose it by the DFT matrix as follows: H ,

(

)

,

jm jm = Λjm Λ =jm Diag N

H F F Fh

for all values of j and m. Let Xk Fck. Since d is obtained from k c by conjugated k time reversal, we have[24, pp. 123-124], * *

k k =

d X

F , 1≤ ≤k K . Hence we can rewrite (3.2) as

(

)

(

)

(

)

(

)

(

)

(

)

(

)

{

}

(

)

(

)

{

}

' ' 2 1 * ' 2 1 1 * ' 1 1 ' * 1 1 (:, ) (:, ) (:, ) (:, ) (:, ) (:, ) (:, ) . t t k k t k k t k k M j K jm B j m M K K jm A k B k j k m M K A jm k B jm k j k m M K k k A jm B jm j k m m m m m m m m = = = = = = = = ⊗ Λ + = ⊗ Λ ⊗ + ⊗ + = ⊗ Λ + ⊗ Λ + = ⊗ Λ + ⊗ Λ +

∑

∑∑

I G I c d c d Χ Χ ΧΧ ΧΧ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Y F W F W F F W W X X Based on (2.2), we have

(

)

(

)

' 1 ' * 1 1 1 (:, ) . (:, ) t k k M K k A jm _NK j j k m NK A jm k m m × = = ×      _{⊗ Λ} _ _        = ___ __{ } + _{⊗ Λ} ____+          

∑∑

G ₀ 0 Y W G X X (3.3)

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(

)

(

1 *

)

1 1 1 (:, ) (:, ) t k k M K k A jm _NK j j k m NK A jm k m m × = = ×      _{⊗ Λ} _ _        = ___ _+__ __+  ⊗ Λ             

∑ ∑

G ₀ 0 Y W G X X 1 * 1 1 1 (:, ) . (:, ) t k k M K A K k jm jm j k m K A m m × = = ×          _               =

∑ ∑

__  ⊗ Λ + ⊗ Λ  __+ 0 0 X G W G (3.4)

Note that the definition of Yj follows from the reversed transmission mode by virtue of

GOSTBC. Let

(

)

(

1

)

1 (:, ) * (:, ) 1 t Ak K K _Ak M m jk jm m jm m × × =  _{⊗ Λ +} _{⊗ Λ}     

∑

0 0 O G G , 1 2 T T T T K       X X X X and 2KN KN 1 2 j j j jK ×  

Λ __O O O __∈ _{. We can then rewrite (3.4) more compactly as,} for 1≤ ≤j Mr,

j = Λj + j

Y _X W. (3.5)

Equation (3.5) represents the frequency-domain input-output relation between the jth receive antenna and the transmit antennas, where Λ Xj, and Wj are embedded with knowledge of channel state information (CSI), of transmitted signal blocks and of channel noise at the jth

receive antenna, respectively. Let 1 2 r

T T T T M   = _ _ Y Y Y Y r 2KNM

∈ . By stacking the received signals in (3.5) across Mr receive antennas, we arrive at the following MIMO channel model in frequency domain

= Λ + Y _X W, (3.6) where ₁T ₂T T_r T M   = _ _ W W W W and 2KNMr KN 1 2 r T T T T M ×   Λ__Λ Λ Λ __ ∈ _is the equivalent frequency-domain MIMO channel matrix for block transmission.

3.1.2 Frequency-Domain Equalization with Perfect CSI

Let _X= Λ+Y and W = Λ+W and assume that rank

( )

Λ =KN . Hence _{Λ Λ is}H invertible, and the Moore-Penrose pseudo-inverse of Λ can be written as Λ = +

(

Λ ΛH

)

−1Λ ∈ H KN 2KNM× r _{. So by pre-multiplying both sides of (3.6) by the linear}

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X=X+ W, (3.7)

where X contains the decoded DFT’s of the transmitted signal blocks. Therefore the transmitted blocks can be recovered through FDE. It is seen from (3.7) that by nonsingularity of Λ Λ , FDE can be achieved up to a zero-mean equalization error term H

W when perfect CSI is available. The proposed BGOSTBC achieves block-level orthogonality in frequency domain, that is, _{Λ Λ is a diagonal matrix and diagonally}H loaded with diversity weightings. More precisely, we have the following results.

Theorem 3.1 1 1 2 r t M M H H K jm jm j= m=  _  _ Λ Λ = _ ⊗ Λ Λ _  I

∑∑

.

(Proof: See Appendix A1.)

Remark 3.1 With perfect CSI, the FDE can be carried out efficiently. By efficiency we mean the involved computation for matrix inverse

(

_{Λ Λ}H

)

−1

reduces to merely computing the reciprocals of each of the diagonal entries of _{Λ Λ . This}H can be seen from Theorem 3.1 that _{Λ Λ is diagonal.}H

Remark 3.2 The double summation of the elementary square block of _{Λ Λ}H implicitly indicates that the full MrMt-fold spatial diversity is achieved through the proposed FDE scheme (See also chapter 5 for an explicit validation of full diversity via PEP analysis). Notice that had the encoder been realized with QO-STBC, the system would enjoy halved spatial diversity by carefully decoupling the received signals upon DFT transform[2]. The QO-STBC is more redundant than OSTBC in nature by inspection and relies on additional arithmetic operations for exploiting quasi-orthogonality at the receiving end, with approximately full transmission rate though.

Remark 3.3 Equation (3.3) is hereby regarded as a building block as we will exploit more of it towards the discussion of training-based channel estimation in chapter 5. Such discussion is as practical as essential to applying the OSTBC to the frequency-selective fading environment. Our proposed scheme can be adopted to

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scenario in a quite straightforward manner, in contrast to [16], with similar conclusion on optimal training design obtained in [3].

Remark 3.4 By adopting the Alamouti scheme for the construction of G with B proper changes in dimension and scalar scaling factor makes the 2-fold diversity FDE (Alamouti-like) scheme a special case.

Remark 3.5 Notice that the FDE behind (3.7) is based on the premise that

( )

rank Λ =KN , and hence

(

H

)

rank Λ Λ =KN . This in turn requires that

1 1 r t M M H jm jm j m rank N = =  _  _{Λ Λ} _₌  _  _



∑ ∑

 be satisfied as we can see from Remark 3.1. Notice that each of the diagonal entries of Λ Λ is nonnegative. To proceed further, let Hjm jm

us define ( )₀ ( )₀ ( ₁)T N jm jm = hjm hjm hjm N −  ∈   h h F , where h jm( )n

is the _nth_{entry of the DFT, 0}_{≤ ≤}_n _N_{− , 1}₁

r

j M

≤ ≤ and 1≤m≤Mt.

Theorem 3.2 If there exist at least one pair of values

(

j m*, *

)

, 1≤ j* ≤Mr, *

1≤m ≤Mt, such that * *( ) 2

0 j m

h n > , 0≤ ≤n N − , then 1 rank

( )

Λ =KN holds.

Let us denote the sufficient condition in Theorem 3.2 as (c1.R). That is, if there is a subchannel whose transfer function is free from zeros at the frequencies

2 n N

j

e π , 0≤ ≤n N − , then 1 _{Λ Λ is nonsingular. The condition is very weak and is}H

generically satisfied. It is interesting to consider the following special case. Suppose

( ) * * 2 0 j m h n > , for 0≤ ≤n N− while 1 * 1, for 1 jm = _N_× ≤ ≠j j ≤M_r h 0 , 1 ≤ * t

m ≠m ≤M . Consequently equation (3.4) becomes

{

}

1 K k j jk j k= =

∑

O + Y _X W with

(

*

)

(

1

)

* * * * * 1 (:, ) (:, ) Ak K K Ak m H jk j m m j m × ×  _{⊗ Λ} ₊ _{⊗ Λ}      0 0 O G _G .

By stacking across all Mr receive antennas, we have Λ with the block-level orthogonality that 2

(

* * * *

)

H H

K j m j m

Λ Λ = I ⊗ Λ Λ , which clearly lacks of spatial diversity, compared to (3.6) whose subchannels are all active. Intuitively, the decoded signal X = Λ Y+ becomes much more erroneous without the spatial

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diversity than with any. The above scenario arises when our MIMO channel model degenerates to a single-input-single-output (SISO) frequency-domain equivalent. So far, even though we consider a scenario which is more likely than (c1.R) to occur, such situation is far from common, not to mention that only one pair of values

(

*_, *

)

_{, 1} * _{and 1} * r t j m ≤ j ≤M ≤m ≤M such that * *( ) 2 0 j m h n > , 0≤ ≤n N − . 1 This is why we call (c1.R) a weak condition as it can be easily satisfied in practice.

Notice that the reason why we are able to recover the transmitted symbols even under the above mentioned harsh situations lies in that the space-time redundancy transmitted a signal block through every transmit antennas over different time epochs. Even though the MIMO channels degenerates to an SISO channel, all the transmitted blocks can still manage to the single receiver, provided that the reception lasts for 2K time epochs.

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Chapter 4 Pairwise Error Probability Analysis

In this chapter, we will derive upper bounds for the average pairwise error probability (PEP) for the FDE-based scheme. The essential assumption under which the derivation is justified would be stated with numerical support. Our perspective reveals that at high SNR, the proposed system dose have the potential of delivering maximum possible spatial as well as multipath diversity.

4.1 PEP Analysis for Suboptimal Detection Problem

First, we derive an upper bound for the average PEP, assuming that the decoding consists of a linear equalization followed by an symbol-wise quantization into the signal constellation A . See also [11] and [17]. To see that, let us formulate the detection problem as follows. Based on (3.6), we have Y=Ωc+W_{with the matrix} Ω

(

)

2KNMr KN

K

×

Λ I ⊗F ∈ being responsible for yielding the frequency-domain output of

the time-domain input signal blocks KN

1 2 T T T T K  _{ ∈}     c c c c _{. Hence we can} address the ML-detection problem as follows:

KN 2 argmin ML ∈ = − c c c A Y Ω _{, (4.1)}

where cML is the optimal decoded symbol block in ML sense. However, this ML exhaustive search yields infeasible computational cost of order AKN for practical values of K and N, where A denotes size of the employed constellation. So we seek a suboptimal linear equalization approach. Note that W is a white Gaussian vector since DFT serves as a unitary operation only. Let us define _Z _ΛH_Y_∈KN

, _ΛH _∈KN

W W and _Θ_ΛH_Ω_∈KN KN× _{. Then we can rewrite (3.6) as}

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We assume rank

( )

Λ =KN . Since IK ⊗F is of full rank, rank( )Θ =KN by definition of Θ . Hence the inverse _Θ−1_{exists and can be used as a linear equalizer.} Let _Z ₌_Θ−1_Z_{denote the output of the linear equalizer}_Θ−1_{, then from (4.2) we} readily arrive at

_Z _{= +}_c _Θ−1_W_{. (4.3)}

Notice that the diversity gain weighting has been normalized at the output of _Θ−1_and hence the detection problem can be formulated with respect to the employed constellation. Based on signal model of (4.3), we obtain an suboptimal detection problem: KN 2 argmin , ∈ = − c c c A Z (4.4)

where c is an estimate of c after equalization. Note that the so obtained estimate minimizing the metric in (4.4) is suboptimal since the underlying noise is no longer white and its covariance depends on the matrix _Θ−1_ΛH_{. We see that minimizing the} metric in (4.4) amounts to a symbol-wise hard-decision into A since

2 − c

Z is the sum of KN nonnegative terms and each of which can be minimized with respect to an entry of c , i.e., minimization on symbol-level. Therefore, using _Θ−1_{as an equalizer} and (4.4) as the detector amounts to having a linear equalization scheme followed by a hard decision on each entry of Z into the constellation A . It is noteworthy that the computational cost of (4.4) is linear in composite block length KN, which is computationally cheaper than that of (4.1).

The PEP analysis considers the probability that a symbol block _c_∈_AKN_is transmitted while another c is detected in the minimum-distance perspective. Given the channel realization h , and hence the matrix Λ , the conditional PEP is defined as all

[

]

2 2

Pr c→ c hall =Pr_c Z- < c-Z Λ_. (4.5)

Note that proving that the average PEP has maximum diversity is equivalent to proving that the error rate performance exhibits maximum diversity, by virtue of the union

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the Euclidean distance between c and c , as well as the corresponding normalized difference, respectively. Let _Re

{

(

1

)

H

}

ξ Θ− W e _{[18, pp. 508], then (4.5) becomes}

[

]

Pr Pr . 2 all ξ     → = > Λ     c c h D _(4.6)

As the variable ξ , when conditioned on channel realizations, is a zero-mean Gaussian random variable, the error probability in (4.6) is completely determined by the variance

2 ξ

σ . To compute 2 ξ

σ , first let us write W =

(

I₂KM_r ⊗F w

)

for notational purpose, where w is related to the white noise '

j w in (3.1) by '(_{0 :} ₁)T T T j KN   =__w − w __ w , ( )* ' mod : 2 1 j KN KN −N = −

w w . Then we can factorize

(

_Θ−1_W

)

H _e _as H e w , where

(

)

(

1

)

2 r H H KM −_Λ _⊗ e I e

Θ F _{. The variance is computed as follows:}

{

}

{

(

)

}

{

}

{

}

{ }

{

}

{ }

{

}

{ }

(

)

(

{ }

)

{

}

2 2 2 1 2 2 2 Re Re Re Re Im Im 2 Re Re Im Im . H H T T T T E E E E E E ξ σ = ξ Λ = _ − Λ_     = Λ   = __ __ Λ +  _{ Λ +}     Λ e e e e e e W Θ w w w w w

Notice that since w is white, the entries of w are i.i.d. circular symmetric Gaussian, with real part and imaginary part of each of the entries being _{(0, 0.5} 2₎

w

σ

N _distributed.

Also notice that Re{ }_w T Re

{ }

e _and_Im_{{ }}T _Im

_{{ }}

e

w are statistically independent of each other by virtue of the circular nature of w . For notational purpose, let [ ]y_i denotes the ith

entry of a vector y . So we can rewrite the above equation as

{ }

{

}

{

{ }

}

{ }

[

]

{ }

[

{ }

]

{ }

{

[

{ }

]

}

{ }

{

[

{ }

]

}

2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 Re Re Im Im Re Re Im Im Re Re Im Im r r r r T T KNM KNM i i i i i i KNM KNM i i i i i i E E E E E ξ σ = = = =     = __ __ Λ + __ __ Λ          _ _ _ _ _ _ _ _  = __ _ _ _ +_ _ _ _ Λ_               = _ _ Λ + _ _ Λ

∑

e e e e e e w w w w w w

(33)

{ }

(

)

(

)

(

)

2 2 2 2 2 1 1 2 2 2 2 2 1 2 2 2 2 2 2 1 Re Im 2 1 2 1 2 1 , 2 r r r r KNM KNM w _i _i i i w H H w KM H H w K KM σ σ σ σ = = − +      __ _ _ _ _ _  =__ __ _ _ + _ _ _      _  =__ __    = __ Λ ⊗ __   = __ ⊗ Λ ⊗ __

∑

e

∑

e e I e I I e F F F Θ

where we have used independence between each of the entries of Re w due to w { } being white and zero-mean. Notice that 1 H

(

H

)

K

−_{Λ =} _I _⊗_F _Λ+

Θ _{. Similar reasoning}

applies to Im w{ }. We know that the PEP in (4.6) can be expressed in terms of the

Q-function, ( ) 1 22 2 t Q _πe dt α α

∫

∞ − , as

[

]

(

)

(

)

(

)

(

)

2 2 2 2 2 2 2 2 Pr 4 2 , 2 r r all H H w K KM H w K KM F Q Q Q ξ σ σ σ + +  _  _  _ → _{= }_ _     _  _  _  _  _  = _ _  _   _⊗ _Λ _⊗  _  _ _ _        _  _  _  _  ≤ _ __    ⊗ Λ ⊗ _   c c h I I e I I F F F F D D D

where we have invoked the inequality that Ae ₂ ≤ A_F e ₂ = A_Ffor any matrix A

and a unit vector e. Since multiplication by unitary matrix dose not change Frobenius norm, we arrive at

[

]

2 2 Pr . 2 all w _F Q σ +  _  _  _  → ≤ _ _  _Λ      c c h D Then by using ( ) 1 2 2exp( 2)

Q _α ≤ −α _{, we have the PEP of interest upper bounded as}

2

(34)

Let us define the following terms:

(

)

(

)

(

)

( ) 1 ( ) 1 (1) (2) ( ) (1) (2) ( ) 1 1 1 2 1 2 2 2 2 1 (:, ) ; ; (:, ) ; ; ; k k t t t T m _A jk _N K T m K jk _N A M jk jk jk jk M jk jk jk jk T H H H H H H j _K _j _K _j _K _jM j _K _j m m × × − − −  _  _{ ⊗}  _      _  _  _ ⊗  _                  _{⊗ Λ Λ Λ} _{⊗ Λ Λ Λ} _{⊗ Λ} _{Λ Λ}      ⊗ Λ Λ A I 0 0 A I A A A A A A A A I I I I G G ϒ ϒ

(

)

1

(

)

1

(

)

1 2 2 2 t . T H H H K j K jM − − −                  _ _  _Λ _{⊗ Λ Λ Λ} _{⊗ Λ} _{Λ Λ}        I I (4.8)

From (4.8), _Λ+

(

_{Λ Λ}H

)

−1_ΛH_{can be rewritten as}

+ ,       Λ = __ __{  }      A A ϒ ϒ (4.9) 11 1 11 1 1 ₁ 1 1 where ; ; ; . r r r t _r _t r _r M M K M M _K _{M M} M _M        _ _ _ _  _ _    _ _     _ _ _ _  _ _    _ _       _ _   _ _    _ _     _ _ _ _  _ _    _ _     _ _ _ _  _ _    _ _ _ _     A A A A A A A A _A _A ϒ ϒ ϒ ϒ ϒ _ϒ Hence, 2 ₂ 2 2 F _F F _F +   _ _ Λ ≤ _ _{ }_ + ___

A A ϒ ϒ . Notice that A A is solely determined by

the encoder structure, and by (4.8) it can be easily verified that 2 2

F = _F ϒ ϒ _and 2 1 2 1 1 1 1 2 . r t r t M M M M H H jm jm jm F j m j m F K − = = = =  _  _ = ⋅ _ Λ Λ _ Λ   

∑∑ ∑∑

ϒ _(4.10)

(35)

With the definition of hjm

in Remark 3.5, (4.10) can be rewritten as

( ) ( ) 2 1 2 2 2 1 1 0 1 1 2 r t r t M M N jm F M M j m n jm j m h n K N h n − = = = = =  _  _  _  _  _     = ⋅ _ _   _{ }    _      _ _   _ _   

∑∑ ∑

∑∑

ϒ ( ) 1 1 ₂ 0 1 1 2 . r t M M N jm n j m K h n N − − = = =  _  _ = ⋅ _ __  

∑ ∑∑

(4.11)

Now, let us define an N by N matrix D as ps

( ) ( ) ( ) 2 1 1 2 1 1 2 1 1 0 1 1 r t r t r t M M jm j m M M jm j m ps M M jm j m h h h N = = = = = =                 =            ₋       

∑∑

D . (4.12)

We assume rank

( )

Λ =KN, or equivalently

1 1 r t M M H jm jm j m rank N = =  _  _{Λ Λ} _₌  _  _ 

∑ ∑

 . Hence, we see that Dps is invertible. Hence, by defining

2 E F C _ _ A A _{and from (4.9) up to (4.12)} we arrive at 2 2 4 ₁ E ps F F K C N + − Λ ≤ ⋅ ⋅ D . (4.13)

Before we proceed further, let us make one more assumption deduced from the rank premise: condition number of Dps, denoted as K

( )

Dps , is upper bounded by a finite real number Ku ∈ for all possible channel realizations, i.e.,

( )

Dps ≤ u for all Dps.

(36)

( )

max 1 1 1 1 min 1 1 1 1 max min r t r t r t r t M M M M H H jm jm jm jm j m j m M M M M H H jm jm jm jm j m j m Diag Diag λ λ = = = = = = = =     _ _ _ __  _{Λ Λ} _ _ _ _{Λ Λ} __  _ _ _       _ _     Λ = =  _     _{Λ Λ} _   _     Λ Λ           __  __

∑∑

K ,

where λ_max

( )

Λ and λ_min

( )

Λ denote the maximal and minimal eigenvalues of matrix Λ respectively. Hence K

( )

Dps =K

( )

Λ . By regarding the simulation results from Fig. 3-1. to Fig. 3-5, we see that

i) For all possible system configurations (N,Mr,Mt,L,K), the probability of singular occurrence rank

( )

Λ <KN is very low in practice. As one can perform as many simulations as possible to find the above highly plausible, our rank premise is reasonable. Hence K

( )

Dps < ∞.

ii) Given known channel order L and desired transmission rate, selecting high values for Mt and Mr can effectively suppress K

( )

Dps .

iii) It agrees with the intuition that as N increases, the probability of K

( )

Dps → ∞ drops. This is because each of the diagonal entries, which are correlated, of Dps is sum of nonnegative terms.

From the above three observations we deduce that K

( )

Dps can be universally upper bounded by a finite real number in practice.

On the other hand, suppose _B_∈n n× _{is a nonsingular diagonal matrix, then for any} full rank _A_∈n m× _{, m<n, such that}

( ) ( ) 2 1 m F ≤ n +n B A _K , 1 2 ( ) 2 F F + − _≤ B BA . Proof: [ ] [ ]

₍

_{( )}

₎

( ) 1 1 2 2 2 2 1 0 0 2 2 2 2 1 . n n ii ii i i F F F F F F n n m − − − + − = = ⋅ + =

∑

≤ ≤ ≤ B B B B BA B B B A K (4.15)

Therefore, by assuming (4.14) and along with (4.15), there exist infinitely many full rank

{ }

Ψ ∞ ∈ N N L× −( )_satisfying _Ψ 2 _≤ m _{such that} _D−1 2 _≤

(

_{D Ψ}

)

+ 2 _for

(37)

all channel realizations. Now let us find such a matrix Ψ among these possible candidates so that any (N-L) of the rows of Ψ are linearly independent. Note that this additional constraint can be very easily satisfied and hence does not in any way contradict (4.14). Then, (4.13) can be written as

(

)

2 2 4 E ps F _F K C N + Λ ≤ ⋅ ⋅ D + Ψ _{. (4.16)}

Let β

{

n n0, ,1,nL−1

}

be the set of indexes corresponding to the smallest L composite frequency responses such that

( )

(

)

2 2 1 1 1 1 2 2 2 0 1 1 1 1 1 1 1 1 for and , r t r t r t r t r t M M M M jm _p jm _q _p _q j m j m M M M M M M jm jm jm _L j m j m j m h n h n n n h n h n h n β β = = = = − = = = = = =  _≤ _∈ _∈    _≤ _≤ _≤  

∑∑

(4.17)

where 0≤ni ≤N−1 and 0≤ ≤ − Define i L 1. Dβ as an N by N diagonal matrix whose entries are given by

( ) 2 1 1 1 , if . 0 , if r t nn M M jm j m nn n h n n β β β β = =    = ∈        = ∈   

∑∑

D D (4.18)

Then we construct an (N-L) by N matrix Ψβ by the following two-fold step:

i) Remove all the rows, whose row indexes belong to set β , of Ψ to obtain an (N-L) by (N-L) matrix Ψ .

ii) Insert L zero columns to Ψ−1_{so that after the insertion, the inserted columns} have indexes belong to set β . Let Ψβ denote the so obtained matrix. Note that the constraint on linear independence of any (N-L) of rows of Ψ ensures existence of Ψ−1_.

正交時空區塊碼之頻域等化及其基於訓練機制之多輸入多輸出的擇頻通道估測

國

立

交

通

大

學

電機與控制工程學系

碩

士

論

文