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
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 has K = 256 subcarriers. The LDPC has code rate R = 0.5 and the iterations of
- 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 )
BER
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), an LDPC codeword can be completely decoded.
1 2 3 4 5 6 7 8 9
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
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 )
BER
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.
3The 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
31
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−1X
k=0
Xkexpj2πkt/K , 0 ≤ t ≤ T (4.1)
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−1X
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 |spn|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
s1, ... , sNt(P AP R) (4.8)
where (s1, . . . , sNt) 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).
For CARI, we partition the Ca and Cb into W subblocks of equal sizes, denoted
as
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 shown in 4.4. Next, four operations for Ca2 and Cb2 are performed and the PAPR
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.