Multicarrier Modulation with Blind Detection Capability Using Cosine Modulated Filter Banks Behrouz Farhang-Boroujeny Department of Electrical & Computer Engineering University of Utah email:
[email protected]
Date: July 17, 2003 ECE Department, Univ of Alberta
1
Outline Introduction o ISI cancellation using multicarrier modulation (MCM)
Discrete Multitone Modulation (DMT/OFDM) o Time and frequency domain equalization
Filter Bank Based Multicarrier Modulation o Structure o Blind Equalization o Bandwidth Efficiency o Computer Simulations
Conclusions
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Introduction Time spreading (Intersymbol Interference – ISI)
s (n)
{h(n)}
x ( n) = ∑i h(i ) s ( n − i )
Solution: Channel equalization s (n)
H (z )
z −∆ ≈ H (z )
s(n − ∆)
Problem: noise enhancement
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Cancellation (reduction) of ISI using MCM By dividing the channel into a large number of sub-channels, each sub-channel will have almost flat gain, thus is free of ISI. H(f)
frequency
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Structure of MCM System TRANSMITTER Serial input
S/P
Encoder
Wave shaping & mod.
P/S
Channel
RECEIVER Serial output
P/S
Decoder + FEQ
Filtering & Demod.
S/P
S/P: Serial-to-Parallel P/S: Parallel-to-Serial
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Discrete Multitone (DMT/OFDM): an elegant
technique for efficient realization of MCM systems TRANSMITTER Serial input
S/P
IFFT + CP
Encoder
P/S
Channel
RECEIVER Serial output
P/S
Decoder + FEQ
S/P: Serial-to-Parallel P/S: Parallel-to-Serial
Remove CP +FFT
S/P + TEQ
CP: Cyclic Prefix TEQ: Time Domain Equalizer FEQ: Frequency Domain Equalizer
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What are TEQ and FEQ? • Time domain equalizer is used to shorten the duration of the channel response. • Frequency domain equalizer is to compensate for gain distortion due to channel response.
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MCM with Critically Sampled Filter Banks (also called discrete wavelet multitone –DWMT) TRANSMITTER
Data in
S/P
Synthesis Filter Bank + P/S
Channel
TEQ
RECEIVER
Data out
P/S
PostProcessing Equalizer
S/P +
Analysis Filter Bank
Synthesis and analysis are based on cosine modulated filter banks (CMFB). 8
Advantages of CMFB over DMT Higher efficiency no cyclic prefix and/or suffix more bandwidth efficient More immune to narrow-band interference More immune to intercarrier interference (some times)
No frame synchronization Simple blind equalization possible
9
Disadvantages of CMFB compared to DMT More Complex? Not compliant with the present standards
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Cosine Modulated Filter Bank (CMFB) d0
⊗
G0 ( − z 2 M )
d1
z −1W −1 / 2
G1 ( − z 2 M )
(a) −1
z W
2M-point DFT
−1 / 2
⊗
Q1a ( z )
d 2 M −1
z −1W −1 / 2
⊗
G 2 M −1 ( − z 2 M )
Q2aM −1
Q0a ( z )
Q0a
Q1a
Q2a
Q2aM −1
Q2aM −1 ( z )
Q0a
(b)
π /M
ω 11
a a In the conventional CMFB the pairs of Qk (z ) and Q2 M −1−k ( z ) are combined together to make an analysis filter
H k ( z ) = Qka ( z ) + Q2aM −1−k ( z )
Q2aM −1−k
(k + 1)π − M
kπ − M
| H k ( e jω ) |
kπ M
Qka
(k + 1)π M
ω
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Synthesis filters are matched to the analysis filters:
Q2aM* −1−k
(k + 1)π − M
kπ − M
| Fk (e jω ) |
kπ M
Qka*
(k + 1)π M
ω
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Vestigial Side-Band Property of CMFB Modulation and Demodulation in CMFB
ω Modulation
− ωc
ω
ωc Demodulation
ω
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a Q Output of k (z ) after demodulation:
π
ω
M
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Distribution of the real and imaginary parts of xk(n):
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Effect of channel and Equalization ASSUMPTION: Over each sunchannel, channel is flat. Hence, it can be modelled as a complex gain, and thus, equalization can be achieved using a complex gain.
Equalizer output:
yk (n) = ℜ{wk* xk (n)} = wk , R xR (n) + wk , I xI (n)
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Blind Equalization We propose an algorithm that works based on the same principal as Godard’s algorithm.
Criterion: minimize
ξ = E [(| yk (n) | p − R) 2 ], R is a constant and p is an integer This is called dispersion function.
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Update equation: We use the algorithm
wk (n + 1) = wk (n) − µ∇Cwk ξˆ where
∇
C wk
∂ ∂ = +j and ξˆ = (| y k (n) | p − R ) 2 ∂wk , R ∂wk , I
We obtain
wk (n + 1) = wk (n) − 2µxk (n)(sign ( yk (n))) p ( yk (n)) p −1 (| yk (n) | p − R)
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The case of interest here is p = 1 :
wk (n + 1) = wk (n) − 2µxk (n)sign ( yk (n))(| yk (n) | − R) The optimum value of R is
[
]
E | s (n) |2 R= E [| s (n) |] For L-ary PAM:
2( L2 − 1) R= 3L
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Blind equalizer is blind to a phase ambiguity of 180o.
Solution => differential encoding There is no loss of 3 dB.
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Bandwidth Efficiency of CMFB-MCM • Each subchannel occupies a bandwidth of π / M and carries one PAM symbol. • In QAM signaling, we need a bandwidth of 2π (M1 + α ) to carry QAM symbols at the same rate as PAM symbols. • Each QAM symbol may be thought as 2 PAM symbols. • We thus find that compared to single carrier modulation, 1 1+α
CMFB requires times less bandwidth. α is the excess bandwidth. • Compared to OFMD, CMFB is more bandwidth efficient because of the absence of cyclic extensions. 22
Computer Simulations Learning Curves • Simulations are done for binary and 4-ary signaling. • Each result is based on 500 independent runs, • Learning curves are based on the error functions
[
]
For binary data:
J = E (| y k ( n) | −1) 2 ]
For 4-ary data:
J = E min{(| yk (n) | −1) 2 , (| y k ( n) | −3) 2 ]
[
]
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An example of learning curves for binary symbols: 0 Subchannel 6 Subchannel 14 Subchannel 27
−2 −4
−8
2
J/σs (dB)
−6
−10 −12 −14 −16 −18
0
100
200 300 NO. OF ITERATIONS
400
500
1
An example of learning curves for 4-ary symbols: −6 Subchannel 6 Subchannel 14 Subchannel 27
−8
J/σs (dB)
−10
2
−12 −14 −16 −18 −20
0
100
200 300 NO. OF ITERATIONS
400
500
2
A comparison of CMFB-MCM (2 taps per subcarrier) and DWMT (33 taps per subcarrier) : 0 CMFB−MCM DWMT −5
MSE/σ2s (dB)
−10
−15
−20
−25
−30
0
20
40
60 80 SUBCHANNEL NO.
100
120
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Bit-Error-Rate (BER) Comparison with OFDM Channel has a multipath Rayleigh fading model
h(t ) = ∑ a (t ,τ i )δ (t − τ i ) i
with power-delay profile 2τ i / T
σ (τ i ) = Kλ 2 a
where λ is a positive constant smaller than one.
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BER comparison of CMFB-MCM and OFDM CMFB-MCM block length = 64 OFDM block length = 32 0
10
−1
10
−2
BER
10
−3
10
−4
10
CMFB−MCM, λ=0.5 CMFB−MCM, λ=0.7 CMFB−MCM, λ=0.9 OFDM, λ=0.5 OFDM, λ=0.7 OFDM, λ=0.9
−5
10
−6
10
0
5
10
15
20 25 SNR (dB)
30
35
40
45
4
BER comparison of CMFB-MCM and OFDM CMFB-MCM block length = 128 OFDM block length = 64 0
10
−1
10
−2
BER
10
−3
10
−4
10
CMFB−MCM, λ=0.5 CMFB−MCM, λ=0.7 CMFB−MCM, λ=0.9 OFDM, λ=0.5 OFDM, λ=0.7 OFDM, λ=0.9
−5
10
−6
10
0
5
10
15
20 25 SNR (dB)
30
35
40
45
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BER comparison of CMFB-MCM and OFDM CMFB-MCM block length = 256 OFDM block length = 128 0
10
−1
10
−2
BER
10
−3
10
−4
10
CMFB−MCM, λ=0.5 CMFB−MCM, λ=0.7 CMFB−MCM, λ=0.9 OFDM, λ=0.5 OFDM, λ=0.7 OFDM, λ=0.9
−5
10
−6
10
0
5
10
15
20 25 SNR (dB)
30
35
40
45
6
BER comparison of CMFB-MCM and OFDM CMFB-MCM block length = 512 OFDM block length = 256 0
10
−1
10
−2
BER
10
−3
10
−4
10
CMFB−MCM, λ=0.5 CMFB−MCM, λ=0.7 CMFB−MCM, λ=0.9 OFDM, λ=0.5 OFDM, λ=0.7 OFDM, λ=0.9
−5
10
−6
10
0
5
10
15
20 25 SNR (dB)
30
35
40
45
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VDSL Applications There are currently two candidates for VDSL, discrete multitone (DMT), and filtered multitone (FMT). • DMT is a DFT based solution and is very similar to OFDM. • FMT is a filterbank-based solution (similar to CMFB-MCM). The difference is that FMT uses filters which are nonoverlapping, thus suffer from bandwidth loss. Moreover, FMT solution requires very complex equalizers (36 taps for each subcarrier). • The fact that FMT has been received by the industry, is a good indication that filterbank solutions have recognized as good alternative solutions to the widely used DFT-based MCM techniques. 25
A thorough comparison of DMT, FMT and CMT (CMFB-based solutions) shows that: • DMT has the lowest computational complexity. • FMT and CMT are significantly superior to DMT in terms of latency and resistant to narrowband noise (HAM radio interference). • CMT offers the highest transmission rate. • While FMT and CMT may be the preferred choices to DMT in the application of VDSL because of much higher resistance to narrowband noise, we believe CMT is a better choice due to its much lower complexity. It is three times less complex.
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Bit rate comparison of DMT, FMT and CMT (cosine modulated multitone): 40 z−DMT CMT FMT
35
Bit Rate (Mbps)
30 25 20 15 10 5 0
0
200
400
600 800 1000 Length of TP1 (m)
1200
1400
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Comparison of DMT, FMT and CMT when channel is corrupted by an RF interference: 45 z−DMT w/o RFC z−DMT with RFC CMT FMT
40 35
SNR(dB)
30 25 20 15 10 5 0
0
0.5
1
1.5 2 Frequency(MHz)
2.5
3
3.5
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Comparison of DMT, FMT and CMT when channel is corrupted by an RF interference: 45 z−DMT w/o RFC z−DMT with RFC CMT FMT
40 35
SNR(dB)
30 25 20 15 10 5 0
0
0.5
1
1.5 2 Frequency(MHz)
2.5
3
3.5
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Conclusions • We proposed CMFB as an alternative MCM to OFDM for wireless applications. • CMFB-MCM was shown to be more bandwidth efficient than OFDM. • CMFB may be thought as a tool for VSB modulation of a number of PAM channels that are packed together in a minimum bandwidth. • Simple blind equalizer was proposed for CMFBMCM.
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Some related research topics • Convergence study of the blind equalizer. • Tracking behavior of the CMFB-MCM in wireless channels. • Diversity combining techniques. • Space-time systems (MIMO). • Peak-to-average ratio controlling methods. • Carrier and timing recovery. • MC-CDMA and MCSS. 28