Chapter 2 OFDM and MIMO Fundamentals
2.3 Signal Detection Algorithms for MIMO Systems
2.3.2 Suboptimal Algorithms
2.3.2.3 The SQRD Detection Method
Although V-BLAST algorithm achieves good performance, it still costs high computation power. The sorted QR decomposition (SQRD) detection algorithm [10,11] is introduced in the sequel based on QR decomposition. SQRD algorithm costs rather lower computation power with small performance degradation. The method’s operation is listed below.
The SQRD algorithm [10,11] is an extension to the modified Gram-Schmidt procedure by reordering the columns of the channel matrix before each orthogonalization step. The basic idea is to minimize |rk,k| in the order it is computed (from 1 to Nt ) instead of to maximize in the order of detection (from Nt to 1). |rk,k| denotes absolute value of the element at k-th row and k-th column in R matrix. This is motivated because the signal detected last has effects on only few other layers through error propagation and therefore has small SNR’s. Since r1,1 is simply the norm of the column vector h1, the first step in the SQRD algorithm is simply to permute the column of H with minimum norm to this position. During the following orthogonalization of the vectors h2, . . . , hNt with respect to the normalized vector h1, the first row of R is obtained. In the second step, r2,2 is determined in a similar fashion from the remaining Nt-1 orthogonalized vectors, et cetera.
BLAST, but in many cases of interest the performance degradation is small compared to the reduced complexity [10,11]
Begin
exchange columns in the first Nr+i-1 rows of Q (2.14.h)
i
Chapter 3
MIMO Application to OFDM
Algorithms introduced for MIMO transmission require frequency-flat fading channels, and it limits its application to narrowband transmissions. For real broadband transmission systems, channel conditions are often frequency-selective fading. A technique alleviating severe effect of frequency-selective fading is demanded. OFDM technique is a good solution for this purpose in wireless transmission owing to its advantages [3,4,5].
3.1 MIMO-OFDM Architecture
According to Section 2.1, OFDM turns frequency-selective fading channel into several frequency-flat fading subchannels, solving the major problem in wideband transmission systems. Boubaker et. al [12] proposed MIMO-OFDM using V-BLAST algorithm to detect transmitted signals on each subcarrier. MIMO-OFDM transceiver architectures are also proposed, as shown in Figure 3.1 and 3.2.
Subchannels are orthogonal to each other in OFDM systems. Hence, in single-input-single-output (SISO) OFDM systems, the received signals are product of channel response and transmitted signal, as shown in equation 2.4. In MIMO systems, signals transmitted from different antenna on a subcarrier simultaneously interfere each other, but signals at different subcarriers are independent. At the receiver antennas, a linear combination of the transmitted signal and channel response on each subcarrier is observed, as equation 2.5 shows. That corresponds to assumptions of MIMO systems. On each subchannel, a space division multiplexing (SDM) like V-BLAST is applied. That is, the task is to recover x from the received signal y and channel state information (CSI) H on each subcarrier.
Figure 3.2 System achitecture of an OFDM and V-BLAST receiver [12]
3.2 Simplified Data Detection Algorithms for MIMO OFDM Systems
On each subchannel, a V-BLAST or SQRD detector is used since we can formulate the problem as equation 2.5. For most practical OFDM systems, there are too many subcarriers to implement a detector for each subcarrier. Fortunately, channel responses of successive subcarriers are often similar. Some simplified algorithms are proposed based on this observation.
3.2.1 Boubaker’s Algorithm [13]
In [13], Boubaker et. al proposed an algorithm based on V-BLAST which divide used subcarriers into groups. Since data detection and channel inversion can be separated to two parts, the same nulling vectors corresponding to a subcarrier of a group is used to
detect data of all the group. That is, the algorithm assumes that the channel conditions on all subcarriers in this group are identical. Therefore, a dramatically reduction of computation cost is obtained. In addition, a new detection algorithm is proposed. A sub-optimal ordering is obtained simply from the first pseudo-inverse by sorting ||(Gl)j ||2 in an ascending order, for j = 1,. . . , M, where G1 = HH. Then, the series of calculations of pseudo-inverse can be replaced by applying Gram-Schmitt Orthogonalization (GSO) once to H, provided that the decoding ordering is known in advance.
3.2.2 Liu’s Algorithm [20]
For good channel conditions, [13] provide a good way for both good performance and low complexity, while for slightly poor channels, this may not be enough. Liu and Yang [20] proposed an algorithm for this situation. They use V-BLAST to determine pseudo inverse matrixes at some subcarrier of a group and then use equation 3.1 to approximate pseudo inverse matrixes at other subcarriers.
(
A−ε)
+≈ A+ +A+εA+ (3.1)With equation 3.1, pseudo inverse at (k+1)-th subchannel is calculated by using pseudo inverse of the k-th subchannel.
+ +
+
+ ≈ + − +
+1) ( ) ( ) ( ( ) ( 1)) ( )
(k H k H k H k H k H k
H (3.2)
where H(k) is channel response of k-th subchannel.
Other pseudo inverse is also calculated.
3.3 Introduction to WWiSE 802.11n Proposal
To examine validity of proposed algorithms in next chapter, WWiSE proposal, a proposal for next generation 802.11n standard, is adopted for simulation. 802.11n standard utilize both MIMO and OFDM techniques for efficient wireless LAN. In the following, brief introduction is described.
WWiSE proposal [21] emphasize backward compatibility with existing installed base, building on experience with interoperability in 802.11g and previous 802.11 amendments which are mainly designed for indoor wireless internet applications. Hence, we review the physical layer of wireless LAN 802.11a [22] system which is based on OFDM technology. The main system parameters of IEEE 802.11a Wireless LAN standard are listed in Table 3.1.
Table 3.1 Parameters and specifications of 802.11a system [22]
Signal bandwidth 20MHz Sample duration 50ns
FFT length 64
Used subcarriers 52 Data subcarriers 48 Pilot subcarriers 4
Symbol period 3.2us (64 samples) Cyclic prefix 0.8us (16 samples) Subcarrier spacing 312.5 kHz
Modulation BPSK, QPSK, 16QAM, 64QAM Channel coding 1/2 convolutional, constrain length 7,
Optional puncturing
Data rate 6, 9, 12, 18, 24, 36, 48, 54 Mbps
In 802.11a standard, a frame is composed of three fields. Figure 3.3 shows the packet format which facilitates synchronization and channel estimation of the receiver. In the preamble field, the preambles are composed of ten repeated short symbols and two repeated long symbols. The total duration of short symbols is 8us and so is that of long symbols. Since the SIGNAL field contains the most important information of the packet, such as frame length and modulation, synchronization and channel estimation must be finished before decoding of the SIGNAL.
Figure 3.3 Frame structure of 802.11a [22]
For the purpose of compatibility with the 802.11 legacy devices, the legacy part
Short Training Symbol Field
• Frame Timing Sync.
• Coarse CFO Sync.
• Symbol Timing Sync.
Long Training Symbol Field
• Fine CFO Sync.
• Channel Estimation.
cyclical delay is adopted in WWiSE proposal, whose format is used for simulation.
Figure 3.4 shows the cyclical delay format in WWiSE. The maximum number of the spatial data streams is four.
Figure 3.4 Cyclical delay format of the preamble in WWiSE [21]
STRN stands for the short training sequence. LTRN represents the long training symbol. GI2 is the guard interval of the long training symbol.
STRN 400 ns cs
STRN
STRN 600 ns cs
GI21 GI2 LTRN
GI23
LTRN 100 ns cs
LTRN 1700 ns cs
GI22
LTRN 1600 ns cs STRN
200 ns cs
Chapter 4
The Proposed Data Detection Algorithms
Since the algorithms to be proposed are for MIMO-OFDM systems to reduce complexity of V-BLAST detection, the technique are also useful to simplify the linear detection and SQRD detection algorithms. In the sequel, some simplification algorithms are introduced based on the designs of [13,20]
4.1 The Simplified Linear Detection Methods
In the linear detection method, a received vector is simply multiplied with pseudo inverse of channel. Direct computation of pseudo inverse is costly. Although pseudo inverse is calculated only once for each subcarrier, an approximation with good performance and low cost is highly required, because it is calculated N times for a symbol.
Based on the concept of [13], we propose two simplified algorithms. The simplified algorithm 1, for each two suncarriers, compute inverse of one subcarrier, and directly
applies it to the data detection of the other subcarrier. Based on the approximation to pseudo inverse in [20], obviously it is also useful in linear detection. Our proposed linear detection algorithm 2 is that for each two suncarriers, we compute the inverse of one subcarrier, while approximate the other using the computed one.
Figure 4.1 Scenario of the proposed linear detection algorithm 1
Begin
4.2 The Simplified V-BLAST Detection Method
Simplified algorithms for V-BLAST detection method are already proposed in [13,20]. [13] assumed successive subcarriers share a channel response so that vectors of a subcarrier is applied to detect signal of other subcarriers.
Figure 4.3 The simplified V-BLAST detection algorithm in [13]
In [20], the simplification, however, miss an important assumption. The subcarrier which is decoded by approximation is assumed to have the same decode order with that of the subcarrier which provides the pseudo inverse. At each step of V-BLAST, a column
of channel matrix is set to zero in order to ignore the effect of the corresponding transmitting antenna. If the pseudo inverse is used to estimate inverse of other subcarriers, the subcarriers have to recognize the nulled columns, which stand for detected signals. In other words, the optimal decoding order is assumed not to change.
Under this assumption, there are some opportunities for further simplification.
The simplified algorithm has identical performance but with lower complexities. Owing to unchanged decoding order, what we need is merely a row of the approximated pseudo inverse which represents the transmitted signal to be detected, rather than the inverse, as shown below.
[
H(k+1,p)+] [
j ≈ H(k,p)++H(k,p)+(H(k,p)−H(k+1,p))H(k,p)+]
j (4.2) where []j denotes the j-th row.[
H(k+1,p)+] [
j ≈ H(k,p)+] [
j+ H(k,p)+(H(k,p)−H(k+1,p))H(k,p)+]
j (4.3) then[
H(k+1,p)+] [
j ≈ H(k,p)+] [
j+ H(k,p)+]
j(H(k,p)−H(k+1,p))H(k,p)+ (4.4) As a result, computation complexity is simplified, but performance is not degraded because the basis approximation remains unchanged.Begin
for(i=1 ; i <= subcarrier number ; i+=2) (4.5.a) calculate weight vectors g according to equation 2.13 iZF (4.5.b)
if algorithm [13] (4.5.c)
i ZF i
ZF g
g+1= (4.5.d)
else //algorithm 2 (4.5.e)
calculate weight vectors g according to equation 4.4 iZF+1 (4.5.f)
end (4.5.g)
end (4.5.k)
4.3 The Simplified SQRD Detection Methods
Like V-BLAST algorithm, transmitted signals are detected one by one in SQRD algorithm. Also, optimal detection order of a subcarrier is assumed to be the same with the subcarrier which provides QR decomposition. Our proposed simplified algorithm 1 for SQRD detection is that for each two subcarriers, we compute QR decomposition of one subcarrier using SQRD algorithms [10,11], and applied it to detect the other subcarrier, too.
Figure 4.5 Scenario of the proposed simplified SQRD detection algorithm 1
Due to simplicity consideration, we assume the channel response of a subcarrier has the same Q matrix and different R matrix with that of neighboring subcarriers.
Assume that
QR
H = (4.6)
is calculated for one subcarrier. According to previous assumption, '
H' QR= (4.7)
is the channel response of the neighboring subcarrier. Therefore, R matrix must be updated by 'R matrix. neighboring subcarrier, respectively. We can simply multiply QH matrix with both sides of equation 4.5.
It is known that each column of Q is orthogonal to each other. That is, I
Equation 4.9 can be re-written as
' ' ' Rx y
QH = (4.12)
Our proposed simplified algorithm 2 for SQRD detection method is that for each
Figure 4.6 Scenario of the proposed simplified SQRD detection algorithm 2
Begin
for(i=1 ; i <= subcarrier number ; i+=2) (4.5.a) calculate Q i R according to [10,11] i (4.5.b)
if algorithm 1 (4.5.c)
i
i Q
Q+1= ,Ri+1=Ri (4.5.d)
else //algorithm 2 (4.5.e)
i
i Q
Q+1= ,Ri+1=
( )
Qi HHi+1 (4.5.f)end (4.5.g)
end (4.5.k)
4.4 Complexity Analysis and Comparison
It is known that the signal detection and calculation of pseudo inverse of a channel are independent. Inverse of a channel only needs to be calculated once when the channel is estimated, and then, as long as the channel does not change, all we need to do is to use the same inverse to detect signal for different data symbols. Accordingly, two parts are separately analyzed.
4.4.1 Number of Complex Multiplications
In these tables, linear denotes direct calculation of pseudo inverse of channel response, APP-Linear denotes the method of using equation 3.1 to estimate pseudo inverse, SQRD denotes direct calculation of QR decomposition according to [10,11], APP-SQRD denotes the method of calculating R matrix according to equation 4.11, ' VBLAST denotes calculation of weight vectors using the technique of [8] and APP-VBLAST corresponds to the calculation based on equation 4.4, which results in less complexity than [20]. All ZAPP algorithms denote the no calculation of pseudo inverse, QR decomposition or weight vectors but adopt the closet neighboring ones.
Table 4.1 Multiplication complexities of various detection algorithms for channel inversion
No. Multiplications Square Root Real Division Nt=4 Nr=6
Linear Nt3+2Nt2Nr 0 0 256
APP 2Nt2Nr 0 0 192
ZAPP 0 0 0 0
VBLAST Nt4+2Nt3Nr+Nt2Nr 0 0 768
APP 2Nt2Nr 0 0 192
ZAPP 0 0 0 0
SQRD Nt2Nr+8NtNr+0.5Nt2+1.5Nt NtNr 2NtNr 302
APP Nt2Nr 0 0 96
ZAPP 0 0 0 0
Table 4.1 shows the required numbers of multiplications for channel inversion. In
Table 4.2 Multiplication complexities of various detection algorithms for data detection
Table 4.2 shows the required numbers of multiplications for data detection. Since signal detection algorithms are the same for the linear detection algorithm, the three conditions have identical complexity, and so are V-BLAST and SQRD. 6 symbols mean the total required numbers of multiplications for 1-symbol channel inversion and data detection for 6 symbols, assuming that the subcarrier channel response is invariant for the duration of 6 symbols. For Nt = 4 , Nr = 6 and 6 symbols, the parameters are chosen for simulation for apparent difference on complexities and BER performance.
4.4.2 Number of Complex Additions
Table 4.3 Addition complexities of various detection algorithms for channel inversion No. additions Nt=4 Nr=6
Table 4.4 Addition complexities of various detection algorithms for data detection No. Multiplications Nt=4 Nr=6 6 symbols
Linear Nt Nr 24 400
APP Nt Nr 24 336
ZAPP Nt Nr 24 144
VBLAST Nt2+ Nt Nr 40 1008
APP Nt2+ Nt Nr 40 432
ZAPP Nt2+ Nt Nr 40 240
SQRD Nt Nr+0.5 Nt2+0.5 Nt 34 368 APP Nt Nr+0.5 Nt2+0.5 Nt 34 300 ZAPP Nt Nr+0.5 Nt2+0.5 Nt 34 204
Table 4.3 and Table 4.4 show the required numbers of addition for channel inversion and data detection, respectively. Owing to the fact that a complex multiplication needs much more computation cost than a complex addition, we think the number of multiplication will dominate computation time.
Chapter 5
Simulation Results
In this chapter, we conduct computer simulations and test the performance of the discussed algorithms in Chapter 3 and 4 by using Matlab program. Those simulations are performed by applying them to WWiSE proposal. Table 5.1 lists the parameter settings og WWiSE in the simulations including frame structure, multi-antenna preambles format, signal bandwidth, subcarrier number, et cetera. Modulation scheme is fixed to QPSK and channel coding is neglected. It is also assumed that channel state information (CSI) is perfectly known during the periods of preambles.
First of all, based on the previously mentioned complexity analysis, simulation time is examined. Then an important part, bit error rate (BER), is simulated. Computation time indicates complexity while BER indicates performance.
Table 5.1 Simulated WWiSE system parameters Signal bandwidth 20MHz
Sample duration 50ns
FFT length 64
Used subcarriers 52 Data subcarriers 48
Symbol period 3.2us (64 samples) Cyclic prefix 0.8us (16 samples) Subcarrier spacing 312.5 kHz
Modulation QPSK Channel Coding No
Transmit antenna 4
Receive antenna 4, 5, or 6 Data symbol 6 symbols
Doppler frequency 150 ( 9m/s at 5GHz )
5.1 Performance – Execution Time
In the following figures, computation time is measured in seconds using Matlab etime functions. Only signal detection is measured and other parts are not, because we are only interested in complexity of detection. 4 transmit and 6 receive antennas are assumed with the theoretically analyzed complexities. In the table, the fractional numbers represent the ratio normalized to the methods of linear, SQRD or V-BLAST, respectively.
Figure 5.2 Computation time of the SQRD detection method and its new simplified methods
Figure 5.3 Computation time of the V-BLAST detection method and its new simplified methods
It shows that the time saving is not apparent, and contradictory to our previous much simplified complexity analysis. There may be some reasons for this. For all the proposed simplified algorithms, we directly compute channel inverse of the representing subcarrier and use it to approximate the other. Take V-BLAST for example, according to Table 4.2 the normalized complexity of the simplified algorithm with respect to the original one is,
5 . 0 71 . 1008 0
* 2
432
1008+ = >
(5.1) We divide the total multiplication numbers of APP-VBLAST by that of the pure
V-BLAST detections. As shown, a complexity saving of more than 0.5 is impossible
because for every two subcarriers, the channel inverse of one subcarrier is directly computed so that there is no saving for this subcarrier. Besides the computer simulation result shows further degradation.
710.79>0. (5.2)
For computer simulations, a program may consist of lots of memory accesses.
Actually, a large memory is essential to run the new algorithms. For example, APP-VBLAST needs pseudo inverse in each operation step for approximation. That is, for the simulated 4 transmit and 6 receive antennas systems, four 6 by 4 pseudo inverse matrixes should be stored in a memory and accessed.
5.2 Performance – Bit Error Rate
In our discussion, correlations between transmit antennas and that between receive antennas are assumed to be independent, and each transmit and receive antenna pair follows the same channel model. In the BER simulations, indoor channel model [23]
is adopted because both 802.11n and 802.11a focus similar on indoor wireless applications, and the simulated channel is generated by a hand-written program using Jake’s model. As shown before, a simulated packet consists of preamble part and 6 data symbols. In our simulation, perfect channel state information (CSI) is adopted.
The first simulated channel, as listed in Table 5.2, is measured in a typical old office environment where partitions are often made of brick. The longest tap has a delay
Table 5.2 Indoor channel model [23] with short delays, office
1 0 0 Rayleigh Classical/Flat
2 36 -5 Rayleigh Classical/Flat
3 84 -13 Rayleigh Classical/Flat
4 127 -19 Rayleigh Classical/Flat
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
BER
L4x4 S4x4 v4x4
Figure 5.4 BER performance versus detection techniques ( 4x4 ), office
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
BER
L4x5 S4x5 v4x5
Figure 5.5 BER performance versus detection techniques ( 4x5 ), office
10-6
802.11n QPSK simulations
BER
L4x6 S4x6 v4x6
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
Figure 5.7 BER performance versus the linear detection method and the proposed approximation method, office
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
Figure 5.8 BER performance versus the SQRD detection method and the proposed approximation method, office
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
Figure 5.9 BER performance versus the V-BLAST detection method and the proposed approximation method, office
Figure 5.4 shows performances of various techniques. L denotes linear detection, S denotes SQRD detection, and V denotes V-BLAST detection. In Figure 5.5, 4x5 means that there are 4 transmit and 5 receive antennas. Similarly and etc for Figure 5.6 It is obvious that V-BLAST has the best performance and linear has the worst, as mentioned in Section 2.3.
Figure 5.7 shows performances of the linear detection method and the proposed approximation method. Here we use similar notations as Section 5.1. There are similar
and Figure 5.9. This is maybe because the channel model has very short delays and thus a very wide coherent bandwidth.
The second simulated channel is measured in an airport representing a typical large hall area. The channel has a few very long delay paths which indicate bad channel conditions and is harmful to communication.
Table 5.3 Indoor channel model [23], large hall Tap
802.11n QPSK simulations
SNR
BER
L4x4 S4x4 v4x4
Figure 5.10 BER performance versus detection techniques ( 4x4 ), large hall
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
BER
L4x5 S4x5 v4x5
Figure 5.11 BER performance versus detection techniques ( 4x5 ) , large hall
10-8
802.11n QPSK simulations
BER
L4x6 S4x6 v4x6
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
Figure 5.13 BER performance versus the linear detection method and the proposed approximation method, large hall
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
Figure 5.14 BER performance versus the SQRD detection method and the proposed approximation method, large hall
0 5 10 15 20 25 30 35
802.11n QPSK simulations
SNR
Figure 5.15 BER performance versus the V-BLAST detection method and the proposed approximation method, large hall
Figure 5.10 shows performances of various techniques under channel model of large hall. It shows the similar performance trends as in Figure 5.4. Surprisingly, Figure 5.13, Figure 5.14 and Figure 5.15 show no difference between original algorithm and the simplified algorithms. The reason may be as follows.
It is known that within coherent bandwidth, channel frequency response can be viewed as flat. Coherent bandwidth is inversely proportional to channel delay spread [24].
rms
Bc τ
≈ 1 (5.1)
Then
Let OFDM signal bandwidth be M, FFT length be N, cyclic prefix length be K N ,
and the subcarrier spacing be N
M . Assume the maximum channel delay equals to the
cyclic prefix length, which stands for the worst channel condition.
M
1 represents sampling period.
Then
By dividing both side by subcarrier spacing N
M , we have
M K
Bc N > (5.6)
Coherent bandwidth divided by subcarrier spacing defines the number of subcarriers which has the same channel response. According to equation 5.6, the number is larger than K, so K consecutive subcarriers can be seen to have the same channel response. For example, 802.11n system has FFT length 64 and cyclic prefix length 16.
Coherent bandwidth divided by subcarrier spacing defines the number of subcarriers which has the same channel response. According to equation 5.6, the number is larger than K, so K consecutive subcarriers can be seen to have the same channel response. For example, 802.11n system has FFT length 64 and cyclic prefix length 16.