FUNDAMENTALS OF OFDM

7/27/2019 FUNDAMENTALS OF OFDM

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Wireless Information Transmission System Lab.

National Sun Yat-sen UniversityInstitute of Communications Engineering

Introduction to Orthogonal FrequencyDivision Multiplexing (OFDM)


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OutlineOFDM Overview

OFDM System Model

Orthogonality

Multi-carrier Equivalent Implementation by Using

Inverse Discrete Fourier Transform (IDFT)Cyclic Prefix (CP)

Summary


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OFDM Overview (2/3)OFDM

Orthogonal Frequency Division Multiplexing

The basic principle of OFDM is to split a high-rate datastream into a number of lower rate streams that are

transmitted simultaneously over a number of sub-carriers.

It eliminates or alleviates the problem ofmulti-path

channel fading effect, andlow spectrum efficiency.


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OFDM Overview (1/3)

Single carrier : data are

transmitted over only

one carrierSelective fading

Multi-carrier : data are sharedamong several carriers and

simultaneously transmittedFlat fading per subcarrier

Single carrier (SC) vs. multi-carrier (MC)


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OFDM Overview (3/3)OFDM modulation

Features

No intercarrier guard bands

Overlapping of bands

Spectral efficiency

Easy implementation by IDFTs

Very sensitive to synchronization

Problems of Peak to Average Power Ratio (PAPR)


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OFDM Transmitter

Figure 1

Data

Encodert

fs

= 1 ( ) ( ){ }njbna +

( )ndS/P MUX

tNffnffn

=+=

1,0

( )tD

ChannelInput

( )tf02cos

( )tf02sin

( )tfN 12cos

( )tfN 12sin

( )0a

( )0b

( )1Na

( )1Nb


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OFDM System Model (1/3)An OFDM system transmitter is shown in Figure 1.

The transmitted waveformD(t) can be expressed as

where

Using a two-dimensional digital modulation format, the datasymbols d(n) can be represented as a(n) +jb(n)

a(n) : in-phase component

b(n) : quadrature component

{ } )1()2sin()()2cos()()(1

0

=

+=N

n

nn tfnbtfnatD

tNffnffn

=+=

1and0


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OFDM System Model (2/3)

DataEncoder

tfs

=

1 ( ) ( ){ }njbna +( )nd S/P MUX ( )tD ChannelInput

( )tf02cos

( )tf02sin

( )tfN 12cos

( )tfN 12sin

( )0a

( )0b

( )1Na

( )1Nb

( )0a

tT

tN1+

( )0a

tT

tN

1+

( )1a ( )1Na1

t


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OFDM System Model (3/3)The serial data elements spaced by are grouped and

used to modulateNcarriers. Thus they are frequency

division multiplexed.

The signaling interval is then increased to , which

makes the system less susceptible to channel delay

spread impairments.

t

tN

Small-scale fading(Based on multipath time delay spread)

Flat Fading

2. Delay spread < Symbol preiod1. BW of signal < BW of channel

Frequency Selective Fading

1. BW of signal > BW of channel2. Delay spread > Symbol preiod


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Orthogonality (1/3)Consider a set of transmitted carriers as follows:

We now show that every two carriers are orthogonal to

each other.

In summary:

(2)1,...,1,0for)(2 0

==

+

Nnett

tN

nfj

n

=== tNabqpqpab

dtttb

aqp

)(andfor0

for)()()( *


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Orthogonality (2/3) --- Proof

tNabqptNqpj

ee

tNqpj

ee

dtedttt

batN

qpjtN

bqpj

tN

aqpj

tN

bqpj

b

a

tN

tqpjb

aqp

==

=

=

=

)(andfor,0)(2

1

)(2

)()(

)(1

)(2)(2

)(2)(2

)(2*


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Orthogonality (3/3)

tN

Very Sensitive to

Frequency Errors


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Mathematical Expression of

OFDM Signal (1/2)From above, we know that is the orthogonal

signal set. An OFDM signal based on this orthogonal

signal set can be written as:

nknknk

n

tfj

n

k

N

n

nnk

jbad

tNTT

n

ff

TtNnet

kTtdtx

n

,,,

0

2

1

0

,

,

01,...,2,1,0for)(where

(3))(Re)(

+=

=+=

==

=

=

=

( ){ }tn


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Mathematical Expression of

OFDM Signal (2/2)T: OFDM symbol duration

dk,,n : transmitted data on the n-th carrier of the k-th symbol

If there is only one OFDM symbol ( i.e. k= 0 ), it can be

simplified as:

( ) ( ){ } (4))(2sin)(2cos

)(Re)(

1

0

,,

1

0

,

=

=

=

=

=

=

k

N

n

nnknnk

k

N

n

nnk

kTtfbkTtfa

kTtCtx

{ } (5))2sin()2cos()(1

0

=

=N

n

nnnn tfbtfatx


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Multi-carrier Equivalent

Implementation by using IDFT (1/6)According to the structure of the transmitter,

transmitting an OFDM signal requiresNoscillators, an

extremely high cost for hardware implementation.

Solution: An equivalent method for transmitting OFDM

signals is using inverse discrete Fourier transform

(IDFT).


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Implementation by using IDFT (2/6)In general, each carrier can be expressed as:

We assume that there areNcarriers in the OFDM signal.

Then the total complex signal Ss(t) can be represented by:

( ) (6))()( )(2 ttfjcc ccetAtS +

=

( )

ly.respectivecarrier,th-of

frequencycarrierphase,amplitude,are),(),(and

where

(7))(1

)(

0

1

0

)(2

n

fttA

fnff

etAN

tS

nnn

n

N

n

ttfj

nsnn

+=

=

=

+


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Implementation by using IDFT (3/6)Then we sample the signal at a sampling frequency ,

andAn(t) andn(t) becomes:

Then the sampled signal can be expressed as:

t1

(9))(

(8))(

nn

nn

AtA

t

=

=

( ) (10)1)(1

0

)(2 0

=

++=N

n

tkfnfj

nsneA

NtkS

( )( ) (11)1)(1

0

22 0

=

+ =N

n

tfnkjtkfj

ns eeAN

tkS n


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Implementation by using IDFT (4/6)The inverse discrete Fourier transform (IDFT) is defined

as the following:

Comparing eq.(11) and eq.(12), the condition must besatisfied in order to make eq.(11) an inverse Fourier

transform relationship:

(12))(1

)( 21

0

NnkjN

n

efnFN

tkf

=

=

(13)1

tNf

=


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Implementation by using IDFT (5/6)If eq.(13) is satisfied,

is the frequency domain signal

is the time domain signal

is the sub-channel spacing

is the symbol duration in each sub-channel

This outcome is the same as the result obtained in the

system of Figure 1. Therefore IDFT can be used to

generate an OFDM transmission signal.

( )ntkfjneA

+02

)( tkSs f

tN


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Implementation by using IDFT (6/6)

tfs

=

1 ( ) ( ){ }njbna +

( )nd

tNffnffn

=+=

1,0

( )tDChannelInput

( )tf02cos

( )tf02sin

( )tfN 12cos

( )tfN 12sin

( )0a

( )0b

( )1Na

( )1Nb

tfs

=

1 ( ) ( ){ }njbna +( )nd

fNffnffn

=+=

1,0

( )tD ChannelInput

( )0d

( )1d

( )1Nd

( )2d


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Cyclic PrefixIn multipath channel, delayed replicas of previous OFDM signal

lead to ISI between successive OFDM signals.

Solution : Insert a guard interval between successive OFDM

signals.


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Cyclic PrefixGuard interval leads to intercarrier interference (ICI) in OFDMdemodulation

In DFT interval, difference between two subcarriers does notmaintain integer number of cycles loss of orthogonality.

Delayed version of subcarrier 2 causes ICI in the process ofdemodulating subcarrier 1.


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Cyclic PrefixCyclic prefix (CP) : A copy of the last part of OFDM signal is

attached to the front of itself.

[ ]nd

[ ]kD~

symbolsdataInput

[ ]0d

[ ]1d

[ ]1Nd

[ ]2d

[ ]0D

[ ]1D

[ ]2D

[ ]1ND

[ ]gNND


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Cyclic Prefix

All delayed replicas of subcarriers always have an

integer number of cycles within DFT interval no ICI


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OFDM Receiver

( )nr~ SymbolsdataOutput

( )nr


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Cyclic Prefix

One of the most important reasons to do OFDM is the efficient

way it deals with multipath delay spread.

To eliminate inter-symbol interference (ISI) , a guard time orcyclic prefix is introduced for each OFDM symbol.

The length of the guard time or cyclic prefix is chosen to be larger

than the delay spread.


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Bandwidth Efficiency

In a classical parallel system, the channel is divided into

Nnon-overlapping sub-channels to avoidinter-carrier

interference (ICI).

The diagram for bandwidth efficiency of OFDM system

is shown below:


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Summary

The advantage of the DFT-based OFDM system :

The use of IDFT/DFT can reduce the computation complexity.

The orthogonality between the adjacent sub-carriers will makethe use of transmission bandwidth more efficient.

The guard interval or cyclic prefix is used to resist the inter-

symbol interference (ISI).The main advantage of the OFDM transmission technique is its

high performance even in frequency selective channels.

The drawbacks of the OFDM system :It is highly vulnerable to synchronization errors.

Peak to Average Power Ratio (PAPR) problems.

Documents

FUNDAMENTALS OF OFDM