Chapter 4 The Proposed Data Detection Algorithms 27
4.4 Complexity Analysis and Comparison
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.
Hence, 4 consecutive subcarriers can be seen to have the same channel response. As a result, undoubtedly the proposed simplified algorithms result in no performance degradation compared to the original algorithms. Furthermore, the proposed algorithms can potentially save more computation complexities by considering every 4 consecutive
subcarriers as a group shares the same channel response. And this better consideration is remained to be verified.
To understand influence of non-perfect channel knowledge, simulations with added channel noise are finished. This adopts the V-BLAST algorithm and the same parameters as the large hall.
Figure 5.16 BER performance versus channel MSE, large hall
The figure shows that the non-perfect channel estimation will degrade the performance very apparently. If channel estimation is not well designed, V-BLAST
Chapter 6 Conclusion
In this thesis, new simplified algorithms for various well known MIMO OFDM detection techniques are proposed, followed by a thorough investigation and verifications in terms of complexity and BER performance by testing WWiSE systems. Although V-BLAST algorithm results in the best performance, it demands the highest computation cost. Since designs of detection methods are trade-off problems between cost and performance, complexity and performance analysis helps a lot to decide a suitable design.
Some extended proporty about coherence bandwidth is proposed, which predicts the minimum number of consecutive subcarriers which share the same channel response. Since when a channel response is shared, channel inversion can be computed for those related subcarriers. It helps to reduce system complexity without loss of BER performance and makes V-BLAST implementation on OFDM systems possible.
It is well known that MMSE criterion results in better performance than ZF criterion and thus is considered as future work. Detection algorithms about MMSE criterion are going to be investigated and extended for lower complexity and better
performance. Besides, data detection is performed in frequency domain because the FFT output can be modeled as linear transformation of IFFT input. It is also future work to search for a method that can perform data detection in time domain. Under slow fading channels, the FFT output is linear transformation of IFFT input, but under fast fading channels, this is not the case. It is originally a challenging task to model the time varying channel. If taking both fast fading channel modeling and signal detection into consideration, the problem is even bigger and is interesting for research. It is also referred to as future work to find out some algorithms reach low complexity and solve this challenging problem.
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