Chapter 4 Proposed Low-Complexity Frequency Domain
4.4 Complexity Analysis
The LMMSE proposed by Xiaodong Cai and Georgios B. Giannakis requires flops, and the LMMSE method proposed by Philip Schniter which is called MMSE method requires flops. The PCG LMMSE method proposed by this paper also requires computations linearly with N. The complexity of the two methods, Partial LMMSE and PCG LMMSE will be analyzed here, and their complexity increases linearly with N.
A flop here is defined as a complex multiplication, and N is the FFT size, P is the bandwidth of the precondition matrix. Table 4.1 shows the complexity of PCG MMSE equalizer, and Table 4.2 shows the complexity of Partial MMSE equalizer
Note that the bandwidth of the precondition matrix not only affects the
complexity per hat how many
for achieving the lowest complexity. The optimum bandwidth of the precondition matrix can be obtained by
simu mputation requir
( 2) O N
the Partial L O N( )
iteration but also the convergence rate. It is a trade-off t bandwidth of the original matrix required to choose
lations. We will discuss the actual co ed by these two methods, Partial MMSE and PCG MMSE, in the next section.
Table 4.1: Complexity analysis of PCG MMSE equalizer
Operation Complexity
(
AA') (
2Q2+ +Q 1)
Nflopsinv(AA'+δCxx) *r
( )
2
4 6 4 number of iteration
Q 2 P N
⎛ + +P + ⎞×
⎜ ⎟
⎜ ⎟
⎝ ⎠ flops
'*inv( '+δ xx) *
H AA C r
(
2Q+1)
NflopsTotal 4 6 P2 4
(
number of iteration)
2 2 4 2Q P Q Q N
⎧⎛ ⎞ ⎫
⎪⎜ + + + ⎟× + + + ⎪
⎨ ⎬
⎜ 2 ⎟
⎪⎝ ⎠ ⎪
⎩ ⎭
flops
: FFT size
Q: The bandwidth of the approximation channel P: The bandwidth of the precondition matrix N
Table 4.2: Complexity analysis of Partial MMSE equalizer
: The bandwidth of the approximation channel : FFT size
Q
N
.5 Computer Simulations
In this section, computer simulations are conducted to evaluate the performance f the OFDM system using PCG LMMSE equalizer. Through out the simulations, we only deal with discrete time signal processing in the baseband, hence pulse-shaping and matched-filtering are removed from consideration for simplicity. Also, channel estimation and timing synchronization are assumed to be perfect. In the simulations, the relationship between SNR and
4
Es
where is the symbol energy, Ts is the symbol duration, B is the system bandwidth, and M is the modulation order. The system transmit bit power is normalized to one, the noise power given by σ2 corresponding to a specific
0
Eb N can be generated by
2= 0 b
N
σ E (4.14)
Table 4.3 lists all parameters used in our simulations. The configuration we consider here is an OFDM system with a bandwidth of 1.5 MHz and 64 subcarriers.
The set of QAM constellation used in the simulations is QPSK. The channel model is the Jakes model [12], [17], [18] and the normalized Doppler spread equals 0.1.
Table 4.3: Parameters of Computer Simulations
Transmit/Receive antennas SISO
Carrier frequency 5.2 GHz
Bandwidth 1 MHz
Number of carriers, FFT size 64
μ
OFDM symbol duration 42 s
Guard interval 5.25μs
Modulation order QPSK
Velocity 250 km/hour
Maximum Doppler frequency 1.2 KHz
Normalized Doppler frequency 0.05
Channel model Jakes Model [17], [18]
: Jakes model simulator
The BE the CG MMSE equalizer with different numbers of iterations are shown in Figure 4.10. It can be shown that the conventional CG algorithm suffers from slow convergence rate problem, and this problem can be solved by the PCG algorithm
precondition is shown in Figures 4.11-4.14. It is shown that the convergence rate is proportional to the bandwidth of the precondition matrix, but a lager b x results in more comp er iteration. It is thus a trade-off in choosing the bandwidth of the preconditio ix. We define the complexity to be the num multiplications per ite ber of iterations. In Table 4.2, we show the complexity of different bandwidths of the precondition matrix and the number of iterations required for the convergence. By the simulations result, we can determine the optimum bandwidth of the precondition
Figure 4.8
R performances of
. The convergence rate of the proposed equalizer with different matrix bandwidths
andwidth of precondition matri utations p n matr
ber of ration multiplied by the num 2 sinβ1
0 5 10 15 20 25 30 35 40 10-4
10-3 10-2 10-1 100
Eb/No
BER
5 iterations 10 iterations 20 iterations 30 iterations
matrix. The optimum bandwidth of the precondition matrix here equals three. By the comp is above, the PCG MMSE only requires 30% the computations of the Partial MMSE.
lexity analys
Figure 4.9: BER performance obtained by using CG based MMSE equalizer.
The performance of different numbers of iterations is shown. It can be seen that it requires about 30 iterations to converge.
P is diag
100
0 5 10 15 20 25 30 35 40
10-4 10-3 10-2 10-1
Eb/No
BER
2 iterations 3 iterations 4 iterations
Figure 4.10: BER performance obtained by using PCG based MMSE equalizer, (BW of precondition matrix in PCG MMSE is zero). The performance of different numbers of iterations is shown. It can be seen that it requires about 4 iterations to converge.
0 5 10 15 20 25 30 35 40 10-4
10-3 10-2 10-1 100
2 iterations 3 iterations 4 iterations
Eb/No
Figure 4.11:
shown. It can be seen that it requires about 3 iterations to converge.
BER
BER performance obtained by using PCG based MMSE equalizer, (BW of precondition matrix in PCG MMSE is one). The performance of different numbers of iterations is
0 5 10 15 20 25 30 35 40 10-4
10-3 10-2 100
10-1
2 iterations 3 iterations
Eb/No
BER
igure 4.12: BER performance obtained by using PCG based MMSE equalizer, F
(BW of precondition matrix in PCG MMSE is two). The performance of different numbers of iterations is shown. It can be seen that it requires about 2 iterations to converge.
Table 4.4: Convergence rate for different precondition bandwidths Precondition Matrix
Bandwidth
Complexity
(4Q+6+1/2P2+4P)N
Number of iterations
P=0 (4Q+6)N 4
P=1 (4Q+10)N 3
P=2 (4Q+16)N 2
P=3 (4Q+22)N 2
Figure 4.13 shows the BER performance of different schemes. The conventional one-tap equalizer scheme has poor performance due to the influence of ICI. The Partial MM
equalizer is also sh PCG MMSE is du
applying a more complicated method such as MMSE-SIC, MMSE-PIC [3], [4]. It shows that with the channel approximation the MMSE-PIC equalizer has better performance than the MMSE equalizer. The MMSE-PIC equalizer can even have better BER performance than the MMSE equalizer BER bound in low SNR region.
SE and PCG MMSE have similar performance. A BER bound of the MMSE own in this figure. The gap between the MMSE BER bound and the e to the channel approximation errors. This gap can be reduced by
100
Figure 4.14 shows the BER performances under different vehicle speeds. The BER performance of the OFDM system degrades with an increasing vehicle speed because the ICI is more significant in the high mobility environments. It will thus require more bandwidth of the approximated channel or a complicated method to mitigate the ICI.
Figure 4.15 shows the BER performance with channel estimation errors. The channel estimation errors are defined as a AWGN noise with variance
Figure 4.13: BER performance of different schemes
2
σe to disturb the e ated channel taps, by the definition in [26]
(4.)
where is the estimated channel impulse response, and stim
[
1 2 -1]
= e e, eL
e represents the error vector. It is assumed that is independent of and is modeled as independent zeros means complex-valued Gaussian noise. It can be shown in Figure 4.18 that the PCG MMSE has similar performance to the Partial MMSE equalizer even if the channel estimation errors are considered. Because the Partial MMSE equalizer only takes parts of the equations, it may be more sensitive to the disturbance of channel.
e h
100
0 5 10 15 20 25 30 35 40
10-4 10-3 10-2 10-1
Eb/No
BER
490 km/hr 370 km/hr 250 km/hr
Figure 4.14: BER performance under different vehicle speeds.
0 5 10 15 20 25 30 35 40 10-4
10-3 10-2 10-1 100
Eb/No
BER
Channel estimation errors variance=1e-2
Partial MMSE PCG MMSE
Perfect Channel Partial MMSE Perfect Channel PCG MMSE
Figure 4.15: BER performance with channel estimation errors
4.7 Summary
In this chapter, we first introduce the channel approximation of mobile channel.
This approximation is based on the concept that we are only concerned with the significant channel coefficients and ignore the trivial parts. With this approximation, the number of coefficients to be processed is reduced, so the computations for the equalizers can also be reduced. Although this approximation is useful, there is still an error floor due to the approximation errors. Furthermore, we introduce and compare several different low-complexity equalizers in Section 4.2, which are important techniques in this subject. By complexity analysis, it is shown that our scheme can achieve lower computation complexity while still have similar BER performance to the Partial MMSE equalizer.
.
ose a PCG based MMSE equalizer for the OFDM system over time-varying channels. Compared with conventional one-tap equalizers, this
eme can achieve better performance in mobile environments. In Chapter 2, the oncept of OFDM system is introduced and the reason why OFDM system can be used fficiency in time-invariant channel is given. Besides, the challenges to OFDM system mobile environments are introduced and mathematically analyzed. The most
portant issue is that the channel variations with time destroy the orthogonality etween subcarriers and produce the intercarrier interference (ICI). Then the characteristics of the ICI are analyzed and the mobile channel matrix is approximated to a band matrix based on this ICI analysis. Furthermore, some basic techniques to cancel the ICI are introduced in this chapter, such as ICI self-cancellation schemes and the frequency domain equalizer schemes, the latter being adopted in this thesis In Chapter 3, we first introduce the idea of orthogonal projection and then derive the conjugate gradient (CG) algorithm. In Chapter 4, several schemes based on the frequency domain equalizer techniques are introduced, which have already been
Chapter 5
Conclusion
In this thesis, we prop
sch c e in im b
proposed to cancel the ICI with moderate complexity. The CG method in Chapter 3 is odified into a preconditioned version and a PCG based low-complexity MMSE
ualizer for canceling the ICI is proposed. Furthermore, the effect of the precondition method with different bandwidths is discussed. Then we f this scheme and compare this scheme with other schemes.
The conventional system adopting the preamble to estimate the channel will have poor performance because of the channel variation with time. Actually, in a mobile o insert sufficient pilot symbols for channel estimation.
e can estimate the channel at some time instance and use linear or non-linear terpolation to obtain the entire time-varying channel estimate. There are still some prob
m eq
matrix on the proposed analyze the complexity o
OFDM system, it is necessary t W
in
lems in the estimation of time-varying channels because of the inaccuratcy of the interpolation. Besides, the condition number of the system decreases as the number of receiver antennas increases. This suggests that the proposed scheme can be better applide to the mobile MIMO-OFDM system.
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