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IERG 4100 Wireless Communications

Part X: OFDM

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Introduction

OFDM: Orthogonal Frequency Division Multiplexing

Converts a wideband frequency selective fading channel into a parallel collection of narrow band frequency flat sub-channels

Reduces the computational complexity associated with high data-rate transmission over frequency-selective channels

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History of OFDM

The basic principles of OFDM was proposed in several publications in the 1960’s.

Since 1966 FDM systems with overlapping spectra were proposed

The next step is a proposal to realize an FDM system with DFT

Finally, in 1971 Weinstein and Ebert proposed a complete OFDM system, which included generating the signal with an FFT and adding a guard interval in the case of multipath channels

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OFDM Applications

Broadcasting DAB (Digital Audio Broadcasting) DVB (Digital Video Broadcasting)

WLAN (Wireless local area network) IEEE 802.11a HiperLan/2

WMAN (Wireless metropolitan area network) IEEE 802.16 (WiMax)

4G LTE (Long Term Evolution)

5G ?

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Motivation

Inter-symbol interference in high-data-rate wireless communications

To avoid ISI, data rate is limited the radio environment – delay spread

Otherwise, equalizer is needed at the receiver to overcome ISI

OFDM can overcome and take advantage of multipath fading and thus eliminate inherent data rate limitations

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Time and Frequency Domain Description of Multipath

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Inter-symbol interference

Single-Carrier Transmission vs. OFDM

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time

frequency

……

Single carrier transmission:

time

……

frequency

OFDM (Multi carrier transmission):

Each symbol sees a frequency selective fading channel

Each symbol on a subcarrier sees a frequency flat fading channel

Single Carrier System

Sequential Transmission of WaveformsWaveforms are of short Duration T Waveforms occupy full system bandwidth 1/T

Multi-Carrier System

Parallel Transmission of waveforms Waveforms are of long duration MT Waveforms occupy 1/Mth of system bandwidth 1/T

Subcarriers in the Time Domain

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Subcarrier Orthogonality

In conventional FDMA The whole bandwidth is divided

into many narrow sub-channels which are spaced apart and not overlapped.

⇒ Low spectral efficiency In OFDM

By using orthogonal carriers with nulls at the center of the other carriers, the subchannels are overlapped.

⇒ Increase spectral efficiency

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In the frequency domain, the orthogonality is seen by zerosAll other subcarriers are zero when one subcarrier peaks

frequency

OFDM Transmitter and Receiver

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Add Cyclic

Prefix & Pulse

Shaping

Serial

to

Parallel

IFFT

Parallel

to

Serial

Parallel

to

Serial

FFT

Serial

to

Parallel

Mixer

fc

Mixer &Filter

fc

FrequencyDomainSamples

TimeDomainSamples

Matched Filter

and Remove Cyclic Prefix

channel

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DFT implementation

Equivalent baseband notation

At a sample rate of Ts/N

Since

(I)DFT can be much more efficiently implemented by (I)FFT

1

0

( ) exp 2 , 0N

n n sn

s t d j f t t T

s(k) =skTsN

⎛

⎝⎜⎞

⎠⎟= dnexp j2nk⋅Δf ⋅

TsN

⎛

⎝⎜⎞

⎠⎟n=0

N−1

∑ , 0 ≤k≤N −1

1

0

( ) exp 2 , 0 1N

n nn

nks k d j IDFT d k N

N

ΔfgTs =1

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DFT implementation

Matrix representation

s=FHd

F: FFT matrix

Each dn, n=0, 1, …, N1 is a modulated

frequency domain sample

Each sn, n=0, 1, …, N1 is a sample of the

OFDM symbol, i.e., time domain sample

OFDM Signal in the Time Domain

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Guard Interval

OFDM deals with ISI within one OFDM symbol (OFDM block)

Inter-block interference still exists Solution: Insert a guard interval that is longer than

the delay spread

Guard interval can consist of no signal. In this case, however the problem inter-carrier interference (ICI) would arise, since sub-carriers are no longer orthogonal

By cyclic prefix in OFDM symbol, ISI and ICI can be eliminated completely

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Cyclic Prefix

When the length of the cyclic prefix is larger than the delay spread, there is no inter-block interference after the cyclic prefix is removed

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Matrix representation of the ISI channel

Assume channel impulse response length is P

Matrix representation

1

0

P

t k t k tk

y h s n

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Circulant Matrix

A Circulant matrix is an n-by-n matrix whose rows are composed of cyclically shifted versions of a length-n list. For example, the circulant matrix on the list l={1, 2, 3, 4} is given by

One important property: a circulant matrix can be diagonalized by the Fourier transformation matrix

1 2 3 4

2 3 4 1

3 4 1 2

4 1 2 3

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Cyclic Prefix

In order to form a circulant matrix, instead of transmitting s, we transmit

Assume P=1, then

1 1, , , ,TT

N P N P Ns s s s s

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Cyclic Prefix

An effective circulant matrix is created using cyclic prefix

Efficiency: with ,since a vector of length will be transmitted for a length-N data vector

When N increases, efficiency increases

H

( )sN N N 1sN P

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Diagonalization of Circulant Matrix

Circulant matrix can be diagonalized aswhere

N parallel flat fading subchannels are created Note, the transmitter can diagonalize

without knowing any information about

HHFHF D

1exp 2kn

knj

NN

F

1

0

exp 2N

H knnk

knh j

N

D Gain of a sub-channel

HH

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Advantages of OFDM

With cyclic prefix, intra and inter OFDM symbol ISI can be eliminated completely

An effective circulant matrix can be created using cyclic prefix, as a result, ICI can be eliminated completely

Implementation complexity is significantly lower than that of a single carrier system with an equalizer

Provide frequency diversity Forward error correcting code such as convolutional

code with interleaver is needed as some sub-carriers will be in deep fade

Fading Across Subcarriers

Example:

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h(τ ) =0.8δ(τ ) + 0.6δ(τ −Ts)

diag(DH ) =FFT 0.8, 0.6, 0, 0, 0, 0, 0, 0[ ]( )=[1.4, 1.22 −0.424i, 0.8 −0.6i, 0.376 −0.424i, 0.2, 0.376 + 0.424i, 0.8 + 0.6i, 1.224 + 0.424i]

Different BERs Across Subcarriers

Compensation technique Coding across subcarriers Adaptive loading (power and rate)

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Variable-Rate Variable-Power MQAM

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γ : Channel to noise ratio | h |2 N0 B

Adaptive Techniques

Variable-rate variable-power techniques Fixed BER, maximize average data rate Fixed data rate, minimize average BER Fixed BER and data rate, minimize

average power

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Formulation

BER in non-fading AWGN channel with MQAM (M>=4) modulation and coherent detection:

Adaptive MQAM for fixed BER

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BER ≤0.2e−1.5γPM−1

Rate Maximization in Single-Carrier Systems

Optimal solution: Water filling

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maxP(γ)

E log2 M (γ)( )⎡⎣ ⎤⎦=maxP(γ )

E log2 1+1.5γP(γ)−ln 5BER( )

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢⎢

⎤

⎦⎥⎥

s.t. E P(γ)[ ] =P

P*(γ) =1γ0

−1γ

γ > γ0

0 otherwise

⎧

⎨⎪

⎩⎪

Power Minimization in Single-Carrier Systems

Practical (suboptimal) solution: Fix M. Transmit at the minimum power that

meets the BER performance

Optimal solution: water filling with a carefully chosen water

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minM (γ)

E P γ( )⎡⎣ ⎤⎦=minM (γ )

EM (γ)−1( ) ln

15BER

1.5γ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

s.t. E M (γ)[ ] =M

Constellation Restriction

M is restricted to {0, …, MN} Carefully design region boundaries Power control maintains target BER

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Adaptive Loading in Multi-Carrier Systems

Pros: Smaller rate and power fluctuation Requires smaller buffer size Channel gains are known

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Rate Maximization

Concave maximization Transmit power per OFDM symbol is

fixed Constellation constraint can be imposed

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maxPk (γ)

log2 Mk(γk)( )k=1

N

∑ =maxPk(γ )

log2 1+1.5γkPk(γk)−ln 5BER( )

⎛

⎝⎜⎞

⎠⎟k=1

N

∑

s.t. Pk(γk)k=1

N

∑ =NP

Power Minimization

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minM k (γ)

Pk γk( )k=1

N

∑ =minMk(γ )

Mk(γk)−1( ) ln1

5BER1.5γk

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥k=1

N

∑

s.t. Mk(γk)k=1

N

∑ =NM

Linear programming Data rate per OFDM symbol is fixed Constellation constraint can be imposed