9. Multi Carrier Modulation and OFDM
Transmission of Data Through Frequency Selective Time
Varying Channels
We have seen a wireless channel is characterized by time
spread and frequency spread.
Frequency Spread
S ( , F )
F
FDMAX
 RMS
 RMS
 MEAN
 Time Spread
Single Carrier Modulation in Flat Fading Channels
• if symbol duration >> time spread then there is almost no Inter
Symbol Interference (ISI).
channel
TS
1
0
time
Problem with this: Low Data Rate!!!
1
0
phase still recognizable
… in the Frequency Domain
• this corresponds to Flat Fading
channel
1 / TS
Frequency
Frequency
Flat Freq. Response
Frequency
Single Carrier Modulation in Frequency Selective Channels
• if symbol duration ~ time spread then there is considerable Inter
Symbol Interference (ISI).
channel
time
1
0
? ?
phase not recognizable
One Solution: we need equalization
channel
equalizer
time
1
time
0
1
0
Channel and
Equalizer
Problems with equalization:
• it might require training data (thus loss of bandwidth)
• if blind, it can be expensive in terms computational effort
• always a problem when the channel is time varying
The Multi Carrier Approach
• let symbol duration >> time spread so there is almost no Inter
Symbol Interference (ISI);
• send a block of data using a number of carriers (Multi Carrier)
“symbol”
1
“symbol”
0
time
0
1
time
1
0
time
channel
Compare Single Carrier and Multi Carrier Modulation
SC
Frequency

1
Frequency
1
One symbol

Frequency
0 1 0 1 1 1
Block of
symbols
1
1

Flat Fading Channel:
Easy Demod
channel
MC
subcarriers
0
0 1 0 1 1 1

Frequency
Each subcarrier sees a
Flat Fading Channel:
Easy Demod
Structure of Multi Carrier Modulation
In MC modulation each “MC symbol” is defined on a time interval and it contains a
block of data
OFDM Symbol

data
data
data

data
time
TSymbol
data
t
Tg
Tb
guard interval
with
data interval
Tg   MAX
MAX channel time spread
Guard Time
We leave a “guard time”
between blocks to allow
multipath
TX
Guard Time
Tg
Data Block
the “guard time” is long
enough, so the multipath in
one block does not affect the
next block
Data Block
Tb


TSymbol
data+guard
RX
RX


Tg

TX
NO Inter Block Interference!
MC Signal
Transmitted Signal:

s(t )  Re e
j 2FC t
x(t )

FC carrierfrequency
Baseband Complex Signal:
k
x(t ) 
NF
2
c e
j 2kF t
k
k 
k 0
NF
2
0  t  TSymbol
kF  subcarrier frequencyoffset
ck  data
“Orthogonal” Subcarriers and OFDM
t
guard interval Tg
Tb data interval
1
Choose: F 
Tb
N F F
F
FC
Orthogonality:
1
Tb
t 0 Tb
e
t0
j 2Fk t  j 2F t
e
F
Fk  FC  kF
1
dt 
Tb
t 0 Tb
e
t0
j 2 ( k   ) Ft
1 if k  
dt  
0 if k  
Orthogonality at the Receiver
Transmitted
subcarrier
j 2Fk t
Channel
(LTI)
e
t
0
Received
subcarrier
Tg  Tb
x(t )   ck e j 2Fk t
k
0  t  Tg  Tb
h(t )
t
H ( F )  FTh(t )
0
Tg  Tb
Tg
transient steady state
response response
y (t )   ck H ( Fk )e j 2 Fk t
k
Tg  t  Tg  Tb
still orthogonal at the
receiver!!!
1  1
ck 
H ( Fk )  Tb

Tg Tb
 y(t )e
Tg
 j 2Fk t

dt 


OFDM symbols in discrete time
Let
• FS be the sampling frequency;
• N  NF
be the number of data samples in each symbol;
• F  1 /N TS   FS / N the subcarriers spacing
Then:
1
x(nTS ) 
N
with
NF
2
c e
k
k 
NF
2
j 2k FF ( n  L )
s
1

N
NF
2
c e
k
k 
Tg  L  TS the guard time.
NF
2
jk 2N ( n  L )
n  0,.., L  N  1
Summary OFDM Symbol
# samples
# subcarriers
guard
L
data
N  NF
TIME:
0
Tg
Tb
Sampling Interval TS  1 / FS
t
Freq spacing F  FS / N
FREQUENCY:
 FS / 2
N F
 F S
2 N
0
N F FS
2 N
FS / 2
F
OFDM Symbol and FFT
1
x[n  L] 
N
NF
2
c e
jk 2N n
k
k 
NF
2
NF
2
1

N
c e
1

N
N 1
k 1
jk 2N n
k
 X [k ]e
1

N
jk 2N n
1
c e
j ( N  k ) 2N n
k
k 
NF
2
IFFTX [k ]
k 0
Where:
X [k ]  ck ,
k  1,..., N F / 2
X [ N  k ]  ck ,
k  1,..., N F / 2
X [k ]  0,
otherwise
positive subcarriers
negative subcarriers
unused subcarriers
Guard Time with Cyclic Prefix (CP)
x[ L],...,x[ L  N 1]  IFFTX [k ], k  0,...,N 1
0
L  N 1
L
CP
IFFT{ X }


N
CP from the periodicity
x[n]  x[ N  n]

x[0]  x[ N ]
x[1]  x[ N  1]
...
x[ L  1]  x[ L  N  1]
OFDM Demodulator
See each block:
y[n]
0
L 1
n
L  N 1
No Inter Block Interference
y[n  L]  h[n]* x[n  L]
2
j kn
1 N 1
 h[n]*  X [k ]e N
N k 0
2
j kn
1 N 1
  H [k ] X [k ]e N  IFFT H [k ] X [k ]
N k 0
H [k ] X [k ]  FFTy[ L],...,y[ L  N 1]
with
H[k ]  FFT h[0],..., h[L 1],0,...,0 k  0,..., N  1
Overall Structure of OFDM Comms System
 X [ 0] 
 X [1] 

X 





X
[
N

1
]


IFFT
N
H [0] X [0]




H
[
1
]
X
[
1
]
 W
Y 





H
[
N

1
]
X
[
N

1
]


+CP
P/S
NL
N
NL
h[n]
w[n]
FFT
-CP
S/P
NL
N
N
NL
Simple One Gain Equalization
To recover the transmitted signal you need a very simple one
gain equalization:
received
transm.
noise
Y [k ]  H [k ] X [k ]  W [k ]
channel
Use simple Wiener Filter:
Xˆ [k ] 
H *[k ]
H [k ]  
2
2
W
Y [k ]
OFDM as Parallel Flat Fading Channels
Significance: a Freq. Selective Channel becomes N Flat Fading
Channels
h(t )
X m [0]


X m [ N 1]
x(t )
w(t )
y(t )
OFDM
Mod
OFDM
Demod
Frequency
Selective
channel

N Flat
Fading
Channels
Wm [0]
X m [0]
H [ 0]

X m [ N 1]

H [ N  1]
Ym [0]
Wm [ N 1]

Ym[ N 1]

Ym [0]

Ym[ N 1]
OFDM Parameters
Summarize basic OFDM Parameters:
• FS
sampling rate in Hz
• N length of Data Field in number of samples
• L length of Cyclic Prefix in number of samples
• NF  N
total number of Data Subcarriers
guard
data
data
t / TS
L
guard
guard
N
time
0
NF / N
frequency
F / FS
IEEE 802.11a:
Frequency Bands: 5.150-5.350 GHz and 5.725-5.825 GHz (12 channels)
Modulation OFDM
Range: 100m
IEEE 802.11g
Frequency Bands: 2.412-2.472GHz
Modulation: OFDM
Range: 300m
Channel Parameters: FCC
Example: the Unlicensed Band 5GHz U-NII (Unlicensed National
Information Infrastructure)
• 8 channels in the range 5.15-5.35GHz
30 MHz
20MHz
30 MHz

5150
5180
5200
5300
FC
• 4 channels in the range 5.725-5.825GHz
5320
5350
F (MHz)
Channel Parameters: Example IEEE802.11
In terms of a Transmitter Spectrum Mask (Sec. 17.3.9.2 in
IEEE Std 802.11a-1999)
0dB
Typical Signal
Spectrum
20dB
28dB
40dB
30
20
11 9
FC
Typical BW~16 MHz
9 11
20
30
F (MHz)
In either case:
FS  20MHz Sampling frequency
N  64
L  16
FFT size
Cyclic Prefix
CP
N  16
Tg  16 / 20  0.8 sec
DATA
N  64
Tb  64 / 20  3.2 sec
Sub-carriers: (48 data + 4 pilots) + (12 nulls) = 64
NULL
c1
c26
N F  52
NULL
c 26
c1
Frequency
Pilots at: -21, -7, 7, 21
0
1
0


26



38


63
63
IFFT
x0
N  64
x63
Time
Frequencies:
F  20 MHz / 64  312.5kHz
38
64  26
63 1
8.125
26
k
8.125
F ( MHz )
Subcarriers index
16 .25 MHz
DATA
F (MHz)
FCARRIER  10
FCARRIER
20MHz  1 / Ts
FCARRIER  10
Time Block:
Ts  TFFT / 64  50109 sec
TG  0.8 sec
TFFT  3.2 sec
TFRAME  4.0 sec
time
Overall Implementation (IEEE 802.11a with 16QAM).
1. Map encoded data into blocks of 192 bits and 48 symbols:
data
Encode
Buffer
Interleave
(192 bits)
…010011010101…
48
Map to
16QAM
…
1110
0111
1000
…
1101
4
4x48=192 bits
a
+1+j3
-1+j
+3-j3
…
+1-j
48
Overall Implementation (IEEE 802.11a with 16QAM).
2. Map each block of 48 symbols into 64 samples
time domain
frequency domain
xm [0]
null
0
0
+1+j3
…
-3-j
+3-j3
…
+1-j






24 data 

2 pilots 


null 

24 data 

2 pilots 
am [ ]


26
27
 27  64
 26  64
X m [k ]
 1: 48
1
2
1
 1 64
IFFT
k  0 : 63

 26
1 1
 
62
63
xm [62]
xm [63]
xm [n]
n  0 : 63

26
xm [1]
k
Channel Parameters: Physical
Frequency Spread
S ( , F )
F
FDMAX  kHz

Time Spread
 MAX  1  10 sec outdoor
 MAX  10  50 n sec indoor
Constraints on OFDM Symbol Duration:
 MAX  Tg  Tb  1/ FD
MAX
106 sec
to minimize CP overhead
103 sec roughly!!!
for channel Time Invariant
Summary of OFDM and Channel Parameters
Channel:
2. Doppler Spread
 MAX sec
FDMAX Hz
3. Bandwidth
BW Hz
1. Max Time Spread
FS Hz
4. Channel Spacing
OFDM (design parameters):
1. Sampling Frequency
FS
2. Cyclic Prefix
L   MAX FS
3. FFT size (power of 2)
4L  N  FS / FDMAX
4. Number of Carriers
NF   N  BW / FS  integer
integer
integer
Example: IEEE802.11a
Channel:
1. Max Time Spread
 MAX  0.5  sec
2. Doppler Spread
FDMAX  50 Hz
3. Bandwidth
BW  16 MHz
4. Channel Spacing
FS  20MHz
OFDM (design parameters):
1. Sampling Frequency
FS  20MHz
2. Cyclic Prefix
L  16  0.5  20  10
3. FFT size (power of 2)
N  64  20 106 / 50 integer
4. Number of Carriers
NF  52  64 16 / 20 integer
Applications: various Area Networks
According to the applications, we define three “Area Networks”:
• Personal Area Network (PAN), for communications within a few meters. This is the typical
Bluetooth or Zigbee application between between personal devices such as your cell phone,
desktop, earpiece and so on;
• Local Area Network (LAN), for communications up 300 meters. Access points at the
airport, coffee shops, wireless networking at home. Typical standard is IEEE802.11 (WiFi) or
HyperLan in Europe. It is implemented by access points, but it does not support mobility;
• Wide Area Network (WAN), for cellular communications, implemented by towers. Mobility
is fully supported, so you can move from one cell to the next without interruption. Currently it
is implemented by Spread Spectrum Technology via CDMA, CDMA-2000, TD-SCDMA,
EDGE and so on. The current technology, 3G, supports voice and data on separate networks.
For (not so) future developments, 4G technology will be supporting both data and voice on the
same network and the standard IEEE802.16 (WiMax) seems to be very likely
More Applications
1. WLAN (Wireless Local Area Network) standards and WiFi. In particular:
• IEEE 802.11a in Europe and North America
• HiperLAN /2 (High Performance LAN type 2) in Europe and North America
• MMAC (Mobile Multimedia Access Communication) in Japan
2. WMAN (Wireless Metropolitan Network) and WiMax
• IEEE 802.16
3. Digital Broadcasting
• Digital Audio and Video Broadcasting (DAB, DVB) in Europe
4. Ultra Wide Band (UWB) Modulation
• a very large bandwidth for a very short time.
5. Proposed for IEEE 802.20 (to come) for high mobility communications (cars,
trains …)