Outline Chapter 4: Orthogonal Frequency Division Multiplexing file¾Orthogonal Frequency Division...

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Outline Chapter 4: Orthogonal Frequency Division Multiplexing

Fading ChannelFlat fading channelFrequency selective channel ISISingle Carrier Equalization

Orthogonal Frequency Division MultiplexingPrinciple of Multi-carrier systemsInter Carrier InterferenceOrthogonal Frequency Division Multiplexing: OFDMCyclic Prefix – EqualizationImpulse Shortening EqualizerOFDM Channel EstimationDigital-to-Analog Interface

4. OFDM

2

Fading Channel

Transmission channel is in general frequency selective

Transmit signal is narrow band (small bandwidth B)Frequency response is almost constant in considered band: H(jω) = H0

Impulse response of channel is given by weighted dirac

1-Path Fading Channel (memoryless channel)

ω

s[k] n[k] r[k]

B

|H(jω)|

h[k] = h0δ[k]

h0

4. OFDM

3

1-Path-Fading Channel

Signal space diagram for QPSK transmissionTransmitted QPSK signal: s[k]Rotation & attenuation by h0: h0·s[k]Receive signal: r[k] = h0·s[k]+ n[k]

For signal detection r[k] is de-rotated

Transmit signal has long time duration small baud rate

s[k] r[k]n[k]

d[k]

b[`]

s[k] =h∗0|h0|2 · r[k] =

h∗0|h0|2 (h0s[k] + n[k])

= s[k] + n[k]

h0 h∗0/|h0|2

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(L+1)-Path-Fading Channel

To increase baud rate, time per symbol has to be reduced Short transmission pulse leads to increased band width BFrequency selectivity of channel becomes effective

Discrete impulse response of channel is given by

L denotes the number of “memory elements” w.r.t. Ts

System does not fulfill 1st Nyquist condition ISI

ω

1z−

1z−

s[k] r[k]h0

h1

hL

n[k]

B

τ

|h(τ )||H(jω)|

h0h1h2 h3 h4 h5 h6

h[k] = h0δ[k] + h1δ[k − 1] + . . .+ hLδ[k − L]

h[k]

r[k] = h0s[k] + h1s[k − 1] + . . .+ hLs[k − L] + n[k]

4. OFDM

5

Single Carrier EqualizationSignal space diagram for QPSK transmission

Transmitted QPSK signal s[k]Influence of frequency selective channel

ISI

Disturbance by noise n[k]ISI-channel plus noise

Estimation of signals becomes demanding task

Equalization schemesLinear equalization (applying linear filter)Decision Feedback Equalizer (DFE)Maximum Likelihood Sequence Estimation, i.e. Viterbi Algorithm

effort increases exponentially with channel memory

PL`=0 h`s[k − `]

r[k] =PL

`=0 h`s[k − `] + n[k]

4. OFDM

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4. OFDM

7

f

B

t

|H(f )|

N · Ts

Principle of Multi-Carrier Systems

t

B

f

τ

|h(τ )|

Intersymbolinterference of single-carrier systems

MLSE equalization using Viterbi algorithmComplexity increases exponentially with the channel memory

Ts

Multi-carrier systems experience non-selective sub-channels

Simple equalization with scalar multiplication

|H(f )|

4. OFDM

8

Multi Carrier Transmission (MC)

Dest. P/S

h t( )

Demod.

Demod.

Demod.

h t( )

h t( )ld( )M

d i ( )1

d i ( )N-1

d i ( )0

e -j2 f tπ N-1

e -j2 f tπ 0

e -j2 f tπ 1

Source S/P

g t( )

Mod.

Mod.

Mod.

g t( )

g t( )ld( )M

d i ( )1

d i ( )N-1

d i ( )0

Σe j2 f tπ N-1

e j2 f tπ 0

e j2 f tπ 1

channel ( )c ta

s t ( )a

s t ( )a

η( )t

^^

^

^

Channel

Receiver

Transmitter

b i ( )b^

b i ( )b

Map.

Map.

Map.

Demap.

Demap.

Demap.

gTX(t)

gTX(t)

gTX(t)

gRX(t)

gRX(t)

gRX(t)

ha(t)

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9

Inter Carrier Interference (ICI)

ICI

Problem of MC:If the frequency bands of different subcarriers overlap, Inter Carrier Interference (ICI) appears.

Solution:A special design of transmit and receive filter leads to orthogonality of the subcarriers.

4. OFDM

10

Orthogonal Frequency Division Multiplexing (OFDM)

Orthogonality of subcarriers

0 fn−1 fn+2fn+1fn f

|Gn−1(f)|

|Gn(f)| |Gn+1(f)|

|Gn+2(f)|

f

|G(f)|

−1/Ts 0 1/TstTs0

|g(t)|

carrier distance:1

Ts

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Mathematical Description of the OFDM-Transmitter

continuous-time representation of anOFDM transmitter:

discrete-timerepresentation ofan OFDM transmitter:

,1 2 /

0

1 2 /0 1 1

0

( , ) ( ) ( ) [0,1,2,..., 1]

/

( ) = ( ), ( ),..., (DFT )I

A SS A

N j n Tnt iT kT

n

S

kT

N

A

N j nkn N

n

s i k s t d i e k N

N T T

d i e N d i d i d i⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

−

= +=

−−

=

= = ∈ −

↓ =

= ⋅

∑

∑

π

π

,

1 2

0

1 2

0

/

( ) ( )

( ) / /

( ) , 1S

nN j f t

n Sn

nS S

N j tn

nS

n

TS

s t d i g t iT e

g t rect t T f n f n T

d i e iT t i T

π

π

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

−

=

−

=

= −

↓ = = ⋅Δ =

= ≤ ≤ +

∑

∑

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Mathematical Description of the OFDM-Receiver

received signal in time domain

received signalafter DFT

Complete System:

Decided Data: dn(i) = Q{xn(i)}

(n = subcarrier index)

xn(i) = DFT©IDFT{dn(i)} + n(i, k)

ª= dn(i) + DFT{n(i, k)}

xn(i) = DFT{r(i, 0), r(i, 1), · · · , r(i, N − 1)}

=N−1Xk=0

r(i, k) exp(j2πkn/N )

r(i, k) = s(i, k) + n(i, k) = IDFT{dn(i) + n(i, k)}

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Symbol Rate Model of an OFDM System

4. OFDM

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Orthogonal Frequency Division MultiplexingPrinciple of Multi-carrier systemsInter Carrier InterferenceOrthogonal Frequency Division Multiplexing: OFDMCyclic Prefix – EqualizationImpulse Shortening EqualizerChannel EstimationDigital-to-Analog Interface

4. OFDM

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Inter-Symbol- (ISI) and Inter-Carrier-Interference (ICI)

OFDM symbolOFDM symbol

t

OFDM symbol OFDM symbol

magnitude of channelimpulse response:

|c (t)|

... fade out (ISI)

OFDM symbol

symbol i

symbol( - )i 1

fade in (ICI)

( - )i 1( - )i 2 ( + )i 0 ( + )i 1 ( + )i 2

a

t

|ha (t)|

4. OFDM

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t|c (t)|

...

fade out

symbol( - )i 1

fade in

( - )i 1 ( + )i 1

cyclic prefix

T T

G G G( + )i 0G

g s

symbol i

magnitude of channelimpulse response:

a

The OFDM Cyclic Prefix / Guard Interval

The OFDM cyclic prefix serves for the suppression of ISI and ICI !

|ha(t)|

4. OFDM

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OFDM Transmitter

channel coding (convolutional codes with Viterbi decoding)IDFT: discrete realized filter bank (very efficient FFT) cyclic prefix / guard interval (GI) prevents intersymbol interference (ISI)

S/Psource

Map.

Map.

Map.

Mod.Map.

IDFTN

DAC

d i1( )

d i2( )

d iN-1( )

d i0( )

PS/GI

.

.

.

CC

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SynchronizationFFT window position (time domain)sample and modulation frequency correction

Pre equalizer (PE) for impulse compressionOFDM: Orthogonal Frequency Division Multiplexing

separate multiplicative channel correction on each subcarrier

equalizer coefficient design: en = 1 / Hn circular convolution

Demap.

Demap.

Mod.Demap.

.

.

.

P/S

e1

e2

eN-1

e0

CC -1 dest.DFTNADC

SYNCGI-1

Viterbidecoder

PE

x0(i)

x1(i)

x2(i)

xN−1(i)

OFDM Receiver

4. OFDM

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OFDM Circular ConvolutionReceived signal in time domain (noiseless case)

Circular convolution due to cyclic prefix iff

Discrete Fourier Transform yields

Equalization by simple division separately for each subcarrier

maxgT > τ

Circular convolution w.r.t.

yn(i) =1

H (n) · xn(i); en =1

H (n)

k*

*

*r(i, k) = s(i, k) h(k);∆=

DFT(k)N

ns(i, k) h(k)

o| {z }

xn(i)

= DFT(k)N

ns(i, k)

o| {z }

dn(i)

·DFT(k)N

nh(k)

o| {z }

H (n)

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Eye Pattern for QPSK Transmission (real part)

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Influence of Guard IntervalBandwidth Efficiency under Guard Interval

SNR Loss caused by Guard Interval

1symbol rate 1β 1bandwidth 1

s G

G

s S

NT T

TNT T

⋅+

= = =⋅ +

( ) ( )

( ) ( )

( ) ( )

2

0

2 20

0

2

0 0 0

SNR

β

S

S S

G

T

RX TXS

T T

RX TXT

S S S S S

S S G S G

g t g t dtEN g t dt g t dt

E T E T EN T T T N T T N

−

⎡ ⎤′ ′ ′−⎢ ⎥⎣ ⎦= ⋅′ ′ ′ ′

= ⋅ = ⋅ = ⋅⋅ + +

∫∫ ∫ e.g. 20% of

1dB lossG G ST T T= +→ ≈

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Impulse Shortening EqualizerLinear pre-filter; impulse response such that

Cost function:

MMSE-solution:

h(k) ∗ epre(k)|k=k0+κ =½g(k), κ = 0, · · · , `g − 1; `g < Ng

ε0(k) else

epre = [Rrr − 1σ2SRrsR

Hrs ]−1rrs

where Rrr∆= autocorr.matrix received signal; Rrs ; rrs

∆= Cross corr. Matrix/vector

P(k)

|ε0(k)|2 ⇒ min, s.t.

½g(0) = 1 MMSE, Kammeyer 1995P |g(k)|2 = 1 Falconer 1973

epre(k)

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Examples| h(k)|→

Ng = 16 Equalizer: n = 32, `g = 8

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Parameters of an OFDM System

Data rate

Bandwidth

Subcarrier spacing

Total symbol duration

Guard time

Core symbol duration

-Sampling frequency

Time discreteTime continuousMeaning

1A

A S

Nf T T= =

STAN T⋅

GT G AN T⋅

S GT T T= + ( )G AT N N T= + ⋅

1S

f TΔ = 1A

f N TΔ = ⋅

B N f≈ ⋅Δ B N f≈ ⋅Δ

ld( )b

S G

N MR T T⋅= +

ld( )( )b

A G

N MR T N N⋅= ⋅ +

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4. OFDM

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Pilot Symbol Constellationfor WLAN

26252423222120

01234567

-1-2-3-4-5-6-7

-26-25-24-23-22-21-20

8

-8

pilot symbols

data symbols

i

n

burst structure of HIPERLAN/2 and IEEE802.11a

short symbols for AGC and raw synchronizationtraining sequence (TS): 2 identical symbols per subcarrier (52)data OFDM symbols with 48 user data and 4 pilot symbols eachpilot symbols for fine synchronization (insufficient for channel estimation)

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Nonblind (reference-based) Channel Estimation

S/P

disc.Prefix

&FFT

y i( )0

y i( )1

y i( )P-1

y k( )

d i( )0,ref

d i( )1,ref

d i( )N ref-1,

~

~

~

Channel estimator Equalizer

C n( ) 1

d i( )0

d i( )1

d i( )N-1

d (0)n ref,~

d (1)n ref,~

+2dn ref,

C n( ) =

d n,refN

N

Averaging over only two identical training symbols• 2 dB loss in SNR compared to „estimator“ with ideal channel knowledge• Perform additional noise reduction (NR) to increase estimation quality

H (n)

H (n)

yn(0) + yn(1)

y0 (i)

y1 (i)

yN−1 (i)

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Noise Reduction Algorithm (NR)Background

a-priori knowledge: limited channel impulse response in time domainchannel impulse response fits into guard interval

lowpass filtering in frequency domain NR algorithm (required operations)

transform the estimated channel transfer function into time domain (IDFT) truncate the estimated impulse response (rectangular window)re-transform into frequency domain (DFT)

CCE C DFTcIDFTNC

windowing in time domain

~c~

noise reduction (NR)

H Hh h

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• estimated and real channel transfer functions (frequency domain)

• ... in time domain

Noise Reduction Algorithm – Example

0 dB

-10 dB

-20 dB| |Cn | |Cn

0 4 8 16 32 64 n

0 4 8 16 32 64 k

| |c(k) | |c(k) |h(k)||h(k)|

|Hn ||Hn |

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• time limited (windowed) impulse response

0 4 8 16 32 64 k

~| |c(k)

• smoothed and real transfer functions (in frequency domain)

0 dB

-10 dB

-20 dB0 4 8 16 32 64 n

| |Cn | |Cn

~

|h(k)|

|Hn ||Hn |

Noise Reduction Algorithm - Example

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10 12 14 16 1810

-3

10-2

10-1

100

Eb/N0

PE

R

only CECE+NR ideal

• simulation of a HIPERLAN/2 system (27 Mbit/s)

• time invariant Rayleigh distributedmultipath channel

• (only CE)Eb/N0 loss: about 1.8 dB

• (CE+NR)Eb/N0 loss: about 0.5 dB

• (ideal)perfectly known channel

Noise Reduction Algorithm – Simulation Result

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Scattererd Pilot Constellation:Time-Frequency Interpolation

i

n

distance of sampling points in time direction

distance of sampling points in frequency direction

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i →

n→

H (nPi, iPi) =xnPi (iPi)

dnPi(iPi)

P (n, i)

bopt = arg minb

©E{|H (n, i) − H (n, i)|2}

ª

Scattererd Pilot Constellation:Time-Frequency Interpolation (2)

Estimation of transfer function at

Choose a set of neighbored pilotsymbols

At sampling points the transferfunction is simply:

∈ P(n, i)

Perform two-dimensional Wiener Interpolation

H(n, i) =X

{n0,i0}∈P(n,i)b(n − n0, i− i0) · H(n0, i0)

Wiener ansatz:

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4. OFDM

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Digital to Analog InterfacefA = N/TSCritical sampling , no guard interval

For correct analog reconstruction: Ideal low-pass (band-pass) necessary, which is not realistic.Solution: Oversampling, overlapping rect-impulses with raised cosine slopesinstead of pure rect impulses.

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Realistic analog transmit filter:

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Intersymbol interference due to overlapping symbols

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Nonlinearity of the Power Amplifier

4. OFDM

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Characteristics of Nonlinear Class A and B Amplifier

4. OFDM

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Input Backoff (IBO)

IBO: Ratio between maximum outputmagnitude and root meansquare of input signal

Clipping Class A Class AB

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Conclusions

OFDM converts frequency selective channel intoN non-selective subchannels.

Very simple equalization by the use of a guard intervalEquivalent structure: Single Carrier Frequency Domain EqualizerPre-equalizer for impulse shortening

Channel estimation: Preamble for slowly varying channels (noise reduction)Scattered pilots with Wiener interpolation for fast varying channels

Digital-to-analog interfaceTime-domain cos-roll-off shapingNonlinear distortions by power amplifier

4. OFDM

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