Communication

1. IV. Orthogonal Frequency Division
Multiplexing (OFDM)

2. © Tallal Elshabrawy
Introduction
2
Evolution of Wireless Communication Standards
OFDM

3. © Tallal Elshabrawy 3
Wireless Communication Channels
 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
From “Wireless Communications” Edfors, Molisch, Tufvesson

4. © Tallal Elshabrawy 4
Multi-Path Propagation Modeling
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
Power
Time
τ0 τ1 τ2
Multi-Path
Components

5. © Tallal Elshabrawy 5
Multi-Path Propagation Modeling
As the mobile receiver (i.e. car) moves in the environment, the strength of each
multi-path component varies
Power
Time
τ0 τ1 τ2
Multi-Path
Components

6. © Tallal Elshabrawy 6
Multi-Path Propagation Modeling
Power
Time
τ0 τ1 τ2
Multi-Path
Components
As the mobile receiver (i.e. car) moves in the environment, the strength of each
multi-path component varies

7. © Tallal Elshabrawy
Multi-Path = Frequency-Selective!
7
1 μs
0.5 0.5
1 μs
0.5 0.5
1 μs
0.5 0.5
1
0.5
1
1
-1
1
-1
0.5
-0.5
1 μs
1 μs
1
-1
1
-1
0.5
-0.5
1 μs
f=0
f=1 MHz
f=500 KHz

8. © Tallal Elshabrawy
Multi-Path = Frequency-Selective!
 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
1 μs
0.5 0.5
0 0.5 1 1.5 2
f (MHz)
|H(f)|
1
h(t)

9. © Tallal Elshabrawy 9
 Subdivide wideband bandwidth into multiple narrowband sub-
carriers
 Bandwidth of each channel is selected such that each sub-carrier
approximately displays Flat Fading characteristics
 The bandwidth over which the wireless channel is assumed to
display flat fading characteristics is called the coherence
bandwidth
Power
Frequency
Frequency Division & Coherence Bandwidth

10. © Tallal Elshabrawy 10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
6
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
H(f)
Example Frequency Response for 3G Channel
Resolv
able
Path
Relative
Delay
(nsec)
Average
Power (dB)
1 0 0.0
2 310 -1.0
3 710 -9.0
4 1090 -10.0
5 1730 -15.0
6 2510 -20.0
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
Power Delay Profile
(Vehicular A Channel Model)
Snapshot for Frequency Response

11. © Tallal Elshabrawy 11
Example Frequency Response for 3G Channel
Resolv
able
Path
Relative
Delay
(nsec)
Average
Power (dB)
1 0 0.0
2 310 -1.0
3 710 -9.0
4 1090 -10.0
5 1730 -15.0
6 2510 -20.0
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
Power Delay Profile
(Vehicular A Channel Model)
Snapshot for Frequency Response
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
6
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
H(f)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
6
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
H(f)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
6
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
H(f)

12. © Tallal Elshabrawy
Frequency Division Multiplexing (FDM)
+
Binary
Encoder
Transmitting
Filter (f1)
Modulation
Binary
Encoder
Transmitting
Filter (f2)
Modulation
Binary
Encoder
Transmitting
Filter (fN)
Modulation
Wireless
Channel
Bandpass
Filter (f1)
Demod.
Bandpass
Filter (f2)
Demod.
Bandpass
Filter (fN)
Demod.

13. © Tallal Elshabrawy
Channel Bandwidth of FDM
13
TS
Tx Filter
Time-Limited Communications
0
-fc
Band-Limited Communications
Rectangular Filter Raised Cosine Filter
Tx Signal in
Time
TS
Tx Signal in
Frequency
fc
2/TS
Signal
Bandwidth Zero-to-Zero Bandwidth = 2/TS
0 fc+ RS/2
-fc- RS/2 -fc+ RS/2 fc- RS/2
Bandwidth = RS = 1/TS

14. © Tallal Elshabrawy
Orthogonal FDM
14
   
S
T
i j
0
cos 2πft cos 2πf t dt 0 i j
  

Is it possible to find carrier
frequencies f1, f2 … fN such that
       
 
S S
T T
i j i j i j
0 0
1
cos 2πft cos 2πf t dt cos2π f f t cos2π f f t dt
2
 
    
 
 
 
 
   
 
 
 
 
S
S
T
T
i j i j
i j
0 i j i j
0
sin2π f f t sin2π f f t
1
cos 2πft cos 2πf t dt
2 2π f f 2π f f
 
 
 
 
 
 
 

   
 
 
 
 
S
T
i j S i j S
i j
0 i j i j
sin2π f f T sin2π f f T
1
cos 2πft cos 2πf t dt
2 2π f f 2π f f
 
 
 
 
 
 
 


15. © Tallal Elshabrawy
Orthogonal FDM
15
   
S
T
i j
0
cos 2πft cos 2πf t dt 0 i j
  

Is it possible to find carrier
frequencies f1, f2 … fN such that
   
 
 
 
 
S
T
i j S i j S
i j
0 i j i j
sin2π f f T sin2π f f T
1
cos 2πft cos 2πf t dt
2 2π f f 2π f f
 
 
 
 
 
 
 

   
   
   
S
T
i j
0
i j S i j S
i j i j
S S
cos 2πft cos 2πf t dt 0
2π f f T nπ n=1,2,3, .... & 2π f f T mπ m=1,2,3, ....
n m
f f n=1,2,3, .... & f f m=1,2,3, ....
2T 2T

    
    


16. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
16
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
4
s
2
f
T

Ts
1
s
1
f
2T

2
s
1
f
T

3
s
3
f
2T


17. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
17
Ts
1
s
1
f
2T

2
s
1
f
T

The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
s s
πt 2πt
sin sin
T T
   
   
   
 
 
 
 
s s s
s
s
T T T
s s s s
0 0 0
T
T
s s
s s s s
0 0
πt 2πt πt 3πt
sin sin dt cos dt cos dt
T T T T
sin πt T sin 3πt T
πt 2πt
sin sin dt 0
T T πt T 3πt T
       
 
       
       
 
   
  
 
   
 
     
  


18. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
Ts
1
s
1
f
2T

3
s
3
f
2T

s s
πt 3πt
sin sin
T T
   
   
   
 
 
 
 
s s s
s
s
T T T
s s s s
0 0 0
T
T
s s
s s s s
0 0
πt 3πt 2πt 4πt
sin sin dt cos dt cos dt
T T T T
sin 2πt T sin 4πt T
πt 3πt
sin sin dt 0
T T 2πt T 4πt T
       
 
       
       
 
   
  
 
   
 
     
  


19. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
Ts
1
s
1
f
2T

4
s
2
f
T

s s
πt 4πt
sin sin
T T
   
   
   
 
 
 
 
s s s
s
s
T T T
s s s s
0 0 0
T
T
s s
s s s s
0 0
πt 4πt 3πt 5πt
sin sin dt cos dt cos dt
T T T T
sin 3πt T sin 5πt T
πt 4πt
sin sin dt 0
T T 3πt T 5πt T
       
 
       
       
 
   
  
 
   
 
     
  


20. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
Ts
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
2
s
1
f
T

3
s
3
f
2T

s s
2πt 3πt
sin sin
T T
   
   
   
 
 
 
 
s s s
s
s
T T T
s s s s
0 0 0
T
T
s s
s s s s
0 0
2πt 3πt πt 5πt
sin sin dt cos dt cos dt
T T T T
sin πt T sin 5πt T
2πt 3πt
sin sin dt 0
T T πt T 5πt T
       
 
       
       
 
   
  
 
   
 
     
  


21. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
Ts
2
s
1
f
T

4
s
2
f
T

s s
2πt 4πt
sin sin
T T
   
   
   
 
 
 
 
s s s
s
s
T T T
s s s s
0 0 0
T
T
s s
s s s s
0 0
2πt 4πt 2πt 6πt
sin sin dt cos dt cos dt
T T T T
sin 2πt T sin 6πt T
2πt 4πt
sin sin dt 0
T T 2πt T 6πt T
       
 
       
       
 
   
  
 
   
 
     
  


22. © Tallal Elshabrawy
Orthogonality of Sub-Carriers
The sinusoid signals with
frequencies f1, f2, f3, f4 are
all mutually orthogonal
over the symbol period Ts
Ts
4
s
2
f
T

3
s
3
f
2T

s s
3πt 4πt
sin sin
T T
   
   
   
 
 
 
 
s s s
s
s
T T T
s s s s
0 0 0
T
T
s s
s s s s
0 0
3πt 4πt πt 7πt
sin sin dt cos dt cos dt
T T T T
sin πt T sin 7πt T
3πt 4πt
sin sin dt 0
T T πt T 7πt T
       
 
       
       
 
   
  
 
   
 
     
  


23. © Tallal Elshabrawy
Orthogonal FDM
23
+
Binary
Encoder
Transmitting
Filter (f1)
Modulation
Binary
Encoder
Transmitting
Filter (f2)
Modulation
Binary
Encoder
Transmitting
Filter (fN)
Modulation
Wireless
Channel
Correlate
with (f1)
Demod.
Correlate
with (f2)
Demod.
Correlate
with (fN)
Demod.
f2=f1+1/2TS
fN=f1+1/2(N-1)TS

24. © Tallal Elshabrawy
Number of Subcarriers in OFDM
 For band-limited FDM if the system bandwidth is
B, number of sub-carriers is given by:
24
   
S
C
S
BT
B
N
1 α / T 1 α
 
 
 For OFDM if the system bandwidth is B, Number
of sub-carriers is given by:
C S
S
B
N 2BT
1/ 2T
 
0 α 1 Rolloff Factor
  
OFDM has the potential to at least double the
number of sub-carriers (i.e., double the total
transmission rate over the system bandwidth)

25. © Tallal Elshabrawy
+
Intersymbol Interference in OFDM
Ts
1
s
1
f
T

2
s
2
f
T

25
Ts
OFDM Symbol
Tx
Signal
Assume OFDM over two subcarriers: f1=1/Ts, f2=2/Ts

26. © Tallal Elshabrawy
Intersymbol Interference in OFDM
Suppose multi-path channel with delay Ts/8
26
h0 h1
Ts/8
Inter-Symbol
Interference (ISI)
Inter-symbol interference (ISI) occurs when one OFDM symbol affects the next
one due to the multi-path channel
OFDM Symbol
Tx
Signal
OFDM Symbol
Rx
Signal

27. © Tallal Elshabrawy
Inserting Guard Time
 Guard Time eliminates ISI between neighboring OFDM symbols
 However each OFDM symbol suffers from inter-carrier interference (ICI)
 Guard time corresponds to a reduction of bit rate
27
OFDM Symbol
Guard
Time
Ts Ts/4
Tx
Signal
No ISI
Ts
Ts/8
Rx
Signal
Suppose multi-path channel with delay Ts/8
h0 h1
Ts/8
Guard
Time
Ts Ts/4
Ts
Ts/8

28. © Tallal Elshabrawy
Guard Time & Inter-Carrier Interference
28
OFDM Symbol
Guard
Time
Ts
Ts/8
+
Tx Signal Rx Signal
OFDM Symbol
+
Guard
Time
Ts
Ts/8

29. © Tallal Elshabrawy
Guard Time & Inter-Carrier Interference
Rx Signal
Correlation at Rx over Ts
Consider the receiver for f1=1/Ts that
correlates over Ts with  
s
sin 2πt T
Ts
29
x
Not
Orthogonal
Intra-Carrier
Interference
+
Guard
Time
Ts
Ts/8
OFDM Symbol

30. © Tallal Elshabrawy
Guard Time & Inter-Carrier Interference
Consider the receiver for f1=1/Ts that
correlates over Ts with  
s
sin 2πt T
Ts
30
x
Orthogonal
No
Interference
Rx Signal
Correlation at Rx over Ts
+
Guard
Time
Ts
Ts/8
OFDM Symbol

31. © Tallal Elshabrawy
Guard Time & Inter-Carrier Interference
Consider the receiver for f1=1/Ts that
correlates over Ts with  
s
sin 2πt T
Ts
31
x
Not
Orthogonal
Inter-Carrier
Interference
Rx Signal
Correlation at Rx over Ts
+
Guard
Time
Ts
Ts/8
OFDM Symbol

32. © Tallal Elshabrawy
Ts
Ts/8
+
Cyclic Prefix
Tx Signal (Guard Time) Tx Signal (Cyclix Prefix)
The cyclic prefix is used to eliminate Inter-carrier interference
Cyclic Prefix
Cyclic Prefix
Cyclic Prefix
OFDM Symbol
Guard
Time
Ts
Ts/8
+
OFDM Symbol
Ts/8

33. © Tallal Elshabrawy
Cyclic Prefix
Rx Signal (Cyclix Prefix)
+
Cyclic Prefix
Correlation at Rx over Ts
Consider the receiver for f1=1/Ts that
correlates over Ts with  
s
sin 2πt T
Ts
x
Ts
Ts/8
Ts/8
Not
Orthogonal
Intra-Carrier
Interference

34. © Tallal Elshabrawy
Cyclic Prefix
Consider the receiver for f1=1/Ts that
correlates over Ts with  
s
sin 2πt T
Ts
x
Orthogonal
No
Interference
Rx Signal (Cyclix Prefix)
+
Cyclic Prefix
Correlation at Rx over Ts
Ts
Ts/8
Ts/8

35. © Tallal Elshabrawy
Cyclic Prefix
Consider the receiver for f1=1/Ts that
correlates over Ts with  
s
sin 2πt T
Ts
x
Orthogonal
No Inter-
Carrier
Interference
Rx Signal (Cyclix Prefix)
+
Cyclic Prefix
Correlation at Rx over Ts
Ts
Ts/8
Ts/8

36. © Tallal Elshabrawy
Cyclic Prefix (Summary)
Assume fi, fj are two OFDM sub-carriers and φij is the phase
shift associated with the cyclic prefix and multi-path channel
36
Cyclic Prefix eliminates Inter-carrier Interference (ICI)
   
         
           
   
s
s
s s
s
T
i j ij
0
T
i j ij j ij
0
T T
ij i j ij i j
0 0
T
i j ij
0
sin 2πft sin 2πf t φ dt
sin 2πft sin 2πf t cos φ cos 2πf t sin φ dt
cos φ sin 2πft sin 2πf t dt sin φ sin 2πft cos 2πf t dt
sin 2πft sin 2πf t φ dt 0

 
 
 
 
  


 


37. © Tallal Elshabrawy
Cyclic Prefix (Summary)
Assume an OFDM sub-carrier fi, and φii is the phase shift
associated with the cyclic prefix and multi-path channel
37
With Cyclic Prefix remains the component cos φii as a
source of Intra-Carrier Interference
   
         
         
         
s
s
s s
s s
T
i i ii
0
T
i i ii i ii
0
T T
2
ii i ii i i
0 0
T T
2
i i ii ii i ii
0 0
sin 2πft sin 2πft φ dt
sin 2πft sin 2πft cos φ cos 2πft sin φ dt
cos φ sin 2πft dt sin φ sin 2πft cos 2πft dt
sin 2πft sin 2πft φ dt cos φ sin 2πft dt 0 cos φ

 
 
 
 
    


 
 

38. © Tallal Elshabrawy
Cyclic Prefix vs Guard Time
38
Guard Time Cyclic Prefix
Eliminates Inter-symbol
Interference
Eliminates Inter-symbol
Interference
Suffers from Inter-carrier
Interference
Eliminates Inter-carrier Interference
Suffers from Intra-carrier
Interference
Suffers from Intra-carrier
Interference
Causes a reduction in data rate as
a result of the increased OFDM
symbol time
Causes a reduction in data rate as
a result of the increased OFDM
symbol time
Does not consume additional
power associated with OFDM
symbol time expansion due to the
guard time
Necessitates additional power
associated with OFDM symbol
expansion due to the introduction
of cyclic prefix

39. © Tallal Elshabrawy
OFDM with Cyclic Prefix System Model
39
Suppose multi-path channel with delay Ts/8
Each sub-carrier is treated as an independent
transmission
h0 h1
Ts/8
Cyclic Prefix
Correlation at Rx over Ts
Tx Signal Rx Signal
multiplied by h0 multiplied by h1
       
 
s s
T T
i 0 i i 1 i ii
0 0
0 1 ii
β sin 2πft h sin 2πft dt sin 2πft h sin 2πft φ dt
β h h cos φ
  
 
 
Fading Effect of the Channel

40. © Tallal Elshabrawy
Mitigation of Fading: Freq. Equalization
 Conduct channel estimation for h0 and h1
 Divide the correlated signal by β=h0+h1cos(φii)
40
 Requires channel estimation
 For low value values of β equalization also results
in noise amplification

41. © Tallal Elshabrawy
Mitigation of Fading: Precoding
 Conduct channel estimation for h0 and h1
 Divide the transmitted signal by β=h0+h1cos(φii)
41
 Requires channel estimation
 Requires channel estimation knowledge at
transmitter
 Does not result in any noise amplification at the
receiver
 For low values of β, excessively high transmission
power might be needed at the transmitter

42. © Tallal Elshabrawy
Mitigation of Fading: Adaptive Loading
 Distribute power over sub-carriers such as to
maximize total system data rate
42
 Requires channel estimation
 Requires channel estimation knowledge at
transmitter