Time
Single Carrier System
T=1/B (1?s)
Frequency
B
(1MHz)
When ?_max>T
?Intersymbol interference (ISI)
Bandwidth = B
Symbol Duration T=1/B
Equaliser is needed to
reduce ISI
Frequency
Time
T (1ms)
B
OFDM Systems (N subcarriers)
?f
(1kHz)
?Available bandwidth
is divided into N
subband
?Each symbol
occupies a narrow
band but longer time
period
For each subcarriers
Bandwidth ?f=B/N
Symbol duration
T=N/B
e
j f t2
0?
e
j f tN2 1?
?
e
j f tN2
1? ?
e
j f t2
0?
e
j f tk2 ?
e
j f tk2 ?
Baseband model of an OFDM communication system
fkf k ??,1,,1,0 ????? Nk
-
-
-
For the uniform subcarrier spacing
? ?
?
?
?
?
???
.,0),(
,0,2e x p),(
o th e r w is ektg
Tttfjktg k?
??
?
?
?
????
???
??
?
TT
T
pkTdtktgdtptgktg
pkdtptgktg
0
2
0
*
0
*
.,),(),(),(
,,0),(),(
The basic orthogonal function for the kth subcarrier
is defined as
The basic functions satisfies the condition of
orthogonality
Structure of OFDM systems using FFT
Encoding
(BPSK…) IFFT
Guard
insertion
Lowpass
filtering
Carrier
modulation
Binary data
Channel
Carrier
demodulation
Lowpass
filtering
A/D
converting
Guard
deleting FFT
decoding
(BPSK…)
Binary data
OFDM symbol
synchronisation
Channel
estimation
X(k) x(n) x’(n) s(t)
y’(k) y(n) y’(n) r(t)
Example of OFDM signals
data Real part,cos(2?fkt) Imaginary part,sing(2?fkt)
1
-1
-1
1
1
OFDM
0 1 2 3 4 5 6
- 0, 4
- 0, 2
0
0, 2
0, 4
0, 6
0, 8
1
S u b ca r r i e r N u m b e r k
S
p
e
ct
r
a
Comparison with FDMA
ch1 ch2 ch3
Frequency
guard
Assume the information data sequence be X(k),k=0,1… N-1,
then the transmitted signal s(t) can be expressed as
? ? ? ? ? ?
? ? ? ?,10;0,2e x p
2e x p
1
0
1
0
???????
?
?
?
?
?
?
?
NpTtftpjpX
tfjpXts
N
p
p
N
p
?
?
fT ??
1and
The received signal r(t) is
? ? ? ? ? ? ? ?twthtstr ?? *
h(t):Channel response; w(t):AWGN signal
OFDM symbol period
The received on subcarrier k0 is
? ? ? ? ? ?dtftkjtr
T
kY
T
? ??? 0 2e x p1 ?
Assume h(t)=1,w(t)=0 (ideal channel),then Y(k) becomes
? ? ? ? ? ? ? ?? ?
?
?
????
T N
p
ftkjftpjpX
T
kY
0
1
0
2e x p2e x p1 ??
Using FFT and IFFT for carrier modulation/demodulation
In practice,such parallel data modulation and coherent
demodulation can be simply done by using inverse fast
Fourier transfer (IFFT) and fast Fourier transfer (FFT)
The basic orthogonal function is
1,.,,1,0,1,.,,1,0,2e x p)( ?????
?
??
?
?? NnNkkn
N
jkg ?
?
?
?
?
?
?
?
???
?
?
?
?
?
??
?
?
?
?
?
???
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
1
0
1
0
,0
2
e x p
2
e x p
,
2
e x p
2
e x p
N
n
N
n
pkkn
N
jpn
N
j
pkNkn
N
jpn
N
j
??
??
??
?
???????
1
0
2e x p)()( N
k
nkNjkXnx ?
Signal representations
10 ??? Nn
The transmitted discrete time domain signal x(n) is
After inserting guard interval,the signal becomes
x’(n) with length N+G
? ?
?
?
?
??
?
?
?
?
?
T
N
f
T
N
f
N
T
fB
2;0
2;
Assume the lowpass filter has rectangular shape
?
?
?
?
?
?? t
T
N
tb s in c)(
or,
The transmitted time domain signal is
? ?
? ?,s in c'
s in c*')(
1
0
1
0
?
?
?
?
?
?
?
?
?
?
?
?
??
??
?
?
??
?
?
?
?
?
?
?
?
??
?
?
??
?
?
?
?
?
?
?
?
??
N
n
N
n
nt
T
N
nx
t
T
N
N
T
ntnxts ?
,*” denotes the convolution of two signals
The received signal is
? ? ? ? ? ?? ? ? ?twthtstr ?? *
h(t),the equivalent baseband channel impulse response
t,time;
w(t),white Gaussian noise,
?r(t) is sampled to obtain a discrete time domain data
sequence y’(n),
?The OFDM symbol synchronisation is needed
?Removing the guard intervals,yield the time domain
OFDM samples y(n),n=0,1..N-1,
The received signal on subcarrier k is calculated as
10),(2e x p)(1)(
1
0
?????
?
??
?
? ?? ? ?
?
NkkWkn
N
jny
N
kY
N
n
?
10,2e x p)(1)(
1
0
????
?
??
?
? ?? ? ?
?
Nkkn
N
jnw
N
kW
N
n
?
OFDM spectrum and ICI
The OFDM subcarrier spectrum can be obtained as follows,
Modulating a unit signal on the subcarrier p,where 0?p?N-1
and N is the total number of subcarriers,the transmitted time
domain baseband OFDM signal can be represented as
? ? ? ?Tpt
T
jtx,0r e c t
2
e x p ?
?
?
?
?
?? ?
T=1/?f is the OFDM symbol period and ?f the
subcarrier separation
rect(0,T) represents the rectangular function of
the time period ?0,T?
The Fourier transform of the standard rectangular
function rect(-T/2,T/2) is the sinc(x) function
Then the time and frequency shifted rectangular function is
? ? ? ?? ?? ? ? ?? ?pTfj
pTf
pTffX ??
?
?? ?
?
? e x ps i n
Denoting the normalised frequency is k=Tf,where k is a real
value,then above equation can be simplified as
? ?? ?
? ? ? ?? ?pkjpk
pkS
pk ???
?? ?
?
? e x ps i n
,
For the integer k,the Sk,p represents the
received signal on subcarrier k,
The derived OFDM spectrum is a sinc(x) function with
phase rotation exp(-j?x),which is different from common
knowledge of sinc(x) function OFDM spectrum,
D OC U M E N T T Y P E
N OKIA R E S E A R C H C E N T E R T y pe D ateH ere
T y pe Y ou r N ameH ere
0 1 2 3 4 5 6 7 8
- 0, 4
- 0, 2
0
0, 2
0, 4
0, 6
0, 8
1
k = f T
Re
O FD M Sp e c t r u m
Sin c ( x )
Real part of
OFDM
spectrum
D O C U M E N T T Y P E
N O K I A R E S E A R C H C E N T E R T y pe D ateH ere
T y pe Y ou r N ameH ere
0 1 2 3 4 5 6 7 8
- 0, 8
- 0, 6
- 0, 4
- 0, 2
0
0.2
0.4
0.6
0.8
k = f T
Im
O F D M S p e ct r u m
Imaginary part of OFDM spectrum
Comparison of Spectrum and ICI
0 1 2 3 4 5 6 7 8
- 0, 4
- 0, 2
0
0, 2
0, 4
0, 6
0, 8
1
k = fT
Re
O F D M S p e c t r u m
I C I s ig n a ls
s in c ( x ) S p e c t r u m
I C I s ig n a ls
-3 -2 -1 0 1 2 3
0
0, 1
0, 2
0, 3
0, 4
0, 5
0, 6
0, 7
0, 8
0, 9
1
Amplitude Spectrum of OFDM
3,Carrier to interference ratio (CIR)
When the subcarriers are modulated by the signals ak,
(k=0,…,N-1),then the received signal on the 0th
subcarrier can be expressed as
? ? ??
?
?
1
0
0
N
p
pp aSY
where Sp can be obtained by substituting k=0
The desired carrier signal is determined by the signal
modulated on the 0th subcarrier,which can be written
as
00 aSC ?
The ICI signal on 0th subcarrier is sum of the
ICI signals generated by transmitting data on
the rest of subcarriers,This sum can be
represented as
?
?
?
?
1
1
N
p
pp aSI
Using the zero mean and independent conditions of data
sequence ap,that is E?ap?=0,E?apap?=?ap?2,E?apak?=0 for
k?p,then the average carrier power can be derived as
? ? ? ?20202 aESCE ?
The average ICI signal then can be calculated as
? ? ? ?201
1
22 aESIE N
p
p?
?
?
?
the CIR of the standard OFDM systems can be expressed as
?
?
?
?
1
1
2
2
0
N
p
pS
S
C I R
0 0, 0 5 0, 1 0, 1 5 0, 2 0, 2 5 0, 3 0, 3 5 0, 4 0, 4 5 0, 5
-5
0
5
10
15
20
25
30
35
40
45
N o rm a l i z e d F re q u e n c y O ffs e t ?
CI R
(d B)
T h e or y
B P S K
4P S K
16Q A M
CIR versus ? for a standard OFDM system
Correlative Coding for OFDM
??
?
??
1
0
L
i
ikik afb
P r e c od ing BP S K
Cor r e la ti on
Cod i ng
I F F T
Lo w pass
F i lt e r i ng
C h a n n e l
Lo w pass
F i lt e r i ng
F F TD e c od ing
B in a r y
D a ta
B in a r y
d a t a
a
k
b
k
? ? ??
?
? 1
0
L
i
i
i DfDF
or
Bandwidth Efficiency is not reduced
0 0,1 0,2 0,3 0,4 0,5
-5
0
5
10
15
20
25
30
35
40
45
N o r m al i ze d F r e que ncy O f f s e t ?
C I R
( dB )
S t a nda rd O F D M
( 1 - D )
( 1 - D )
2
( 1 - D )
3
S i m ul a t i ons
CIR for correlative coding
Time Domain Window
C o di n g P o l y n o mi a l s W i ndo w F u n c t i o ns W i ndo w S h a pe s
S t a n da r d OF DM,L =1; ? ?B n ? 1
P ( D) =( 1+ D ),L =2
? ?B n
N
n?
?
?
?
?
?
?co s
?
P ( D) =( 1- D ),L =2
? ?B n
N
n?
?
?
?
?
?
?s i n
?
P ( D) =( 1- D )
2
,L =3
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
n
N
n
N
nB
?
?
2
c o s1
2
1
s i n
2
( Ha nni ng w i n do w)
P ( D) =( 1- D )
3
,L =4 ? ?B n
N
n?
?
?
?
?
?
?s i n
?
3
Timing of OFDM symbols
Representation of the problems in timing
When the timing changes,the phase of the received subcarriers
change,The relation between the phase ?i of the subcarrier i and
the timing offset ? is given by
??? ii f2?
Synchronisation using the cyclic extension
? ? ? ? ? ?? ???? GT dTtrtrtx 0 ???
Because the number of independent samples is proportional to the number of
subcarriers,the extension correlation techniques is only effective when a larger
number of subcarriers are used
192 subcarriers,20% guard 48 subcarriers,20% Guard
Exercises
problems 12-3,12-4