IV. Orthogonal Frequency Division Multiplexing (OFDM)

© Tallal Elshabrawy

Introduction

2

Evolution of Wireless Communication Standards

OFDM

© 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


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



As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies

Power

Timeτ0 τ1 τ2




Power

Timeτ0 τ1 τ2


As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies


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


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)


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


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


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

)


Frequency Division Multiplexing (FDM)

+

BinaryEncoder

Transmitting Filter (f1)

Modulation

BinaryEncoder


Modulation

BinaryEncoder

Transmitting Filter (fN)

Modulation

WirelessChannel

BandpassFilter (f1)

Demod.

BandpassFilter (f2)

Demod.

BandpassFilter (fN)

Demod.


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


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


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



16

Ts

1s

1f

2T

2s

1f

T


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




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




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


Orthogonality of Sub-CarriersTs


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




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




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


Orthogonal FDM

22

+

BinaryEncoder


Modulation

BinaryEncoder


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


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)


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

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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



26

OFDM Signal over 4 Sub-carriers 1 sf cos πt T 2 sf cos 2πt 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


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


We need to compute the composite spectrum in the frequency domain to be able to compute the 4 samples used by the IDFT



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


The separation between carriers guarantee that samples from individual spectrum of sub-carriers correspond to samples from the composite spectrum



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

IV. Orthogonal Frequency Division Multiplexing (OFDM)

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