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Lecture # I
N. BOUBAKER [email protected]
Orthogonal Frequency Division Multiplexing
(OFDM): Basic OFDM System
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Outline
Motivations
Diagonalization of Vector Channels
Transmission of one OFDM Symbol
Transmission of sequence of OFDM Symbols
Sample and Symbol Time Synchronization
Carrier Frequency Synchronization
Peak-to-Average Power Ratio (PAPR) Issue
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OFDM: Motivations
Treat a Wideband FS fading channel as Multiple Narrowband Flat fading channels
Change in TX so that RX does not suffer from ISI Use FEC with codeword span across all sub-channels
achieve Frequency Diversity, but with no ISI problem
f
H(f)
Realization of Frequency
Selective Fading Channel
Each Sub-channel is a
Flat Fading channel
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OFDM : Motivations
Motivation:
± Split a frequency selective fading channel into multiple, say
N=1024, narrowband flat fading sub-channels
± Send the bits over these sub-channels in parallel
Serial
to
Parallel
modulator Mixer
f 1Com-
biner
modulator Mixer
f 1024
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OFDM: Motivation
Problems:
± Multiple transmitter front ends (mixer, modulator, etc)
± require guard bands
Solutions: ± Do all these in digital domain using a wide baseband signal
± Use DFT (discrete Fourier transform) to create the baseband
equivalent of the transmit signal and then up-covert it to the
center frequency using one front end
± As DFT is an orthogonal transformation, no guard band is
needed
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Diagonalization of Vector Channels
Consider a Vector Channel
with input x and output y
Want to diagonalize it such
that
i
N
j j jii n x H y !
!
§ !1 ,
nHxy
N in sd z iiii ,,1~
~
-!!
!
!!
nsD
UnU VsUyz
n
x yUV
s z
di
ni
si zi
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Maintaining the same SNR during
Diagonalization U has to be Unitary to prevent noise enhancement
V has to be Unitary to maintain the same transmit
power
unitaryisi
~~
0
0
U
UU
UUnnnnE
N
N H
H H H
!
!
!
unitaryisif VsEs
VsVEsxEx
H
H H H
!
!
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Advantage and Issues with
Diagonalization Need to find U and V s.t. UHV = DH is diagonal
± Decompose the vector channels into parallel channels with
different gain (allow adaptive modulation, and TX
optimization to be discussed in 2nd part of OFDM notes)
V depends on H (TX needs to know the vector
channel)
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Diagonalization of ISI channel
For channel is ISI, H is Toeplitz
is a circulant matrix if ¼¼¼
¼¼¼
½
»
¬¬¬
¬¬¬
-
«
!!
01
1
01
0
,
00
0
0
00
00
hh
h
hh
h
H H H i j ji
.
111/
/11
.
..
¼¼¼¼¼
¼
½
»
¬¬¬¬¬
¬
-
«
¼¼¼¼¼
¼
½
»
¬¬¬¬¬
¬
-
«
!!
01
1
01
101
,
00
0
000
0
0
0
0
hh
h
hh
hhh
H H H N ji ji
.
111/
/11
.
..
/
H~
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Diagonization of Circulant Matrix
If H is circulant, then
Note that THE W that diagonalized is independent
of ! ± Hence, TX does not need to know !!
Use cyclic prefix to create an effective circulant matrix
§
!
!
¹ º
¸©ª
¨!!
1
0
exp1
;~
N
m
N mn j
mnn
mn
H
eh
N
mn j
N
T
T
H
H WWHW
FT of the channelimpulse response:
Gains of the
subchannels
H~
H
~
H~
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Cyclic Prefix
Instead of transmitting x=Ws, transmit
Then,
Transmit a length N+Ns vector for a length-N datavector. Efficiency = N/(N+Ns) with Ns>L (ISI length)
For N=210=1024, Ns=10, Efficiency ~ 99%.
? AT
N x xx !~
xHxH ~
0
00
0
00
000
00
0
0
00
~ 1
01
01
01
10
1
01
01
01
01
!
¼¼¼
¼
½
»
¬¬¬
¬
-
«
¼¼¼¼
¼¼
½
»
¬¬¬¬
¬¬
-
«
!
¼¼¼
¼
½
»
¬¬¬
¬
-
«
¼¼¼¼
¼¼
½
»
¬¬¬¬
¬¬
-
«
!
N N
N
x
x
hh
hh
hh
hh
x
x
x
hh
hh
hh
hh
/
..
/
//
.
.
/
..
/
//
/.
..
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Add
Cyclic
Prefix
& Pulse
Shaping
OFDM Transmission
Serial
to
Parallel
IFFT
Parallel
to
Serial
Parallel
to
Serial
FFT
Serial
to
Parallel
Mixer
f c
Mixer &
Filterf c
Frequency
Selective
Channel
Frequency
omain
Samples
Timeomain
Samples
A B C
Matched
Filter
and
RemoveCyclic
Prefix
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Transmission of a sequence of
OFDM symbols Using a block of N samples to create an OFDM
symbol (x=Ws) and the cyclic prefix, ISI between
samples within an OFDM symbol is eliminated
What happens to the intersymbol interference between
OFDM symbols?
tNNs Ns
tNNs Ns N
TX:
RX:
No ISI
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Cyclic Prefix
Insert a Cyclic Prefix before every OFDM symbol
± Cycle Prefix length > Xmax
± Overhead is Xmax/(NTs) where Ts=1/B is the sampling period,
B is the bandwidth and N is the number of sub-carriers or
points in the IFFT
the larger the N, the smaller the overhead!!
If Xmax/Ts=Ns. Then, there will have N+Ns sampled
points for every OFDM symbol
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Cyclic Prefix
If we just take the last N points out of the N+Ns pointsto do the FFT at the receiver,
Then
where denotes circular convolution
and Yk = Hk Xk
where Yk , Xk , and Hk are the DFT of yn, xn, and hn, resp
nnn h x y !
NTs NsTs
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Advantages of OFDM
With cyclic prefix, we can eliminate ISI completely
Provide frequency diversity
± Forward error correcting code such as convolutional code
with interleaver is needed as some sub-carriers will be in
deep fade
Potential
± If the transmitter knows the channel conditions
can select only the good sub-carriers to transmit or
transmit different numbers of bits based on the sub-
carriers¶ gains
If the transmitter knows the channel, OFDM with bit
allocation is better than the best equalizer (e.g. MLSE)
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Sample and Symbol
Synchronization Sampling time synchronization
± Sampling Frequency needs to be correct, but sampling
instance offset only leads to linear phase shift in the sub-
channels¶ gains. (which will be handled by channel
estimation)
OFDM Symbol Synchronization
± Determine the beginning of the OFDM symbol and the
beginning of the cyclic prefix (to avoid Inter-OFDM symbol
interference) ± Use Cyclic Prefix
Compute Correlation between
two intervals separated by NTs
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Carrier Frequency Synchronization
Carrier Frequency offset can cause significant inter-
subcarrier interference
As there is no guard band, very small frequency offset
can lead to large inter-subcarrier (or inter-subchannel)interference
Very important and performance is sensitive to this
f
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Peak-to-Average Power Ratio
The data symbol, si, may be QPSK modulated
(constant magnitude), but the transmitted samples, xi,
is the output of the IFFT and hence takes values over a
wide range.
Statistically, as si are independent and has random
phase, xi approaches an Gaussian distribution when N
is large
Hence, high Peak-to-Average Power Ratio
2
2
max
i
ii
x
x PAPR
E!
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Disadvantages of OFDM
Overheads
± Cyclic Prefix: can be reduced by increasing N
± Power to transmit cyclic prefix: can be lower by increasing N
Implementation issues
± sensitivity to frequency offsets
especially when N is large and sub-carrier spacing issmall
± require highly linear power amp
high peak-to-average-power (PAP) ratio, especially when N is large
Typical value for N is 27 to 211