OFDM

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OFDM

• Adaptive Modulation
• Reduction of Peak-to-Average Power Ratio
• Channel estimation
• OFDM in frequency selective fading channel

Puja Thakral Silvija Kokalj-Filipovic
Youngsik Lim Sadhana Gupta
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OUTLINE

• Introduction to OFDM
• Adaptive Modulation
• Reduction of Peak-to-Power Average Ratio
• OFDM in Frequency Selective Fading Channel
• Channel Estimation
• Conclusions

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OFDM SYSTEM
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Baseband Transmitter
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Baseband Ideal Receiver
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Adaptive Modulation

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

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OFDM Block StructureWith Adaptive Modulation
S/P
IFFT
FFT
P/S
MODULATOR 1
DEMODULATOR 1
MODULATOR 2
DEMODULATOR 2
FREQUENCY SELECTIVE CHANNEL

MODULATOR N
DEMODULATOR N
CHANNEL ESTIMATION
ADAPTIVE BIT AND POWER ALLOCATION
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Various Algorithms in Adaptive Modulation

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

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ALGORITHM

• Compute the subchannel signal to noise ratios.
• Compute the number of bits for the ith subchannel
  based on the formula, b(i)log2(1SNR(i))
• Round the value of b(i) down to b(i).
• Restrict b(i) to take the values 0,1,2,4,6,8
• Compute the energy for the ith subchannel based
  on the number of bits initially assigned to it
  using the formula e(b(i))(2b(i)-1)/SNR

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RESULTS
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FUTURE WORK

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

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Peak To Average Power Ratioin OFDM

• Causes, Effects and Reduction Methods

Silvija Kokalj-Filipovic
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Summary

• Goal reducing maximum output power to near
  average power by limiting the set of transmitted
  signals through coding
• Complementary Golay Sequences have
  peak-to-average power less then 2
• Reed-Muller Coding used to produce these
  sequences out of information sequence

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Stochastic Structure

• In accordance with CLT, when large number of
  modulated carriers (N) are combined into a
  composite time-domain signal by means of IFFT
  (they are assumed to be independent, since the
  assigned data symbols are iid (µ0, s0 )), it
  leads to near Gaussian pdf of amplitude, where
  the amplitude value exceeds certain threshold
  value A with probability Q(A-µ/s), and
• µ Nµ0 s Ns0

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• Since we have N independent points in the
  composite time signal
• For BPSK modulation well have Gaussian
  distribution of the amplitude
• For MPSK and M-QAM modulations (which both have
  2-dimensional space I and Q component ) we have
  a Rayleigh distribution (square root of the sum
  of squares of I Q Gaussian random variables).
• Cumulative distribution of power F (z) 1-e-z

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Definition of PAPR (PMEPR)

• PAPR PAR Peak-To-Average Power Ratio
• PMEPR Peak-To-Mean Envelope Power Ratio
• Crest factor of x(t) square root of PAR
• Definition PAR (x8)2 / E(x2) 2

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Crest Factor - notation
The crest factor of u(t) square root of
PMEPR where is the maximum absolute
value of u(t) and is the rms of u(t)  
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Effects of PAPR

• The power amplifiers at the transmitter need to
  have a large linear range of operation.
• nonlinear distortions and peak amplitude limiting
  introduced by the High Power amplifier (HPA) will
  produce inter-modulation between the different
  carriers and introduce additional interference
  into the system.
• additional interference leads to an increase in
  the Bit Error Rate (BER) of the system.
• one way to avoid non-linear distortion is by
  forcing the amplifier to work in its linear
  region. Unfortunately such solution is not power
  efficient and thus not suitable for wireless
  communication.
• The Analog to Digital converters and Digital to
  Analog converters need to have a wide dynamic
  range and this increases complexity.
• if clipped, it leads to in-band distortion
  (additional noise) and ACI (out-of-band
  radiation)

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Classification of PAR reduction methods

• BLOCK CODING (Golay sequences)
• CLIP EFFECT TRANSFORMATION
• PROBABILISTIC TECHNIQUES
• Selective Mapping (SLM) and Partial Transmit
  Sequences (PTS)
• Tone Reduction (TR) and Tone Injection (TI)

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Representation of OFDM signal

• In the bandpass with the multi-carrier
  (multitone) signal can be represented as
• where corresponds to initial phase of the
  tones, i.e. the effect of modulating data.

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Representation of OFDM signal


assuming t is the frequency and 1/T is the
sampling period of sequence
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
Observation OFDM has somewhat inverted logic
we are looking for flat PSD in time domain, while
autocorrelation is taken in frequency domain
Choosing
to be complementary Golay sequence
crest factor of less than 6dB (PAPR of 3 dB) can
be obtained
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Proof

• Aperiodic correlation Cx(z) of some sequence
• The Fourier transform Sx(f) of sequence
• Definition Two sequences and of the
  length N form a complementary pair if
• Golay complementary sequences have that property.
  

where Ts is the sampling period of sequence
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• N carrier OFDM H-PSK modulation
• Information-bearing sequence
  is
• in fact an OFDM codeword and is the
  primitive H-root of unity (j in QPSK case)
• Instantaneous Envelope Power

For complementary sequences
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Theory behind Reed-Muller codes

• An rth order Reed-Muller code R(r,m) is the set
  of all binary strings (vectors) of length n 2m
  associated with the Boolean polynomials p(x1, x2,
  , xm) of degree at most r.
• 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 expression of the form
• P x1r1 x2r2 , xmrm where ri 0,1,2.. and 1
  i m.
• Degree of a monomial is deduced from it reduced
  form (after rules xixj xjxi and xi2 xi are
  applied), and it is equal to the number of
  variables. This rule extends to polynomials
• Ex. of a polynomial of degree 3
• q x1 x2x1 x2 x1 x2 x3
• How to associate Boolean monomial in m variables
  to a vector with 2m entries
• a vector associated with monomial of degree 0 (1)
  is a string of length 2m where each entry is 1.
• a vector associated with monomial x1 is 2m-1 ones
  followed by 2m-1 zeros.
• a vector associated with monomial x2 is 2m-2 ones
  followed by 2m-2 zeros, then another 2m-2 ones
  followed by 2m-2 zeros.
• a vector associated with monomial xi is a pattern
  of 2m-i ones followed by 2m-i zeros, repeated
  until 2m values are defined.

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Example of RM generator matrix

• m 5 RM(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 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 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 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 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 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 1

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Relationship between Reed-Muller codes and
Complementary Golay Sequences

• 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 code (cosets).
• Each coset has assigned coset representative of
  the form
• where is any permutation of the sequence
  of generator matrix rows see graph with rows as
  hypercube vertices
• number of elements in the
  Galois field

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Simulation
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Conclusions and Further Work

• Result complete elimination of clipping noise
• Drawback serious overhead (low bandwidth
  utilization 17/32)
• Further work
• implementation of Tone Reservation Algorithm and
  Comparison with Golay Sequences
• Extension of the method to Golay sequences that
  do not form complementary pairs but have
  satisfying PAR (coset representatives of
  different forms)

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Conclusions and Further Work

• Result complete elimination of clipping noise
• Drawback serious overhead (low bandwidth
  utilization 17/32)
• Further work
• implementation of Tone Reservation Algorithm and
  Comparison with Golay Sequences
• Extension of the method to Golay sequences that
  do not form complementary pairs but have
  satisfying PAR (coset representatives of
  different forms)

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Cyclic prefix of OFDM in frequency selective
fading channel
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Problem Description

• 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.

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Transmission over frequency selective fading
channel()
?(t)
Pulse Shaping ?Tx
Receive Filter ?Rx
Channel ?ch

tnTs
?(n)
?(n)
H0H1z-1
h(n)


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

• Channel response

Dispersive in time, Static over block interval
Selective in frequency
Black Average , Gray a realization of the
channel
() Frequency selective Flat fading
channel(Naftali Chayat in IEEE P802.11-97/96)
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What is H0 and H1?
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How is IBI deleted ?
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Cyclic prefix effect on OFDM
OFDM
Input bits
S/P
Mapping
IFFT
FFT
Demapping
P/S
Output bits
. . .
. . .
. . .
. . .
. . .

No IBI plus simpler equalizer
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• Simulation configuration
• Perfect channel estimation , QPSK, Fixed
  sub-channel power
• Zero Forcing equalization
• 64 sub-carriers
• Simulation Results

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802.11a Pilot subcarrier placement

Subcarrier
-
21
-
7 0 7
21

numbers
-
31 to 32

Pilot subcarrier placement used

L 7 14
22
14 7

Subcarr
ier
0 7 21
43 57
64

numbers 1 to 64

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