PAR_OFDM by panniuniu

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```									                             OFDM

Reduction of Peak-to-Average Power
Ratio
Channel estimation
channel

Puja Thakral   Silvija Kokalj-Filipovic   Youngsik Lim   Sadhana Gupta
OUTLINE

• Introduction to OFDM
• Reduction of Peak-to-Power Average Ratio
• OFDM in Frequency Selective Fading
Channel
• Channel Estimation
• Conclusions
OFDM SYSTEM
Baseband Transmitter
set the modulation level in each frequency
band such that a predefined total number
of bits are transmitted with minimum
optimizes the modulation scheme to each
sub carrier so that spectral efficiency is
maximized,while maintaining a target Bit
Error Rate(BER).
OFDM Block Structure
MODULATOR 1                                             DEMODULATOR 1

MODULATOR 2                                             DEMODULATOR 2
FREQUENCY
S/P                 IFFT    SELECTIVE     +             FFT                   P/S
CHANNEL

MODULATOR N                                             DEMODULATOR N

CHANNEL
ESTIMATION

BIT AND POWER
ALLOCATION
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.
ALGORITHM
• Compute the subchannel signal to noise ratios.
• Compute the number of bits for the ith
subchannel based on the formula,
b`(i)=log2(1+SNR(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))=(2^b(i)-1)/SNR
RESULTS
FUTURE WORK

• Feasibility study of MIMO OFDM systems
• Simulation of MIMO OFDM system with
transmit power control.
Peak To Average Power Ratio
in OFDM
Causes, Effects and Reduction
Methods
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
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, σ0 )), it leads
to near Gaussian pdf of amplitude, where the
amplitude value exceeds certain threshold value
A with probability Q(A-µ/σ), and
•   µ ~ Nµ0     σ ~ Nσ0
• Since we have N independent points in the
composite time signal:
– For BPSK modulation we’ll 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
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 = (||x||∞)2 / E[(||x||2) 2]
Crest Factor - notation
The crest factor of u(t): square root of PMEPR
where || u ||  is the maximum absolute value of u(t) and
|| u || 2 is the rms of u(t):
|| u || 
CF (u ) 
|| u || 2
1/ 2
1   T

|| u || 2    u (t ) dt 
2

T o          
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.
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.
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)
Representation of OFDM signal
• In the bandpass with fc = f 0 the multi-carrier
(multitone) signal can be represented as
N
u (t )   cos(2kf 0t  k  1), f 0  1 / T
k 1

• where k  1 corresponds to initial phase of the
tones, i.e. the effect of modulating data.

u (t )  Re e                j 2 f 0t
N 1

e
k 1
 j 2 kf0 t  k


u (t )  Re e    j 2 t / T
N 1

s e
k 1
k
 j 2 kt / T
 Ree          j 2 t / T
S (t )
Representation of OFDM signal
N 1
S (t )   sk e              sk 
 j 2 kt / T   Fourier
    Transform
k 0
assuming t is the frequency and 1/T is the sampling period of sequence   s 
k

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

Choosing s k  to be complementary Golay sequence
crest factor of less than 6dB (PAPR of 3 dB) can be obtained
Proof:
• Aperiodic correlation Cx(z) of some sequence xi 
N 1 z
C x ( z )   xi xi* z
i 0

The Fourier 1transform Sx(f) of sequence xi 
N
S x ( f )   xi e           j 2 f iTs
| S( f ) |        
2 F
C (o )
i 0
where Ts is the sampling period of sequence xi 

•    Definition: Two sequences ai  and bi  of the length N
form a complementary pair if

Ca ( z )  Cb ( z )  0 , z 0             2 N , z o

– Golay complementary sequences have that property.
• N carrier OFDM; H-PSK modulation
N 1                                                                         N 1 z

S a (t )    e             ai      j 2  ( f 0  if ) t
Ca ( z )                      a i  ai  z

i 0                                                                             i 0

• Information-bearing sequence a  a   is
;                                                  j             j                     H

in fact an OFDM codeword and ξ is the primitive
H-root of unity (j in QPSK case)
•   Instantaneous Envelope Power
Pa (t ) | S a (t ) |                                             n   a a                            n
2                ai  a j  H ( i  j ) ft                        i     i u  Huft

i, j                                      u 0 i

n 1

 C (u)
u 0
a
 Huft
 n   | Ca (u ) | *1  n  2 (n  u )  n
u 0                                        u 1
2

For complementary sequences:

Pa (t )  Pb (t )  2n                                               Pa (t )  2n
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+ x2+x1 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.
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
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:
m 1
2   h 1

k 1
 x    (k )   x ( k 1)
• where      
is any permutation of the sequence of generator
matrix rows – see graph with rows as hypercube vertices

h
2       : number of elements in the Galois field
Simulation
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)
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)
Cyclic prefix of OFDM in frequency
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.
Transmission over frequency selective fading channel(*)
(t)               
x(t )     u (n)h(t  nT )   (t ) 
n  
s                rc   (t )
x (t )                   x (n)

u (n)      Pulse Shaping                     Channel                            Receive Filter
Tx                            ch                +                  Rx
t=nTs
h(t )   Tx (t )  ch (t )  Rx (t )

(n)                                                                   (n)

u (n)                                              x (n)                                                                      X (i )
U (i )
h(n)                    +                                 H0+H1             z-1                 +
L
x(n)   h(l )u (n  l )   (n)
l 0

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

 Channel response
magnitude

Dispersive in time,
Static over block interval

Selective in frequency
0        Ts   2Ts   3Ts       4Ts   5Ts   6Ts   7Ts   8Ts   9Ts   10Ts   time

Black : Average , Gray : a realization of the channel

hi  N (0, 1  k2 )  jN (0, 1  k2 )
2                 2
hi : Channel impulse responseof ith block
 k2   0 e kT / T
2            s   RMS
 k2 : Variance of kth complex sample
 0  1  e T / T
2               s       RMS

(**) Frequency selective Flat fading channel(Naftali Chayat in IEEE P802.11-97/96)
What is H0 and H1?

x0  hLu L  hL 1u L 1    h0u0  0                    X (i )  H 0U (i )  H1U (i  1)   (i )
L                     x1  hLu L 1  hL 1u L  2    h0u1  1
xk   hnu k  n                                                                         U (i ) : ith transmitted block
n 0                   xL  hLu0  hL 1u1    h0u L   L                        U (i  1) : (i - 1)th transmitted block
                                                          X(i ) : ith received block
xN 1  hLu L  hL 1u L 1    h0u N 1   N 1            (i ) : noise vector

 hL hL 1       h0       0          0   0 
                                                            h0    0     00  
 0 hL         hL 1  h0        0        0                                 
      h0 0  0 
X  H U , H                                           0 
H 0   hL        
                 
                                              
                                   0                                 
                                                                    0 
0            0     0 hL hL 1            h0                               
0       hL  h0 

N+L
0           hL  h1 
                     
Memory                     0   
N                                                  from the   H1   0            hL 
                     
past                         0 
block                                
0           0  0 

H                              U
How is IBI deleted ?
 (i )

U (i)                                                                                                              X (i )  H (i )U (i )   (i )
Tcp                          H0+H1      z-1            +                         Rcp

H

X (i )  Rcp H 0TcpU (i )  Rcp H1TcpU (i  1)  Rcp (i )                              h0                   hL                 h1 
                                              
 Rcp H 0TcpU (i )   (i )                                                                 h0                            
 H (i )U (i )   (i )                                                 H (i )   hL                                     hL 
                                              
Tcp : [ I cp I N ]T , Guard - inserting matrix , (N  L)  N
T    T                                                                                                               
                                              
I cp : Last L rows of an N  N identity matrix I N
0             hL                         h0 
Rcp : [0 N L I N ]Guard - discarding matrix, N  ( N  L)
T

F H F 1  diag[ H (e j 0 ), H (e j 2 / N ),, H (e j 2 ( N 1) / N )]
H (i ) becomes a circulant matrix
Fk ,n  N 1/ 2 exp( j 2kn / N )
L
H (e j 2f )   h(n) exp( j 2fn)
n 0
Cyclic prefix effect on OFDM

OFDM
S (i )                                             (i )
Input                                                                                                                      Output
bits           .                                                                                .                .         bits
.          U (i)                         .
S/P    .       Mapping   .   IFFT               H   +             .       FFT           .    Demapping   .   P/S
.                 .                                        .                     .                .

X (i )  H (i )U (i )   (i )

                        
F X (i )  F H (i )U (i )   (i )  F H (i ) F 1S (i )  F (i )  S (i )  F (i )

No IBI plus simpler equalizer
• Simulation configuration
– Perfect channel estimation , QPSK, Fixed sub-channel power
– Zero Forcing equalization
– 64 sub-carriers
• Simulation Results
Evaluation of Pilot-symbol based channel estimation

Guard
Bit                                                                          Multipath   Tapped
mapping                      IFFT                                            channel     delay line
insertion
Random                                                                                   model       H(n)
bits

Channel                                         Guard                                      AWGN
FFT
compensation                                    extraction

Bits
Symbol-
to-bit          Pilot extraction
mapping
Channel
estimation and
interpolation
Naftali Multipath Channel Model

Magnitude

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

hk = N(0, 1/2k2) + jN(0,1/2k2)
k2 = 0 e-kTs/TRMS
02 = 1 – e –Ts/TRMS
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))
802.11a Pilot subcarrier placement

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

Pilot subcarrier placement used

L=   7         14            22                   14            7

0        7          21                   43               57        64   Subcarrier
numbers 1 to 64
BER Performance in AWGN

Constellation in AWGN
Frequency Selective Fading, Naftali model Trms = 50 ns

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

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