lect7-i

8/7/2019 lect7-I

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Lecture # I

N. BOUBAKER [email protected]

Orthogonal Frequency Division Multiplexing

(OFDM): Basic OFDM System

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Outline

Motivations

Diagonalization of Vector Channels

Transmission of one OFDM Symbol

Transmission of sequence of OFDM Symbols

Sample and Symbol Time Synchronization

Carrier Frequency Synchronization

Peak-to-Average Power Ratio (PAPR) Issue

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OFDM: Motivations

Treat a Wideband FS fading channel as Multiple Narrowband Flat fading channels

Change in TX so that RX does not suffer from ISI Use FEC with codeword span across all sub-channels

achieve Frequency Diversity, but with no ISI problem

f

H(f)

Realization of Frequency

Selective Fading Channel

Each Sub-channel is a

Flat Fading channel

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OFDM : Motivations

Motivation:

± Split a frequency selective fading channel into multiple, say

N=1024, narrowband flat fading sub-channels

± Send the bits over these sub-channels in parallel

Serial

to

Parallel

modulator Mixer

f 1Com-

biner

modulator Mixer

f 1024

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OFDM: Motivation

Problems:

± Multiple transmitter front ends (mixer, modulator, etc)

± require guard bands

Solutions: ± Do all these in digital domain using a wide baseband signal

± Use DFT (discrete Fourier transform) to create the baseband

equivalent of the transmit signal and then up-covert it to the

center frequency using one front end

± As DFT is an orthogonal transformation, no guard band is

needed

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Diagonalization of Vector Channels

Consider a Vector Channel

with input x and output y

Want to diagonalize it such

that

i

N

j j jii n x H y !

!

§ !1 ,

nHxy

N in sd z iiii ,,1~

~

-!!

!

!!

nsD

UnU VsUyz

n

x yUV

s z

di

ni

si zi

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Maintaining the same SNR during

Diagonalization U has to be Unitary to prevent noise enhancement

V has to be Unitary to maintain the same transmit

power

unitaryisi

~~

0

0

U

UU

UUnnnnE

N

N H

H H H

!

!

!

unitaryisif VsEs

VsVEsxEx

H

H H H

!

!

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Advantage and Issues with

Diagonalization Need to find U and V s.t. UHV = DH is diagonal

± Decompose the vector channels into parallel channels with

different gain (allow adaptive modulation, and TX

optimization to be discussed in 2nd part of OFDM notes)

V depends on H (TX needs to know the vector

channel)

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Diagonalization of ISI channel

For channel is ISI, H is Toeplitz

is a circulant matrix if ¼¼¼

¼¼¼

½

»

¬¬¬

¬¬¬

-

«

!!

01

1

01

0

,

00

0

0

00

00

hh

h

hh

h

H H H i j ji

.

111/

/11

.

..

¼¼¼¼¼

¼

½

»

¬¬¬¬¬

¬

-

«

¼¼¼¼¼

¼

½

»

¬¬¬¬¬

¬

-

«

!!

01

1

01

101

,

00

0

000

0

0

0

0

hh

h

hh

hhh

H H H N ji ji

.

111/

/11

.

..

/

H~

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Diagonization of Circulant Matrix

If H is circulant, then

Note that THE W that diagonalized is independent

of ! ± Hence, TX does not need to know !!

Use cyclic prefix to create an effective circulant matrix

§

!

!

¹ º

¸©ª

¨!!

1

0

exp1

;~

N

m

N mn j

mnn

mn

H

eh

N

mn j

N

T

T

H

H WWHW

FT of the channelimpulse response:

Gains of the

subchannels

H~

H

~

H~

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Cyclic Prefix

Instead of transmitting x=Ws, transmit

Then,

Transmit a length N+Ns vector for a length-N datavector. Efficiency = N/(N+Ns) with Ns>L (ISI length)

For N=210=1024, Ns=10, Efficiency ~ 99%.

? AT

N x xx !~

xHxH ~

0

00

0

00

000

00

0

0

00

~ 1

01

01

01

10

1

01

01

01

01

!

¼¼¼

¼

½

»

¬¬¬

¬

-

«

¼¼¼¼

¼¼

½

»

¬¬¬¬

¬¬

-

«

!

¼¼¼

¼

½

»

¬¬¬

¬

-

«

¼¼¼¼

¼¼

½

»

¬¬¬¬

¬¬

-

«

!

N N

N

x

x

hh

hh

hh

hh

x

x

x

hh

hh

hh

hh

/

..

/

//

.

.

/

..

/

//

/.

..

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Add

Cyclic

Prefix

& Pulse

Shaping

OFDM Transmission

Serial

to

Parallel

IFFT

Parallel

to

Serial

Parallel

to

Serial

FFT

Serial

to

Parallel

Mixer

f c

Mixer &

Filterf c

Frequency

Selective

Channel

Frequency

omain

Samples

Timeomain

Samples

A B C

Matched

Filter

and

RemoveCyclic

Prefix

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Transmission of a sequence of

OFDM symbols Using a block of N samples to create an OFDM

symbol (x=Ws) and the cyclic prefix, ISI between

samples within an OFDM symbol is eliminated

What happens to the intersymbol interference between

OFDM symbols?

tNNs Ns

tNNs Ns N

TX:

RX:

No ISI

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Cyclic Prefix

Insert a Cyclic Prefix before every OFDM symbol

± Cycle Prefix length > Xmax

± Overhead is Xmax/(NTs) where Ts=1/B is the sampling period,

B is the bandwidth and N is the number of sub-carriers or

points in the IFFT

the larger the N, the smaller the overhead!!

If Xmax/Ts=Ns. Then, there will have N+Ns sampled

points for every OFDM symbol

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Cyclic Prefix

If we just take the last N points out of the N+Ns pointsto do the FFT at the receiver,

Then

where denotes circular convolution

and Yk = Hk Xk

where Yk , Xk , and Hk are the DFT of yn, xn, and hn, resp

nnn h x y !

NTs NsTs

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Advantages of OFDM

With cyclic prefix, we can eliminate ISI completely

Provide frequency diversity

± Forward error correcting code such as convolutional code

with interleaver is needed as some sub-carriers will be in

deep fade

Potential

± If the transmitter knows the channel conditions

can select only the good sub-carriers to transmit or

transmit different numbers of bits based on the sub-

carriers¶ gains

If the transmitter knows the channel, OFDM with bit

allocation is better than the best equalizer (e.g. MLSE)

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Sample and Symbol

Synchronization Sampling time synchronization

± Sampling Frequency needs to be correct, but sampling

instance offset only leads to linear phase shift in the sub-

channels¶ gains. (which will be handled by channel

estimation)

OFDM Symbol Synchronization

± Determine the beginning of the OFDM symbol and the

beginning of the cyclic prefix (to avoid Inter-OFDM symbol

interference) ± Use Cyclic Prefix

Compute Correlation between

two intervals separated by NTs

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Carrier Frequency Synchronization

Carrier Frequency offset can cause significant inter-

subcarrier interference

As there is no guard band, very small frequency offset

can lead to large inter-subcarrier (or inter-subchannel)interference

Very important and performance is sensitive to this

f

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Peak-to-Average Power Ratio

The data symbol, si, may be QPSK modulated

(constant magnitude), but the transmitted samples, xi,

is the output of the IFFT and hence takes values over a

wide range.

Statistically, as si are independent and has random

phase, xi approaches an Gaussian distribution when N

is large

Hence, high Peak-to-Average Power Ratio

2

2

max

i

ii

x

x PAPR

E!

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Disadvantages of OFDM

Overheads

± Cyclic Prefix: can be reduced by increasing N

± Power to transmit cyclic prefix: can be lower by increasing N

Implementation issues

± sensitivity to frequency offsets

especially when N is large and sub-carrier spacing issmall

± require highly linear power amp

high peak-to-average-power (PAP) ratio, especially when N is large

Typical value for N is 27 to 211

lect7-i

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