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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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)
Discrete-time linear time-invariant (LTI) dispersive channel
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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h(3)h(2)h(1)h(0)
s(n)
r(n)
Discrete-time linear time-invariant (LTI) dispersive channel
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
Challenge: synchronisation
“Rattling the grid”
multicarrier No. (time) −→←−
subcarrierNo.(frequency)
∆FC
symbol timing offset (shown above)
carrier frequency/phase offset
sampling clock offset
Challenge: synchronisation
“Rattling the grid”
multicarrier No. (time) −→
←−
subcarrierNo.(frequency)
(N + L)∆TS
∆FS
symbol timing offset (shown above)
carrier frequency/phase offset
sampling clock offset
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