ofdm
TRANSCRIPT
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Multi-Carrier Transmission over Mobile Radio Channels
Jean-Paul M.G. Linnartz
Nat.Lab., Philips Research
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Outline
• Introduction to OFDM• Introduction to multipath reception • Discussion of receivers for OFDM and MC-CDMA• Introduction to Doppler channels• Intercarrier Interference, FFT Leakage• New receiver designs• Simulation of Performance• Conclusions
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OFDM
OFDM: a form of MultiCarrier Modulation. • Different symbols are transmitted over different subcarriers• Spectra overlap, but signals are orthogonal.• Example: Rectangular waveform -> Sinc spectrum
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I-FFT: OFDM Transmission
Transmission of QAM symbols on parallel subcarriers
Overlapping, yet orthogonal subcarriers
cos(ct+ st)
cos(ct)
cos(ct+ ist)
cos(ct+ (N-1)st)
User symbols
Ser
ial-
to-
par
all
el = Ser
ial-
to-
Par
alle
l
I-F
FT
Par
alle
l-to
-S
eria
l
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OFDM Subcarrier Spectra
OFDM signal strength versus frequency.
Rectangle <- FFT -> Sinc
before channel
after channel
Frequency
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Applications
Fixed / Wireline:• ADSL Asymmetric Digital Subscriber Line
Mobile / Radio:• Digital Audio Broadcasting (DAB)• Digital Video Broadcasting - Terrestrial (DVB-T)• Hiperlan II• Wireless 1394• 4G (?)
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The Wireless Multipath Channel
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The Mobile Multipath Channel
Delay spread Doppler spread
Frequency Time
FT
Frequency
FT
Frequency
Time
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Effects of Multipath Delay and Doppler
Frequency
Tim
e
Narrowband
Frequency
Tim
e
OFDMWideband QAM
Frequency
Tim
e
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Effects of Multipath (II)
Frequency
Tim
e
+-+--+-+
DS-CDMA
Frequency
Tim
e +
-
-
FrequencyHopping
Frequency
Tim
e + - + -
+ - +-
+ - +-
MC-CDMA
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Multi-Carrier CDMA
Various different proposals.• (1) DS-CDMA followed by OFDM • (2) OFDM followed by DS-CDMA• (3) DS-CDMA on multiple parallel carriers
First research papers on system (1) in 1993:– Fettweis, Linnartz, Yee (U.C. Berkeley)– Fazel (Germany) – Chouly (Philips LEP)
System (2): Vandendorpe (LLN)
System (3): Milstein (UCSD); Sourour and Nakagawa
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Multi-Carrier CDM Transmitter
What is MC-CDMA (System 1)?
• a form of Direct Sequence CDMA, but after spreading a Fourier Transform (FFT) is performed.
• a form of Orthogonal Frequency Division Multiplexing (OFDM), but with an orthogonal matrix operation on the bits.
• a form of Direct Sequence CDMA, but the code sequence is the Fourier Transform of the code.
• a form of frequency diversity. Each bit is transmitted simultaneously (in parallel) on many different subcarriers.
P/SI-FFT
NS/P N
B
Code Matrix
C
N
A
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MC-CDM (Code Division Multiplexing) in Downlink
In the ‘forward’ or downlink (base-to-mobile): all signals originate at the base station and travel over the same path.
One can easily exploit orthogonality of user signals. It is fairly simple to reduce mutual interference from users within the same cell, by assigning orthogonal Walsh-Hadamard codes.
BS
MS 2MS 1
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Synchronous MC-CDM receiver
The MC-CDM receiver • separates the various subcarrier signals (FFT)• weights these subcarriers in W, and
• does a code despreading in C-1:(linear matrix over the complex numbers)
Compare to C-OFDM:
W := equalization or AGC per subcarrier
C-1 := Error correction decoder (non-linear operation)
S/P P/SI-CodeMatrix
C-1
FFTN N N
Y
WeighMatrix
W
N
A
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Synchronous MC-CDM receiver
Receiver strategies (How to pick W ?)• equalization (MUI reduction) w = 1/• maximum ratio combining (noise reduction) w = • Wiener Filtering (joint optimization) w = /( c
Next step: W can be reduced to an automatic gain control, per subcarrier, if no ICI occurs
S/P P/SI-CodeMatrix
C-1
FFTN N N
Y
WeighMatrix
W
N
A
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Synchronous MC-CDM receiver
• Optimum estimate per symbol B is obtained from B = EB|Y = C-1EA|Y = C-1A.
• Thus: optimum linear receiver can implement FFT - W - C-1
• Orthogonality Principle: E(A-A)YH = 0N, where A = WYH
• Wiener Filtering: W = E AYH (EYYH)-1 • EAYH diagonal matrix of signal power• EYYH diagonal matrix of signal plus noise power• W can be reduced to an AGC, per subcarrier
S/P P/SI-CodeMatrix
C-1
FFTN N N
Y
WeighMatrix
W
N
A B
s
*
*
T
Nββ
βw
0
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MC-CDM BER analysis
Rayleigh fading channel– Exponential delay spread– Doppler spread with uniform angle of arrival
Perfect synchronisation
Perfect channel estimation, no estimation of ICI
Orthogonal codes
Pseudo MMSE (no cancellation of ICI)
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Composite received signal
Wanted signal
Multi-user Interference (MUI)
Intercarrier interference (ICI)
000,
1
0,,
1
0,00 ][][
mnn
N
nnmnn
N
nnn
s mncncwwN
Tbx
][][0,
1
0,
1
1ncncwbTx knn
N
nnn
N
kksMUI
00n,n,n
1
0n ][ncwaTx n
N
nsICI
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Composite received signal
Wanted signal
Multi-User Interference (MUI)
Intercarrier interference (ICI)
1
0
2,ch
1
0
2,ch
0
20
1
1
22 EE)()(EE1 N
nnn
N
nnmk
N
kkICI wmncncb
N
2
,,,,ch
1
1
22
2*
ch2 EEEE
nn
Annnnn
Annn
N
kk
sMUIMUIMUI wwb
N
Txx
nn
N
nnn
s wβN
Tbx ,
1
0,00
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BER for MC-CDMA
BER for BPSK versus Eb/N0
(1) 8 subcarriers
(2) 64 subcarriers
(3) infinitely many subcarriers
(4) 8 subc., short delay spread
(5) 8 subc., typical delay spread1 0 -5
1 0 -4
1 0 -3
1 0 -2
1 0 -1
5 1 0 1 5L o cal-m e an E n /N 0
E b /N 0 E b /N o (d B )
(1 )
(2 )
(3 )
(4 )
(5 )
A v g. B E R
A W G N
O F D M
Local-mean Eb/N0
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Capacity relative to non-fading channel
Coded-OFDM
same as N fading channels
For large P0Ts/N0 on a Rayleigh fading channel, OFDM has 0.4 bit less capacity per dimension than a non-fading channel.
MC-CDMData Processing Theorem:
COFDM = CMC-CDM
In practise, we loose a little.
In fact, for infinitely many subcarriers,
CMC-CDM = ½ log2(1 + P0Ts/N0).
where is MC-CDM figure of merit, typically -4 .. -6 dB.
022
1
0
0
0
0 21logexp2 dxxxTP
N
TP
NC
ssOFDM
ssOFDM TP
NE
TP
NC
0
01
0
0
22exp
2ln
1
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Capacity
Capacity per dimension versus local-mean EN/N0,
no Doppler.
-5 0 5 10 15 20 25 30 35 400
1
2
3
4
5
6
7
Local-mean En/N0 (dB)
Cap
acity
: B
its p
er S
ubca
rrie
r
-* : Rayleigh
* : MC-CDMA
- : LTI
Non-fading, LTI
Rayleigh
MC-CDM
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OFDM and MC-CDMA in a rapidly time-varying channel
Doppler spread is the Fourier-dual of a delay spread
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Doppler Multipath Channel
Describe the received signal with all its delayed and Doppler-shifted components
Compact this model into a convenient form, based on time-varying amplitudes.
Make a (discrete-frequency) vector channel representation
Exploit this to design better receivers
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Mobile Multipath Channel
Collection of reflected waves, each with • random angle of arrival• random delay
Angle of arrival is uniform
Doppler shift is cos(angle)
U-shaped power density
spectrum
Doppler Spectrum
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ICI caused by Doppler
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10
-4
10-3
10-2
10-1
100
Normalized Doppler [fm/fsub]
Pow
er, V
aria
nce
of IC
I
P0
P1 P2 P3
Po
we
r o
r va
ria
nce
of I
CI
Doppler spread / Subcarrier Spacing
Neighboring subcarrier2nd tier subcarrier
3rd tier subcarrier
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BER in a mobile channel
0 5 10 15 20 25 30 35 4010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Antenna Speed (m/s)
Lo
cal-M
ea
n B
ER
for
BP
SK
OFDM, 10 dB
MC-CDMA, 20 dB 30 dB
MC-CDMA, 10 dB
OFDM, 20 dB
OFDM, 30 dB
• Local-mean BER for BPSK, versus antenna speed.
• Local mean SNR of 10, 20 and 30 dB.
• Comparison between MC-CDMA and uncoded OFDM for fc = 4 GHz
• Frame durationTs= 896s
• FFT size: N = 8192.
•Sub. spacing fs = 1.17 kHz
•Data rate 9.14 Msymbol/s.
Antenna Speed [m/s]
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Doppler Multipath Channel
Received signal r(t)
Channel model:
Iw reflected waves have
the following properties:
• Di is the amplitude
I is the Doppler shift
• Ti is the delay
OFDM parameters:•N is the number of subcarriers•Ts is the frame duration•an is the code-multiplexed datac is the carrier frequencys is the subcarrier spacing
)(}))((exp{)(1
0
1
0tntjTtnjDatr
wI
iiiscin
N
n
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Taylor Expansion of Amplitude
Rewrite the Channel Model as follows
Tayler expansion of the amplitude
Vn(t) = vn(0)+ vn
(1) (t-t) + vn(2) (t-t)2/2 + .. .
vn(q) : the q-th derivative of amplitude wrt time, at instant t = t.
vn(p) is a complex Gaussian random variable.
)(})(exp{)()(1
0tntnjtVatr sc
N
nnn
1
0
)( })(exp{wI
itiisci
qi
qn jTnjDjv
)(}))((exp{)(1
0
1
0tntjTtnjDatr
wI
iiiscin
N
n
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Random Complex-Gaussian Amplitude
It can be shown that for p + q is even
and 0 for p + q is odd.
• This defines the covariance matrix of subcarrier amplitudes and derivatives,
• allows system modeling and simulation between the input of the transmit I-FFT and output of the receive FFT.
srms
qpqqp
Dq
mp
n Tmnj
j
qp
qpfvv
)(1
)1(
!)!(
!)!1(2E )*()(
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DF Vector Channel Model
Received signal Y = [y0, y1, … yN-1 ],
Let’s ignore
f : frequency offset
t : timing offset
We will denote = (0) and = (1)
• For integer , :: 0 (orthogonal subcarriers)
models ICI following from derivatives of amplitudes
0 does not carry ICI but the wanted signal
m
N
n q
qqmn
qn
ftsnm nq
Tvnjay f
1
0 0
)()(
!exp
Complex amplitudes and derivatives
System constants (eg sinc) determined by waveform
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DF-Domain Simulation
Simulation of complex-fading amplitudes of a Rayleigh channel with Doppler and delay spread
• Pre-compute an N-by-N matrix U, such that UUH is the channel covariance matrix with elements n,m = Evn
(0)vm*(0)
– Simply use an I-FFT, multiply by exponential delay profile and FFT
• Generate two i.i.d vectors of complex Gaussian random
variables, G and G’, with unity variance and length N.
• Calculate V = U G.
• Calculate V(1) = 2fT U G’.
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DF Vector Channel Model
Received signal Y = [y0, y1, … yN-1 ],
models ICI following from derivatives of amplitudes
0 does not carry ICI but the wanted signal
NTAV
AVIχY N
*'.
*.30
021
201
110
3
..
........
..
..
NN
N
N
FFT leakage
Amplitudes & Derivatives
User data
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Possible Receiver Approaches
Receiver
1) Try to invert adaptive matrix (Alexei Gorokhov)
2) See it as Multi-user detection: (J.P. Linnartz, Ton Kalker)
– try to separate V .* A and V(1).* A
3) Decision Feedback (Jan Bergmans)– estimate iteratively V, V (1) and A
)DIAG()DIAG( )1(0 VVχ
NATV
AVIχY N
*.'
*.0
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Receiver 1: Matrix Inversion
• Estimate amplitudes V and complex derivatives V (1)
• create the matrix Q1 = DIAG(V)+ T DIAG(V(1))
• Invert Q1 to get Q1-1 (channel dependent)
• Compute Q1-1Y
Zero-forcing: For perfect estimates V and V (1), Q1
-1Y = A + Q1-1N,
i.e., you get enhanced noise.
MMSE Wiener filtering inversion W
ChannelEstimator
SlicerQ-1Y
3
X1 X2
X3
+x
x
V
V’
A
N
V’
A
V
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Receiver 1: MMSE Matrix Inversion
Receiver sees Y = Q A + N, with Q=DIAG(V)+ T DIAG(V(1))
Calculate matrix Q = DIAG(V)+ T DIAG(V(1))
Compute MMSE filter W = QH [Q QH + n
2 IN]-1.
Performance evaluation:
• Signal power per subcarrier• Residual ICI and Noise enhancement from W
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Receiver 1: Matrix Inversion
Simulation of channel for N = 64, v = 200 km/h fc = 17 GHz, TRMS = 1 s, sampling at T = 1s. fDoppler = 3.14 kHz, Subcarrier spacing fsr = 31.25 kHz, signal-to-ICI = 18 dB
0 10 20 30 40 50 60 70-80
-70
-60
-50
-40
-30
-20
-10
0
10
Subcarrier number
Ma
gn
itud
e in
d
B
Amplitudes Derivatives
Amplitudes
First derivatives
Determined by speed of antenna, and carrier frequency
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Receiver 1: Matrix Inversion
SNR of decision variable. Simulation for N = 64, MMSE Wiener filtering to cancel ICI
0 10 20 30 40 50 60 700
5
10
15
20
25
30
Subcarrier number
Ou
tpu
t S
INR
Conventional OFDM MMSE equalization
MMSE ICI canceller
Conventional OFDM
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Performance of (Simplified) Matrix Inversion
N = 64, v = 200 km/h, fc = 17 GHz, TRMS = 1 s, sampling at T = 1s.
fDoppler = 3.15 kHz, Subc. spacing fsr = 31.25 kHz:
Compare to DVB-T: v = 140 km/h, fc = 800MHz: fdoppler = 100 Hz while fsr = 1.17 kHz
5 10 15 20 25 300
5
10
15
20
25
30
Input SNR
Conventional OFDM MMSE equalization simplified MMSE
k = 4
ConvOFDM
MMSE
Ou
tpu
t S
INR
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Receiver 1: Subconclusion
• Performance improvement of 4 .. 7
• Complexity can be reduced to ~2kN, k ~ 5 .. 10.
• Estimation of V(1) to be developed, V is already being estimated
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Receiver 3: Decision Feedback
Estimate
• data, • amplitudes and • derivatives
iteratively
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Receiver 3: Decision Feedback
Iteratively do the following:• Compare the signal before and after the slicer• Difference = noise + ICI + decision errors• Invert to retrieve modulated derivatives from ICI
– V(1).*A = -1 ICI– MMSE to minimize noise enhancements
• Remove modulation 1/A• Smooth to exploit correlation in V(1)
• Modulate with A• Feed through to estimate ICI• Subtract estimated ICI
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Receiver 3: DFE
Estimate V(1) in side chain
Pilot
Slicer
M6
+
x
x
+
Cancel Doppler
Estimated Amplitudes
3
ICI
-+
M7
Z10
Z8
Z7
Z6
Y2Y0
3
X1 X2
X3
+x
x
V
V’
A
N
Z9 =V’
A
1/A
A.*V
V
-
A
INT
Channel Model
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Performance of Receiver 3: DFE
Variance of decision variable after iterative ICI cancellation versus variance in conventional receiver
10-2
10-1
100
101
102
10-3
10-2
10-1
100
101
102
Variance Conventional
Va
ria
nce
Ne
w S
yste
m 3
Variance decision variable in conventional receiver
Var
ianc
e of
dec
isio
n va
riabl
e in
DF
E r
ecei
ver
afte
r IC
I ca
ncel
ling
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Error Count
Receiver 3: DFE
N = 64 out of 8192 subcarriers, v = 30 m/s, fc = 600 MHz TRMS / NT= 0.03, fDoppler = 60 Hz, Subcarrier spacing fsr = 1.17 kHz
0 10 20 30 40 50 60 70-30
-25
-20
-15
-10
-5
0
5
10
Decision FeedbackSample run N=64 9 errors -> 4 errors
Subcarrier Number
Am
plit
ud
e
Amplitudes
Derivatives
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Conclusions
Modeling the Doppler channel as a set of time-varying subcarrier
amplitudes leads to useful receiver designs.
Estimation of V(1) is to be added, V is already being estimated
Basic principle demonstrated by simulation
Gain about – 3 .. 6dB, – factor of 2 or more in uncoded BER, – factor 2 or more in velocity.
Promising methods to cancel FFT leakage (DVB-T, 4G)
More at http://wireless.per.nl
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Further Research Work
Optimise the receiver design and estimation of derivatives
Can we play with the waveform (or window) to make the tails of the filter steeper?
Can we interpret the derivatives as a diversity channel?
Can estimation of derivatives be combined with synchronisation?
Isn’t this even more promising with MC-CDMA?
Apply it to system design.