# OFDM sp

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```IV. Orthogonal Frequency Division
Multiplexing (OFDM)
Introduction
Evolution of Wireless Communication Standards
OFDM
2
Wireless Communication Channels
From “Wireless Communications” Edfors, Molisch, Tufvesson



Communications over wireless channels suffer from multi-path
propagation
Multi-path channels are usually frequency selective
OFDM supports high data rate communications over frequency
selective channels
3
Multi-Path Propagation Modeling
Power
Multi-Path
Components
τ0
τ1
τ2
Time
Multi-path results from reflection, diffraction, and scattering off environment
surroundings
Note: The figure above demonstrates the roles of reflection and scattering only on multi-path
4
Multi-Path Propagation Modeling
Power
Multi-Path
Components
τ0
τ1
τ2
Time
As the mobile receiver (i.e. car) moves in the environment, the strength of each
multi-path component varies
5
Multi-Path Propagation Modeling
Power
Multi-Path
Components
τ0
τ1
τ2
Time
As the mobile receiver (i.e. car) moves in the environment, the strength of each
multi-path component varies
6
Multi-Path = Frequency-Selective!
f=0
0.5
0.5
1
1
0.5
1 μs
1 μs
f=1 MHz
1
0.5
-1
0.5
1 μs
1
0.5
-0.5
-1
1 μs
1
f=500 KHz
-1
0.5
0.5
1 μs
1
0.5
-0.5
-1
1 μs
7
Multi-Path = Frequency-Selective!
h(t)
|H(f)|
0.5
0.5
1
f (MHz)
1 μs
0
0.5
1
1.5
2
 A multi-path channel treats signals with different
frequencies differently
 A signal composed of multiple frequencies would
be distorted by passing through such channel
8
Frequency Division & Coherence Bandwidth
Power
Frequency



Subdivide wideband bandwidth into multiple narrowband subcarriers
Bandwidth of each channel is selected such that each sub-carrier
The bandwidth over which the wireless channel is assumed to
display flat fading characteristics is called the coherence
bandwidth
9
Example Frequency Response for 3G Channel
Power Delay Profile
(Vehicular A Channel Model)
10
Snapshot for Frequency Response
9
Resolv Relative
able
Delay
Path
(nsec)
0
8
7
6
0.0
H(f)
1
Average
Power (dB)
5
2
310
-1.0
3
710
-9.0
3
4
1090
-10.0
2
5
1730
-15.0
1
6
2510
-20.0
0
0
4
0.5
1
Simulation Assumptions
 Rayleigh Fading for each resolvable path
 System Bandwidth = 5 MHz
 Coherence Bandwidth = 540 KHz
 Number of Sub-Carriers = 64
 Sub-Carrier Bandwidth = 78.125 KHz
1.5
2
2.5
Frequency (Hz)
3
3.5
4
4.5
5
x 10
10
6
Example Frequency Response for 3G Channel
Power Delay Profile
(Vehicular A Channel Model)
10
Snapshot for Frequency Response
9
1
0
Average
Power (dB)
8
7
0.0
6
H(f)
Resolv Relative
able
Delay
Path
(nsec)
5
2
310
-1.0
3
710
-9.0
3
4
1090
-10.0
2
5
1730
-15.0
1
6
2510
-20.0
0
0
4
0.5
1
1.5
2
2.5
Frequency (Hz)
Simulation Assumptions
 Rayleigh Fading for each resolvable path
 System Bandwidth = 5 MHz
 Coherence Bandwidth = 540 KHz
 Number of Sub-Carriers = 64
 Sub-Carrier Bandwidth = 78.125 KHz
3
3.5
4
4.5
5
x 10
11
6
Frequency Division Multiplexing (FDM)
Binary
Encoder
Transmitting
Filter (f1)
Modulation
Bandpass
Filter (f1)
Demod.
Binary
Encoder
Transmitting
Filter (f2)
Modulation
Bandpass
Filter (f2)
Demod.
Bandpass
Filter (fN)
Demod.
+
Binary
Encoder
Transmitting
Filter (fN)
Modulation
Wireless
Channel
Orthogonal FDM
Is it possible to find carrier
frequencies f1, f2 … fN such that
TS
 cos  2πf t  cos  2πf t  dt  0
i
j
i  j
0
T

1 S
cos
2πf
t
cos
2πf
t
dt

cos
2π
f

f
t

cos
2π
f

f
t
dt



j
i
j
i
j
0  i 
2  0

TS











TS
1  sin2π fi  fj t sin2π fi  f j t 
0 cos  2πfit  cos 2πfjt dt  2  2π f  f  2π f  f 
i
j
i
j

0
TS














1  sin2π fi  fj TS sin2π fi  f j TS 
0 cos  2πfit  cos 2πfjt dt  2  2π f  f  2π f  f 
i
j
i
j


TS


13
Orthogonal FDM
Is it possible to find carrier
frequencies f1, f2 … fN such that
TS
 cos  2πf t  cos  2πf t  dt  0
i
j
i  j
0








1  sin2π fi  fj TS sin2π fi  f j TS 
0 cos  2πfit  cos 2πfjt dt  2  2π f  f  2π f  f 
i
j
i
j


TS


TS
 cos  2πf t  cos  2πf t  dt  0
i
0

j

 2π fi  f j TS  nπ


 fi  fj 
n
2TS

n=1,2,3, .... &

2π fi  f j TS  mπ
n=1,2,3, .... &
m
 f  f   2T
i
j
m=1,2,3, ....
m=1,2,3, ....
S
14
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f1 
1
2Ts
f2 
1
Ts
f3 
3
2Ts
f4 
2
Ts
15
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f1 
1
2Ts
f2 
1
Ts
 πt   2πt 
sin   sin 

T
T
 s  s 
s
s
 πt   2πt 
 πt 
 3πt 
sin
sin
dt

cos
dt

cos






0  Ts   Ts  0  Ts  0  Ts  dt
Ts
T
T
 sin  πt Ts  sin  3πt Ts  
 πt   2πt 
sin
sin
dt





0  Ts   Ts    πt Ts    3πt Ts    0

0
Ts
Ts
16
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f1 
1
2Ts
f3 
3
2Ts
 πt   3πt 
sin   sin 

T
T
 s  s 
s
s
 πt   3πt 
 2πt 
 4πt 
sin
sin
dt

cos
dt

cos






0  Ts   Ts  0  Ts  0  Ts  dt
Ts
T
T
 sin  2πt Ts  sin  4πt Ts  
 πt   3πt 
sin
sin
dt





0  Ts   Ts    2πt Ts    4πt Ts    0

0
Ts
Ts
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f1 
1
2Ts
f4 
2
Ts
 πt   4πt 
sin   sin 

T
T
 s  s 
s
s
 πt   4πt 
 3πt 
 5πt 
sin
sin
dt

cos
dt

cos






0  Ts   Ts  0  Ts  0  Ts  dt
Ts
T
T
 sin  3πt Ts  sin  5πt Ts  
 πt   4πt 
sin
sin
dt





0  Ts   Ts   3πt Ts    5πt Ts    0

0
Ts
Ts
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f2 
1
Ts
f3 
3
2Ts
 2πt   3πt 
sin 
 sin 

T
T
 s   s 
s
s
 2πt   3πt 
 πt 
 5πt 
sin
sin
dt

cos
dt

cos






0  Ts   Ts  0  Ts  0  Ts  dt
Ts
T
T
 sin  πt Ts  sin  5πt Ts  
 2πt   3πt 
sin
sin
dt





0  Ts   Ts    πt Ts   5πt Ts    0

0
Ts
Ts
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f2 
1
Ts
f4 
2
Ts
 2πt   4πt 
sin 
 sin 

T
T
 s   s 
s
s
 2πt   4πt 
 2πt 
 6πt 
sin
sin
dt

cos
dt

cos






0  Ts   Ts  0  Ts  0  Ts  dt
Ts
T
T
 sin  2πt Ts  sin  6πt Ts  
 2πt   4πt 
sin
sin
dt





0  Ts   Ts    2πt Ts    6πt Ts    0

0
Ts
Ts
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
f3 
3
2Ts
f4 
2
Ts
 3πt   4πt 
sin 
 sin 

T
T
 s   s 
s
s
 3πt   4πt 
 πt 
 7πt 
sin
sin
dt

cos
dt

cos






0  Ts   Ts  0  Ts  0  Ts  dt
Ts
T
T
 sin  πt Ts  sin  7πt Ts  
 3πt   4πt 
sin
sin
dt





0  Ts   Ts    πt Ts    7πt Ts    0

0
Ts
Ts
Orthogonal FDM
Binary
Encoder
Transmitting
Filter (f1)
Modulation
Correlate
with (f1)
Demod.
Modulation
Correlate
with (f2)
Demod.
Correlate
with (fN)
Demod.
f2=f1+1/2TS
Binary
Encoder
Transmitting
Filter (f2)
+
Wireless
Channel
fN=f1+1/2(N-1)TS
Binary
Encoder
Transmitting
Filter (fN)
Modulation
22
Number of Subcarriers in OFDM
 For band-limited FDM if the system bandwidth is
B, number of sub-carriers is given by:
BTS
B
NC 

1 α / TS 1 α
0  α  1  Rolloff Factor
 For OFDM if the system bandwidth is B, Number
of sub-carriers is given by:
NC 
B
 2BTS
1/ 2TS
OFDM has the potential to at least double the
number of sub-carriers (i.e., double the total
transmission rate over the system bandwidth)
23
OFDM a New Idea?





The idea of OFDM has been out there since the 1950s
OFDM was first used in military HF radios in late 1950s and
early 1960s
Early use of OFDM has been limited in commercial
communication systems due to the high costs associated with
the requirements for hundreds/thousands of oscillators
The use of OFDM has experienced a breakthrough in the 1990s
Currently, OFDM has been adopted in numerous wire-line and
wireless communications systems, such as:





Digital subscriber lines (DSL)
Wireless LAN 802.11
WiMAX 802.16
LTE (Long term Evolution), 4G Cellular Networks
24
OFDM & DFT (Discrete Fourier Transform)
OFDM Signal over 4 Sub-carriers f1  cos  πt Ts  f2  cos  2 πt Ts 
f3  cos  3πt Ts  f4  cos  4πt Ts 
Ts
f1 
1
2Ts
f2 
1
Ts
f3 
-f1
-f2
f2
3
2Ts
-f3
f4 
f1
f3
2
Ts
-f4
OFDM Signal:
Time Domain
f4
OFDM Signal:
Freq. Domain
25
OFDM & DFT (Discrete Fourier Transform)
OFDM Signal over 4 Sub-carriers f1  cos  πt Ts  f2  cos  2 πt Ts 
f3  cos  3πt Ts  f4  cos  4πt Ts 
OFDM Signal:
Time Domain
OFDM Signal:
Freq. Domain
DFT is means to generate samples of the OFDM signal in the frequency
and time domain without the use of oscillators
At the transmitter OFDM uses IDFT to convert samples of the spectrum of
the OFDM signal into a corresponding equal number of samples from the
OFDM signal at the time domain
At the receiver OFDM uses DFT to restore the signal representation in the
frequency domain and proceed with symbols detection
26
OFDM & DFT (Discrete Fourier Transform)
OFDM Signal over 4 Sub-carriers f1  cos  πt Ts  f2  cos  2 πt Ts 
f3  cos  3πt Ts  f4  cos  4πt Ts  (Separated by 1/2Ts)
We need to compute the
composite spectrum in the
frequency domain to be able to
compute the 4 samples used
by the IDFT
f1 
1
1
3
2
f2 
f3 
f4 
2Ts
Ts
2Ts
Ts
27
OFDM & DFT (Discrete Fourier Transform)
OFDM Signal over 4 Sub-carriersf1  cos  2πt Ts  f2  cos  4 πt Ts 
f3  cos  6πt Ts  f4  cos  8πt Ts  (Separated by 1/Ts)
The separation between
carriers guarantee that
samples from individual
spectrum of sub-carriers
correspond to samples from
the composite spectrum
f1 
1
Ts
f2 
2
Ts
f3 
3
Ts
f4 
4
Ts
28
Number of Subcarriers in OFDM with DFT
 For band-limited FDM if the system bandwidth is
B, number of sub-carriers is given by:
BTS
B
NC 

1 α / TS 1 α
0  α  1  Rolloff Factor
 For OFDM if the system bandwidth is B, Number
of sub-carriers is given by:
B
NC 
 BTS
1/ TS
OFDM with DFT has the potential to at increase the number of subcarriers compared to FDM for α>0 (remember that α=0 filter is not
physically realizable )
DFT implementation of OFDM avoids the needs for oscillators to
generate the OFDM signal
29
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