用於增進MIMO-OFDM 系統性能之編碼與調變－子計畫二：具低均峰比率及錯誤率之MIMO OFDM系統(1/3)

用於增進 MIMO-OFDM 系統性能之編碼與調變--子計畫二：

具低均峰比率及錯誤率之 MIMO OFDM 系統(1/3)

期中進度報告(完整版)

中 華 民 國 96 年 05 月 25 日

編碼與調變

-子計畫:二具低均峰比率及

錯誤率之

MIMO OFDM 系統(1/3)

MIMO OFDM Systems with Low

PAPR and Error rates(1/3)

計畫編號　 : NSC 95-2219-E-002-026

執行期限　 : 95/08/01 ~ 96/7/31

計畫主持人　：　林茂昭

摘要

關鍵詞

We consider the problem of space-frequency (SF) codes design for multiple-input-multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) mod-ulation in frequency-selective Rayleigh fading channel. In particular, we investigate a space-frequency code with two transmit antennas that is constructed by the con-catenation of binary LDPC code and the Alamouti space-time coding. Based on this efficient space-frequency code, we propose a low complexity selective-mapping type PAPR reduction technique. In the proposed technique, the candidates are generated in the time-domain instead of the frequency domain. Thus, only two IFFT operations are needed in the proposed technique while for the selective mapping using frequency domain many IFFT operations are needed. In case the number of candidates is not great (no more than 16), the proposed technique can significantly reduce the com-plexity without sacrificing the PAPR reduction capability and error rates. Simulation results verify the advantage of the proposed PAPR reduction technique.

Index Term _{–Alamouti code, Low-density parity-check (LDPC) codes,}

1 INTRODUCTION 1 2 SYSTEM MODEL 5 2.1 Space-Time Codes . . . 5 2.2 Space-Frequency (SF) Codes . . . 7 2.2.1 System Model . . . 7 2.2.2 Design Criteria . . . 10

3 LDPC Coded Alamouti Scheme for MIMO OFDM 14 3.1 LDPC Codes . . . 15

3.2 Alamouti Space-Time Code . . . 17

3.3 The Space-Frequency Code Under Investigation . . . 23

3.4 Simulation Results . . . 26

4 PAPR Reduction in MIMO OFDM Systems 31

4.3 Time-Domain Circular Shift Scheme . . . 38

4.3.1 Time-Domain Circular Shift Scheme . . . 38

4.4 Simulation Results . . . 40

4.4.1 CCDF Performance . . . 41

4.4.2 BER Performance . . . 42

5 CONCLUSIONS AND SELF EVALUATION 47

Bibliography 49

3.1 Numeric power delay profile of six paths and two paths. . . 27

4.1 Four operations of each subblock. . . 36

4.2 Comparison of information bit S and the number of IFFT needed ξ. . 41

3.1 A block diagram of the Alamouti space-time encoder. . . 17

3.2 A simply model of the Alamouti space-time codes. . . 21

3.3 Alamouti space-time codes on flat fading channel. . . 22

3.4 A block diagram of the space-frequency code under investigation. . . 23

3.5 Power delay profile of six paths and two paths. . . 27

3.6 BER of considered SF codes, Nt = Nr = 2, QPSK modulation . . . . 28

3.7 BER of considered SF codes, Nt = Nr = 2, 8PSK modulation . . . 29

3.8 BER of considered SF codes, Nt = 2, Nr = 1, QPSK modulation . . . 29

3.9 BER of considered SF codes, Nt = Nr = 1, 8PSK modulation . . . . 30

4.1 Partition of the Ca and Cb. . . 35

4.2 Cross-Antenna Rotation and Inversion (CARI) scheme. . . 37

4.3 Block diagram of SS-CARI scheme. . . 38

4.4 Successive Suboptimal Cross-Antenna Rotation and Inversion (SS-CARI)

scheme. . . 38

4.6 SS-CARI and TDCS scheme for different value of candidates Q. . . . 42

4.7 SS-CARI BER performance for W = 4. . . 43

4.8 TDCS BER performance for Q = 16. . . 44

4.9 SS-CARI BER performance for W = 4 with side information embedded. 45

4.10 TDCS BER performance for Q = 16 with side information embedded. 46

INTRODUCTION

It has been well recognized that Multiple-input-multiple-output (MIMO) systems em-ploying multiple transmit and receive antennas can play a significant role in the broad-band wireless communications. By employing the diverse characteristics of channels between each pair of transmit and receive antennas, the MIMO system can provide a large potential capacity increase as compared with the conventional single antenna systems. To exploit this capacity increase, many space-time (ST) codes have been proposed, such as [1, 2, 3, 4, 5, 6]. These ST codes were basically designed for flat fading channels. In case of broadband wireless communication systems, the channels are frequency selective channel, resulting in inter symbol interference (ISI)[7]. Or-thogonal frequency division multiplexing (OFDM) is a technique, which is effective in combating the problem of ISI. [8].

In order to combine the advantages of both the MIMO systems and the OFDM, space-frequency (SF) coded MIMO-OFDM systems have been proposed [9], where two-dimensional coding is applied to distribute channel symbols across space (trans-mit antennas) and frequency (OFDM tones). In [10], an SF coding was obtained by exchanging the time domain arrangement for the frequency domain arrangement in the existing ST coding.

In the ST coding, the achievable diversity advantage is bounded by the product of the number of transmit antennas and the number of receive antennas [2]. Usually, multiple delayed paths will deteriorate the error performance in the digital trans-mission. However, SF codes can turn the negative effect of multiple delayed paths into advantage. In fact, SF coding can have additional multipath diversity in case the transmission is over the frequency-selective fading channel. Using SF coding, the maximum diversity is product of the number of transmit antennas, the number of receive antennas and the number of channel delay paths [9].

A well known problem for the OFDM system is the occasionally occurred high peak-to-average power ratio (PAPR), that is due to its approximately Gaussian dis-tributed output signal samples. An OFDM system with high PAPR requires a costly linear power amplifier with large dynamic range for the transmitter, otherwise signif-icant out-of-band energy and signal distortion will occur. By now, many techniques

have been proposed for relieving the PAPR problem in the OFDM, such as Ampli-tude Clipping, Coding, Selective Mapping (SLM) [11, 12] and Active Constellation Extension (ACE) [13]. For MIMO OFDM, the problem of PAPR is similar to the conventional OFDM system. The techniques used for mitigating the PAPR effect in the conventional OFDM system can also be applied to the MIMO OFDM systems. However, the usage of multiple transmit antennas may somewhat deepen the prob-lem of PAPR while also may provide additional room for executing the operation of PAPR reduction.

In this first-year report of the three-year project, we have completed the investiga-tion of an efficient MIMO-OFDM system, which is constructed by the concatenainvestiga-tion of LDPC coding and Alamouti coding. By insuring the good error performance of this concatenated SF code, we further investigate the associated PAPR problem. We pro-pose a low-complexity PAPR reduction technique for the investigated SF code, which is a kind of selective mapping technique implemented on the time-domain. Compared to a known PAPR reduction technique for MIMO-OFDM [14] that is a selective map-ping technique implemented on the frequency domain, the proposed PAPR reduction technique is more effective in PAPR reduction in case the number of candidates is not large, even though the proposed technique has lower complexity.

This report is organized as follows. The review of some MIMO-OFDM properties is given in Chapter 2. The SF code constructed from the concatenation of a binary

LDPC code and the Alamouti code and its related performances are described in Chapter 3. In Chapter 4, a new PAPR reduction technique for SF codes is shown. Finally, Conclusions are given in Chapter 5.

SYSTEM MODEL

2.1

Space-Time Codes

Throughout this report, we consider the MIMO system with Nt transmit antennas

and Nr receive antennas signaling over the fading channel. The received signal at

time t (t = 1, 2, · · · , T ) at the jth receive antenna (j = 1, 2, . . . , Nr) is given by

rj(t) = r Es Nt Nt X i=1 hj,i(t)si(t) + nj(t). (2.1)

where si(t) is the symbol transmitted from the ith transmit antenna at time t, hj,i is

the complex fading gain from the ith transmit antenna to receive jth receive antenna,

and nj(t) denotes the additive complex Gaussian noise with zero mean and unit

variance at time t. The received signal in Eqs.(2.1) can be rewritten in matrix form 5

as R = r Es Nt HS + N. (2.2)

where R, H, N and S are respectively

R =             R11 R12 · · · R1T R21 R22 · · · R2T .. . ... . .. ... RNr1 RNr2 · · · RNrT             , H =             H11 H12 · · · H1Nt H21 H22 · · · H2Nt .. . ... . .. ... HNr1 HNr2 · · · HNrNt             (2.3) N =             N11 N12 · · · N1T N21 N22 · · · N2T ... ... . .. ... NNr1 NNr2 · · · NNrT             , S =             s11 s12 · · · s1T s21 s22 · · · s2T ... ... . .. ... sNt1 sNt2 · · · sNtT             (2.4)

The matrix S represents a codeword of the space time code. The ith row of S is composed of the symbols transmitted from the ith transmit antenna over a period

of T . The jth column of S is composed of the symbols transmitted from all the Nt

2.2

Space-Frequency (SF) Codes

For a space time code, its codeword consists of symbols transmitted from all the Nt

antennas over a period of time. In contrast, for a space frequency code, its

code-word consists of symbols transmitted from all the Nt antennas over a frequency band

comprising many subcarriers.

2.2.1

System Model

Consider an SF-coded MIMO-OFDM system with Nt transmit antennas, Nr receive

antennas, and K subcarriers. Suppose that all the frequency-selective fading channels, each represents a pair of transmit antenna and receive antenna, have L independent delay paths and the same power delay profile. The MIMO channel is assumed to re-main unchanged over the period of each OFDM block. The channel impulse response from the ith transmit antenna to the jth receive antenna can be modelled as

hj, i(τ ) = L−1

X

l=0

αj, i(l)δ(τ − τl). (2.5)

where τl is the delay of the lth path, and αj, i(l) is the complex amplitude of the

lth path between transmit antenna i to receive antenna j. Each αj, i(l) is a complex

Gaussian random variable with zero mean and variances E|αj, i(l)|2 = δl2, where E

δ2

l are the same for all the pairs represented by (i, j), i = 1, 2, · · · , Nt, j = 1, 2, · · · , Nr.

The variances of the L paths are normalized such thatPL−1_l=0 δ2

l = 1. From Eqs.(2.5),

the frequency response of the channel represents the pair (i, j) is given by

Hj,i(f ) = L−1 X l=0 αj,i(l)e−j 2πf τl, j = √ −1. (2.6)

We assume that the MIMO channel is spatially uncorrelated, i.e., all the αj, i(l)’s

are statistically independent. The space frequency codeword can be expressed as an N t × K matrix C =             c1(0) c1(1) . . . c1(K) c2(0) c2(1) . . . c2(K) ... ... . .. ... cNt(0) cNt(1) . . . cNt(K)             (2.7)

where ci(k) denotes the channel symbol transmitted over kth subcarrier by transmit

antenna i. Each space frequency codeword is assumed to satisfy the energy constraint

of E[kCk2

F] = NtK, where kCkF is the Frobenius norm 1 of C.

The OFDM transmitter applies a K-point IFFT (inverse fast Fourier transform) to each row of the matrix C. After appending a cyclic prefix (CP), the OFDM symbol corresponding to the ith row of C is transmitted by the ith transmit antenna. At the receiver, after matched filtering, removing the CP, and applying FFT, the received

1

The matrix norm of an m × n matrix A is defined as the square root of the sum of the absolute squares of its elements.

signal at the kth subcarrier at the jth receive antenna is given by yj(k) = r Es Nt Nt X i=1 Hj, i(k)ci(k) + nj(k). (2.8) where Hj,i(k) = L−1 X l=0 αj, i(l)e−j 2πk∆f τl. (2.9)

is the channel frequency response at the nth subcarrier between the ith transmit antenna and the jth receive antenna, ∆f = 1/T is the subcarrier separation in the frequency domain, and T is the OFDM symbol period. Let

Λj,i =  αj, i(1) αj, i(2) · · · αj, i(L) H (2.10) ωk =  e−j2πk∆f τ1 _e−j2πk∆f τ2 _{· · · e}−j2πk∆f τL T (2.11)

Eqs.(2.9) can be written as

Hj,i(k) = (Λj,i)Hωk (2.12)

We assume that the CSI (channel state information), Hj,i(k), is known at the

receiver. In Eqs.(2.8), nj(k) denotes the additive complex Gaussian noise with zero

mean and unit variance at the kth subcarrier at the jth receive antenna. The noise

samples nj(k)’s are assumed to be uncorrelated for different j and k. Using the factor

p

and is independent of the number of transmit antennas.

2.2.2

Design Criteria

Consider the maximum likelihood decoding [15, 16] of the space frequency code by

ˆ C = arg min ˆ C Nr X j=1 K X k=1     yj(k) − Nt X i=1 Hj, i(k)ci(k)      2 (2.13)

where the minimization is performed over all possible space frequency codewords. Let d2 H(C, ˆC) be described as d2H(C, ˆC) = Nr X j=1 K X k=1      Nt X i=1 Hj, i(k)[ci(k) − ˆci(k)]      2 (2.14) = Nr X j=1 K X k=1      Nt X i=1 (Λj,i)Hωk[ci(k) − ˆci(k)]      2 (2.15) = Nr X j=1 K X k=1 | ΓjWkek|2 (2.16)

where Eqs.(2.15) is derived from Eqs.(2.12) and in Eqs.(2.16), the new matrix Γj,Wk

and ek are shown below

Γj =  (Λj,1)H (Λj,2)H · · · (Λj,Nt) H  1×LNt Wk =             wk 0 · · · 0 0 wk · · · 0 ... ... ... ... 0 0 _{· · · w}k             LNt×Nt ek =             c1(k) − ˆc1(k) c2(k) − ˆc2(k) ... cNt(k) − ˆcNt(k)             Nt×1 (2.17)

Assuming that perfect CSI is available at the receiver, the pairwise error probability

of the transmitted codeword C and the erroneously decoded codeword ˆC conditioned

on a fixed H is given by P (C, ˆ_{C|H) ≤ exp}  −d2H(C, ˆC) Es 4N0  (2.18)

where Es is the average symbol energy, N0 is the one-sided power spectral density of

Eqs.(2.16) can be written as d2H(C, ˆC) = Nr X j=1 K X k=1 ΓjWkekeHkW H k Γ H j
= Nr X j=1 Γj K X k=1 WkekeHkW H k ! ΓHj = Nr X j=1 ΓjDH(C, ˆC)ΓHj (2.19)

where DH(C, ˆC) is an LNt × LNt matrix given by

DH(C, ˆC) = K X k=1 WkekeHkW H k (2.20)

Note that the matrix DH(C, ˆC) is concerned about the codeword difference and the

power delay profile of the channels. Denote the rank of DH(C, ˆC) by γ. In Eqs.(2.20),

we know rank(ekeHk) = 1 if and only if ekis not a zero vector, otherwise, rank(ekeHk ) =

0. Assume the number of nonzero vector in e1, e2, . . . , eK is bγ. Thus, DH(C, ˆC) is the

sum of bγ rank one matrices. It can be derived that

rank(DH(C, ˆC)) = γ ≤ min(bγ, LNt) ≤ min(K, LNt). (2.21)

since bγ ≤ K. For the nonnegative definite Hermitian matrix DH(C, ˆC), its eigenvalues

can be ordered as

By averaging Eqs.(2.18) with respect to the channel H, the pairwise error probability can be derived from [17] as

P (C, ˆ_{C) ≤} γ Y j=1 λj !−Nr Es 4N0 −γNr (2.23)

Thus, the maximum achievable diversity is at most min(KNr, LNtNr). As a

conse-quence, we can formulate the performance criteria as follows

• Diversity (rank) criterion: The minimum rank of DH(C, ˆC), i.e., γ, over all

pairs of distinct signals C and ˆC should be as large as possible.

• Product criterion: The minimum value of the product Qγj=1λj over all pairs of

LDPC Coded Alamouti Scheme for

MIMO OFDM

In the last chapter, we are aware that to achieve good error performance of a

space-frequency code, we need to maximize γ, the rank of D(C, ˆC) , which is determined

by ˆγ and LNt, where gamma is the column distance of the codeword differenceˆ

matrix C − ˆC and L is the number of multiple delayed paths. In the following,

we investigate a space-frequency code with Nt = 2, which is the concatenation of a

binary LDPC code , signal mapper and the Alamouti space-time code. The reason for using this concatenation is that (i) the binary LDPC code has large binary Hamming distance which can still yield a significant amount of column distance; (ii) the full rank characteristics of Alamouti space-time coding will enhance the column distance

of the concatenated coding.

3.1

LDPC Codes

Low-density parity-check (LDPC) codes were originally proposed in 1962 by Robert Gallager[18]. LDPC is a linear block codes with parity check matrices H and generator matrices G. The parity check matrix H is N × K and the generator matrix G is (N − K) × N, such that HG = 0. Suppose that we use an M × N matrix H which

has weight wc in each column and weight wr in each row. The constructed LDPC

is call a regular LDPC, denote as a (wc, wr, N ) code. The associated rate of this

regular LDPC code is R = 1 − wc/wr. Gallager showed that the minimum distance

of a regular LDPC code increases linearly with N provided that wc≥ 3. The parity

check matrix H needs not be regular. That means we can consider LDPC codes with varying column weights. In case of very large code lengths, there exist irregular LPDC codes with error performances superior to regular LDPC codes of similar code lengths. In most LDPC codes, N is a large number (at least several hundreds) while

wc is usually less than 10, so the density of 1s in H is quite low. That is reason for

the name of low density parity check codes.

Since H is sparse, it can be represented by the lists of its nonzero locations. The mth column of H represents the mth parity check, 1 ≤ m ≤ M and the nth column of H represents the nth code bit of a codeword, 1 ≤ n ≤ N. Hence, code bits can be

indexed by n (e.g. cn) and the parity checks can be indexed by m (e.g.zm). The set

of code bits that participate in the parity check zm (i.e. the nonzero elements in mth

row of H) is denoted

Nm = {n : Hmn = 1}. (3.1)

Thus we can write mth parity check as

zm =

X

n∈Nm

cn. (3.2)

The set of code bits that participate in the parity check zm except for the code bit n

is denoted

Nm,n = Nm\ n. (3.3)

Let |S| denote the size of a set S. We see that |Nm| is the number of nonzero elements

in mth row of H or the number of code bits that participate in the mth parity check.

Similarly, The set of parity checks in which bit cn participates (i.e. the nonzero

elements of the nth column of H) is denoted

Mn = {m : Hmn= 1}. (3.4)

that check on the nth code bit. Let

Mn,m = Mn\ m. (3.5)

be the set of parity checks in which code bit cnparticipates except for check m. LDPC

can be regarded as the concatenation of repeatition codes (the number of repetition is the number of checks for each code bit) and signle parity check codes (each check checks on several code bits). Then, iterative decoding between the repetition codes and single parity check codes can be implemented as follows.

3.2

Alamouti Space-Time Code

Alamouti space-time code is a simple design to transmit two orthogonal sequences respectively through the two transmit antennas in the space-time coding system with two transmit antennas. With this, full diversity, i.e., two, can be achieved, A block diagram of the Alamouti space-time coding is shown in Fig.(3.1).



₋



= 











Algorithm 1 Iterative Log Likelihood Decoding Algorithm for Binary LDPC Codes Input:

Parity check matrix HM ×N, the maximum number of iterations L, and vector of the

channel value Lc (Lc is the LLR value).

Initialization:

Set ηm,n[0] = 0 for all (m, n) with H(m, n) = 1.

Set λ[0]n = Lc[n].

Set the loop counter l = 1.

Check node update: For each (m, n) with H(m, n) = 1: Compute

η[l]m,n = −2 tanh −1   Y j∈Nm,n tanh(−λ [l−1] j − η [l−1] m,j 2 )   (3.6)

Bit node update: For m = 1, 2, . . . , N : Compute

λ[l]_n = Lc+

X

m∈Mn

η[l]_m,n (3.7)

Hard Decision: Set ˆcn= 1 if λ

[l]

n > 0, else set ˆcn= 0.

If H ˆc = 0, then Stop.

Else if iterations< L, go to Check node update. Else declare a decoding failure and Stop.

The information bits are first modulated using an 2M_{-ary modulation scheme.}

The encoder then takes a block of two modulated symbols s1 and s2 as input in each

encoding operation and send its output to the transmit antennas according to the code matrix[5], S =     s1 −s∗2 s2 s∗1     (3.8)

In Eqs.(3.8), the first column represents the two symbols transmitted through the two tranmist antennas in first transmission period and the second column represents the two symbols transmitted through the two tranmist antennas in the second trans-mission period. The first row corresponds to the symbols transmitted through the first antenna and the second row corresponds to the symbols transmitted through the second antenna.

Consider the Alamouti scheme with two transmit antennas and two receive an-tennas in flat fading channel. We can rewrite Eqs.(2.2) in chapter 2.1 as

    r11 r12 r21 r22     =     h11 h12 h21 h22         s1 −s∗2 s2 s∗1     +     n11 n12 n21 n22     =     h11s1+ h12s2 + n11 −h11s∗2+ h12s∗1+ n12 h21s1+ h22s2 + n21 −h21s∗2+ h22s∗1+ n22     (3.9)

the additive complex Gaussian noise with zero mean and variance σ2_.

The receiver constructs two decision statistics based on the linear combination of

the received signals. The decision statistics, denoted by es1 and es2, are given by

es1 = h∗11r11+ h12r12∗ + h ∗ 21r21+ h22r22∗ = kHk2Fs1+ h∗11n11+ h12n∗12+ h ∗ 21n21+ h22n∗22 = kHk2Fs1+ n (3.10) es2 = h∗12r11− h11r21∗ + h ∗ 22r12− h21r∗22 = kHk2Fs2+ h∗12n11− h11n∗21+ h ∗ 22n12− h21n∗22 = kHk2Fs2+ n (3.11)

where n is the additive complex Gaussian noise with zero mean and variance kHk2

Fσ2,

kHk2

F is the Frobenius norm of H.

From Eqs.(3.10) and Eqs.(3.11), we observe that es1 is only concerned about s1

but independent of s2. Similarly, es2 is only concerned about s2 but independent of

s1. Hence we can simplify the Alamouti MIMO model to two independent system,

which is shown as Fig.(3.2).

=

Figure 3.2: A simply model of the Alamouti space-time codes.

signal modulation constellation $ to minimize the Euclidean Distance

ˆ s1 = arg min ˆ s1∈$ d2_(es1, ˆs1). ˆ s2 = arg min ˆ s2∈$ d2_(es2, ˆs2). (3.12)

In case that the Alamouti space-time coding is only a part of a concatenated coding system, the soft-in-soft-out (SISO) decoding of Alamouti space-time coding is de-sired. We can use the demapper formula for modulation as in [19]. For an arbitrary

number of M modulated bits b0,··· , M −1 per symbol es (es1 or es2) we obtain the LLR

(log likelihood ratio) of a posteriori probability of bit bk as

L(bk | es) = La(bk) + Le(bk) = La(bk) + ln P2M −1₋₁ i=0 exp(−kes − kHk 2 F · map ([(ci)1: k−1 1 (ci)k: M −1])k2 kHk2F σ2 ) exp(ciLa) P2M −1₋₁ i=0 exp(−kes − kHk 2 F · map ([(ci)1: k−1 0 (ci)k: M −1])k2 kHk2F σ2 ) exp(ciLa) (3.13)

where map(·) denotes the modulation of information bits from the signal modulation

constellation $, [ci] is a row vector having the values 0 or 1 according to the binary

decomposition1 _{of i, and (c}

i)a: b denotes the part of the vector [ci] consisting of the ath

element to bth element, La(bk) and Le(bk) is are the LLR of the a priori probability

of bit bk and the extrinsic LLR value of bit bk respectively. Using the demapper,

Alamouti decoding can pass the soft value Le(bk) instead of hard value to other

decoder in the concatenated coding system for further process.

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00 E b / N o ( d B ) B E R A l a m o u t i s c h e m e A l a m o u t i 2 x 2 A l a m o u t i 2 x 1

Figure 3.3: Alamouti space-time codes on flat fading channel.

Fig.(3.3) shows the bit error rate (BER) performance of Alamouti space-time code using coherent QPSK modulation.

1 If we set i = 21 = 1 · 20 + 0 · 21 + 1 · 22 + 0 · 23 + 1 · 24

, the decomposition of i is a row vector [ci] = [c1c2c3c4c5] = [1 0 1 0 1]

3.3

The Space-Frequency Code Under

Investiga-tion

Figure 3.4: A block diagram of the space-frequency code under investigation.

By studying [20, 21, 22, 23] and Chapter 2, we believe that LDPC coded Alamouti Scheme is a good candidate to meet the design criteria shown in Chapter 2. The transmitter and receiver structure of an LDPC coded Alamouti Scheme for MIMO OFDM systems is illustrated in Fig.(3.4). We assume that the receiver has perfect channel state information (CSI).

In Fig.(3.4), the 2N M R information bits are first encoded by a rate R LDPC encoder into 2N M coded bits and then the binary LDPC coded bits are modulated

into 2N 2M_PSK2_{symbols. We split these 2N symbols into 2 streams, and each stream}

has N symbols. The N symbols of each stream are transmitted from one transmit antenna which has K subcarrier over an OFDM slot. Usually, we set N < K in order to reserve subcarriers for side information (SI) or some other purposes. We have two

streams denoted Ca and Cb which will be transmitted trough two transmit antennas

respectively. The Alamouti encoder converts these two streams into a space frequency

codeword represented by [C1, C2], where C1 and C2 are described as

C1 =     Ca Cb     =     c1(0) . . . c1(N − 1) c1(N ) c1(N + 1) . . . c1(K) c2(0) . . . c2(N − 1) c2(N ) c2(N + 1) . . . c2(K)    (3.14) C2 =     −C∗ b C∗ 1     =     −c∗ 2(0) ... −c∗2(N −1) −c∗2(N ) −c∗2(N +1) . . . −c∗2(K) c∗ 1(0) . . . c∗1(N − 1) c∗1(N ) c∗1(N + 1) . . . c∗1(K)    (3.15)

each ci(j) is an 2MPSK symbol, and c1(N + 1) = . . . = c1(K) = c2(N + 1) = . . . =

c2(K) = 0. The symbols represented by C1 is transmitted in the first OFDM time

slot and the symbols represented C2 is transmitted in the following OFDM time slot.

The transmitter applies an K-point IFFT to each row of the matrix Ci and then

appends a cyclic prefix (CP), which is then used for transmission.

It is assumed that the fading process remains static during two consecutive OFDM time slots and the fading at every two consecutive OFDM time slots is independent

2

of any other two consecutive OFDM time slots.

At the receiver, we have receives signals from two receive antennas. After matched filtering and sampling, the FFT is applied to the discrete-time signal to obtain

Y1 =     y1 1(0) y11(1) . . . y11(K) y1 2(0) y21(1) . . . y21(K)     Y2 =     y2 1(0) y12(1) . . . y12(K) y2 2(0) y22(1) . . . y22(K)     (3.16) where yi

k denotes the received signal at the kth subcarriers for ith OFDM time slot,

i = 1, 2, and yi

k can be obtained from Eqs.(2.8). The decoding consists two stages,

i.e., the soft Alamouti combing and the soft LDPC decoder and the so-called extrinsic information passed from first stage to second.

For the first stage of Alamouti combing, Alamouti decoder takes y1

1(k), y21(k),

y2

1(k) and y22(k) to a matrix as in Eqs.(3.9), which can be written as

    r11 r12 r21 r22     =     y1 1(k) y21(k) y1 2(k) y22(k)     (3.17)

and Alamouti soft decoding can be obtained from equations as Eqs.(3.10), Eqs.(3.11)

and Eqs.(3.13) for k = 0, 1, . . . , N − 1. For each k, there is a pair of symbols (es1, es2).

LLR value corresponds to an LDPC coded bit. A total of 2N M LLR values are used

as the vector Lc in Algorithm 1. After LDPC decoding by using Algorithm 1, we

obtain 2N M hard value (1 or 0) for coded bits. Then, 2N M R information bits are detected. The LDPC decoding can produce soft output, which can be fed back to the demapper of Alamouti coding for outer iterations (other than the inner iterations inside the LDPC decoding operation). Simulation shows that such outer iterations will be significantly reduce the BER. Hence, in the rest of this report, we only consider the decoding without outer iterations.

3.4

Simulation Results

In this section, we provide computer simulation results to examine the performance of the LDPC-coded Alamouti scheme. The characteristics of the fading channels are described in Section 2.2.1. In the following simulations, the available bandwidth is 1MHz and the number of subcarriers is K = 256. Thus, the OFDM word duration is T = 256µs without the cyclic prefix. We set the length of cyclic prefix to 5µs to combat the effect of inter-symbol-interference, since the delay is no more than 5µs.

We simulated the space frequency codes with different power delay profiles: (i) a two-ray equal power delay profile and (ii) COST207[24] typical urban six-ray power delay profile. The subcarrier path gains are generated according to Eqs.(2.9), inde-pendently for different transmit and receive antennas. The power delay profile of the

0 1 2 3 4 5 0 0 . 1 0 . 2 0 . 3 0 . 4 T i m e D e l a y ( u s ) Fr ac tio n of P ow er P o w e r D e la y P r o fil e L = 6 0 1 2 3 4 5 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 T i m e D e l a y ( u s ) Fr ac tio n of P ow er P o w e r D e la y P r o fi le L = 2

Figure 3.5: Power delay profile of six paths and two paths.

channel is shown in Fig.(3.5) and Table(3.1). All the LDPC codes used in simulation

Six paths Two paths

Delay Fractional Doppler Delay Fractional Doppler

(us) Power Category (us) Power Category

0.0 0.189 CLASS 0.0 0.500 CLASS 0.2 0.379 CLASS 5.0 0.500 CLASS 0.5 0.239 CLASS 1.6 0.095 GAUS1 2.3 0.061 GAUS1 5.0 0.037 GAUS1

Table 3.1: Numeric power delay profile of six paths and two paths.

are regular LDPC codes with column weight wc = 3 in the parity-check matrix and

with appropriate block lengths and code rates. The modulation under consideration are QPSK or 8PSK constellation respectively. Simulation results are shown in terms

of the information bit-error rate (BER) versus Eb/N0. The simulation MIMO system

- 1 0 1 2 3 4 5 6 7 8 9 1 0- 4 1 0- 3 1 0- 2 1 0- 1 E b / N o ( d B ) B E R L D P C ( R a t e = . 5 , c o d e le n g t h = 1 0 0 8 ) + A l a m o u t i Q P S K 2 x 2 + O F D M ( S u b c a r r ie r = 2 5 6 ) N o L D P C L = 1 L = 2 L = 6

Figure 3.6: BER of considered SF codes, Nt = Nr = 2, QPSK modulation

LDPC codes is 30.

In order to span LDPC coded bits in an OFDM word, the LDPC code length varies with modulation. That is, the LDPC code lengths are 1008 and 1512 with respect to QPSK and 8PSK respectively. Both cases have the same number of modulation symbols, which is 1008/2 = 1512/3 = 504. The 504 symbols are transmitted by two transmit antennas. Hence, there are 252 modulated symbols to be transmitted by each antenna. Thus, 252 of the 256 subcarriers will be used to represent the 252 symbols, while 4 of the 256 subcarriers are free subcarriers which can be used to carry

side information or some other purpose. When the LDPC decoder receive (Y1, Y2),

1 2 3 4 5 6 7 8 9 1 0- 4 1 0- 3 1 0- 2 1 0- 1 E b / N o ( d B ) B E R L D P C ( R a t e = . 6 6 7 , c o d e l e n g t h = 1 5 1 2 ) + A l a m o u t i 8 P S K 2 x 2 + O F D M ( S u b c a r r i e r = 2 5 6 ) N o L D P C L = 1 L = 2 L = 6

Figure 3.7: BER of considered SF codes, Nt = Nr = 2, 8PSK modulation

2 4 6 8 1 0 1 2 1 4 1 6 1 8 1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00 E b / N o ( d B ) B E R L D P C ( R a t e = . 5 , c o d e l e n g t h = 1 0 0 8 ) + A l a m o u t i Q P S K 2 x 1 + O F D M ( S u b c a r r i e r = 2 5 6 ) N o L D P C L = 1 L = 2 L = 6

2 4 6 8 1 0 1 2 1 4 1 6 1 8 1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00 E b / N o ( d B ) B E R L D P C ( R a t e = . 6 6 7 , c o d e le n g t h = 1 5 1 2 ) + A la m o u t i 8 P S K 2 x 1 + O F D M ( S u b c a r r i e r = 2 5 6 ) N o L D P C L = 1 L = 2 L = 6

Figure 3.9: BER of considered SF codes, Nt = Nr = 1, 8PSK modulation

Fig.(3.6), Fig.(3.7), Fig.(3.8) and Fig.(3.9) depict the performances of the consid-ered space-frequency (SF) codes under the condition of various power delay profiles and various modulation methods. We use the BPSK modulation for the case of not using LDPC and QPSK for the case of using LDPC codes in Fig.(3.6) and Fig.(3.8). Also, we use the QPSK modulation for the case of not using LDPC and 8PSK for the case of using LDPC codes in Fig.(3.7) and Fig.(3.9). Thus, the transmission rate

of both cases are the same. Clearly, the performance of the SF codes without LDPC3

codes, i.e., pure Alamouti coding is much worse than the case of using LDPC codes.

3

The performance of the case without LDPC is independent of the condition of the existence or nonexistence of delay paths.

PAPR Reduction in MIMO OFDM

Systems

In OFDM systems, peak-to-average power ratio (PAPR) is an important issue for the transmitter because a system with a large PAPR requires the linear power amplifier with a large dynamic range. The problem of PAPR is due to the fact that in OFDM systems, the summation of various signals of many subcarriers will lead to the prob-able occurrence of high peak power as compared to the average power. For MIMO OFDM, the problem of PAPR is similar to the conventional OFDM system. The techniques used for mitigating the PAPR effect in the conventional OFDM system can also be applied to the MIMO OFDM systems. However, the usage of multiple transmit antennas may somewhat deepen the problem of PAPR while the usage of

multiple transmit antennas may provide additional room for executing the operation of PAPR reduction.

By now, many techniques have been proposed for relieving the PAPR problem in the OFDM systems, such as Amplitude Clipping, Coding, Selective Mapping (SLM) [11, 12] and Active Constellation Extension (ACE) [13]. In particular, there is a tech-nique, called the “cross-antenna rotation and inversion”[14] (CARI) which addresses on PAPR reduction for the MIMO-OFDM systems. Our research team has recently devised a there time-shifted PAPR reduction technique for the ordinary OFDM sys-tem that is a modified form of selective mapping technique which has the advantage of low complexity and will appear in [25]. We find that after some modification, the technique is very suitable for the MIMO-OFDM system. We call this new PAPR re-duction technique for MIMO-OFDM systems “Time-domain circular shift (TDCS)” technique. We will compare the cross-antenna rotation and inversion[14] (CARI) with TDCS. The numeric result will show the advantage of TDCS in Chapter 4.4.

4.1

Basics of PAPR

After the IFFT operation, the resulting complex baseband OFDM signal is

s(t) = _√1

K

K−1_X k=0

where T is the duration of an OFDM symbol and Xk is the data symbol considered

in the frequency domain. The PAPR of this OFDM system can be defined as 1. DEFINITION 1 For any baseband OFDM signal, the PAPR of the OFDM symbol can be expressed as

PAP R = max0≤t<T |s(t)|

2

E{ |s(t)|2_} . (4.2)

where E{x} denotes the expectation function of x.

In case of the discrete-time OFDM system, enough oversampling on the OFDM symbol is required to preserve the accurate PAPR value [26]. Suppose that oversam-pling factor is J. The discrete-time OFDM signal can be written as

sn = 1 √ K K−1_X k=0 Xkexpj2πkn/JK , n = 0, · · · , JK − 1. (4.3)

Thus, the PAPR of the discrete-time OFDM signal is shown in Definition 2. DEFINITION 2 The PAPR of the OFDM symbol in discrete-time signals is

PAP R = max0≤n<JK|sn| 2 1 JK PJK−1 n=0 |sn|2 . (4.4)

The case of J = 1 is called Nyquist rate sampling or critical sampling. The case of J > 1 is called oversampling. For J = 4, the peak of continuous-time value can be estimated sufficiently.

The complementary cumulative distribution function (CCDF) of the PAPR can be used to evaluate the capability of PAPR reduction. Here, CCDF is defined as the

probability of the occurrence of PAPR exceeding a given threshold P AP R0, which is

CCDF = Pr(P AP R > P AP R0). (4.5)

Consider MIMO OFDM systems with Nt transmit antennas and Nr receive

an-tennas. Denote P AP Rp as the PAPR of the pth transmit antenna. Then for multiple

transmit antennas, the PAPR is defined in Definition 3.

DEFINITION 3 The PAPR of the MIMO OFDM symbol in discrete-time signals is as below PAP R = max(P AP R1, P AP R2, . . . , P AP RNt). (4.6) PAP Rp = max0≤n<JK|spn|2 1 JK PJK−1 n=0 |s p n|2 , p = 1, 2, . . . , Nt. (4.7)

where Nt is transmit antenna, P AP Rp is the PAPR of transmit antenna p, and spn

is the discrete OFDM signal of transmit antenna p.

In case that selective mapping technique is used for PAPR reduction. We assume that the number of candidates for the selective mapping operation scheme is Q. The selector choose the candidate with the lowest P AP R. In other words, a minimax

Figure 4.1: Partition of the Ca and Cb.

criterion is used, which can be described as

(s₁, . . . , sNt) = arg min_s

1, ... , sNt

(P AP R) (4.8)

where (s₁, . . . , s_Nt) are transmitted OFDM symbols with lowest PAPR, and by Eqs.(4.7)

sp = (sp,0, . . . , sp,JK) which is time-domain signal for transmit antenna p.

4.2

CARI Scheme

We now describe the CARI scheme[14] with Nt = 2 based on the MIMO OFDM

investigated in Section (3.3). We consider the space-frequency code described by

Eqs.(3.14) and Eqs.(3.15). It is easy to show that Ci and ±Ci∗(i = a, b) have the

same PAPR properties. Therefore, for the LDPC coded Alamouti scheme, the PAPR

reduction needs to be done only for C1 in Eqs.(3.14).

as Ca =  Ca1 Ca2 · · · CaW  (4.9) Cb =  Cb1 Cb2 · · · CbW  (4.10)

Each subblock has K/W elements in it as shown in Fig.(4.1). Now we perform anti-clockwise rotation and inversion across 2 antennas, which is shown in Fig.(4.2) and

Table(4.1). That is, there are four kinds of candidates for each pair of Cai and Cbi.

With W subblocks and 2 antennas, 4W _{candidates can be obtained. In case that}

Operation (i = 0, 1, . . . , W )

I Cai and Cbi are unchanged

II Cai and Cbi are swapped

III Cai and Cbi are inverted, i.e.,−Cai and −Cbi

IV Cai and Cbi are swapped and inverted, i.e., −Cai and −Cbi are swapped

Table 4.1: Four operations of each subblock.

M is large, this method will be impractical, since we have to search a large number of candidates to obtain the best PAPR. Hence, a suboptimal method is considered in [14], which is called Successive Suboptimal CARI (SS-CARI) scheme. The block

diagram is shown in Fig.(4.3). At the beginning, the operations for Ca1 and Cb1 in

Table(4.1) are executed. Then, PAPR values of the four candidates are calculated. For example, if the operation II has the lowest PAPR, the fist subblock is fixed as

Figure 4.2: Cross-Antenna Rotation and Inversion (CARI) scheme.

values of the four candidates are calculated. Again, the second subblock is fixed with the lowest PAPR. By proceeding this to all W subblocks, a total of 4W candidates

can be obtained which is less than 4W _{candidates in CARI scheme. Note that S = 2W}

bits for side information are still needed for the SS-CARI.

In Section (4.3), we will propose a novel method to reduce the PAPR of MIMO-OFDM in time domain. To compare to the proposed time-domain scheme, we set the partition number to be W = Q/4 for SS-CARI, where Q = 8, 16, . . . is the total number of candidates. In Fig.(4.3), we observe that the transmitter needs 2Q IFFT computations in order to obtain Q candidates. The proposed method described in Section (4.3) will need only two IFFT computations.

Figure 4.3: Block diagram of SS-CARI scheme.

Figure 4.4: Successive Suboptimal Cross-Antenna Rotation and Inversion (SS-CARI) scheme.

4.3

Time-Domain Circular Shift Scheme

Now we will show our main result of this research that is Time-Domain Circular Shift Scheme (TDCS) for PAPR reduction in the MIMO-OFDM system.

4.3.1

Time-Domain Circular Shift Scheme

The time-domain circular shift (TDCS), which produces candidates in time-domain instead of in frequency domain, is depicted in Fig.(4.5). The time-domain OFDM

 

 ₋ 

 

Figure 4.5: Time-domain circular shift (TDCS) scheme.

symbols for two transmit antennas are denoted as

Sa =  Sa1 Sa2 · · · Sa, JK  (4.11) Sb =  Sb1 Sb2 · · · Sb, JK  (4.12)

We apply a circular shift1 _{with parameter τ}

i on Sb (i = 1, . . . , Q). We denote the

circular-shifted signals for parameter τi by bSbi, where

b Si b =  b Si b1 Sbb2i · · · bSb, JKi  (4.13)

Then, we multiply the combination of Sa and bSbi by a unitary matrix U to obtain a

1

A circular shift is a permutation of the entries in a tuple where the last element becomes the first element and all the other elements are shifted, or where the first element becomes the last element and all the other are shifted.

candidate for parameter τi. The ith candidate will be Vi =     Vi a Vi b     = U     Sa1 Sa2 · · · Sa, JK b Si b1 Sbb2i · · · Sbb, JKi     (4.14) where Vi

a and Vbiare the OFDM symbols for the first and the second transmit antennas

respectively and, U = √1 2     1 1 1 −1     (4.15)

We have one candidate for each shift. Hence there are Q candidates in total. We use minimax criterion to find the lowest PAPR and transmitted OFDM symbols according to Eqs.(4.6) and Eqs.(4.8).

From Fig.(4.5) we observe that only two IFFT computations are needed and

the number of bits for side information is S = log₂Q. Thus, TDCS can reduce

the implementation complexity as compared to SS-CARI and is more applicable to practical systems.

4.4

Simulation Results

In this section, we provide the simulation results so that we can compare the perfor-mances of SS-CARI and the proposed TDCS regarding CCDF and BER (bit error rates).

4.4.1

CCDF Performance

In our simulation, 105 _{random OFDM sequences are generated to obtain the CCDF.}

We use Nt = 2 and K = 128 subcarriers. The modulation is QPSK. Oversampling

factor J is set to 4..

Q = 8 Q = 16

PAPR Side IFFT Side IFFT

Scheme Information Number Information Number

S bits ξ S bits ξ

TDCS 3 2 4 2

SS-CARI 4 16 8 32

Table 4.2: Comparison of information bit S and the number of IFFT needed ξ.

Fig.(4.6) shows the CCDF of PAPR for the TDCS and SS-CARI scheme using Q = 8 and Q = 16 candidates respectively. The proposed TDCS scheme for Q = 8 achieves better performance than the SS-CARI scheme for W = 2. The TDCS and SS-CARI schemes for Q = 16 perform almost the same. We list the number of needed side information bits and the number of needed IFFT computations for both TDCS and SS-CARI in Table(4.2). We can observe that both the side information S and the number of IFFT computations needed ξ for TDCS scheme are less than that for SS-CARI scheme.

6 6 . 5 7 7 . 5 8 8 . 5 9 9 . 5 1 0- 3 1 0- 2 1 0- 1 1 00 P A P R 0 ( d B ) P r( P A P R >P A P R 0) S u b c a r r i e r = 1 2 8 , M o d u l a t i o n i s Q P S K T D C S Q = 8 S S - C A R I W = 2 ( Q = 8 ) T D C S Q = 1 6 S S - C A R I W = 4 ( Q = 1 6 )

Figure 4.6: SS-CARI and TDCS scheme for different value of candidates Q.

4.4.2

BER Performance

In many PAPR research works, the effect on BER is neglected. In fact, the effect on

BER may be great in some cases. In our simulation, we use Nt = 2 and K = 256

subcarriers. The available bandwidth is 1MHz and the subcarrier K = 256. We consider the channel with power delay profiles: COST207[24] typical urban six-ray power delay profile. The subcarrier path gains are generated according to Eqs.(2.9), independently for different transmit and receive antennas. The oversampling factor is J = 4. All the other parameters are just the same as what we use in Section (3.4). Fig.(4.7) and Fig.(4.8) show the performance of SS-CARI and TDCS using the

- 1 0 1 2 3 1 0- 4 1 0- 3 1 0- 2 1 0- 1 E b / N o ( d B ) B E R S S - C A R I( W = 4 ) C l i p r a t i o = 6 C l i p r a t i o = 7 C l i p r a t i o = 8 C l i p r a t i o = 9 N o C l i p p i n g

Figure 4.7: SS-CARI BER performance for W = 4.

space frequency code investigated in Section (3.3). In the simulation, we assume that CSI and side information can be recovered correctly by the receiver. Take 7dB clipping ratio case as example. We can observe that the BER performance of both

schemes are around 10−4 _{at E}

b/N0 = 3dB.

In Eqs.(4.14), multiplying unitary matrix U to the left side of a space-frequency codeoword will not effect the BER performance. Hence the U can be designed by any unitary matrix other than that shown in Eqs.(4.15)

- 1 - 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 1 0- 4 1 0- 3 1 0- 2 1 0- 1 E b / N o ( d B ) B E R T i m e - d o m a i n s c h e m e ( Q = 1 6 ) C l i p r a t io = 6 C l i p r a t io = 7 C l i p r a t io = 8 C l i p r a t io = 9 N o C li p p i n g

Figure 4.8: TDCS BER performance for Q = 16. Side Information Embedded

In some situations, the side information is embedded into the system. There are many methods to embed the side information into the system. A major concern is that the side information must be well protected. Otherwise, serious error propagation will occur. Here, we consider a simple method which is obtained by inserting the side information into the zero terms of Eqs.(3.14) and Eqs.(3.15) and each reserved subcarrier contains one side information bit. In fact, we can insert more than one bit to one subcarrier if the system needs a large number of the side information bits. In order to protect the side information bit, the power of side information signals is transmitted four times of original signals. The performance of SS-CARI scheme

remain similar to TDCS scheme, shown in Fig.(4.9) and Fig.(4.10). Take the 7dB clipping ratio condition as example, we can observe that the BER performance of

both schemes are around 10−4 _{at E}

b/N0 = 3dB. That is, the system suffers no BER

performance degradation by inserting the side information bits in the simple methods described above. - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 1 0- 4 1 0- 3 1 0- 2 1 0- 1 E b / N o ( d B ) B E R S S - C A R I( W = 4 ) C li p r a t i o = 6 C li p r a t i o = 7 C li p r a t i o = 8 C li p r a t i o = 9 N o C l i p p i n g

- 1 0 1 2 3 1 0- 4 1 0- 3 1 0- 2 1 0- 1 E b / N o ( d B ) B E R T i m e - d o m a in s h c e m e ( Q = 1 6 ) C l i p r a t io = 6 C l i p r a t io = 7 C l i p r a t io = 8 C l i p r a t io = 9 N o C li p p in g

CONCLUSIONS AND SELF

EVALUATION

In this three-year project, our goal is to investigate MIMO-OFDM systems so that both low PAPR and error rates can be achieved. In the first year, we investigate a space-frequency code with two transmit antennas that is constructed by the concate-nation of binary LDPC code and the Alamouti space-time coding. The reason for choosing such a design is that this construction can achieve large column distance and full rank of the codeword difference matrix, which will ensure large diversity for combating the multi-path fading MIMO-OFDM channel. Simulation results ver-ify that the construct space-frequency code does perform well in the MIMO-OFDM channel. Based on this efficient space-frequency code, we propose a low complexity

selective-mapping type PAPR reduction technique. In the proposed technique, the candidates are generated in the time-domain instead of the frequency domain. Thus, only two IFFT operations are needed in the proposed technique while for the selec-tive mapping using frequency domain many IFFT operations are needed. In case the number of candidates is not great (no more than 16), the proposed technique can significantly reduce the complexity without sacrificing the PAPR reduction capabil-ity and error rates. Simulation results verify the advantage of the proposed PAPR reduction technique.

Some of the results of this research comes from the PhD thesis of S.K. Deng [25] and the master thesis of Y. H. Lo [27]

As a summary, we have a very significant research result in this first-year term, i.e., the time-domain PAPR reduction technique for the MIMO-OFDM system. The idea is novel and the advantage is obvious in case the number of selective mapping is not large. We believe that this result can be published in prestigious academic conferences and journals.

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