OFDM Basics.ppt

Multi-Carrier Transmission
over Mobile Radio Channels
Jean-Paul M.G. Linnartz
Nat.Lab., Philips Research

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

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

I-FFT: OFDM Transmission
Transmission of QAM symbols on parallel subcarriers
Overlapping, yet orthogonal subcarriers
cos(wct+ wst)
cos(wct)
cos(wct+ iwst)
cos(wct+ (N-1)wst)
User
symbols
Serial-to-
parallel
=
Serial-to-
Parallel
I-FFT
Parallel-to-
Serial

OFDM Subcarrier Spectra
OFDM signal strength versus
frequency.
Rectangle <- FFT -> Sinc
before channel
after channel
Frequency

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 (?)

The Wireless Multipath Channel

The Mobile Multipath Channel
Delay spread Doppler spread
Frequency Time
FT
Frequency
FT
Frequency
Time

Effects of Multipath Delay and Doppler
Frequency
Time
Narrowband
Frequency
Time
OFDM
Wideband
QAM
Frequency
Time

Effects of Multipath (II)
Frequency
Time
+
-
+
-
-
+
-
+
DS-CDMA
Frequency
Time
+
-
-
Frequency
Hopping
Frequency
Time
+ - + -
+ - +
-
+ - +
-
MC-CDMA

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

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/S
I-FFT
N
S/P N
B
Code
Matrix
C
N
A

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 2
MS 1

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/S
I-Code
Matrix
C-1
FFT
N N N
Y
Weigh
Matrix
W
N
A

Receiver strategies (How to pick W ?)
• equalization (MUI reduction) w = 1/b
• maximum ratio combining (noise reduction) w = b
• Wiener Filtering (joint optimization) w = b/(b2 + c)
Next step: W can be reduced to an automatic gain control, per
subcarrier, if no ICI occurs
S/P P/S
I-Code
Matrix
C-1
FFT
N N N
Y
Weigh
Matrix
W
N
A

• 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/S
I-Code
Matrix
C-1
FFT
N N N
Y
Weigh
Matrix
W
N
A B
s
*
*
T
N
β
β
β
w
0
+


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)

Composite received signal
Wanted signal
Multi-user Interference (MUI)
Intercarrier interference (ICI)









+
  





 0
0
0
,
1
0
,
,
1
0
,
0
0 ]
[
]
[
m
n
n
N
n
n
m
n
n
N
n
n
n
s
m
n
c
n
c
w
w
N
T
b
x b
b






 





]
[
]
[
0
,
1
0
,
1
1
n
c
n
c
w
b
T
x k
n
n
N
n
n
n
N
k
k
s
MUI b





+

+

+



+

0
0
n
,
n
,
n
1
0
n ]
[n
c
w
a
T
x n
N
n
s
ICI b

Composite received signal
Wanted signal
Multi-User Interference (MUI)
Intercarrier interference (ICI)
 









 











1
0
2
,
ch
1
0
2
,
ch
0
2
0
1
1
2
2
E
E
)
(
)
(
E
E
1 N
n
n
n
N
n
n
m
k
N
k
k
ICI w
m
n
c
n
c
b
N
b

2
,
,
,
,
ch
1
1
2
2
2
*
ch
2
E
E
E
E










 




+



n
n
A
n
n
n
n
n
A
n
n
n
N
k
k
s
MUI
MUI
MUI w
w
b
N
T
x
x b
b

n
n
N
n
n
n
s
w
β
N
T
b
x ,
1
0
,
0
0 




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 spread
10-5
10-4
10-3
10-2
10-1
5 10 15
Local-mean En/N0
Eb/N0Eb/No (dB)
(1)
(2)
(3)
(4)
(5)
Avg. BER
AWGN
OFDM
Local-mean Eb/N0

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-CDM
Data 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.
 )


+










0
2
2
1
0
0
0
0
2
1
log
exp
2 dx
x
x
T
P
N
T
P
N
C
s
s
OFDM

















s
s
OFDM
T
P
N
E
T
P
N
C
0
0
1
0
0
2
2
exp
2
ln
1

Capacity
Capacity per dimension versus local-mean EN/N0,
no Doppler.
-5 0 5 10 15 20 25 30 35 40
0
1
2
3
4
5
6
7
Local-meanEn/N0(dB)
Capacity:
Bits
per
Subcarrier
-* : Rayleigh
* : MC-CDMA
- : LTI
Non-fading,
LTI
Rayleigh
MC-CDM

OFDM and MC-CDMA in a
rapidly time-varying channel
Doppler spread is the Fourier-dual of a delay
spread

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

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

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
10
0
Norm
alizedDoppler [fm
/fsub]
Power,
Variance
of
ICI
P0
P1 P2 P3
Power
or
variance
of
ICI
Doppler spread / Subcarrier Spacing
Neighboring subcarrier
2nd tier subcarrier
3rd tier subcarrier

BER in a mobile channel
0 5 10 15 20 25 30 35 40
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
AntennaSpeed(m
/s)
Local-Mean
BER
for
BPSK
OFDM,10dB
MC-CDMA,20dB 30dB
MC-CDMA,10dB
OFDM,20dB
OFDM,30dB
• 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= 896ms
• FFT size: N = 8192.
•Sub. spacing fs = 1.17 kHz
•Data rate 9.14 Msymbol/s.
Antenna Speed [m/s]

Doppler Multipath Channel
Received signal r(t)
Channel model:
Iw reflected waves have
the following properties:
• Di is the amplitude
• wI 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 data
•wc is the carrier frequency
•ws is the subcarrier spacing
)
(
}
)
)(
(
exp{
)
(
1
0
1
0
t
n
t
j
T
t
n
j
D
a
t
r
w
I
i
i
i
s
c
i
n
N
n
+
+

+
 





w
w
w

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
0
t
n
t
n
j
t
V
a
t
r s
c
N
n
n
n +
+
 


w
w
 )




+
+


1
0
)
(
}
)
(
exp{
w
I
i
t
i
i
s
c
i
q
i
q
n j
T
n
j
D
j
v w
w
w
w
)
(
}
)
)(
(
exp{
)
(
1
0
1
0
t
n
t
j
T
t
n
j
D
a
t
r
w
I
i
i
i
s
c
i
n
N
n
+
+

+
 





w
w
w

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.
 )
s
rms
q
p
q
q
p
D
q
m
p
n
T
m
n
j
j
q
p
q
p
f
v
v
w

)
(
1
)
1
(
!
)!
(
!
)!
1
(
2
E )
*(
)
(

+

+

+

+
+

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
q
q
m
n
q
n
f
t
s
n
m n
q
T
v
n
j
a
y
f
+



  







1
0 0
)
(
)
(
!
exp

w
Complex amplitudes
and derivatives
System constants
(eg sinc) determined
by waveform

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) = 2fT U G’.

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
  N
T
A
V
A
V
I
χ
Y N +








*
'.
*
.
3
0














+

+




0
2
1
2
0
1
1
1
0
3
..
..
..
..
..
..
..









N
N
N
N
FFT leakage
Amplitudes & Derivatives
User data

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 V
V
χ 
+
  N
AT
V
A
V
I
χ
Y N +








*
.
'
*
.
0

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
Channel
Estimator
Slicer
Q
-1
Y
3
X1 X2
X3
+
x
x
V
V’
A
N
V’
A
V

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

Simulation of channel for N = 64, v = 200 km/h fc = 17 GHz, TRMS =
1 ms, sampling at T = 1ms. fDoppler = 3.14 kHz, Subcarrier spacing
fsr = 31.25 kHz, signal-to-ICI = 18 dB
Amplitudes
First derivatives
Determined by speed of antenna,
and carrier frequency

SNR of decision variable. Simulation for N = 64, MMSE Wiener
filtering to cancel ICI
MMSE ICI canceller
Conventional OFDM

Performance of (Simplified) Matrix Inversion
N = 64, v = 200 km/h, fc = 17 GHz, TRMS = 1 ms, sampling at T = 1ms.
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 30
0
5
10
15
20
25
30
Input SNR
Conventional OFDM
MMSE equalization
simplified MMSE
k = 4
Conv
OFDM
MMSE
Output
SINR

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

Receiver 3: Decision Feedback
Estimate
• data,
• amplitudes and
• derivatives
iteratively

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

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
Y2
Y0
3
X1
X2
X3
+
x
x
V
V’
A
N
Z9 =V’
A
1/A
A.*V
V
-
A
INT
Channel Model

Performance of Receiver 3: DFE
Variance of decision variable after iterative ICI
cancellation versus variance in conventional receiver
10
-2
10
-1
10
0
10
1
10
2
10
-3
10
-2
10
-1
10
0
10
1
10
2
V ariance C onventional
V
ariance
New
S
ystem
3
Variance decision variable in conventional receiver
Variance
of
decision
variable
in
DFE
receiver
after
ICI
cancelling

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 Feedback
Sample run N=64 9 errors -> 4 errors
Subcarrier Number
Amplitude
Amplitudes
Derivatives

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

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.

OFDM Basics.ppt

More Related Content

Similar to OFDM Basics.ppt

Recently uploaded

OFDM Basics.ppt