應用於高速動正交分頻多工系統分組子載波干擾消除及資料偵測之研究

國 立 交 通 大 學

電 信 工 程 學 系

碩 士 論 文

應用於高速移動正交分頻多工系統分組子載波干擾

消除及資料偵測之研究

A Study on

Group Based ICI Cancellation and Data Detection for

High Mobility OFDM Systems

研 究 生：邱麟凱

指導教授：黃家齊 博士

應用於高速移動正交分頻多工系統群組化子載波干

擾消除及資料偵測之研究

A Study on

Group Based ICI Cancellation and Data Detection for

High Mobility OFDM Systems

研 究 生：邱麟凱 Student：Li-Kai Chiu

指導教授：黃家齊 Advisor：Dr. Chia-Chi Huang

國 立 交 通 大 學

電 信 工 程 學 系 碩 士 班

碩 士 論 文

A Thesis

Submitted to Department of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Communication Engineering

July 2007

Hsinchu, Taiwan, Republic of China

應用於高速移動正交分頻多工系統分組子載波干擾

消除及資料偵測之研究

學生：邱麟凱 指導教授：黃家齊

國立交通大學電信工程學系 碩士班

摘

要

由移動所造成的頻道變化會使得一個正交分頻多工系統的符元有頻率

選擇性的衰減，這樣的現象使傳統一級等化器的方法不能被利用，且此現

象更進一步的會破壞子載波之間的正交性使得正交分頻多工系統的符元遭

受到內部子載波互相干擾的現象，若是在高速移動的情況下會造成系統效

能嚴重的被降低，為了減輕此現象帶來的問題，這篇文章提出一個運用球

形解碼或列表式球形解碼反覆式的分組子載波干擾消除方法，在接收端子

載波會被分成許多組，並使用球形解碼或列表式球形解碼去產生資訊再將

這些資訊傳送到其它組做子載波干擾消除，分組的方法是用來減輕使用球

形解碼或列表式球形解碼的計算複雜度，而子載波干擾消除的方法是用來

使系統效能更好。

A Study on

Group Based ICI Cancellation and Data Detection for

High Mobility OFDM Systems

Student：Li-Kai Chiu Advisor：Dr. Chia-Chi Huang

Department of Communication Engineering

National Chiao Tung University

ABSTRACT

The channel variation due to vehicle mobility produces frequency selective fading among OFDM symbols which makes the traditional one-tap equalizer can not be utilized. Moreover, the orthogonal property of OFDM subcarriers is destroyed and OFDM symbol experiences inter-carrier interference (ICI) that severely degrades the performance in high vehicle mobility environment. To reduce the problem, an iterative group based ICI cancellation method which applies sphere decoding and list sphere decoding (LSD) is proposed. At the receiver, subcarriers are partitioned into several groups, each group uses SD or LSD to generate the message information and pass it to other groups for ICI cancellation. The grouping procedure is used to reduce the computation complexity of the SD and LSD, and ICI cancellation procedure is used to make BER performance better.

誌 謝

感謝黃家齊教授這兩年的教導，碩士論文上提供許多的建議和概念

使得這一篇論文的完成更加的順利與完整，並且使我在這一段時間內對通

訊領域有更進一步的了解。

這一篇論文的完成要特別感謝早安學長古孟霖，學長這兩年來不辭辛

苦的解答我的問題與疑惑，並指導我論文的方向和細節以及可能需要注意

的問題，更重要的讓我了解此研究在通訊領域上的相關問題和應用。

感謝這兩年一同患難的實驗室同學鄒鎧駿、吳其珍、杜欣憶，在研究

及課業之餘，大家能以輕鬆詼諧的談話使我將生活上壓力拋開。

最後感謝在我求學路程上不斷的支持的父母及家人。

Contents

中文摘要 i ABSTRACT ii 誌 謝 iii Contents iv List of Tables vi List of Figures vii Chapter 1 Introduction 1 Chapter 2 OFDM System in High Mobility Environment and Sphere Decoding 4 2.1 High Mobility Environment... 4

2.2 ICI on OFDM System... 6

2.3 Sphere Decoding... 10

2.3.1 Real Sphere Decoding ... 10

2.3.2 Complex Sphere Decoding ... 17

2.4 List Sphere Decoding... 21

Chapter 3 Grouped Based ICI Cancellation Method 24 3.1 System Model ... 24

3.2 Group based ICI Cancellation Method with Sphere Decoding... 25

3.3 Group based ICI Cancellation Method with List Sphere Decoding ... 28

3.3.1... Soft Symbol ... 28

3.3.2 Conditional Probability... 30

3.4 Radius of the Sphere ... 33

4.1 Simulation Environment ... 35

4.2 Simulation Result Discussions... 38

4.2.1 Simulation in Environment I... 38

4.2.2 Simulation in Envirnoment II ... 50

4.3 Computational Complexity ... 58

Chapter 5 Conclusions 60

List of Tables

Table 4.1 Simulation parameters of the first kind of simulation environment..…………....

..

.36

Table 4.2 Simulation parameters of the second kind of channel simulation environment

..

…

.

37

Table 4.3 Brief computational complexity comparison………

…

………

……..

59

List of Figures

Fig. 2.1 Base-band OFDM System... 6

Fig. 2.2 A sphere of radius r and centered at AX ... 13 ˆ Fig. 2.3 Sample tree generated to determine points in a N-dimensional sphere. ... 16

Fig. 2.4 Searching disk in 16-QAM. ... 18

Fig. 2.5 Sphere of SD or LSD in received signal domain. ... 22

Fig. 3.1 Group Method for ICI with window length Q=2q+1... 26

Fig. 3.2 Block diagram of group based ICI cancellation method with Sphere Decoding. ... 28

Fig. 3.3 Block diagram of group based ICI cancellation with List Sphere Decoding... 32

Fig. 3.4 signal space in transmitter and receiver... 34

Fig. 4.1 Comparison of BER in different iteration number (I)... 41

Fig. 4.2 Comparison of BER in different speed (I). ... 42

Fig. 4.3 Comparison of BER in different group size (I)... 43

Fig. 4.4 Comparison of BER in different method (I). ... 44

Fig. 4.5 Comparison of BER in different speed and group size (I). ... 45

Fig 4.7 Comparison of BER in different group sizes with LSD. ... 47

Fig. 4.8 Comparison of BER in different speed and group sizes with SD (I)... 48

Fig. 4.9 Comparison of BER in different speed and group sizes with LSD (I). ... 49

Fig. 4.10 Comparison of BER in different numbers of iteration (II). ... 53

Fig. 4.11 Comparison of BER in different sizes of group (II). ... 54

Fig. 4.12 Comparison of BER in different f T (II)... 55 _{d s} Fig. 4.13 Comparison of BER in LSD and SD (II)... 56

Chapter 1

Introduction

OFDM is widely used in many wireless communication systems for high-bit-rate

transmission over a frequency-selective fading channel. The concept of Orthogonal Frequency

Division Multiplexing (OFDM) is initiated from that of multi-carriers systems [1] [2]. Data

are transmitted through multiple carriers simultaneously to achieve high data rate transmission.

In OFDM, the computationally efficient fast Fourier transform (FFT) is used to transmit data

in parallel over a large number of orthogonal subcarriers. A cyclic prefix is inserted before

each transmitted data block to eliminate the inter symbol interference (ISI). For time-invariant

multipath channels, a single tap equalizer in frequency domain can be employed to recover

the transmitted symbol on each subcarrier. However, due to the demand for orthogonality

between each subcarrier, OFDM systems are sensitive to synchronization.

In high mobility environment, multipath channel is time varying. Channel variations

may also arise the presence of an unknown carrier frequency offset, so the orthogonal

property of OFDM is destroyed and result in the effect of inter-carrier interference among

subcarriers which makes the performance of OFDM systems degrades severely [3] [4]. To

self-cancellation scheme [5], Sphere Decoding (SD) [6], minimum mean-squared error

(MMSE) and MMSE with successive detection (MMSED) [7]. In [5], the method proposed

sharps the signal in frequency domain using the windowing operation in time domain to make

subcarriers has approximate nulls around the location of others subcarriers and, therefore,

creates less ICI. In [6], Sphere Decoding which solves the ML criterion is used to reduce the

effect of ICI. In [7], it introduce a high performance equalization method by using MMSE

with successive interference cancellation, but the computational complexity is very high.

According to Cai.et al., [8] shows that ICI effect on a sucarrier comes from neighboring

subcarrieres, [9] and [10] propose a group based ICI cancellation method to lower the

complexity and utilize successive ICI cancellation to get better performance. In this thesis, a

parallel liked group based ICI cancellation method combined with Sphere Decoding or List

Sphere Decoding is proposed to improve the system performance and lower the complexity of

method in [6] which uses Sphere Decoding to reduce the ICI effect of OFDM system in high

vehicle mobility environment. At the receiver, subcarriers are partitioned into several groups,

each group uses SD or LSD to generate the message information and pass it to other groups

for ICI cancellation. The operation of solving Sphere Decoding or List Sphere Decoding and

passing message information for ICI cancellation will repeat iteratively to make performance

better and better. In our proposed method, we assume that the channel state information are

The organization of this thesis is as following. In chapter 2, ICI effect on OFDM in high

mobility environment, Sphere Decoding and List Sphere Decoding are introduced. In chapter

3, at first, the system model used in thesis is introduced then the group based ICI cancellation

method with Sphere Decoding and List Sphere Decoding are introduced. In the end of the

chapter 3, the parameter setting of Sphere Decoding is introduced. Computer simulation

results along with some discussions are showed in chapter 4. Finally, in chapter 5, brier

Chapter 2

OFDM System in High Mobility

Environment and Sphere Decoding

2.1

High Mobility Environment

In wireless communication, received signals come from multiple paths due to reflection

effects. Such environment is called a multipath channel. The equivalent baseband of a

multipath channel impulse response can be described as [11]

1 0 ( , ) ( ) ( ). l L l l l h tτ a t δ t τ = − = =

∑

− (2.1) Where ( )a t and _l τ_l are the time-varying complex fading gain and the path delay of the lth path, L is the total number of multipath, and δ is the delta function. The variation speed of

path gain a t depends on maximal Doppler frequency or Doppler spread which is _l( )

proportioned to the vehicle speed and carrier frequency. Maximal Doppler frequency is

defined as eq. (2.2). The larger the Doppler spread is, the faster variation of the path gains are.

[12] c d f v f c = , (2.2) where f is the central frequency and _c v is the vehicle speed and c is the speed of light. In

OFDM system the parameter f T is used to measure the effect of the ICI where _{d s} T is the _s

OFDM symbol period. We can observe that the inversion of T is the subchannel bandwidth _s

and we can treat Doppler frequency as the frequency offset of a single tone signal after passing through the channel, so f T is the fraction between frequency offset and subchannel _{d s}

bandwidth. If f T is very small, it means that frequency offset relative to the subchannel _{d s}

bandwidth is too small which can be neglect, so the frequency of the signal can be seemed as

the same as the original one. Otherwise, the mismatch of the frequency will occur at the

receiver in OFDM systems. For fixed f and _c T , we can see from eq. (2.2) that _s f goes _d

larger and larger when the relative velocity v goes faster and faster and it will cause the ICI effect goes severer and severer.

In the computer simulation, channel gains a t are generated by Jakes model, the _l( )

introduction of Jakes model is as following:

In the multipath Rayleigh fading channel without line of sight (LOS), the angel of the arrival signal in a plane is assumed to be uniformly distributed in the interval [0, 2 )π . Jakes modeled the Rayleigh fading channel by a bank of oscillators with the maximal Doppler frequency and

its fractions, as eq. (2.3) showed.

0

0 1

1

( ) 2 cos cos 2 cos cos

( ) 2 sin cos 2 sin cos

N I n n d n N Q n n d n f t t t f t t t β ω α ω β ω α ω = = = ⋅ + ⋅ = ⋅ + ⋅

∑

∑

(2.3) where

0 0

2(2 1), 8

N = N + N ≥

0

cos Doppler shifts, 1, 2,...,

n d n n N

ω =ω α = =

0

2

the arrival angel of the n-th arrival signal, 1, 2,...,

n n n N N π α = = = 0

the phase of the n-th arrival signal, 1, 2,...,

n n N

β = =

In eq. (2.3), N₀ must be large enough to approximate to the central theorem. β_n are chosen properly such that the arrival phases are close to uniform distribution in [0, 2 )π .

2.2

ICI on OFDM System

i

w

( )i l

h

0 X 1 X 1 N X ₋ 0 Y 1 Y 1 N Y ₋ 0 y 1 y 1 N y ₋ 0 x 1 x 1 N x ₋ cp xK yK_cp

Fig. 2.1 Base-band OFDM System.

Fig 2.1 is the block diagram of base-band OFDM system. X_k, k=0,...,N− the 1

inputs of the Inverse Fast Fourier Transform (IFFT) represent the frequency domain data on

the k-th subcarrier. x_k, k=0,...,N− the outputs of the IFFT can be represented as 1

2 1 0 0 1 j ik N N i k k x X e i N π − = =

∑

≤ ≤ − . (2.4) cp

xK represents the cyclic prefix (CP) with length N_G and is related to time domain sequence,

x as follows：

( ) 0 1

cp i =xN C i− + ≤ ≤i NG−

xK . (2.5)

Let T be the sampling period. Then h_l( )i is the lth channel tap at time instant t= ×i T. We

assume that the maximum delay spread of the channel is always less than or equal to N_G.

Then the channel output y can be expressed as follows：

( ) (( )) 0 0 1 G N N i i l i l i l y h x ₋ w i N = =

∑

+ ≤ ≤ − . (2.6) In eq. (2.6), (( ))_N represents a cyclic shift in the base of N and w_i represents a sample of additive white Gaussian noise. Then the fast Fourier transform (FFT) of sequence y , will be

as follows： 2 1 0 0 1 j ki N N k i i Y y e k N π − ₋ = =

∑

≤ ≤ − . (2.7) If ( )i l

h is constant during one OFDM symbol time (N N+ G)×T, then eq. (2.6)

becomes (( )) 0 0 1 G N N i l i l i l y h x ₋ w i N = =

∑

+ ≤ ≤ − , (2.8) which is the circular convolution of h and x . By using eq. (2.7), eq. (2.4) and basic DFT

concept [13], we will find that the relationship between X_k and Y_k which is derived as

follows： 2 1 (( )) 0 0 G N j ki N N N k l i l i i l Y h x w e π − ₋ − = = ⎛ ⎞ = _⎜ + _⎟ ⎝ ⎠

∑ ∑

(

)

2 2 1 1 (( )) 0 0 0 2 (( )) 2 1 1 0 0 0 2 (( )) 2 1 1 0 0 0 G N N G N G j ki j ki N N N N N l i l i l i i j i l n j ki N N N N N l n k l i n j i l n j ki N N N N N l n k l n i h x e w e h X e e W h X e e W π π π π π π − ₋ − ₋ − = = = − − − ₋ = = = − − − ₋ = = = = + = + = +

∑ ∑

∑

∑ ∑∑

∑ ∑ ∑

2 2 2 1 1 0 0 0 G j ln j in j ki N N N N N N l n k l n i h X e e e W π π π − − − ₋ = = = =

∑ ∑

∑

+ (2.9) 2 1 0 0 [ ] G j ln N N N l n k l n h X e n k W π δ − − = = =

∑ ∑

− + (2.10) 2 0 G j lk N N k l k l X h e W π − = =

∑

+ k k k H X W = + . (2.11) Because of the orthogonal property of the subcarrier, we can derive eq. (2.9) to eq. (2.10). At last, from eq. (2.11) we can see that Y_k only depends on X_k.

Unfortunately, impulse response of the channel is not a constant during one OFDM

symbol time in high mobility environment. In section 2.1, we know that the channel impulse

response is change in time, and frequency offset occurs due to the Doppler frequency. All the

thing happened above destroy the orthogonal property of subcarriers. In this case, from eq.

(2.7) we can find that [14]

2 1 ( ) (( )) 0 0 2 2 1 1 ( ) (( )) 0 0 0 2 1 ( ) (( )) 0 0 2 2 2 1 1 ( ) 0 0 0 G N G N G N G j ki N N i _N k l i l i i l j ki j ki N N N i _{N N} l i l i l i i j ki N N i N l i l k l i ln j in j ki N N N _j i _{N N N} l n i l n Y h x w e h x e w e h x e W h X e e e π π π π π π π − ₋ − = = − ₋ − ₋ − = = = − ₋ − = = − − _{− −} = = = ⎛ ⎞ = _⎜ + _⎟ ⎝ ⎠ = + = + ⎛ ⎞ = _{⎜ ⎟} ⎝ ⎠

∑ ∑

∑∑

∑

∑∑

∑∑

∑

2 ( ) 2 1 1 ( ) 0 0 0 G k j k n i ln N N N _j i _{N N} n l k n l i W X h e e W π − π − − _{− −} = = = + ⎛ ⎞ = _{⎜ ⎟} + ⎝ ⎠

∑ ∑ ∑

2 2 ( ) 1 1 ( ) (( )) 0 0 0 2 2 2 ( ) 1 1 1 ( ) ( ) (( )) 0 0 1 0 0 G N G G N j di l k d N N N _j i _{N N} k d l k d l i lk j di l k d N N _j N N N _j i _N i _{N N} l k l k d k l i d l i ICI X h e e W h e X h e e X W π π π π π − − − _{− −} − = = = − − ₋ − − _{− −} − = = = = = ⎛ ⎞ = _{⎜ ⎟} + ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ =⎜ ⎟ + ⎜_⎜ ⎜ ⎟ ⎟_⎟ + ⎝ ⎠ ⎝ ⎝ ⎠ ⎠

∑

∑ ∑

∑∑

∑ ∑ ∑

  2 2 ( ) 1 1 ( ) (( )) 0 0 1 0 ( ) G G N lk l k d N N _j N N _j i _{N N} l k l k d l i d l ICI h e X F d e X W π π − − ₋ − ₋ − = = = = ⎛ ⎞ ⎛ ⎞ =_{⎜ ⎟} + _{⎜ ⎟} + ⎝

∑∑

⎠

∑ ∑

_⎝ ⎠ 1 ,0 , (( )) 1 N N k k k d k d k d ICI H X H X W − − = = +

∑

+   , (2.12) where 2 1 ( ) 0 ( ) ,0 & 0 1 j di N i _N l l G i F d h e l N d N π − ₋ = =

∑

≤ ≤ ≤ ≤ − . (2.13)

Then H_{k d,} can be defined as

2 ( ) , 0 ( ) 0 , 1 G l k d N _j N k d l l H F d e k d N π − − = =

∑

≤ ≤ − . (2.14)

The second term of eq. (2.12) represents ICI which is the combination of other subcarriers and

can’t be neglected as the maximum Doppler frequency increases [15]. These ICI term causes

the performance of OFDM degraded severely.

We treat eq. (2.13) as frequency response of the lth path and H_{k d,} is the weight

coefficient from the ((k−d))_Nth subcarrier on the kth one. to the eq. Now, by the notion

above, we try to explain why eq.(2.6) results in eq. (2.12). We can see from eq. (2.6) that at

any time instant t= ×i T , the received signal y_i is the summation of the result of path

coefficient multiply with delayed sample of OFDM symbol, so we can expect that subcarrier

k

We uses the notion above to rederive Y_k.

{

}

{ }

{

}

(( )) 0 ( ) (( )) 0 ( ) (( )) 0 2 0 2 ( ) 1 (( )) 0 0 2 ( ) 1 ( 0 0 { } ( ) ( ) ( ) G N G N G N G G N G k i N l i l i l N i l i l k l N i l i l k l lk N j N l k k l l k d N N _j N l k d k l d l k d N N _j N l d l Y FFT y FFT h x w FFT h x W FFT h FFT x W F k X e W F d X e W F d e X π π π − = − = − = − = − − ₋ − = = − − ₋ = = = ⎧ ⎫ = _⎨ + _⎬ ⎩ ⎭ = + = ⊗ + = ⊗ + = + =

∑

∑

∑

∑

∑∑

∑∑

( )) 1 , (( )) 0 N N k d k N k d k d k d W H X W − − − = + =

∑

+

We can find that the result of above equation is the same as eq.(2.12), so time variant channel

causes frequency response and subcarrier do the circular convolution operation in OFDM

systems.

2.3

Sphere Decoding

2.3.1

Real Sphere Decoding

In communication, Sphere Decoding (SD) [16] ,is used to solve the ML problem as

2

ˆ _{arg min}

ML = _∈Λ −

X

X Y AX , (2.15)

where A is a M-by-N matrix where M >N, Y is a N-by-1 vector, and X is a N-by-1

vector. Λ is the set which includes all possible X . We can derive from eq. (2.15) as

follows ：

(

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

(

(

)

)

2 1 1 1 1 ˆ _{arg min} arg min arg min arg min arg min ML T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T ∈Λ ∈Λ ∈Λ − ∈Λ − − − ∈Λ = − = − − = − − + = − − + + − = − − + + − X X X X X X Y AX Y AX Y AX Y Y Y AX X A Y X A AX Y A A A A Y Y AX X A Y X A AX Y Y Y A A A A Y Y A A A IA Y Y AIX X IA Y X A AX Y I A A A A Y

(

) (

)(

)

(

(

) (

)

(

)(

)

)

(

)

(

)

( )

( )

(

)

(

)

(

)

1 1 1 1 1 † † † † 1 † arg min arg min arg min T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T − − ∈Λ − − − ∈Λ − ∈Λ = − − + + − = − − + + − = X X X Y A A A A A A A A Y Y A A A A A X X A A A A A Y X A AX Y I A A A A Y Y A A AA Y Y A A AX X A AA Y X A AX Y I A A A A Y A A Y

(

)

(

)

T

(

(

_†

)

)

(

(

)

1

)

T T − T −X A A Y−X +Y I−A A A A Y arg min

(

ˆ

)

(

ˆ

)

T T α ∈Λ = − − + X X X A A X X . (2.16)

In eq. (2.16), α is a constant and does not change when different X is chosen.

† 1

( T )− T

=

A A A A is the pseudo inverse of A , so ˆX is the least square solution of

=

AX Y .As the same as [17], to solve eq (2.15) is equivalent to solve as follows：

(

)

(

)

ˆ _{arg min} ˆ T T ˆ

ML = _X∈Λ − −

X X X A A X X . (2.17)

X causes the minimum. However, the computational complexity of above exhaustive search

method is really high So sphere decoding is brought up to avoid the exhaustive search and

searches only over the possible X which lies in a certain sphere centered at the given vector

with radius r. In this notion eq. (2.18) can be written as follows in Sphere decoding

(

)

(

)

2

ˆ _{arg min} ˆ T T ˆ

ML = _X∈Λ − − ≤r

X X X A A X X . (2.18)

It is clear that the closest point inside the sphere will also be the closest point for the whole

point. However, close scrutiny of this basic ideal leads to two key questions [16].

1) How do you choose radius r? Clearly, if radius is too large, we obtain too many points,

and the search remains exponential in size, whereas if radius too small, we obtain no

points inside the sphere.

2) How can we tell which points are inside the sphere? If this requires testing the distance of

each point from ˆX , then there is no point in sphere decoding, as we will still need an

exhaustive search.

Sphere decoding does not really address the first question. However it does propose an

efficient way to answer the second. The basic observation is the following. Although it is

difficult to determine the points inside a general N-dimensional sphere, it is trivial to do so in

the one-dimensional case. The reason is that a one-dimensional sphere reduces to the

endpoints of an interval, and so, the desired points will be the integer values that lie in this

we have determined all k-dimensional points that lie in a sphere of radius. Then, for any such

k-dimensional points, the set of admissible values of the (k+1)th dimensional coordinate that

lie in the higher dimensional sphere of the same radius forms an interval. So we can determine

all points in a sphere of dimension N and radius r by successively determining all points in spheres of lower dimensions 1,..., N and the same radius r.

In eq. (2.18), we choose the least square solution as the center of sphere which is the

optimum unconstrained solution in this problem. As Fig 2.2 we will find the minimal solution

lies in the sphere with radius r.

Fig. 2.2 A sphere of radius r and centered at AX . ˆ

To solve this problem efficiently Cholesky factorization is employed to find an upper

triangular U with u real and positive such that _ii U UT =A A . So eq. (2.18) can be written T

as

(

)

(

)

2

ˆ _{arg min} ˆ T T ˆ

ML = _X∈Λ − − ≤r

(

)

(

)

(

)

(

)

2 1 2 2 1 1 ˆ arg min ˆ 0, ˆ ˆ arg min T N N i i ij j j ij i j i N N ii i i ij j j i j i where

d where d u X X and u for i j u X X u X X r ∈Λ = = ∈Λ _{= = +} = = − = = − = < ⎡ ⎤ = _⎢ − + − _⎥ ≤ ⎣ ⎦

∑

∑

∑

∑

X X D D D U X X 　

(

)

2 2 2 1 1 ˆ ˆ arg min N N ij ii i i j j i j i ii u u X X X X r u ∈Λ _{= = +} ⎡ ⎤ = _⎢ − + − _⎥ ≤ ⎣ ⎦

∑

∑

X . (2.19)

In eq. (2.19), the sphere decoder establishes bounds on X₁,...,X by examining these terms _N

in subsets.

Starting with i=N, and throwing out the terms i=1,...,N− , we obtain from eq. 1

(2.19)

(

)

2 2 _ˆ 2 NN N N u X −X ≤r ˆ ˆ N N N NN NN r r X X X u u ⎡ ⎤ ⎢ ⎥ ⇒_⎢ − _⎥≤ ≤_⎢ + _⎥ ⎢ ⎥ ⎣ ⎦. (2.20)

(⎡ ⎤_{⎢ ⎥}. and ⎢ ⎥_{⎣ ⎦}. means the ceiling function and the floor function operators return the smallest

integer greater than or equal to, and the largest integer less than or equal to their respective

arguments; these functions are applied in the case which the constellation is a set of

consecutive integers such as QPSK or QAM.) After computing the lower and upper bounds in

eq. (2.20), the sphere decoder chooses a candidate value for X and computes the _N

implication of this choice on X_N−1. To find the influence of the choice of Xˆ_N and Xˆ_N−1

the sphere decoder looks at the two terms i=M −1 in eq. (2.19), throws out the remaining

(

)

(

)

2 2 1, 2 2 2 1, 1 1 ˆ 1 ˆ ˆ N N N N N N N N NN N N NN u u X X X X u X X r u − − − − − ⎡ ⎤ − + − + − ≤ ⎢ ⎥ ⎣ ⎦

which yields the upper bound

(

)

2

₍

₎

2 2 1, 1 1 1, 1 ˆ ˆ NN N N N N ˆ N N N N N N NN r u X X _u X X X X u u − − − − − ⎢ _{− −} ⎥ ⎢ ⎥ ≤_⎢ + − − _⎥ ⎢ ⎥ ⎣ ⎦

and the lower bound

(

)

2

₍

₎

2 2 1, 1 1 1, 1 ˆ ˆ NN N N N N ˆ N N N N N N NN r u X X _u X X X X u u − − − − − ⎡ _{− −} ⎤ ⎢ ⎥ ≥_⎢ − − − _⎥ ⎢ ⎥ ⎢ ⎥ .

The sphere decoder now chooses a candidate for X_N−1 within the range given by the upper

and lower bounds, and proceeds to X_N−2, and so on.

There are two things happen during the algorithm operating.

1) The decoder reaches X and chooses a value within the computed range. ₁

2) The decoder finds that no point in the constellation fall within the upper and lower bounds obtained for some X . _j

In the first case, the sphere decoder has a candidate solution for the entire vector X ,

computes its radius which cannot exceed r, and starts the search process over, using this new

smaller radius to find any better candidates. In the second case, the decoder must have made at least one bad candidate choice for X_j+1,...,X_N. The decoder revises the choice for X_j+1

which immediately preceded the attempt for X by finding another candidate value within _j

decoder backtracks to X_j+2, and so on.

The algorithm for determining the points in an N-dimensional sphere essentially

constructs a tree in Sphere Decoding (see Fig 2.3). Let’s use this tree structure to briefly

explain the operation of Sphere Decoding.

Fig. 2.3 Sample tree generated to determine points in a N-dimensional sphere.

The node at the top of the fig. 2.3 is treated as the start node and others with numbers inside

represent the point in the set S where S is equal to N Λ (take BPSK as an example,

{ 1,1}

S = − ). Choosing different node at layer N makes eq. (2.19) generates different upper

and lower bound for layer N-1, so the nodes can be chosen at layer N-1 depend on the node

which is chosen at the layer N. Now, if node choosing order from layer N to N-2 is 1、2、3,

then we can find that there is no node can be chosen at layer N-3. So the algorithm backs to

can be chosen. Then the algorithm backs again and goes to layer N-1. At this layer, it remains

two nodes (excluding 2) can be chosen, so 3 is chosen and algorithm keep going until it

finishes the choosing operation at layer 1. After finishing the choosing operation at layer 1, a

feasible solution is generated and this point will be used to generate a new radius. New radius

replaces the initial radius for generating the upper and lower bound of every layer, so

algorithm keeps going with new lower and upper bound at layer 1. Under this procedure, the

tree structure may change again and again that is the sphere becomes smaller and smaller. If

the algorithm is terminated, the last feasible solution is the best solution.

2.3.2

Complex Sphere Decoding

The Sphere Decoding algorithm described above applies on a real system where X is

chosen from a real lattice, but in communication systems we face to deal with complex

system because of the modulation scheme we used such as QPSK. In this case, eq. (2.18)

becomes as follows：

(

)

(

)

2 ˆ _{arg min} ˆ H H ˆ ML = _∈Λ − − ≤r X X X X A A X X (2.21)

where A , X , and ˆX are complex value. Here two kinds of methods are introduced to deal

with this problem. The first method applies the algorithm on the complex system by

equations with twice the dimension of the original system [18]. For example, eq. (2.21) can be

transformed into a real equation in matrix form as follows

(

)

(

)

2 arg min T T ML = _∈Λ LS − LS − ≤r X X _{ } X X A A X   X (2.22) where

{ }

{ }

Re T Im T ⎡ ⎤ =_⎣ − _⎦ X X X

{ }

ˆ

{ }

ˆ Re T Im T LS =⎡⎣ − ⎤⎦ X X X

{ }

{ }

{ }

{ }

Re Im Im Re T T T T ⎡ − ⎤ ⎢ ⎥ = ⎢ ⎥ ⎣ ⎦ A A A A A 

and Λ =

{

Re( ), Im( )Λ Λ

}

. If X belongs to QPSK then each entry of X belongs to BPSK.

Thus eq. (2.22) can be solved via SD which we introduced in section 2.3.1.

Fig. 2.4 Searching disk in 16-QAM.

The second method [19] uses eq. (2.21) directly without decoupling, but the searching domain Λ is not integer set anymore (see Fig 2.4). As the same as eq. (2.19) we can derive from eq.

(2.21) as follows

(

)

(

)

2 ˆ _{arg min} ˆ H H ˆ ML = _∈Λ − − ≤r X X X X U U X X

(

)

2 2 2 1 1 ˆ ˆ arg min N N ij ii i i i i i j i ii u u X X X X r u ∈Λ _{= = +} ⎡ ⎤ = _⎢ − + − _⎥ ≤ ⎣ ⎦

∑

∑

X (2.23)

As the real case, Sphere Decoding algorithm starts at i=N and lets iN

N c X =r eθ and ˆ ˆ _ˆ i N N c X =r eθ , where

(

)

2 2 1 2 0, ,..., 2 2 c c c M k M M π π θ _{∈ ⎨}⎧⎪ − ⎫⎪_⎬ ⎪ ⎪ ⎩ ⎭ (2.24) c

M is the number of bits per symbol, for example M_c = for QPSK. Then, we get 2

2 2 2 2 2 ˆ ˆ _{ˆ ˆ} | _{N N} | _{c c} 2 _{c c}cos( _{N N}) NN r X X r r r r u θ θ − = + − − ≤ 2 2 2 2 1 ˆ _ˆ cos( ) : ˆ 2 N N c c c c NN r r r r r u θ θ ⎡ ⎤ η ⇒ − ≥ _⎢ + − _⎥= ⎣ ⎦ (2.25) 1 1 2 _ˆ 2 2 _ˆ ( cos ) ( cos ) 2 2 2 c c c M M M N N N θ η θ θ η π − π π − ⎡ ⎤ ⎢ ⎥ ⇒_⎢ − _⎥≤ ≤_⎢ + _⎥ ⎢ ⎥ ⎣ ⎦. (2.26)

We can see from eq. (2.26) the searching domain change to eq. (2.24) now. From eq. (2.26),

if 1η> , then the search disk does not contain any point of the constellation (because the

range of cosine function is [ 1,1]− , it is impossible to find a value makes cos(θN −θˆN)>1). If

1

η< − , then the search disk includes the entire constellation (_cos( ˆ _{) 1}

N N

θ −θ > − is always

right no what value is in the cosine function). For M-QAM there are different values of r , so _c

solving for the points within the search disk simply requires solving the inequality eq. (2.25)

computing of the algorithm, the recursive equations is develop [19]. Let ξi =Xˆi−Xi, 2

ii ii

q =u , q_ij =u_ij/u_ii and substitute into eq. (2.23) gets

2 2 1 1 N N ii i ij i i j i q ξ q ξ r = = + + ≤

∑

∑

. For k=l, 2 2 2 1 1 1 2 2 2 1 1 1 2 2 ₂ 1 1 1 1 ˆ 1 K K K ll l lj l ii i ij i j l i l j i K K K l lj l ii i ij i j l ll i l j i K K l l l ii i ij i i l j i ll l l ll q q q q r q r q q q s s S r q q q T R q ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ = + = + = + = + = + = + = + = + + + + ≤ ⎡ ⎤ ⎢ ⎥ ⇒ + ≤ − + ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⇒ − + ≤ − + ⎢ ⎥ ⎣ ⎦ ≤ =

∑

∑

∑

∑

∑

∑

∑

∑

where 2 2 1 1 M M i ll l lj j l i j l T r q ξ q ξ = + = + = −

∑

+

∑

2 2 1 1 1 M M i ll l lj j l i j l ii R r q q q = + ξ = + ξ ⎛ ⎞ ⎜ ⎟ = − + ⎜ ⎟ ⎝

∑

∑

⎠ i i ii T R q = . Let ' ' ' , ˆ ji i i i c i s = + =s S r eθ , 1 M i ij j j i S q ξ = + =

∑

, we can get 2 1 1, 1 1 1 i i i i i i T =T₊ −q_{+ +} ξ₊ +S₊ , (2.27) and the ith summation term in eq. (2.23) can be written as follows：

2

' 2 '2 ' '

, 2 , cos( )

i i c c i c c i i i i

' 2 '2 , ' , 1 cos( ) 2 i i c c i i i c c i r r R r r θ θ− ≥ ⎡_⎣ + − ⎤_⎦=η (2.28) ' 1 ' 1 2 2 2 ( cos ) ( cos ) 2 2 2 c c c M M M i i i i i θ η θ θ η π − π π − ⎡ ⎤ ⎢ ⎥ − ≤ ≤ + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦. (2.29)

The distance between point AX and obtained point by Sphere Decoding in received signal ˆ

domain can be computed by the formula below：

2 2

0 1 11 1 1

K K

d =T − =T T − +T q ξ +S . (2.30) For every stage, eq. (2.27) is used to compute the parameter which is needed in eq. (2.28) and

eq. (2.29) to calculate the upper and lower bound of next stage. The second method is easy to

use in computer programming, so it is adopted in this thesis.

2.4

List Sphere Decoding

List Sphere decoding (LSD) is modified from sphere decoding [17]. LSD gives a list

L which contains N_cand candidates of X with smaller values in eq. (2.15) for generating

the soft information. In order to generate L , the sphere decoder needs to be modified in two

ways. Every time it finds a point inside the initial radius r：

1) it does not decrease r to correspond to the radius of this new point;

2) adding this point to L if the list is not full; or if L is full, it compares this point with

the point in L with the largest radius and replaces this point if the new point has smaller

By the changes above, List Sphere Decoding searches all the points which are inside of the sphere with given initial radius, and gives the N_cand best point. Sphere Decoding changes to search the point in new sphere (the center of the sphere is still the same) when its find a

new point whose distance to the center of the sphere is smaller than initial radius (mentioned

in section 2.3.1). So the points which are inside the old sphere and outside the new sphere will

not be considered in Sphere Decoding and the searching tree of SD goes smaller and smaller

(actually, another smaller tree) which is not the case in LSD. The searching tree of LSD is still

the same and all the branches must be gone through.

Fig. 2.5 Sphere of SD or LSD in received signal domain.

Here, we use Fig. 2.5 to explain the different between SD and LSD. In Fig.2.5, the

circle mark represents the center of the sphere and cross mark means the feasible solution in

initial radius r , if the cross mark 4 is first found and radius ₁ r is used to be the new radius ₂

of the searching tree then cross mark 1、2 and 3 will not be presented in new searching tree

with radius r . Again, if the cross mark 6 is found in the new searching tree with radius ₂ r ₂

then cross mark 5 will not be considered for next searching tree. However, for LSD with

initial radius r , if the cross mark 4 is first found, cross mark 1、2、3、5、6 are still in the ₁

searching tress with radius r because LSD will not change the radius. Back to fig. 2.3, when ₁

LSD finishes the searching operation at layer 1 successfully, it puts this point into the list,

backs to layer 2, and keeps the algorithm going. Then all the feasible points in the sphere will

be found out successively (the finding order is independent of the distance to the center of the

Chapter 3

Grouped Based ICI Cancellation

Method

3.1

System Model

In section 2.2, we have known the ICI effect on OFDM systems and its mathematic

representation. Now, we change the mathematic representation in section 2.2 into matrix form

which is convenient to be used in proposed method. Eq. (2.12) can be rewrote as follows：

H N

=

x F X (3.1) where x=[ ,...,x₀ x_N−1]T, X=[X₀,...,X_N−1]T as showed in Fig. 2.1 and F_N is the N point FFT matrix which can be represented as follows：

2 0 0 2 0 1 2 0 ( 2) 2 0 ( 1) 2 1 0 2 1 ( 1) 2 ( 2) 0 2 ( 2) ( 1) 2 ( 1) 0 2 ( 1) 1 2 ( 1) ( 2) 2 ( 1) ( 1) N N j j j j N N N N N j j N N N N N N j j N N N N N N N N j j j j N N N N e e e e e e e e e e e e π π π π π π π π π π π π ⋅ ⋅ ⋅ − ⋅ − − − − − ⋅ ⋅ − − − − ⋅ − ⋅ − − − − ⋅ − ⋅ − ⋅ − − ⋅ − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ F " # % # " (3.2)

Assume that the maximum delay spread of the channel always less than or equal to the length

= + y Hx w (3.3) N = Y F y

=F HxN +W =AX+W

(3.4)

with w is an N×1 AWGN noise and A=F HF_{N N}H. The channel impulse response matrix

H is defined as follow [21] (0) (0) (0) 0 1 1 ( 1) ( 1) ( 1) 1 2 0 ( 1) ( 1) 1 0 0 0 0 0 0 0 0 0 L L L L L L N N L h h h h h h h h − − − − − − − − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ H " " # % % % % % # " " % % % % % # # % % % % % " " " (3.5)

where h_k( )i is the k-th channel tap at time instant t= ×i T and T is the sampling period. Now, we want to find a ˆX such that

2 ˆ _{arg min} ∈Λ = − X X Y AX (3.6)

where Λ is the set includes all possible of X . Clearly, we can apply Sphere Decoding to

solve eq. (3.6) [6], but the complexity of this method is very large.

3.2

Group based ICI Cancellation Method

with Sphere Decoding

subcarriers, so we assume the window length of the ICI effect is Q=2q+1 that is X_i causes interference to _{(( ))} ~ _{(( ))} N N i q i q Y ₋ Y ₊ . Let [ ,..., _{( 1) 1}] x x T i = Xi N⋅ Xi+ N −

X be a segment of X and for

any i≠ , {0}j X_i∩X_j = where N_x is the size of one segment. Accordingly, X_i will

induce the interference on [ _{(( ))} ,..., ,..., _{((( 1) 1 ))} ]

x N x x N

T i = Y i N⋅ −q Yi N⋅ Y i+ N − +q

Y . From Fig 3.1, Y_i also

suffers the interference from X_i−1 and X_i+1 or even more segments (depend on how large

x

N and q are), so performance of solving X_i by Y_i directly may not be acceptable.

Fig. 3.1 Group Method for ICI with window length Q=2q+1.

The basic ideal of our method is to cancel the component of X_j, i≠ in j Y_i. Again, form [8], most of the ICI effect on a subcarrier comes from neighboring subcarriers. We use

2 1

x

n > q− so that Y_i is affected by X_i−1, X_i and X_i+1 only. The relationship of Y_i,

1 i−

1 1 1 1 [ i ] [ i] [ i ] i = Ai− i− + Ai i+ Ai+ i+ Y X X X (3.7) where (( )) , (( )) ,( 1) 1 ((( 1) 1 )) , ((( 1) 1 )) ,( 1) 1 [ ] x N x x N x x N x x N x j n q i n j n q i n j i j n q i n j n q i n a a A a a ⋅ − ⋅ ⋅ − + − + − + ⋅ + − + + − ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ " # % # " (3.8) ij

a is the element of A on i-th row and j-th column in eq. (3.4). First, solving X_i, X_i−1

and X_i+1 individually by Y_i, Y_i−1 and Y_i+1 by using Sphere Decoding. Eq. (2.21) applied here becomes as follow：

ˆ _, arg min

(

ˆ

)

[ ] [ ]

(

ˆ

)

2 i H i H i i ML = _∈Λ i− i Ai Ai i− i ≤r X X X X X X (3.9)

where Xˆi is the least square solution of [ ]

i i = Ai i

Y X . After solving the eq. (3.9), we use the

result X_i−1 and X_i+1 to do ICI cancellation by following equation：

1 1 1 1

[ i ] [ i ]

i = i− Ai− i− − Ai+ i+

Y Y X X . (3.10)

Then using Y to do Sphere Decoding, because the ICI effect comes from _i X_i−1 and X_i+1

have been cleaned, we expect that the more correct X_i can be obtained. Using the procedure

as described above iteratively, we expect that the recover performance will be better and better. Fig 3.2 is the block diagram of the algorithm for one group Y_i. At initial state, switch links to the path which input a zero vector to do ICI cancellation (i.e. no ICI cancellation). At the

(j+1)th iteration, switch links to the path which input [A_ii₋₁](Xˆ_i−1)_j and [A_ii₊₁](Xˆ_i+1)_j where

1

ˆ

(X_i−)_j represents the output made from the (i-1)th group at the jth iteration for doing ICI cancellation. After ICI cancellation, Sphere Decoding generates (Xˆ_i−1)_j+1 can be used in ICI

cancellation for next iteration. i Y 1 ˆ (Xi− )j 1 ˆ (X_i+ )_j 1 [A_ii₋] 1 [A_ii₊] 1 ˆ (Xi)j+

Σ

×

×

0

0

−

Fig. 3.2 Block diagram of group based ICI cancellation method with Sphere Decoding.

3.3

Group based ICI Cancellation Method

with List Sphere Decoding

Different from section 3.2, we apply List Sphere Decoding to generate some candidates of X_i and use these candidates to compute the soft symbols for doing ICI cancellation.

3.3.1

Soft Symbol

, , , 1 ˆ ₍ ˆ ˆ ₎ 2 I Q k i k i k i X = X + jX (3.11) , , , ˆI _[ I _{1| ] [} I _{1| ]} k i k i i k i i X =P X = + Y −P X = − Y (3.12) , , , ˆQ _[ Q _{1| ] [} Q _{1| ]} k i k i i k i i X =P X = + Y −P X = − Y (3.13)

where Xˆ_{k i}I_, is an expectation of bit corresponding to the real part of the k-th symbol in X_i.

, ,

[ 1| ]

x

Q

k i i N

P X = + Y is the probability of Xˆ_{k i}Q_, equals to 1 when c is given and can be obtained by follows equation：

, , , ( | ) , _{( | )} , _{( | )} [ 1| ] 1 1 [ 1| ] 1 I D k i i I D k i i I D k i i L X I k i i _{L X} I k i i _{L X} e P X e P X e = + = + = − = + Y Y Y Y Y (3.14) where , , 1 , , 1 , , , [ 1| ] ( | ) ln [ 1| ] ( | ) ln ( | ) I i k i I i k i I k i i I D k i i I k i i i i L i i L P X L X P X p p + − ∈ ∈ = + = = − =

∑

∑

X X Y Y Y X Y X Y X X (3.15)

and LI_{k i, , 1+} is the set of all possible X_i that real part of the k-th symbol equal to 1. Here, X_i

is selected form the list generated by List Sphere Decoding [22]. Assume that the prior

probability of every bit is equal probability and is independent to each other, so we can go

further from eq. (3.15) as follows：

, ( | ) x I D k i N L X Y , , 1 , , 1 , 0, 0, , 1, 1, , 0, 0, , 1, 1, 1 ( | ) ( 1) ( , ,..., ,..., , ) ( ) ln 1 ( | ) ( 1) ( , ,..., ,..., , ) ( ) x x I i k i x x I i k i I I Q Q I Q i i k i i i k i N i N i L i I I Q Q I Q i i k i i i k i N i N i L i p p X p X X X X X p p p X p X X X X X p + − − − ∈ − − ∈ = = = −

∑

∑

X X Y X Y Y X Y X X

, , 1 , , 1 ( | ) ln ( | ) I i k i I i k i i i L i i L p p + − ∈ ∈ =

∑

∑

X X Y X Y X X X (3.16)

3.3.2

Conditional Probability

From above derivations, if we want to compute the soft symbol then we need to find ( _i| _i)

p Y X . For every subcarrier Y , we can use equation below to represent _i

1 i i N i ii i ij j i j j i ICI ij j ij j i j J j J Y a X a X W a X a X W = ≠ ∈ ∉ = + + = + +

∑

∑

∑

  i i i ij j ij j i j J j J I i Qi i Y a X a X W n jn n ∈ ∉ ⇒ − = + = + =

∑

∑

assume n and _{I i} n are independent Gaussian random variable [4], so we need compute the _Qi

mean and variance of n in order to get _i P Y X ( |_{i i})

[ ] [ ] 0 [ ] [ ] 0 i i ij j i Ii Qi j J E n E a X W E n E n ∉ =

∑

+ = ⇒ = = (3.17) 2 2 [ ] [| | ] i i i ij j i j J Var n E a X W σ ∉ = =

∑

+ (3.18)

because X and _j W are independent, eq. (3.18) becomes _i

2 2 1 2 [ ] | | [ ] [ ] 2 i i av ij w Ii Qi i j J

Var n P a σ Var n Var n σ

∉

=

∑

+ ⇒ = = . (3.19)

( )

( )

2 2 1 2 1 2 2 ₂ 2 ₂ 1 1 ( | ) exp I . exp Q i i i i i i n n P Y σ σ πσ πσ ⎡ ⎤ ⎡ ⎤ = _⎢− _{⎥ ⎢}− _⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ X

( )

( )

2 2 2 2 2 2 2 1 exp 1 exp I Q i i i i n n n σ πσ σ πσ ⎡ + ⎤ = _⎢− _⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ = − ⎢ ⎥ ⎣ ⎦

( )

2 2 2 1 exp i i ij j j J i i Y a X σ πσ ∈ ⎡ ⎤ ⎢ − ⎥ ⎢ ⎥ = _⎢− _⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

∑

. (3.20)

At last, because every n in _j Y_i can be seen as independent (the correlation of term

i

jl l l J

a X

∉

∑

between each of n is small under the assumption that ICI effect comes from _j

neighboring subcarrier.), we can get p Y X as follows： ( _i| _i)

( )

(

1

)

(

)

1

(

)

( | ) exp [ ] [ ] det x H i i i i N i i i i i i i i P A A π − ⎡ ⎤ = _⎢− − Σ − _⎥ ⎣ ⎦ ∑ Y X Y X Y X (3.21) where ( 2,..., 2 ₁) x i diag σi σi N+ − ∑ = .

Above is the case of initial state which ICI cancellation has not been done yet. There are some

changes to be done after ICI cancellation. Fortunately, we only need to change the mean and variance of n . Again, let’s see the subcarrier _j Y after ICI cancellation, _i

(

)

ˆ ˆ i i i i i ij j i j J ij j ij j j i j J j J Y Y a X W a X a X X W ∉ ∈ ∉ = − + = + − +

∑

∑

∑



(

ˆ

)

i i i i ij j ij j j i j J j J n Y a X a X X W ∈ ∉ ⇒ = − 

∑

=

∑

− + . (3.22)

[ ]

(

ˆ

)

ˆ i i ij j j i ij j j J j J E n E a X X W a X ∉ ∉ ⎡ ⎤ = _⎢ − + _⎥= − ⎣

∑

⎦

∑

 (3.23)

[ ]

(

)

[ ]

2 2 ˆ i i i ij j j i i ij j i j J j J Var n E a X X W E n E a X W ∉ ∉ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = − + − = + ⎢ ⎥ ⎢ ⎥ ⎣

∑

⎦ ⎣

∑

⎦   2 2 | | i n av ij j J P a σ ∉ = +

∑

. (3.24) Apply eq. (3.23) and eq. (3.24) into eq. (3.21) then we can get p Y X to compute the soft ( _i| _i) symbol after ICI cancellation.

i Y 1 ˆ (Xi−)j 1 ˆ (X_i+)_j 1 [A_ii₋] 1 [A_ii₊] 1 ˆ (X_i)_j+

Σ

×

×

0

0

−

Fig. 3.3 Block diagram of group based ICI cancellation with List Sphere Decoding

The block diagram of the group based ICI cancellation with List Sphere Decoding is

showed in fig. 3.3. The procedure is as the same as the one with Sphere Decoding except that

followed to compute the soft symbol for ICI cancellation here.

3.4

Radius of the Sphere

In chapter 2, there are two key questions of Sphere Decoding are mentioned. The

second one can be solved by the algorithm of Sphere Decoding, but the first one how to

choose the radius does not have exactly solution. Here, we use a simple method to decide the

initial radius for each application of Sphere Decoding in our group based ICI cancellation

method. From eq. (2.21), we use least square solution to be the center of the sphere. We know

that least square solution is the best solution to satisfy the ML criterion, but it may not be the

feasible solution that is, it may not be a point on the signal constellation. Because of the

channel effect, the hard decision of the least square solution may not be the best solution

within the all feasible solutions. We can use Fig 3.4 to interpret the problem above. In Fig 3.4,

left hand side is the feasible point in transmitted signal space and right hand side is the

transformation of left hand side by the channel matrix. On the left hand side, square is the

least square solution and the circle is the hard decision of the least square solution. On the

right hand side, square Yˆ_i is the transformation of the least square solution and circle is the transformation of the hard decision value. Hard decision of the least square is the nearest

transform hard decision is not the nearest point of least square solution’s transformation.

Fig. 3.4 signal space in transmitter and receiver.

If hard decision of least square solution after transformation is not the nearest feasible to the

center of the sphere Yˆ_i in the received signal space then the solution which satisfies eq.

(2.21) must be inside of the sphere with radius which is equal to the distance between Yˆ_i

and the hard decision of least square solution after transformation. By the notion above, it is

reasonable to set the initial radius of the sphere to be the distance between Yˆ_i and the hard decision of least square solution after transformation.