適用於多重輸出入正交分頻多工調變之空頻碼設計

國 立 交 通 大 學

電信工程學系

碩 士 論 文

適用於多重輸出入正交分頻多工調變之空

頻碼設計

Design of Space-Frequency Codes for

MIMO-OFDM Systems

研究生：賴俊池

指導教授：王忠炫博士

適用於多重輸出入正交分頻多工調變之空頻碼設計

Design of Space-Frequency Codes for

MIMO-OFDM Systems

研究生 ：賴俊池 Student：C. C. Lai

指導教授：王忠炫

Advisor：C. H. Wang

國立交通大學

電信工程學系碩士班

碩士論文

Submitted to Department of Communication Engineering College of Electrical and Computer Engineering

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master of Science

in

Electrical Engineering January 2007

Hsinchu, Taiwan, Republic of China

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Design of Space-Frequency Codes for

MIMO-OFDM Systems

Student: C. C. Lai

Advisor: C. H. Wang

Department of Communication Engineering

National Chiao Tung University

Abstract

A space-frequency coded multiple-input multiple-output orthogonal fre-quency division multiplexing (MIMO-OFDM) system on frefre-quency-selective fading channel is considered. By exploiting the central limit theory that the random variables can approach Gaussian distribution, which is used when the number of random variables is large enough, we will analyze the pairwise error probability of space-frequency codes and provide the corresponding de-sign criteria, which mean rank criterion and distance criterion. According to the channel variation speed, rapid fading channel and quasi-static fading channel are distinguished, the individual design criteria are also presented. From the code design of these case, it is deduced that the code construction of space-frequency codes is similar to the code construction of space-time codes by applying the central limit theory. The simulation results are presented to compare the various trellis code designed from various criteria in the channel environments.

Acknowledgement

I am deeply grateful to my advisor Dr. C. H. Wang for his enthusiastic guidance and great patience in research. I also appreciate my friends for their ardent assistance. Finally, I feel an immense gratitude to my parents for their encouragement and love.

Contents

List of Tables vii

1 Introduction 1

2 The Space-Frequency Codes 4 2.1 Space-Frequency Coded Systems . . . 4 2.2 Pairwise Error Probability and Design criteria . . . 7 3 Proposed Design Criteria of Space-Frequency Codes 11 3.1 System Model . . . 11 3.2 Proposed Design Criteria of Space-Frequency Codes for

Spe-cial Case: One OFDM Symbol . . . 13 3.3 Proposed Design Criteria of Space-Frequency Codes for

Gen-eral Cases: Rapid Fading Channel and Quasi-Static Fading Channel . . . 19

3.3.1 Rapid Fading Channel . . . 19 3.3.2 Quasi-Static Fading Channel . . . 24

4 Simulation 28

5 Conclusion 31

List of Figures

2.1 The space-frequency coded MIMO-OFDM system. . . 5 2.2 The equivalent space-frequency MIMO-OFDM system. . . 7 4.1 Simulation results of the various design criteria for one OFDM

symbol. . . 29 4.2 Simulation results of the various design criteria for several

OFDM symbols on rapid fading channel. . . 30 4.3 Simulation results of the various design criteria for several

List of Tables

Chapter 1

Introduction

In future wireless communication, the demand for safer and higher high data rate is rapidly increasing. Recently there has been much interest in applying multiple transmitter and receiver antennas in broadband wireless commu-nication techniques, and this is also called multiple-input multiple-output (MIMO) system, since the unfavorable effects of the wireless propagation environments can be largely reduced by using multiple transmitter and re-ceiver antennas. In order to raise security of communication, numbers of coding and modulation are adopted. One popular method of them is space-time coding. It is a coding technique executed in space and space-time domains to introduce related transmitted signals between various antennas at vari-ous time slots. There are two kinds of design standards of space-time codes commonly [1]. When the product of the codeword distance matrix and the number of the receiver antennas does not exceed 4 [2], the rank and determi-nant criteria are used. Oppositely, the rank and trace criteria will be applied.

In wireless environments, the channel is almost frequency-selective, so intersymbol interference (ISI) would exist. Orthogonal frequency division multiplexing (OFDM) is one of the techniques used to lower the effect of ISI and can transform the frequency-selective fading channel into the set of

correlated flat fading channels. It is clear that the correlation between the dif-ferent subcarriers in one OFDM symbol for the same transmitter and receiver antenna pair is connection with variances of the channel taps and the dif-ference of the frequencies. OFDM not only provides high data transmission rate and owns high bandwidth efficiency, but it also can effectively solves ISI by adding sufficient cyclic prefix (CP). In order to obtain the benefits of MIMO system and OFDM modulation, MIMO-OFDM has been acquired lots of attention, and coding for MIMO-OFDM system on frequency-selective fading channel will be required.

Generally speaking, coding across OFDM subcarriers and multiple trans-mitter antennas is called space-frequency coding, both space and frequency diversity are expected in frequency-selective fading channels. [3] has not only told us how to design the full diversity in the space-frequency scheme, but it has also shown that most of space-time codes designed to achieve full spatial diversity over frequency-nonselective channels are not validly applied in MIMO-OFDM system, since these codes do not exploit the frequency diversity. By the design criteria of [3], one construction of space-frequency codes was presented, the codewords were generated by multiplying particular columns of DFT-matrix with the modulated sample vectors [4]. Even though the method can achieve the maximum diversity, it is not easy to construct the codewords and the code rate is a little low. Later, many papers designed to achieve maximum diversity gain and own high code rate were submitted, such as [5]-[7].

In my thesis, I will exploit the spirit of the central limit theory presented in [1] to gain the design criteria of space-frequency codes. The organization of the thesis is shown below. In chapter II, the common space-frequency coded system, it’s pairwise error probability and the corresponding design criteria

will be introduced. Chapter III describes how to get the design criteria by using the central limit theory in great detail. In chapter IV, some simulation results are presented. Finally, we give some conclusions.

Notation: Superscripts T and H denote transpose and conjugate trans-pose respectly.

Chapter 2

The Space-Frequency Codes

2.1

Space-Frequency Coded Systems

In this section, the MIMO-OFDM system, the channel model and the rela-tion between the transmittd and received signals will be introduced.

The space-frequency coded OFDM-based multi-antenna system is repre-sented in Fig 2.1. This system consists of MT transmitter antennas and MR

receiver antennas, and N subcarriers is considered in each OFDM scheme. Suppose that frequency-selective fading channels have L independent delay paths and the same power delay profile for each transmitter and receiver antenna link. The fading gains are constant over each OFDM symbol.

The impulse response of the channel can be represented as the tap matrix Hl, which is given by Hl =         h0,0(l) h0,1(l) · · · h0,MT−1(l) h1,0(l) h1,1(l) · · · h1,MT−1(l) .. . ... . .. ... hMR−1,0(l) hMR−1,1(l) · · · hMR−1,MT−1(l)         (2.1)

Figure 2.1: The space-frequency coded MIMO-OFDM system.

where hj,i(l) is the value of the impulse response from the i-th transmitter

antenna to the j-th receiver antenna at time l (l = 0, 1, · · · , L − 1). Each hj,i(l) is modeled as the circularly symmetric complex Gaussian random

vari-able with zero mean and variance one. Assume that the MIMO channel is spatially uncorrelated, i.e., hj,i(l) is independent for different indices i and

j. After defining the time matrix of the channel, we can know that the frequency response matrix of the MIMO channel Hej2πNk

 is Hej2πNk  = L−1 P l=0 Hle−j 2π Nkl =         H0,0(_Nk) H0,1(_Nk) · · · H0,MT−1( k N) H1,0(_Nk) H1,1(_Nk) · · · H1,MT−1( k N) .. . ... . .. ... HMR−1,0( k N) HMR−1,1( k N) · · · HMR−1,MT−1( k N)         (2.2)

where the element Hj,i(_Nk) (k = 0, 1, · · · , N − 1) denotes the frequency

re-sponse of the channel from the i-th transmitter antenna to the j-th receiver antenna.

In the MIMO-OFDM system, the data streams are all OFDM-modulated and OFDM-demodulated. The OFDM modulator adopts an N -point IFFT to the consecutive source samples and then adds sufficient CP. After passing the channel, the OFDM demodulator eliminates CP and applies an N -point

FFT to recovery the consecutive source samples. Assume that the transmit-ted codeword matrix in one OFDM symbol is

C =         c(0)₀ c(0)₁ · · · c(0)_{N −1} c(1)₀ c(1)₁ · · · c(1)_{N −1} .. . ... . .. ... c(MT−1) 0 c (MT−1) 1 · · · c (MT−1) N −1         = [ c₀ c₁ · · · c_{N −1}] (2.3)

where each element of the transmitted matrix c(i)_k denotes the data trans-mitted from the i-th antenna on the k-th tone and is taken from a finite complex alphabet set in which the power of each alphabet is unity. In the second equality, c_k = [c(0)_k c(1)_k · · · c(MT−1)

k ]

T _{represents the k-th frequency}

vector, and all of the Mt samples are sent to the channel at the same time.

On the basis of the described framework, the reconstructed data vector at the k-th tone, i.e., r_k = [r_k(0) r_k(1) · · · r(MR−1)

k ]T, is given by r_k =pEsH  ej2πNk  c_k+ z_k, k = 0, 1, · · · , N − 1 (2.4) where z_k = [z_k(0) z_k(1) · · · z(MR−1) k ]

T _{is an addtive complex-valued Gaussian}

noise vector and satisfies

E[z_k zH_k0] =    σ2 nIMR k = k 0 0 k 6= k0 (2.5) with IMR denoting the MR-by-MRidentity matrix and σ

2

n is the noise power.

We also assume that the noise is uncorrelated for different receiver antennas. From (2.4), the space-frequency system is therefore equivalent to Fig2.2.

If we assume that the channel state information is only perfectly known at the receiver, the maximum likelihood decoder of the space-frequency system is ˆ C = arg min C N −1 X k=0 ||r_k−pEsH(ej 2π Nk)c k||2 (2.6)

Figure 2.2: The equivalent space-frequency MIMO-OFDM system. where Es is the energy of every transmitted signal. (2.6) tells us that the

decoder deals with that the minimization which is over all the likely code-words.

2.2

Pairwise Error Probability and Design

cri-teria

Based on the previously mentioned system, the pairwise error probability and the corresponding design criteria of the space-frequency system are derived in this section.

It is assumed that the codeword C is transmitted, and E = [e₀e₁ · · · e_{N −1}] is another legal codeword. Given that the channel state information is known at the receiver, the pairwise error probability that the codeword E is decoded is P  C → E|H  ej2πNk  = Q s Es 2σ2 n d2_{C, E|H}_ej2π_Nk ! (2.7)

where Q(x) is the complementary error function and d2  C, E|H  ej2πNk  = N −1 X k=0 k Hej2πNk  (c_k− e_k) k2 (2.8) represents the squared Euclidean distance between C and E. Then using the approximation

Q(x) ≤ 1

2exp{− x2

2 }, x ≥ 0 (2.9) to (2.7), so the upper bound of the pairwise error probability conditioned on the channel condition is

PC → E|Hej2πNk  ≤ 1 2e −Es 4σ2nd 2  C,E|H  ej 2πNk  . (2.10) We proceed to define y k= H  ej2πNk 

(c_k− e_k) for k = 0, · · · , N − 1 and the vector y is constituted as y = h yT 0 y T 1 · · · y T N −1 iT . (2.11)

From (2.11), we can gain d2_{C, E|H}_ej2π_Nk _{= | y |}2 _{and (2.10) can be}

rewrited as P C → E|Hej2πNk  ≤ 1 2e −Es 4σ2n| y | 2 . (2.12) Because all the channel tap matrices Hl are i.i.d. complex gaussian random

variables, all the Hej2π_Nk _{are jointly gaussian random variables. After}

av-eraging all the random variables, we can get the upper bound of the pairwise error probability is P (C → E) ≤ rank(Cy)−1 Y i=0  1 + λi(Cy) Es 4σ2 n −1 (2.13) in which λi(Cy) and rank(Cy) respectively denote the i-th nonzero eigenvalue

and the rank of Cy which represents the covariance matrix of y and is

where ⊗ denotes the Kronecker product and Ry = F (C, E) FH(C, E) = L−1 X l=0 h Dl(C − E)T(C − E)∗DlHi. (2.15) In (2.15), the matrix D and F (C, E) respectively indicate

D = diag n e−j2πNk oN −1 k=0 (2.16) and F (C, E) =(C − E)T D1(C − E)T · · · DL−1_{(C − E)}T_{ . (2.17)}

We can easily know that at high SNR, the upper bound of the pairwise error probability of the space-frequency coding (2.13) can be simplified as

P (C → E) ≤ Es 4σ2 n −rank(Cy)   rank(Cy)−1 Y i=0 λi  Cy    −1 . (2.18)

From (2.18), we can define the performance criteria as follows.

• Rank (diversity gain) criterion: Maximize the minimum rank over all pairs of any two different codewords as large as possible.

• Product (coding gain) criterion: The minimum value of the prod-uct rank(Cy)−1 Q i=0 λi  Cy  !

for any pairs of distinct codewords should be maximized.

Generally speaking, N > MTL is assumed, which is common in the

space-frequency coded MIMO-OFDM system. According to the property of Kro-necker product, we can know

rank(Cy) = rank(IMR) · rank(Ry)

≤ MRMTL.

So the maximum diversity gain of the method for this design is the product of number of transmitter antennas, receiver antennas and channel taps. The proposed rank criterion can efficiently take advantage of both the spatial diversity and the channel diversity. The coding gain can be obtained by means of the theorem of the Kronecker product, which is that each eigenvalue of Cy is one of the eigenvalues Ry with multiplicity MR.

Chapter 3

Proposed Design Criteria of

Space-Frequency Codes

3.1

System Model

In order to easily state the following content, we must define newly the space-frequency system, so the detailed architecture is established in this section. Because we want to encode and decode during several OFDM symbols, the channel environments and the corresponding codeword matrices will be ex-panded in the suitable form.

It is assumed that frequency-selective fading channel is constant over every OFDM symbol. In terms of the fading amplitude variation speed, rapid fading channel and quasi-static fading channel are distinguished, which are called general cases. For rapid fading channel, the channel gains change randomly from one OFDM symbol to another. It is referred to as quasi-static fading if the fading gains are maintained during several symbols. Specifically, coding for one OFDM symbol is an exception, which is special case, for it has nothing to do with the fading gain variation speed. At each OFDM symbol,

the L independent channel taps and the delay power profile are the same in each link of transmitter and receiver antenna. Let hm_j,i(l) be the l-th channel tap of the link transmitted from transmitter antenna i to receiver antenna j during the m-th OFDM symbol, so the corresponding impulse response of the channel is modeled as

hm_j,i(τ ) =

L−1

X

l=0

hm_j,i(l)δ(τ − τl) (3.1)

where τl is the l-th delay time in the real channel environment. All the fading

gain hm

j,i(l)’s are modeled as circularly symmetric complex Gaussian random

variables with zero mean and variance δ2_l, i.e., E|hm_j,i(l) |2 = δ_l2. In the following contents, it is assumed that the powers of the L coefficients in each link are equal powers and

L−1

P

l=0

δ_l2 = 1. The MIMO channel is also spatially uncorrelated as Chapter 2, so the channel taps are independent for different transmitter antenna index i and receiver antenna index j. After (3.1) is well defined, the frequency response of the channel is

H_j,im(f ) = L−1 X l=0 hm_j,i(l)e−j2πf τl _(3.2) where j =√−1.

In the MIMO-OFDM system, each transmitted codeword during M sym-bols can be defined as

C =c1₀ · · · c1 N −1c 2 0 · · · c 2 N −1 · · · c m k · · · c M 0 · · · c M N −1  (3.3) where cm_k denotes the MT-by-1 vector transmitted over the k-th subcarrier

in the m-th OFDM symbol. At the receiver, the reconstructed signal vector rm_k is given by

rm_k =pEsHˆm(k)cmk + z m

where ˆ Hm(k) =         Hm 0,0(k) H0,1m(k) · · · H0,Mm T−1(k) H_1,0m(k) H_1,1m(k) · · · H_1,Mm _T−1(k) .. . ... . .. ... H_Mm_R−1,0(k) H_Mm_R−1,1(k) · · · H_Mm_R−1,MT−1(k)         . (3.5)

Each element of the matrix ˆHm_{(k) is the corresponding frequency response}

at the k-th subcarrier transmitted from the i-th transmitter antenna to the j-th receiver antenna during the m-th symbol, which is expressed as

H_j,im(k) =

L−1

X

l=0

hm_j,i(l) e−j2πk∆f τl _(3.6)

where ∆f = 1/T is the subcarrier spacing and T is the OFDM symbol du-ration.

It is assumed that the channel state information is also correctly known at the receiver but not at the transmitter. Therefore, when we encode the codewords over M OFDM symbols and decode them, the ML decoder can be described as ˆ C = arg min C M X m=1 N −1 X k=0 ||rm k − p EsHˆm(k)cmk|| 2 _(3.7)

3.2

Proposed Design Criteria of Space-Frequency

Codes for Special Case: One OFDM

Sym-bol

In this section, coding over one OFDM symbol is discussed, and the de-sign criteria will be presented. The time index m will be omitted in order to make the analysis convenient, so the corresponding codeword form is (2.3).

The ML decoder select as its estimate an erroneous codeword E when the codeword C is transmitted, and this holds if

N −1 X k=0 ||r_k−pEsH(k)cˆ k|| 2 _≥ N −1 X k=0 ||r_k−pEsH(k)eˆ k|| 2 _(3.8) where ˆ H(k) =         H0,0(k) H0,1(k) · · · H0,MT−1(k) H1,0(k) H1,1(k) · · · H1,MT−1(k) .. . ... . .. ... HMR−1,0(k) HMR−1,1(k) · · · HMR−1,MT−1(k)         . (3.9) Then (3.8) is rewrited to N −1 X k=0 2RenpES(zk) H _ˆ H (k) (c_k− e_k)o≥ N −1 X k=0 |pESH (k) (cˆ k− ek) | 2 _(3.10)

where Re{·} represents the real part of a complex value. So the pairwise error probability conditioned on the channel state information is

P C → E|all ˆH (k)≤ Q   v u u t Es 2σ2 n N −1 X k=0 |pESH (k) (cˆ k− ek) |2  . (3.11) Then using the property (2.9), the conditional pairwise error probability becomes P C → E|Hej2πNk  ≤ 1 2e −Es 4σ2n N −1 P k=0 MR−1 P j=0 |MT −1P i=0 Hj,i(k) 

c(i)_k −e(i)_k |2

. (3.12) Let us adapt Hj,i(k) for

Hj,i(k) = L−1

X

l=0

hj,i(l)e−j2πk∆f τl = hTj,iG(k) (3.13)

where

and G(k) = e−j2πk∆f τ0 _e−j2πk∆f τ1 · · · e−j2πk∆f τL−1T = wkτ0 _wkτ1 _{· · · w}kτL−1T _{, w = e}−j2π∆f_. (3.15) It is apparent that | MT−1 P i=0 Hj,i(k)  c(i)_k − e(i)_k |2 _{= |}MT −1 P i=0

hT_j,iG(k)c(i)_k − e(i)_k |2

= | MT−1 P i=0 hT_j,iG(k, i)|ˆ 2 (3.16) where ˆ G(k, i) = G(k)c(i)_k − e(i)_k , (3.17) so | MT −1 P i=0 Hj,i(k)  c(i)_k − e(i)_k |2 ₌ _MT−1 P i=0 hT_j,iG(k, i)ˆ  _MT−1 P i=0 hT_j,iG(k, i)ˆ H = h˜hT_jG(k)i h˜h˜ T_jG(k)˜ i H (3.18) where ˜ h_j =hT j,0 h T j,1 · · · h T j,MT−1 T (3.19) and ˜ G(k) =h ˆG(k, 0)T G(k, 1)ˆ T · · · ˆG(k, MT − 1)T iT (3.20) are two MT-by-1 vectors. Therefore, the pairwise error probability

condi-tioned on the channel is

P C → E|all h_j,i
≤ 1 2e −Es 4σ2n N −1 P k=0 MR−1 P j=0 ˜ hT_jG(k) ˜˜ G(k)H˜_hT j H . (3.21) In order to use Gaussian approximation easily, we must modify the exponent of (3.21) as P C → E|all h_j,i
≤ 1 2e −Es 4σ2_n MR−1 P j=0 ˜ hTj " N −1 P k=0 ˜ G(k) ˜G(k)H # ˜_hT j H . (3.22)

Because ˜G(k) ˜G(k)H _{is Hermitian, so} N −1P

k=0

˜

G(k) ˜G(k)H _{is also Hermitian and}

nonegative definite [8]. Hence, there exists a unitary matrix V that diag-onalizes

N −1

P

k=0

˜

G(k) ˜G(k)H_{, which is also called Spectral Theorem [9], and we}

can gain N −1 X k=0 ˜ G(k) ˜G(k)H = MTL−1 X n=0 λnvnv H n (3.23)

where λn, n = 0, . . . , MTL−1 are the eigenvalues of the matrix N −1

P

k=0

˜

G(k) ˜G(k)H

and assumed that λ0 ≥ λ1 ≥ · · · ≥ λMTL−1≥ 0. The corresponding

eigenvec-tors v₀, v₁, · · · , v_M

TL−1 develop an orthonormal basis of MTL-dimensional

vector space, i.e.,

v_n· v_n0 =    1, for n = n0 0, for n 6= n0 . (3.24)

Then, the pairwise error probability can be replaced by P C → E|all h_j,i
≤ 1 2e −Es 4σ2n MR−1 P j=0 ˜ hTj " MT L−1 P n=0 λnvnvHn # ˜_hT j H = 1 2e −Es 4σ2_n MR−1 P j=0 MT L−1 P n=0 λnh˜ T jvnvHn˜h T j H = 1₂e −Es 4σ2_n MR−1 P j=0 MT L−1 P n=0 λn|βj,n|2 (3.25) where βj,n = ˜h T

jvn are independent zero mean complex Gaussian random

variables with variance _2L1 per dimension. Let r denote the rank of the matrix

N −1 P k=0 ˜ G(k) ˜G(k)H_{. A =} Es 4σ2 n and d = MR−1 P j=0 r−1 P n=0

λn|βj,n|2 are also considered.

If rMR is large enough to use Gaussian approximation, d can be close to

Gaussian random variable with mean µd = MR−1 P j=0 r−1 P n=0 λnE [|βj,n|2] = MR−1 P j=0 r−1 P n=0 λn_L1 = MR L r−1 P n=0 λn ∆ = MR L x1 (3.26)

and variance σ2 d = MR−1 P j=0 r−1 P n=0 λ2 nvar [|βj,n|2] = MR−1 P j=0 r−1 P n=0 λ2 n 1 L2 = MR L2 r−1 P n=0 λ2 n ∆ = MR L2 x2. (3.27)

Using the equation

∞ Z d=0 e−Adp(d)d(d) = e12A 2_σ2 d−Aµd_Q Aσ 2 d− µd σd  , (3.28) the whole pairwise error probability can be upper-bound by

P (C → E) ≤ 1₂exp  1 2  Es 4σ2 n 2 MR L2 x2−  Es 4σ2 n  MR L x1  · Q   Es 4σ2 n  q MR L2 x2− MR L x1 q MR L2x2  (3.29) For reasonably high SNR, which satisfies

 Es 4σ2 n  r MR L2 x2− MR L x1 q MR L2 x2 ≥ 0 ⇒ Es 4σ2 n ≥ x1 x2 L, (3.30) we use the inequality (2.9) to gain

P (C → E) ≤ 1₄exp n −Es 8σ2 n MR L x1 o = 1₄exp  −Es 8σ2 n MR L r−1 P n=0 λn  . (3.31)

Assume that ρ (C, E) is the set of frequency indices such taht |c_k− e_k|2 _{6= 0}

and δ is the corresponding number of the frequency indices in which the two codewords differ, it can be easily known that

Further, the sum of the eigenvalues x1 is expressed as r−1 P n=0 λn = tr _{N −1} P k=0 ˜ G(k) ˜G(k)H  = N −1 P k=0 trn ˜G(k) ˜G(k)Ho = N −1 P k=0 trn[(c_k− e_k) ⊗ G(k)] [(c_k− e_k) ⊗ G(k)]Ho = N −1 P k=0 trn(c_k− e_k) (c_k− e_k)Ho· trG(k)G(k)H = N −1 P k=0 L|c_k− e_k|2_. (3.33)

The upper bound of the pairwise error probability can be finally approxi-mated by P (C → E) ≤ 1 4exp ( −Es 8σ2 n MR N −1 X k=0 |c_k− e_k|2 ) . (3.34) From (3.34), the Euclidean distance dominates the pairwise error probability at high SNR, so the code design are listed below

• Rank criterion: Maximize the minimum rank r over all pairs of any two different codewords as large as possible for Gaussian approxima-tion.

• Distance criterion: The minimum value of the Euclidean distance for any pairs of distinct codewords should be maximized.

From the design criteria, we can recognize that the method of encoding and decoding one OFDM symbol is similar to the codeword design for space-time codes when the channel distribution can approach a Gaussian random variable.

3.3

Proposed Design Criteria of Space-Frequency

Codes for General Cases: Rapid Fading

Channel and Quasi-Static Fading

Chan-nel

3.3.1

Rapid Fading Channel

In this section, the code design of space-frequency codes over several OFDM symbols will be derived on rapid fading channel, which means that the fading gains are different during the various OFDM symbols. The system model of this case is organized in section 3.1 without any correction.

From (3.7), when the codeword C is transmitted, another codeword E is mistaken for the correct codeword by the ML decoder if

M X m=1 N −1 X k=0 ||rm_k −pEsHˆm(k)cmk|| 2 _≥ M X m=1 N −1 X k=0 ||rm_k −pEsHˆm(k)emk|| 2 . (3.35) Then, the pairwise error probability is

P  C → E|all ˆHm(k)  = Q s Es 2σ2 n d2_{C, E||all ˆH}m_(k) ! (3.36) where d2C, E|all ˆHm(k)= M X m=1 N −1 X k=0 | ˆHm(k) (cm_k − em k) | 2_{. (3.37)}

Substituting (2.9) into (3.36), we obtain the upper bound of the pairwise error probability on rapid fading channel as

PC → E|all ˆHm_(k) _≤ 1 2e −Es 4σ2n M P m=1 N −1 P k=0 | ˆHm_(k)(cm k−emk)|2 = 1₂e −Es 4σ2n M P m=1 N −1 P k=0 MR−1 P j=0 |MT −1P i=0 Hm j,i(k) 

cm,(i)_k −em,(i)_k |2

where cm k −emk = h cm,(0)_k − em,(0)_k cm,(1)_k − em,(1)_k · · · cm,(MT−1) k − e m,(MT−1) k iT . Similar to (3.13), Hm j,i(k) is modified to H_j,im(k) = L−1 X l=0 hm_j,i(l)e−j2πk∆f τl _{= h}m j,i
T G(k) (3.39) where

hm_j,i=hm_j,i(0) h_j,im(1) · · · hm_j,i(L − 1)T . (3.40) So we can consequently rewriten |

MT−1 P i=0 Hm j,i(k) 

cm,(i)_k − em,(i)_k |2 _{as follows,}

| MT−1 P i=0 Hm j,i(k)  cm,(i)_k − em,(i)_k |2 _{= |}MT −1 P i=0

hm_j,i
T G(k)cm,(i)_k − em,(i)_k |2

= | MT −1 P i=0 hm_j,i
T G(k, m, i)|ˆ 2 (3.41) where ˆ G(k, m, i) = G(k)cm,(i)_k − em,(i)_k . (3.42) As a result, we can get

| MT−1 P i=0 Hm j,i(k)  cm,(i)_k − em,(i)_k |2 ₌ " MT −1 P i=0 hm_j,i
T _ˆ G(k, m, i) # " MT −1 P i=0 hm_j,i
T _ˆ G(k, m, i) #H =  ˜hm j T ˜ G(k, m)   ˜hm j T ˜ G(k, m) H = ˜hm_j  T ˜ G(k, m) ˜G(k, m)H  ˜hm j TH (3.43) where ˜ hm_j =h hm_j,0
T hm_j,1
T · · · hm j,MT−1
TiT (3.44) and ˜ G(k, m) = h ˆG(k, m, 0)T G(k, m, 1)ˆ T · · · ˆG(k, m, MT − 1)T iT (3.45) are all MTL-by-1 vectors. So the pairwise error probability conditioned on

the channel is P C → E|all hm_j,i
≤ 1 2e −Es 4σ2_n M P m=1 N −1 P k=0 MR−1 P j=0 ( ˜ hm_{j )}TG(k,m) ˜˜ G(k,m)Hh_(h˜m j) TiH , (3.46)

and ˜G(k, m) ˜G(k, m)H _{can be expressed as} ˜ G(k, m) ˜G(k, m)H _{= [(c}m k − emk) ⊗ G(k)] [(cmk − emk) ⊗ G(k)] H = [(cm k − emk) ⊗ G(k)] h (cm k − emk) H _{⊗ G(k)}Hi = h(cm_k − em k) (c m k − e m k) Hi ⊗ G(k)G(k)H_. (3.47)

According to (3.22), the right side of (3.46) can be modified as P C → E|all hm_j,i
≤ 1 2e −Es 4σ2_n M P m=1 MR−1 P j=0 ( ˜ hm_{j )}T ( N −1 P k=0 ˜ G(k,m) ˜G(k,m)H ) h (˜_hm j ) TiH (3.48) As N −1 P k=0 ˜

G(k, m) ˜G(k, m)H is Hermitian and nonegative definite, it can be equal to N −1 X k=0 ˜ G(k, m) ˜G(k, m)H = MTL−1 X n=0 λm_nvm_n [vm_n]H (3.49) where λm

n and vmn are the n-th eigenvalue and the corresponding eigenvector

of the m-th OFDM symbol respectively. It is also assumed that λm

0 ≥ λm1 ≥

· · · ≥ λm

MTL−1 ≥ 0 and all the eigenvectors of the m-th transmitted symbol

satifies vm_n · vm n0 =    1, for n = n0 0, for n 6= n0 . (3.50)

Afterwards, substituting (3.49) into (3.48), so (3.48) becomes P C → E|all hm_j,i
≤ 1 2e −Es 4σ2n M P m=1 MR−1 P j=0 MT L−1 P n=0 λm n(h˜ m j) T vm n[vmn] Hh (_h˜m j) TiH = 1₂e −Es 4σ2n M P m=1 MR−1 P j=0 MT L−1 P n=0 λm n|βj,nm|2 (3.51) where βm j,n = ˜h m j T vm

n are i.i.d. zero mean complex Gaussian random

variables with variance _L1. If rr is the rank of N −1 P k=0 ˜ G(k, m) ˜G(k, m)H _and dr = M P m=1 MR−1 P j=0 rr−1 P n=0 λm_n|βm

variable with mean µdr µdr = M P m=1 MR−1 P j=0 rr−1 P n=0 λm nE|βj,nm|2 = M P m=1 MR−1 P j=0 rr−1 P n=0 λm n 1 L = MR L M P m=1 rr−1 P n=0 λm_n ∆ = MR L x1,r (3.52) and variance σ2 dr σ2 dr = M P m=1 MR−1 P j=0 rr−1 P n=0 (λm n) 2 var|βm j,n|2 = M P m=1 MR−1 P j=0 rr−1 P n=0 (λm_n)2 1_L2 = MR L2 M P m=1 rr−1 P n=0 (λm n) 2 ∆ = MR L2 x2,r. (3.53)

Using the equation similar to (3.28), the pairwise error probability on rapid fading channel is P (C → E) ≤ 1₂exp  1 2  Es 4σ2 n 2 MR L2 x2,r−  Es 4σ2 n  MR L x1,r  · Q   Es 4σ2 n  q MR L2 x2,r− MR L x1,r q MR L2x2,r  (3.54) At high SNR, which means

 Es 4σ2 n  r MR L2 x2,r− MR L x1,r q MR L2 x2,r ≥ 0 ⇒ Es 4σ2 n ≥ x1,r x2,r L, (3.55) the pairwise error probability can be further gained by (2.9)

P (C → E) ≤ 1₄expn−Es 8σ2 n MR L x1,r o = 1₄exp  −Es 8σ2 n MR L M P m=1 rr−1 P n=0 λm_n  . (3.56)

The sum of the eigenvalues is farther equal to M P m=1 rr−1 P n=0 λm n = tr  _M P m=1 N −1 P k=0 ˜ G(k, m) ˜G(k, m)H  = M P m=1 N −1 P k=0 trn ˜G(k, m) ˜G(k, m)Ho = M P m=1 N −1 P k=0 trn[(cm k − emk) ⊗ G(k)] [(cmk − emk) ⊗ G(k)] Ho = M P m=1 N −1 P k=0 trn(cm k − cmk) (cmk − emk) Ho · trG(k)G(k)H = M P m=1 N −1 P k=0 L|cm k − emk|2. (3.57)

Assume that ρr(C, E) is the set of all symbol and frequency indices such

taht |cm

k − emk|2 6= 0 and δr is the corresponding number of the set ρr(C, E)

in which the two codewords differ, it can be easily known that rr

∆

= min {MTL, δr} . (3.58)

Finally, the pairwise error probability is upper-bound by P (C → E) ≤ 1 4exp ( − Es 8σ2 n MR M X m=1 N −1 X k=0 |cm k − e m k| 2 ) . (3.59) Similar to (3.34), the pairwise error probability is dominated by the Eucldean distance and the analogous code design are

• Rank criterion: The minimum rank rr over all pairs of any two

dif-ferent codewords is maximized as large as possible.

• Distance criterion: Maximize the minimum value of the Euclidean distance for any pairs of distinct codewords.

We can conclude that the design criteria on rapid fading channel is iden-tical to the design criteria for one OFDM symbol.

3.3.2

Quasi-Static Fading Channel

Quasi-static fading, which means that the fading gains are the same during the various OFDM symbols, will be discussed. The system model of this channel environment is analogous to section 3.1, but all channel matrices

ˆ

Hm(k) must be replaced by (3.9).

When M OFDM symbols are encoded and transmitted on quasi-static fading channel, the ML decoder can be described as

ˆ C = arg min C M X m=1 N −1 X k=0 ||rm_k −pEsH(k)cˆ mk|| 2 (3.60) The wrong codeword E is decided by the decoder when the correct codeword is C if M X m=1 N −1 X k=0 ||rm k − p EsH(k)cˆ mk|| 2 _≥ M X m=1 N −1 X k=0 ||rm k − p EsH(k)eˆ mk|| 2_{. (3.61)}

Therefore, the specific pairwise error probability conditioned on quasi-static fading channel is P C → E|all ˆH(k)= Q s Es 2σ2 n d2_{C, E| ˆH(k)} ! (3.62) where d2  C, E|all ˆH(k)  = M X m=1 N −1 X k=0 | ˆH(k) (cm_k − em_k) |2. (3.63) We also use (2.9) to get

PC → E| ˆH(k) ≤ 1 2e −1 2  Es 2σ2nd 2_{(C,E| ˆH(k))}  = 1₂e− Es 4σ2n M P m=1 N −1 P k=0 | ˆH(k)(cm k−emk)|2 = 1₂e −Es 4σ2n M P m=1 N −1 P k=0 MR−1 P j=0 |MT −1P i=0 Hj,i(k) 

cm,(i)_k −em,(i)_k |2

So we can rewriten |

MT−1

P

i=0

H_j,i(k)cm,(i)_k − em,(i)_k |2 _{as follows,}

|

MT−1

P

i=0

H_j,i(k)cm,(i)_k − e_km,(i)|2 _{= |}MPT−1

i=0

h_j,i
T G(k)cm,(i)_k − em,(i)_k |2

= | MT −1 P i=0 h_j,i
T G(k, m, i)|ˆ 2. (3.65) Then, | MT−1 P i=0

H_j,i(k)cm,(i)_k − em,(i)_k |2 ₌

 ˜h_jT ˜ G(k, m)   ˜h_jT ˜ G(k, m) H = ˜h_j T ˜ G(k, m) ˜G(k, m)H  ˜h_jTH(3.66) Therefore, the pairwise error probability can be amended to

P C → E|all h_j,i
≤ 1 2e −Es 4σ2_n M P m=1 N −1 P k=0 MR−1 P j=0 ( ˜ hj) T ˜ G(k,m) ˜G(k,m)H h (_h˜ j) TiH . (3.67) In order to simplify the complexity of computation and easily design, assume that λn and vn are the n-th eigenvalue and the corresponding eigenvector of

the matrix M P m=1 N −1 P k=0 ˜

G(k, m) ˜G(k, m)H_{. Note that these eigenvectors also form}

an orthonormal set. Accordingly, we can decompose

M P m=1 N −1 P k=0 ˜ G(k, m) ˜G(k, m)H as M X m=1 N −1 X k=0 ˜ G(k, m) ˜G(k, m)H = MTL−1 X n=0 λnvn[vn] H (3.68) Hence, the pairwise error probability can be further expressed as

P C → E|all_j,i
≤ 1 2e −Es 4σ2_n MR−1 P j=0 MT L−1 P n=0 λn(˜hj) T vn[vn] Hh (_h˜ j) TiH = 1₂e −Es 4σ2_n MR−1 P j=0 MT L−1 P n=0 λn|βj,n|2 (3.69) where βj,n = ˜hj T

v_n and all βj,n are i.i.d. complex Gaussian random

variables with zero mean and variance _L1 per dimension. If rq is the rank

of the matrix M P m=1 N −1 P k=0 ˜ G(k, m) ˜G(k, m)H _{and d} q = MR−1 P j=0 MTL−1 P n=0 λ_n|βj,n|2 =

MR−1 P j=0 rq−1 P n=0

λ_n|βj,n|2, when rqMR is large enough to approach to Gaussian

dis-tribution, dq owns mean µdq

µdq = MR−1 P j=0 rq−1 P n=0 λ_nE {|βj,n|2} = MR L rq−1 P n=0 λ_n ∆ = MR L x1,q (3.70) and variance σ2 dq σ2 dq = MR−1 P j=0 rq−1 P n=0 (λ_n)2var {|βj,n|2} = MR L2 rq−1 P n=0 (λ_n)2 ∆ = MR L2 x2,q. (3.71)

We also use (3.28) to compute the pairwise error probability, and the pairwise error probability becomes

P (C → E) ≤ 1₂exp  1 2  Es 4σ2 n 2 MR L2 x2,q−  Es 4σ2 n  MR L x1,q  · Q   Es 4σ2 n  q MR L2 x2,q− MR L x1,q q MR L2x2,q  . (3.72)

It is assumed that the space-frequency codes are operated in a somewhat high SNR case, which means

 Es 4σ2 n  r MR L2 x2,q− MR L x1,q q MR L2 x2,q ≥ 0 ⇒ Es 4σ2 n ≥ x1,q x2,q L. (3.73) By using (2.9) again, the bound can be approximated as

P (C → E) ≤ 1₄expn−Es 8σ2 n MR L x1,q o = 1₄exp  −Es 8σ2 n MR L M P m=1 rq−1 P n=0 λm_n  . (3.74) If ρq(C, E) and δq represent the set of the indices fulfilling |cmk −emk|2 6= 0 and

the number of this set elements respectively, the possible maximum value of rq is

rq ∆

Using the trace property, we can turn the sum of the eigenvalues into rq−1 P n=0 λ_n = tr  _M P m=1 N −1 P k=0 ˜ G(k, m) ˜G(k, m)H  = M P m=1 N −1 P k=0 trn ˜G(k, m) ˜G(k, m)Ho = N −1 P k=0 trn[(cm k − emk) ⊗ G(k)] [(cmk − emk) ⊗ G(k)] Ho = N −1 P k=0 trn(cm k − emk) (cmk − emk) Ho · trG(k)G(k)H = M P m=1 N −1 P k=0 L|cm_k − em k|2, (3.76)

and in consequence the pairwise error probability would be built into P (C → E) ≤ 1 4exp ( − Es 8σ2 n MR M X m=1 N −1 X k=0 |cm k − e m k| 2 ) . (3.77) Finally, the corresponding design criteria on quasi-static fading channels are arranged to

• Rank criterion: Achieve the largest rank rq between two unlike

code-words as possible.

• Distance criterion: The minimum value of the Euclidean distance for any sets of any two distinct codewords should be maximized. We can easily know that all the discussed channel environments have the same methods to design codewords when the central limit theory is exploited.

Chapter 4

Simulation

In this section, the arguments will be evidenced by way of the simulation results. We apadt an MIMO-OFDM system with two transmitter antennas and three receiver antenna, and we assume that the OFDM bandwidth is 800kHz and divided into 128 subcarriers. So the corresponding subcarrier spacing is 6.25kHz and the symbol duration is 160µs. A CP of 40µs is ap-pended to each OFDM symbol. A two equal-power tap fading channel with the delay spread of 5µs is used, and the sum of two tap powers is normalized to 1. QPSK modulation is used, and the adapted generator sequences for simulation is listed in table 4.1.

Table 4.1: The simulated generator sequences.

code MT generator sequences rank distance

BBH[11] 2 g1 _{= [(2, 2), (1, 0)], g}2 _{= [(0, 2), (3, 0)] 2 6}

TSC[10] 2 g1 _{= [(0, 2), (2, 0)], g}2 _{= [(0, 1), (1, 0)] 2 4}

YT[12] 2 g1 _{= [(2, 1), (2, 0)], g}2 _{= [(0, 2), (3, 2)] 2 10}

The simulation results of designing for one OFDM symbol in Figure 4.1. Figure 4.2 and Figure 4.3 are the simulation results of the various design criteria for two OFDM symbols on rapid fading channel and quasi-static fading channel, respectively. The performance of VY is the same to YT in all simulation environment, and both methods are better than BBH and TSC.

Figure 4.1: Simulation results of the various design criteria for one OFDM symbol.

Figure 4.2: Simulation results of the various design criteria for several OFDM symbols on rapid fading channel.

Figure 4.3: Simulation results of the various design criteria for several OFDM symbols on rapid fading channel.

Chapter 5

Conclusion

By making use of the central limit theory, the pairwise error probability and the corresponding design criteria of space-frequency coded MIMO-OFDM system are presented in this thesis. It is easily known that the method for codeword construction of space-frequency codes is similar to the method of space-time codes by way of the same analysis, and the corresponding trellis codes are taken to compare with the other trellis codes designed from various methods. Most space-frequency trellis codes are designed for one OFDM symbol, and therefore we discuss how to encode and decode during several OFDM symbols. Based on the channel variation speed, rapid fading channel and quasi-static fading channel are recognized, and it is proved that the specific design criteria of two distinct channel environments are also similar to the way designed for one OFDM symbol by applying the central limit theory. Simulation results are presented to support the arguments.

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