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FILTER BANK MULTICARRIER WITH LAPPED TRANSFORMS
Maurice Bellanger, CNAM
Davide Mattera, Mario Tanda, Univ.Napoli
March 2015
Objectives
A multicarrier approach to
• improve on OFDM for future wireless systems- asynchronous multi-user access
- spectral separation for coexistence
- robustness to channel impairments – CFO
• keep most of OFDM features- spectral efficiency
- minimum delay
- simplicity of concept
- low computational complexity
Outline
• Lapped Transform: - definition
- implementation
• Transmission system performance- signal characteristics
- channel equalization
• Complex lapped transform for FBMC- implementation
- channel equalization
- carrier frequency offset compensation
• Open issues
Lapped transform
Introduced decades ago to improve the discrimination of critical spectral
components in signal compression
n: time domain ; k: frequency domain
• perfect decomposition-reconstruction
• overlapping factor: K=2
• real processing
novelty in communications: frequency domain equalization
])21)(
221cos[()(),(
MkMnnhknT π−+−=
MkMn ≤≤≤≤ 1;21
]2
)21sin[()(
Mnnh π−−=
LT in communications
• Real lapped transform – QAM modulation
Lapped-OFDM
• Complex lapped transform – PAM modulation
FBMC-PAM
MnkenhknTc MkMnj
2,1;)(),()
21)(
221( ≤≤= −+− π
Frequency response
Response of sine filter h(n):
• M pairs of symmetrical carriers instead of M carriers for the DFT
2)22(1)2cos(2)(
MfMffH L −= π
π
0 5 10 15 20 25 30-90
-80
-70
-60
-50
-40
-30
-20
-10
0Amplitude
dB
Frequency (unit=sub-carrier spacing)
OFDM
Lapped-OFDM
LT in transmission
Multi-carrier transmission with T(n,k)
• QAM modulation can be used- independent real processing of real and imaginary parts of data
• FBMC scheme with overlapping K=2- delay: 2 M
• equalization in the receiver can be performed in the frequency domain (no additional delay)
• frequency domain residual CFO compensation in
multi-user scenario
Implementation
Objective: use a 2M-DFT for frequency domain equalization
Expression of the transform
and
2M-DFT + frequency domain filtering + phase shifts
( coefficients [1 –1] )
]][[4
),()
21)(
221()
21)(
221(
2)
21(
2)
21(
MkMnj
MkMnj
Mnj
Mnj
eeeejknTππππ −+−−−+−−−− +−=
]][
][[4
),(
)1()1(222
)2
1(
)1()1(222
)2
1(
Mkjnk
Mj
Mjnkk
Mjkj
Mkjnk
Mj
Mjnkk
Mjkj
eeeee
eeeeej
knT
πππππ
πππππ
−−−−−−
−−−−−
−−
−=
Transceiver structure
• emitted symbols of 2M samples overlap by M samples
• symbol rate: 1/M
• equalization at FFT output
data
out
S/P
+
QAM
Transpose
Lapped
Transform
Overlap
/ add
+
P/S
S
/
P
FFT(2M)
Equalization
Sine
filter
Post
process.
QAM
detect+P/S
data
in
channel
Transmitter Receiver
Lapped Transform
Emitted spectrum
• The lapped transform defines 2M sub-carriers- a sub-channel consists of 2 parts: k and 2M-k
• Spectrum: continuous / fragmented
0 11/2
A
fk/2M 1-k/2M
f1 f2 1-f2 1-f2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Amplitude
Frequency 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Amplitude
Frequency
Transmission system performance
Impact of timing offset
Envelop of emitted signal
Timing offset: to ; signal-to-interference ratio
OFDM:(GT: guard time)
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 00
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
M T O
t i m e
a m p li tu d e
ππ 2/)2/2sin(2/2/1
MtoMtoSIRL −=
MGTtoSIROFDM /)(
2/1−=
Half rate schemes
Signal-to-interference ratio – half rate
Emitted signal envelop
full rate half rate
)/2sin(161)/sin(2
183
2/1MtoMto
Mto
SIR LHRππππ +−
=
0 1 00 2 00 30 0 4 00 5 00 60 00
0 .2
0 .4
0 .6
0 .8
1
1 .2A m p litud e
tim e
re a l im a g ina ry
0 100 200 300 400 500 6000
0.2
0.4
0.6
0.8
1
1.2A m plitude
tim e
M
SIR curves
Signal-to-interference ratio
Max. to = M/2 SIR = 7.4 dB BER = 0.015 (4-QAM)
0 20 40 60 80 100 1200
5
10
15
20
25
30
35
40SIR
dB
Timing offset (M=256)
OFDM-GT=16 (1/16)
OFDM-GT=32 (1/8)
Lapped-OFDM-half rate
Lapped-OFDM
Multipath channel equalization
channel transfer function
interference power
SNR: multiply interference+noise by equalizer response
iP
ii ZcZC −
=∑=
0
)(
)]1()([2
1
−−= ∑ ∑= =
jfjfcPP
i
P
ijjiP
ππ 2/)2/2sin(2/)( MiMiif −=
Bit error rate
• M=256 sub-channels
• Channel: ITU-R veh.B – max.delay: 0.22M (< M/4)
• Profile: delay: 0 1 25 36 48 56
ampl.: 0.75 1 0.23 0.316 0.055 0.16
4-QAM 64-QAM
Asynchronous access
• OFDM – CP = 64 (1/4)
• One-tap FBMC: OQAM ; single tap equalizer ; K=2
• FS-FBMC: OQAM ; frequency domain equalization
• OFDM-lap: QAM ; lapped transform
Channel ITU-R
veh.B
Eb/No=20 dB
4-QAM
Symmetry
PAPR
• Peak-to-average power ratio
• Complementary cumulative distribution function
1.5 2 2.5 3 3.5 4 4.5 5-30
-25
-20
-15
-10
-5
0
5
amplitude
CCDF
dB
L-OFDMreal data
L-OFDMreal/imaginary data
L-OFDM complex data(full rate)
OFDM
Complex lapped transform for FBMC
Complex transform and implementation
Definition
Factorization
Implementation
• Phase shifts by multiples of π/2
• Frequency domain filtering , coefficients: [1 –1]
• Multiply by (time shift: ½)
• Inverse FFT of size 2M
• Overlap and add (overlapping factor K=2)
MnkeM
nknTc MkMnj
2,1;]2
)21sin[(),(
)21)(
221(
≤≤−=−+− ππ
2)
23(
22)1(
2)1(
22
2 ][21),(
πππππ −−−−− −= kjM
kjnM
kjM
jnkM
jkeeeeeknTc
Mkj
e 2)1( π−−
Transmitter structure
Multicarrier transmitter
• PAM modulation
• Multicarrier symbol length:2M
• Symbol rate: M
Phase
shifts
Filter
iFFT
(2M)
overlap+add
d(k)
(real)
S/P
(2M)
P/S
(M)
y(n)
(channel)
Emitted spectrum
M=256 ; Number of used sub-channels: 230x2 ;
binary data ; 460 bits per symbol ; rate: 1/M
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Amplitude
Frequency
Receiver structure
Multicarrier receiver
• Frequency domain equalization
• Sub-channel filtering after equalization
• CFO compensation: interpolated filter coefficients
phase
shifts
filter
FFT
(2M)
detection
S/P
(M)
y(n) d(k)
input
buffer
equalizer
P/S
(2M)
System impulse response
frequency
time
Total imaginary interference power: unity
-0.021 j0.106 j-0.25 j0.318 j-0.25 j0.106 j-0.021 jn+1
000.5 j1-0.5 j00 n
0.021 j-0.106 j-0.25 j-0.318 j-0.25 j-0.106 j0.021 jn-1
k+4k+2k+1kk-1k-2k-4
SIR curves
Signal-to-interference ratio / timing offset
Maximum timing offset: M/2 ; BER = 0.015 (binary data)
asynchronous access
0 20 40 60 80 100 1200
5
10
15
20
25
30
35
40SIR
Timing offset (M=256)
dB OFD M -GT=16 (1/16)
OFD M-GT=32 (1/8)
FBMC -PAM
Bit error rate
• M=256 sub-channels
• Channel: ITU-R veh.B – max.delay: 0.22M (< M/4)
• Profile: delay: 0 1 25 36 48 56
ampl.: 0.75 1 0.23 0.316 0.055 0.16
4-QAM / 2-PAM 64-QAM / 8-PAM
Carrier frequency offset
Compensation at sub-channel level in multi-user scenario
• CFO = δf ; Filter output at time n0
• for n0 = Mm0 + (M+1)/2
• Receiver filter coefficients (time domain)
In the frequency domain: interpolation of initial set [1 –1]
)(20
12
00
0)()( infjr
M
irir einxhny −
−
=
−= ∑δπ
ifjr
M
Miri
nfjr einxheny δπδπ 2
0
2/1
2/1
20 )()( 0 −= ∑
−
+−=
Mnnhenh fMnjCFO 21;)()( )2/1(2 ≤≤= −− δπ
CFO compensation
Compensation per sub-channel or group of sub-channels
Phase shift + interpolated filter coefficients
0 0.05 0.1 0.15 0.2 0.250
5
10
15
20
25
30
35
40
45
50 SIR
dB
CFO (unit:sub-carrier spacing)
interpolation:6 coefficients
interpolation:4 coefficients
no filter coefficient interpolation
OFDM
BER versus CFO
Performance of OFDM, lapped OFDM, FBMC-PAM
4-QAM/2-PAM
Eb/No = 8dB
Normalized CFO
C: full compensation
C3: 3 coefficients
C5: 5 coefficients
C7: 7 coefficients
Open issues
Algorithmic aspects
• Generalization – extended lapped transform
• Other system options and parameter selection
• Optimization of the structure
• Efficient implementation – minimal complexity
• Performance analysis – multiple asynchronous users
• Comparison with enhanced OFDM techniques
(filtered OFDM, universal filtered multicarrier, generalized FDM)
Open issues
Networking aspects
• Single carrier techniques
• Preamble and pilots for burst transmission
• Duplexing: TDD, FDD, full duplex
• MIMO and massive MIMO
• Compatibility with OFDM
• Capability to meet 5G performance objectives(100 µs time budget for PHY, 55 dB ACLR, short bursts, …)