OFDM what, why, how?

OFDM—what, why, how?

EIT 140, tom<AT>eit.lth.se

OFDM applications

Terrestrial digital TV broadcasting. DVB-T (Digital Video Broadcasting -Terrestrial) EN 300 744

Terrestrial radio broadcasting. DAB (Digital Audio Broadcasting), DRM(Digital Radio Mondial)

Wireless high-speed local networking and wireless Internet access. WLAN(Wireless Local Area Network) or WiFi IEEE 802.11a/g, WiMAX IEEE802.16e

Inhome high-speed networking over home wiring (powerline, phone-line,coax). ITU-T G.hn

High-speed Internet access via phone line. Digital Subscriber Line (ADSL,VDSL)

Wireless personal area networks. WiMedia/ecma-368

Mobile radio. LTE (Long Term Evolution) 3GPP Release 7,8

Elements of a generic communication system

Source

Source

SourceEncoder Encoder

Decoder Decoder

Channel

ChannelChannel

Encryption

Decryption

Modulator

Demodulator

Synchronization Equalization

Sink

source encoding: compression of the message as close to its entropy aspossible (lossless) or below (lossy)encryption: protection of the message from being understood or falsifiedby a third partychannel encoding: controlled introduction of redundancy in order toincrease reliability of transmissionmodulation: conversion of symbols into waveforms appropriate for thegiven channelsynchronization: retrieve block boundariesequalization: mitigate the distortion introduced by the channel

Time dispersion in a wireless channel

delay delay

power powerpower-delay profiletransmitted signal

τmax

τ0 τ1 τ2

p0p0p1p2

Multipath propagation leads to dispersion of the receive signal in time

Time dispersion in a wireline channelchannel input

0

0

0

0

channel output

0

0

0

0

voltage voltage

delaydelay

impulse responsetransmitted signal

τmax

0 0

v0v0

Different frequency-components experience different attenuation and different

delay −→ dispersion (“smearing”) in time

Discrete-time linear time-invariant (LTI) dispersive channel

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00

00

nn

n

n

n

n

n

n

δ(n)

δ(n− 1)

δ(n− 2)

s(0)δ(n)

s(1)δ(n− 1)

s(2)δ(n− 2)

h(n)

h(n− 1)

h(n− 2)

s(0)h(n)

s(1)h(n− 1)

s(2)h(n− 2)

s(n) = ∑k

s(k)δ(n− k) r(n) = ∑k

s(k)h(n− k)


A discrete-time, linear, time-invariant, causal, time-dispersive channel isdescribed by its impulse response h(n) 6= 0, n = 0, 1, . . . ,M of lengthM + 1

The channel performs linear convolution:

r(n) = ∑k

s(k)h(n− k)

Tapped delay-line model

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��z−1 z−1z−1

h(3)h(2)h(1)h(0)

s(n)

r(n)


Linear convolution, denoted by ∗, of a length-N input sequences(n), n = n0, . . . , n0+N−1 and h(n) results in an output sequence

r(n) = h(n)∗s(n) = ∑k

h(k)s(n− k) =

= s(n)∗h(n) = ∑k

s(k)h(n− k), n = n0, . . . , n0+N+M−1

of length N +M.

Since each input sample is “spread out” over M = ⌈τmax/TS⌉ additionalsamples, we refer to M as the dispersion of the channel (TS is thesampling period)

Time-dispersion←→ frequency selectivity

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delay

delay

amplitude

amplitude magnitude

magnitude

impulse response frequency response

frequency

frequency

short response in time

long response in time

flat frequency response

selective frequency response

τmax

τmax

Variation in frequency causes dispersion in time over τmax seconds

Communication: wideband versus narrowband

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delaydelay

amplitudeamplitude

magnitudemagnitude

wideband: Tsym ≪ τmax narrowband: Tsym ≫ τmax

BBfrequencyfrequency

τmaxτmax

Tsym Tsym

Effect of dispersion on block transmission: ISI

Length-N symbol: s(0), s(1), . . . , s(N − 1)

block No. i−1 block No. i block No. i+1

block No. i−1 block No. i block No. i+1

h(n)

MM M

NNN

Transmission of length-N symbols over an LTI channel with dispersion M

(length M + 1) causes inter-symbol interference (ISI)

Dilemma: data rate and intersymbol interference

Moderate symbol rate (with respect to dispersion)

moderate data rate

moderate ISI

High symbol rate (with respect to dispersion)

high data rate

severe ISI!

Real-world examples

Example

GSM

Guess: max. delay spread ≈

Guess: symbol duration ≈

Compute: ISI affects

Conclusion: ISI is manageable (5-tap equalizer is standard)

Example

ADSL (assume: data rate = 8Mbit/s, uncoded 16QAM, single-carrier)

Guess: max. impulse response duration ≈

Compute: symbol duration =

Compute: ISI affects symbol(s)

Conclusion: ISI is severe! (equalizer is not an option)

Single-carrier transmission

frequency domain

time domain constellation diagram

0

0

Tsym

Tsym

2Tsym

2Tsym

3Tsym

3Tsym

−5

0

0

B

5{x

(i)0 , x

(q)0 }

{x(i)1 , x

(q)1 }

{x(i)2 , x

(q)2 }

−3

−3

−1

−1

1

1

3

3

Idea: transmit several longer symbols simultaneously

frequency domain

time domain constellation diagram

0

0

3Tsym = TMC

3Tsym = TMC

−5

0

0

B/3

2B/3

B

5{x

(i)0 , x

(q)0 }

{x(i)1 , x

(q)1 }

{x(i)2 , x

(q)2 }

−3

−3

−1

−1

1

1

3

3

Idea (cont’d): N narrowband channels

magnitude

B

∆f frequency

Multicarrier-symbol duration TMC ≫ delay spread τmax

N narrowband channels with bandwidth ∆f = B/N instead of onewideband channel with bandwidth B

A multicarrier symbol in baseband is N = ⌈TMC/TS⌉ samples long

Subcarrier (tone) spacing ∆f = B/N = 1/TMC

Each subchannel is quasi frequency-flat

Idea (cont’d): cyclic extension

We make each transmit symbol at least τmax seconds (or L ≥ M samples,where M = ⌈τmax/TS⌉) longer

At the receiver, we remove this extension, which eliminates the ISI

If TMC ≫ τmax, the loss is bearable

block No. i−1 block No. i

block No. i−1 block No. i

h(n)

M M

LLL N N

Cyclic extension

We use sinusoidal transmit signal components

We extend the beginning of each multicarrier symbol by L ≥ M samples

Since each transmit signal component extends over an integer number ofperiods, this extension corresponds to copying the last L samples to thebeginning

The same holds for a linear combination of transmit signal components,i.e., for the complete transmit signal

Cyclic extension is a cheap operation (copying)

00 −L−L

00

N−1N−1

r( q )2 (n)s

( q )2 (n)

The “O” of OFDM: an experiment

spectrum transmission errors

0

0

0

0

0

0

0

0

0

0

0

0

−2

−2

−2

−2

−2

−2

2

2

2

2

2

2

4

4

4

4

4

4

6

6

6

6

6

6

8

8

8

8

8

8

We transmit awhite image(black spotsmark receiveerrors; noise-freechannel, nodispersion)

Certain values ofthe carrierspacing do notlead tointer-carrierinterference

Carrierwaveforms withsuch a spacingin frequency aremutuallyorthogonal

Why sinusoids?

Small reasons: easy to generate (simple hardware), offer an intuitivelyappealing control over the spectrum, etc.

BIG reason: infinitely-long sinusoids are eigen-functions of every lineartime-invariant (LTI) system!

We exploit this when transmitting finite-length sinusoids over a dispersivechannel:

channel input

0

0

0

0

0

channel output

0

0

0

0 M

0

Why sinusoids?

Input: sine wave −→ output: transient (τmax = MTS seconds long)followed by shifted and scaled sine wave

The transient is eliminated by removing the cyclic extension at the receiver

The influence of the channel reduces to scaling and shifting in time (2parameters = 1 complex coefficient per subcarrier)

We need the cyclic extension to produce the transient so that the channelanswers to our symbols as if they were infinitely long sinusoids

If we use orthogonal transmit signal components, a cyclic extension of atleast τmax seconds (L ≥ M) avoids not only ISI but also inter-carrierinterference (ICI)

Choice of multicarrier symbol length TMC — limits and

design criteria

τmax ≪ TMC ≪ Tcoh

Lower limit: TMC ≫ τmax keeps loss due to cyclic extension low

Upper limit:

The channel properties should not change during a symbol −→TMC ≪ Tcoh (otherwise our linear time-invariant channel model isinvalid)Latency

The coherence time Tcoh = 1/Bdop is a measure for the duration thechannel properties remain quasi constant

Variation in time causes dispersion in frequency over Bdop Hz

Dispersion in frequency occurs for example due to motion: Doppler effect

Dispersion in frequency due to motion: Doppler effect

When the fixed terminal (FT) transmits a signal with frequency f = c/λ, themobile terminal (MT) receives this signal at frequency f + ν = f + v ′/λ,where v ′ is the relative velocity of the MT with respect to the FT and c is thepropagation speed (for light c ≈ 3 · 108 m/s).

Note that v ′ is a signed quantity

FT

MT1

MT2

v1

v2

v ′1 = v1 cos α1

v ′2 = −v2 cos α2

α1

α2

ν is called the Doppler shift (example GSM: maximum Doppler shift for100 km/h and 900MHz: ν = v ′f /c ≈ 83Hz)

Choice of multicarrier symbol length TMC — limits and

design criteria

Example

HiperLAN2 (pedestrian mobility: v ≤ 15 km/h, f = 5.2 GHz, B = 20MHz,N = 64):

Guess: max. dispersion in time τmax ≈

Compute: max. dispersion in frequency (due to motion): Tcoh =

Compute: symbol duration TMC =

Conclusion: τmax TMC Tcoh

−→ HiperLAN2 should / should not work

Challenge: peaky transmit signal

Problem at receiver: clip noise limits performance

amplitude

amplitude

time

time

clip noise

unclipped

clipped signal

Problem at transmitter: clipping power-amplifier violates PSD specs

frequency

PSDmask clipping

no clipping

Challenge: synchronisation

“Rattling the grid”

multicarrier No. (time) −→

←−

subcarrierNo.(frequency)

symbol timing offset (shown above)

carrier frequency/phase offset

sampling clock offset



multicarrier No. (time) −→←−


∆FC






multicarrier No. (time) −→

←−


(N + L)∆TS

∆FS




Summary

Motivation: high data rate → high symbol rate → time-dispersive channelcauses severe ISI

Multicarrier modulation: simultaneous transmission of many, long, andmutually orthogonal (sinusoidal) symbols

Cyclic extension

avoids ISIavoids ICI

Multicarrier-system design limits: τmax ≪ TMC ≪ Tcoh

Documents

OFDM what, why, how?