par_ofdm (1)

OFDMOFDM

Adaptive ModulationAdaptive Modulation

Reduction of Peak-to-Average Reduction of Peak-to-Average Power Ratio Power Ratio

Channel estimation Channel estimation

OFDM in frequency selective OFDM in frequency selective fading channel fading channel

Puja Thakral Silvija Kokalj-Filipovic Youngsik Lim Sadhana Gupta

OUTLINEOUTLINE

• Introduction to OFDMIntroduction to OFDM

• Adaptive ModulationAdaptive Modulation

• Reduction of Peak-to-Power Average Reduction of Peak-to-Power Average RatioRatio

• OFDM in Frequency Selective Fading OFDM in Frequency Selective Fading ChannelChannel

• Channel EstimationChannel Estimation

• ConclusionsConclusions

OFDM SYSTEMOFDM SYSTEM

Baseband TransmitterBaseband Transmitter

Baseband Ideal ReceiverBaseband Ideal Receiver

Adaptive ModulationAdaptive Modulation

In OFDM ,adaptive bit loading algorithms In OFDM ,adaptive bit loading algorithms set the modulation level in each set the modulation level in each frequency band such that a predefined frequency band such that a predefined total number of bits are transmitted with total number of bits are transmitted with minimum power.Adaptive Modulation minimum power.Adaptive Modulation independently optimizes the modulation independently optimizes the modulation scheme to each sub carrier so that scheme to each sub carrier so that spectral efficiency is maximized,while spectral efficiency is maximized,while maintaining a target Bit Error Rate(BER).maintaining a target Bit Error Rate(BER).

OFDM Block StructureOFDM Block StructureWith Adaptive ModulationWith Adaptive Modulation

S/P IFFT FFT P/SFREQUENCYSELECTIVECHANNEL

MODULATOR 1

MODULATOR 2

MODULATOR N

DEMODULATOR 1

DEMODULATOR 2

DEMODULATOR N

CHANNELESTIMATION

ADAPTIVEBIT AND POWER

ALLOCATION

+

Various Algorithms in Various Algorithms in Adaptive ModulationAdaptive Modulation

• For a given target BER and bit-rate, the For a given target BER and bit-rate, the total transmit power can be minimized by total transmit power can be minimized by optimally distributing the power and bit-optimally distributing the power and bit-rate across the sub channels. rate across the sub channels.

• For a given target BER and power For a given target BER and power transmitted,the total bit-rate can be transmitted,the total bit-rate can be maximized.maximized.

• For a given target power and bit rate,the For a given target power and bit rate,the total BER can be minimized.total BER can be minimized.

ALGORITHMALGORITHM

• Compute the subchannel signal to noise Compute the subchannel signal to noise ratios.ratios.

• Compute the number of bits for the ith Compute the number of bits for the ith subchannel based on the formula, subchannel based on the formula, b`(i)=log2(1+SNR(i))b`(i)=log2(1+SNR(i))

• Round the value of b`(i) down to b(i).Round the value of b`(i) down to b(i).• Restrict b(i) to take the values 0,1,2,4,6,8Restrict b(i) to take the values 0,1,2,4,6,8• Compute the energy for the ith subchannel Compute the energy for the ith subchannel

based on the number of bits initially assigned based on the number of bits initially assigned to it using the formula e(b(i))=(2^b(i)-1)/SNRto it using the formula e(b(i))=(2^b(i)-1)/SNR

RESULTSRESULTS

FUTURE WORKFUTURE WORK

• Feasibility study of MIMO OFDM Feasibility study of MIMO OFDM systemssystems

• Simulation of MIMO OFDM system Simulation of MIMO OFDM system with adaptive modulation and with adaptive modulation and multilevel transmit power control.multilevel transmit power control.

Peak To Average Power Peak To Average Power RatioRatio

in OFDMin OFDM

Causes, Effects and Reduction Causes, Effects and Reduction MethodsMethods

Silvija Kokalj-Filipovic

SummarySummary

• Goal: reducing maximum output power Goal: reducing maximum output power to near average power by limiting the to near average power by limiting the set of transmitted signals through set of transmitted signals through codingcoding

• Complementary Golay Sequences have Complementary Golay Sequences have peak-to-average power less then 2peak-to-average power less then 2

• Reed-Muller Coding used to produce Reed-Muller Coding used to produce these sequences out of information these sequences out of information sequence sequence

Stochastic StructureStochastic Structure

• In accordance with CLT, when large number of In accordance with CLT, when large number of modulated carriers (N) are combined into a modulated carriers (N) are combined into a composite time-domain signal by means of composite time-domain signal by means of IFFT (they are assumed to be independent, IFFT (they are assumed to be independent, since the assigned data symbols are iid since the assigned data symbols are iid (µ(µ00, , σσ00 ) )), it leads to near ), it leads to near Gaussian pdfGaussian pdf of of amplitude, where the amplitude value exceeds amplitude, where the amplitude value exceeds certain threshold value A with probability Q(A-certain threshold value A with probability Q(A-µ/σ), andµ/σ), and

• µ ~ Nµµ ~ Nµ0 0 σ ~ Nσσ ~ Nσ00

• Since we have N independent points in the Since we have N independent points in the composite time signal:composite time signal:– For BPSK modulation we’ll have ~ Gaussian For BPSK modulation we’ll have ~ Gaussian

distribution of the amplitudedistribution of the amplitude– For MPSK and M-QAM modulations (which For MPSK and M-QAM modulations (which

both have 2-dimensional space: both have 2-dimensional space: I and QI and Q component ) we have a Rayleigh component ) we have a Rayleigh distribution (square root of the sum of distribution (square root of the sum of squares of I & Q Gaussian random squares of I & Q Gaussian random variablesvariables).).

– Cumulative distribution of power: Cumulative distribution of power: F (z) = 1-F (z) = 1-ee-z-z

Definition of PAPR (PMEPR)Definition of PAPR (PMEPR)• PAPR & PAR: Peak-To-Average Power PAPR & PAR: Peak-To-Average Power

RatioRatio

• PMEPR: Peak-To-Mean Envelope PMEPR: Peak-To-Mean Envelope Power RatioPower Ratio

• Crest factor Crest factor of x(t): square root of PAR of x(t): square root of PAR

• Definition:Definition: PAR = (||x|| PAR = (||x||∞∞))22 / E[(||x|| / E[(||x||22)) 2 2]]

Crest Factor - notationCrest Factor - notation

The crest factor of u(t): square root of PMEPRwhere is the maximum absolute value of u(t) and is the rms of u(t):

2||||

||||)(

u

uuCF

|||| u2|||| u

2/1

22 )(

1||||

T

o

dttuT

u

Effects of PAPREffects of PAPR• The power amplifiers at the transmitter need to have a The power amplifiers at the transmitter need to have a

large linear range of operation. large linear range of operation. • nonlinear distortions and peak amplitude limiting nonlinear distortions and peak amplitude limiting

introduced by the High Power amplifier (HPA) will produce introduced by the High Power amplifier (HPA) will produce inter-modulation between the different carriers and inter-modulation between the different carriers and introduce additional interference into the system. introduce additional interference into the system.

• additional interference leads to an increase in the Bit Error additional interference leads to an increase in the Bit Error Rate (BER) of the system. Rate (BER) of the system.

• one way to avoid non-linear distortion is by forcing the one way to avoid non-linear distortion is by forcing the amplifier to work in its linear region. Unfortunately such amplifier to work in its linear region. Unfortunately such solution is not power efficient and thus not suitable for solution is not power efficient and thus not suitable for wireless communication.wireless communication.– The Analog to Digital converters and Digital to Analog The Analog to Digital converters and Digital to Analog

converters need to have a wide dynamic range and this converters need to have a wide dynamic range and this increases complexity.increases complexity.

• if clipped, it leads to in-band distortion (additional noise) if clipped, it leads to in-band distortion (additional noise) and ACI (out-of-band radiation)and ACI (out-of-band radiation)

Classification of Classification of PAR reduction methodsPAR reduction methods

• BLOCK CODING (Golay BLOCK CODING (Golay sequences)sequences)

• CLIP EFFECT TRANSFORMATIONCLIP EFFECT TRANSFORMATION

• PROBABILISTIC TECHNIQUESPROBABILISTIC TECHNIQUES: : – Selective Mapping (SLM) and Selective Mapping (SLM) and

Partial Transmit Sequences (PTS)Partial Transmit Sequences (PTS) – Tone Reduction (TR) and Tone Tone Reduction (TR) and Tone

Injection (TI)Injection (TI)

Representation of OFDM signalRepresentation of OFDM signal• In the bandpass with = the multi-In the bandpass with = the multi-

carrier (multitone) signal can be carrier (multitone) signal can be represented asrepresented as

• where corresponds to initial phase where corresponds to initial phase of the tones, i.e. the effect of modulating of the tones, i.e. the effect of modulating data.data.

cf 0f

TftkftuN

k

k /1),2cos()( 0

1

10

1k

1

1

22 00Re)(N

k

tkfjtfj keetu

)(ReRe)( /21

1

/2/2 tSeesetu TtjN

k

Tktj

k

Ttj

Representation of OFDM signalRepresentation of OFDM signalTktj

N

kkestS /2

1

0

)(

k

TransformFourier

s

assuming t is the frequency and 1/T is the sampling period of sequence k

s

ks is the discrete complex sequence of information data (phase-mapped).

Crest factor depends on the maximum absolute value of the multicarrier signal, and that one depends on the “amplitude spectrum” of the complex sequence

ks

Choosing ks to be complementary Golay sequence

crest factor of less than 6dB (PAPR of 3 dB) can be obtained

Observation: OFDM has somewhat inverted logic – we are looking for flat PSD in time domain, while autocorrelation is taken in frequency domain

Proof:Proof:• Aperiodic correlation CAperiodic correlation Cxx(z) of some sequence(z) of some sequence

The Fourier transform SThe Fourier transform Sxx(f) of sequence(f) of sequence

• Definition: Definition: Two sequences and of the length N Two sequences and of the length N form a complementary pair ifform a complementary pair if

– Golay complementary sequences have that Golay complementary sequences have that property. property.

i

x

zN

iziix

xxzC1

0

*)(

i

x

1

0

2)(N

i

Tfj

ixsiexfS

where Ts is the sampling period of sequence i

x

i

a i

b

ozN

zbazCzC

,2

0,0)()(

)(|)(| 2 oCfSF

• N carrier OFDM; H-PSK modulationN carrier OFDM; H-PSK modulation

• Information-bearing sequence Information-bearing sequence is is

in fact an OFDM codeword and is the in fact an OFDM codeword and is the primitive H-root of unity (primitive H-root of unity (jj in QPSK case) in QPSK case)

• Instantaneous Envelope PowerInstantaneous Envelope Power

1

0

)(2 0)(N

i

tfifja

aetS i

ξ

Hjjaa ;

zN

i

aa

aziizC

1

0

)(

1

1

2

00

0

)(

,

2

)(21*|)(|)(

|)(|)(

n

uua

u

ftHu

a

u i

ftHuaaftjiHaa

jiaa

nunnuCnuC

nntStP uiiji

ntPtPba

2)()( ntPa

2)( For complementary sequences:

Theory behind Reed-Muller codesTheory behind Reed-Muller codes• An rth order Reed-Muller code R(r,m) is the set of all binary strings (vectors) An rth order Reed-Muller code R(r,m) is the set of all binary strings (vectors)

of length n= 2of length n= 2mm associated with the Boolean polynomials p(x1, x2, …, xm) of associated with the Boolean polynomials p(x1, x2, …, xm) of degree at most r. degree at most r.

• A Boolean polynomial is a linear combination of Boolean monomials with A Boolean polynomial is a linear combination of Boolean monomials with coefficients in F2. A Boolean monomial p in the variables x1, x2, …, xm is the coefficients in F2. A Boolean monomial p in the variables x1, x2, …, xm is the expression of the form:expression of the form:

• P = xP = x11rr11 x x22rr22 …, x …, xmmrrmm where r where ri i {0,1,2..} and 1 ≤ i ≤ m. {0,1,2..} and 1 ≤ i ≤ m. • Degree of a monomial is deduced from it reduced form (after rules xDegree of a monomial is deduced from it reduced form (after rules x iixxjj = x = xjjxxii

and xand xii22 = x = xii are applied), and it is equal to the number of variables. This rule are applied), and it is equal to the number of variables. This rule

extends to polynomialsextends to polynomials• Ex. of a polynomial of degree 3:Ex. of a polynomial of degree 3:

– q = x1+ x2+x1 x2+ x1 x2 x3q = x1+ x2+x1 x2+ x1 x2 x3

• How to associate Boolean monomial in m variables to a vector with 2How to associate Boolean monomial in m variables to a vector with 2mm entries:entries:– a vector associated with monomial of degree 0 (1) is a string of length 2a vector associated with monomial of degree 0 (1) is a string of length 2mm

where each entry is 1.where each entry is 1.– a vector associated with monomial x1 is 2a vector associated with monomial x1 is 2m-1m-1 ones followed by 2 ones followed by 2m-1m-1 zeros. zeros.– a vector associated with monomial x2 is 2a vector associated with monomial x2 is 2m-2m-2 ones followed by 2 ones followed by 2m-2m-2 zeros, zeros,

then another 2then another 2m-2m-2 ones followed by 2 ones followed by 2m-2m-2 zeros. zeros.– a vector associated with monomial xi is a pattern of 2a vector associated with monomial xi is a pattern of 2m-im-i ones followed by ones followed by

22m-im-i zeros, repeated until 2 zeros, repeated until 2mm values are defined. values are defined.

Example of RM generator Example of RM generator matrixmatrix

• m = 5: m = 5: RM(1,5) has six rowsRM(1,5) has six rows

• X0: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X0: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1• X1: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1X1: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1• X2: 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1X2: 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1• X3: 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1X3: 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1• X4: 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1X4: 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1• X5: 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1X5: 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Relationship between Reed-Muller Relationship between Reed-Muller codes and Complementary Golay codes and Complementary Golay SequencesSequences• In the binary case, Golay pairs and sets occur in the first-order In the binary case, Golay pairs and sets occur in the first-order

Reed-Muller code RM(1,m) within the second-order Reed-Muller Reed-Muller code RM(1,m) within the second-order Reed-Muller code (cosets). code (cosets).

• Each coset has assigned coset representative of the form:Each coset has assigned coset representative of the form:

• where is any permutation of the sequence of generator where is any permutation of the sequence of generator matrix rows – see graph with rows as hypercube verticesmatrix rows – see graph with rows as hypercube vertices

: number of elements in the Galois field: number of elements in the Galois field

1

1)1()(

12m

kkk

h xx

h2

SimulatioSimulationn

Conclusions and Further WorkConclusions and Further Work

• Result: Result: complete elimination of clipping complete elimination of clipping noisenoise

• Drawback: serious overhead (low Drawback: serious overhead (low bandwidth utilization – 17/32) bandwidth utilization – 17/32)

• Further work:Further work:– implementation of Tone Reservation Algorithm implementation of Tone Reservation Algorithm

and Comparison with Golay Sequencesand Comparison with Golay Sequences– Extension of the method to Golay sequences Extension of the method to Golay sequences

that do not form complementary pairs but have that do not form complementary pairs but have satisfying PAR (coset representatives of satisfying PAR (coset representatives of different forms)different forms)

Cyclic prefix of OFDM in Cyclic prefix of OFDM in frequency selective fading frequency selective fading

channelchannel

Signal distortion in frequency selective fading channel What is the cyclic prefix ? How is the interference eliminated with cyclic prefix? How is its performance without the cyclic prefix.

Problem Description

Transmission over frequency selective fading Transmission over frequency selective fading channel(*) channel(*)

Pulse ShapingTx +

(t)

h(n)

)(nu Channelch

Receive FilterRx

)(tx

t=nTs

)(nx

)(nu

+

(n)

)(nxH0+H1z-1)(iU +

(n)

)(iX

)()()()( tttth RxchTx

n

rcs ttnTthnutx )()()()()(

L

l

nlnulhnx0

)()()()(

(*) Z. Wang, G.B. Giannakis, Wireless Multicarrier Communications. IEEE 2000 Signal Processing Magazine

magnitude

time10Ts9Ts8Ts7Ts6Ts5Ts4Ts3Ts2TsTs0

RMSs

RMSs

TT

TkTk

kki

e

e

jNNh

/20

/20

2

2212

21

1

),0(),0(

(**) Frequency selective Flat fading channel(Naftali Chayat in IEEE P802.11-97/96)

Black : Average , Gray : a realization of the channel

Channel response

Dispersive in time,Static over block interval

Selective in frequency

samplecomplex kth of Variance :

blockith of response impulse Channel :2k

ih

Channel Model (**)

L

nnknk uhx

0

01

01

01

000

0

0

000

000

,

hhh

hhh

hhh

HUHX

LL

LL

LL

110111

0110

1102111

000110

NNLLLLN

LLLLL

LLLL

LLLL

uhuhuhx

uhuhuhx

uhuhuhx

uhuhuhx

0

0

0

0

0

0

00

000

hh

h

h

h

H

L

L

000

0

0

0

0 1

1

L

L

h

hh

H

N+L

NMemory from the past block

H U

vector noise : )(

block receivedith : )(X

block mitted1)th trans-(i : )1(

block ittedith transm : )(

)()1()()( 10

i

i

iU

iU

iiUHiUHiX

What is H0 and H1?

How is IBI deleted ?

L)N(N matrix, discarding-Guard]0[ :

Imatrix identity NNan of rows LLast :

NL)(N ,matrix inserting-Guard ,][ :

)()()(

)()(

)()1()()(

N

T

0

10

TNLNcp

cp

TTNcpcp

cpcp

cpcpcpcpcp

IR

I

IIT

iiUiH

iiUTHR

iRiUTHRiUTHRiX

0

0

10

0

)(

hh

hh

h

hhh

iH

L

LL

L

L

n

fj

nk

NNjNjj

fnjnheH

NknjNF

eHeHeHdiagFHF

0

2

2/1,

/)1(2/201

)2exp()()(

)/2exp(

)](,),(),([

Tcp H0+H1z-1 + Rcp

)()()()( iiUiHiX )(iU

)(i

H

matrixcirculant a becomes )(iH

Cyclic prefix effect on OFDM

S/P Mapping +

)()()()( iiUiHiX

)(iU

)(i

H

OFDM

.

.

.

.

.

.

IFFT...

FFT

Input bits

Demapping...

P/S...

Output bits

)()()()()()()()()( 1 iFiSiFiSFiHFiiUiHFiXF

)(iS

No IBI plus simpler equalizer

• Simulation configurationSimulation configuration– Perfect channel estimation , QPSK, Fixed sub-channel powerPerfect channel estimation , QPSK, Fixed sub-channel power

– Zero Forcing equalization Zero Forcing equalization

– 64 sub-carriers64 sub-carriers

• Simulation ResultsSimulation Results

Guard insertion

IFFT Multipath channel model

AWGN Guard extraction

FFT

Symbol-to-bit mapping

Bit mapping

Random bits

Bits

Pilot extraction Channel estimation and interpolation

Channel compensation

Evaluation of Pilot-symbol based channel estimation

Tapped delay line H(n)

Magnitude

Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts

hk = N(0, 1/2k2) + jN(0,1/2k

2)

k2 = 0 e

-kTs/T

RMS

02 = 1 – e –T

s/T

RMS

Naftali Multipath Channel Model

Time

Channel Estimate: He = Yp/Xp k = 0,1,2,3 Interpolation techniques:

1. Linear – based on responses of 2 neighbouring pilots placed at p1 and p2

He(k) = (Hp(m+1) –Hp(m)) *l/L + Hp(m) p1<k<p2, l = k-p1

2. Second order – based on 3 neighbouring pilots placed at p1, p2 and p3 He(k) = c1*Hp(m-1) + c0 * Hp(m) + c-1*Hp(m+1) C1 = ( k-p2)* (k-p3)/ ((p1-p2)*(p1-p3)) C2 = (k-p1) *(k-p3)/ ((p2-p1) * (p2 – p3)) C3 = (k-p1)*(k-p2)/ ((p3-p1) * (p3-p2))

L = 7 14 22 14 7

0 7 21 43 57 64 Subcarrier numbers 1 to 64

802.11a Pilot subcarrier placement

-21 -7 0 7 21 Subcarrier numbers -31 to 32

Pilot subcarrier placement used

BER Performance in AWGN

Constellation in AWGN

Frequency Selective Fading, Naftali model Trms = 50 ns

Constellation in frequency-selective fading

Selected Results/Plots : Trms = 30ns, Modulation : QPSK

1. Linear interpolation SNR = 35db

2. Second order interpolation SNR = 30db

3. Cubic spline interpolation , SNR = 20dB

Eb/No vs BER in multipath channel

Conclusions and future work:

Low pass filtering interpolation shows best performance among evaluated

interpolation methods as reported in literature, especially for larger values of

Trms.

Future work:

Evaluation of performance of differential modulation

Evaluation with Doppler frequency shift

Primary Reference:

Channel Estimation Techniques based on Pilot Arrangement in OFDM

Systems

Coleri, et al, IEEE Transactions on Broadcasting, p223 -229 September 2002

par_ofdm (1)

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