Post on 02-Jan-2016
IV. Orthogonal Frequency Division Multiplexing (OFDM)
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Introduction
2
Evolution of Wireless Communication Standards
OFDM
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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
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Multi-Path Propagation Modeling
Multi-path results from reflection, diffraction, and scattering off environment surroundingsNote: The figure above demonstrates the roles of reflection and scattering only on multi-path
Power
Timeτ0 τ1 τ2
Multi-Path Components
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Multi-Path Propagation Modeling
As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies
Power
Timeτ0 τ1 τ2
Multi-Path Components
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Multi-Path Propagation Modeling
Power
Timeτ0 τ1 τ2
Multi-Path Components
As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies
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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
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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)
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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
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 106
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
H(f
)
Example Frequency Response for 3G Channel
Resolvable 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 -20.0Simulation 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
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Example Frequency Response for 3G Channel
Resolvable 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 -20.0Simulation 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 106
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 106
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 106
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
H(f
)
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Frequency Division Multiplexing (FDM)
+
BinaryEncoder
Transmitting Filter (f1)
Modulation
BinaryEncoder
Transmitting Filter (f2)
Modulation
BinaryEncoder
Transmitting Filter (fN)
Modulation
WirelessChannel
BandpassFilter (f1)
Demod.
BandpassFilter (f2)
Demod.
BandpassFilter (fN)
Demod.
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Orthogonal FDM
13
ST
i j
0
cos 2πf t cos 2πf t dt 0 i j
Is it possible to find carrier frequencies f1, f2 … fN such that
S ST T
i j i j i j
0 0
1cos 2πf t cos 2πf t dt cos2π f f t cos2π f f t dt
2
SS
TT
i j i j
i j
0 i j i j0
sin2π f f t sin2π f f t1cos 2πf t cos 2πf t dt
2 2π f f 2π f f
STi j S i j S
i j
0 i j i j
sin2π f f T sin2π f f T1cos 2πf t cos 2πf t dt
2 2π f f 2π f f
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Orthogonal FDM
14
ST
i j
0
cos 2πf t cos 2πf t dt 0 i j
Is it possible to find carrier frequencies f1, f2 … fN such that
STi j S i j S
i j
0 i j i j
sin2π f f T sin2π f f T1cos 2πf t cos 2πf t dt
2 2π f f 2π f f
ST
i j
0
i j S i j S
i j i jS S
cos 2πf t cos 2πf t dt 0
2π f f T nπ n=1,2,3, .... & 2π f f T mπ m=1,2,3, ....
n mf f n=1,2,3, .... & f f m=1,2,3, ....
2T 2T
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Orthogonality of Sub-Carriers
15
The sinusoid signals with frequencies f1, f2, f3, f4 are all mutually orthogonal over the symbol period Ts
4s
2f
T
Ts
1s
1f
2T
2s
1f
T
3s
3f
2T
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Orthogonality of Sub-Carriers
16
Ts
1s
1f
2T
2s
1f
T
The sinusoid signals with frequencies f1, f2, f3, f4 are all mutually orthogonal over the symbol period Ts
s s
πt 2πtsin sin
T T
s s s
ss
T T T
s s s s0 0 0
TTs s
s s s s0 0
πt 2πt πt 3πtsin sin dt cos dt cos dt
T T T T
sin πt T sin 3πt Tπt 2πtsin sin dt 0
T T πt T 3πt T
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Orthogonality of Sub-Carriers
The sinusoid signals with frequencies f1, f2, f3, f4 are all mutually orthogonal over the symbol period Ts
Ts
1s
1f
2T
3s
3f
2T
s s
πt 3πtsin sin
T T
s s s
ss
T T T
s s s s0 0 0
TTs s
s s s s0 0
πt 3πt 2πt 4πtsin sin dt cos dt cos dt
T T T T
sin 2πt T sin 4πt Tπt 3πtsin sin dt 0
T T 2πt T 4πt T
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Orthogonality of Sub-Carriers
The sinusoid signals with frequencies f1, f2, f3, f4 are all mutually orthogonal over the symbol period Ts
Ts
1s
1f
2T
4s
2f
T
s s
πt 4πtsin sin
T T
s s s
ss
T T T
s s s s0 0 0
TTs s
s s s s0 0
πt 4πt 3πt 5πtsin sin dt cos dt cos dt
T T T T
sin 3πt T sin 5πt Tπt 4πtsin sin dt 0
T T 3πt T 5πt T
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Orthogonality of Sub-CarriersTs
The sinusoid signals with frequencies f1, f2, f3, f4 are all mutually orthogonal over the symbol period Ts
2s
1f
T
3s
3f
2T
s s
2πt 3πtsin sin
T T
s s s
ss
T T T
s s s s0 0 0
TTs s
s s s s0 0
2πt 3πt πt 5πtsin sin dt cos dt cos dt
T T T T
sin πt T sin 5πt T2πt 3πtsin sin dt 0
T T πt T 5πt T
© 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
2s
1f
T
4s
2f
T
s s
2πt 4πtsin sin
T T
s s s
ss
T T T
s s s s0 0 0
TTs s
s s s s0 0
2πt 4πt 2πt 6πtsin sin dt cos dt cos dt
T T T T
sin 2πt T sin 6πt T2πt 4πtsin sin dt 0
T T 2πt T 6πt T
© 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
4s
2f
T
3s
3f
2T
s s
3πt 4πtsin sin
T T
s s s
ss
T T T
s s s s0 0 0
TTs s
s s s s0 0
3πt 4πt πt 7πtsin sin dt cos dt cos dt
T T T T
sin πt T sin 7πt T3πt 4πtsin sin dt 0
T T πt T 7πt T
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Orthogonal FDM
22
+
BinaryEncoder
Transmitting Filter (f1)
Modulation
BinaryEncoder
Transmitting Filter (f2)
Modulation
BinaryEncoder
Transmitting Filter (fN)
Modulation
WirelessChannel
Correlate with (f1)
Demod.
Correlate with (f2)
Demod.
Correlatewith (fN)
Demod.
f2=f1+1/2TS
fN=f1+1/2(N-1)TS
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Number of Subcarriers in OFDM
For band-limited FDM if the system bandwidth is B, number of sub-carriers is given by:
23
S
CS
BTBN
1 α / T 1 α
For OFDM if the system bandwidth is B, Number of sub-carriers is given by:
C SS
BN 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)
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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 with advancements in DSP hardware
Currently, OFDM has been adopted in numerous wire-line and wireless communications systems, such as: Digital audio and video broadcasting Digital subscriber lines (DSL) Wireless LAN 802.11 WiMAX 802.16 LTE (Long term Evolution), 4G Cellular Networks
24
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OFDM & DFT (Discrete Fourier Transform)
25
OFDM Signal over 4 Sub-carriers 1 sf cos πt T 2 sf cos 2πt T
3 sf cos 3πt T 4 sf cos 4πt T
4s
2f
T
Ts
1s
1f
2T
2s
1f
T
3s
3f
2T
OFDM Signal:Time Domain
-f1 f1
-f2 f2
-f3 f3
-f4 f4
OFDM Signal:Freq. Domain
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OFDM & DFT (Discrete Fourier Transform)
26
OFDM Signal over 4 Sub-carriers 1 sf cos πt T 2 sf cos 2πt T
3 sf cos 3πt T 4 sf cos 4πt T
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
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4s
2f
T1
s
1f
2T 2
s
1f
T 3
s
3f
2T
OFDM Signal over 4 Sub-carriers
(Separated by 1/2Ts) 1 sf cos πt T 2 sf cos 2πt T
3 sf cos 3πt T 4 sf cos 4πt T
We need to compute the composite spectrum in the frequency domain to be able to compute the 4 samples used by the IDFT
OFDM & DFT (Discrete Fourier Transform)
© Tallal Elshabrawy
4s
4f
T1
s
1f
T 2
s
2f
T 3
s
3f
T
28
OFDM Signal over 4 Sub-carriers
(Separated by 1/Ts) 1 sf cos 2πt T 2 sf cos 4πt T
3 sf cos 6πt T 4 sf cos 8πt T
The separation between carriers guarantee that samples from individual spectrum of sub-carriers correspond to samples from the composite spectrum
OFDM & DFT (Discrete Fourier Transform)
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Number of Subcarriers in OFDM with DFT
For band-limited FDM if the system bandwidth is B, number of sub-carriers is given by:
29
S
CS
BTBN
1 α / T 1 α
For OFDM if the system bandwidth is B, Number of sub-carriers is given by:
C SS
BN BT
1/ T
0 α 1 Rolloff Factor
OFDM with DFT has the potential to at increase the number of sub-carriers 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