# Lecture 16

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E225C – Lecture 16

OFDM Introduction

EE225C

Introduction to OFDM

l Basic idea

» Using a large number of parallel narrow-band sub-

carriers instead of a single wide-band carrier to

transport information

l Advantages

» Very easy and efficient in dealing with multi-path

» Robust again narrow-band interference

l Disadvantages

» Sensitive to frequency offset and phase noise

» Peak-to-average problem reduces the power

efficiency of RF amplifier at the transmitter

l Adopted for various standards

– DSL, 802.11a, DAB, DVB

1

Multipath can be described in two domains:

time and frequency

Time domain: Impulse response

time time

time

Impulse response

Frequency domain: Frequency response

time time

time f time

Sinusoidal signal as input Frequency response Sinusoidal signal as output

Modulation techniques:

monocarrier vs. multicarrier

Channel

Channelization N carriers

Similar to

Guard bands

FDM technique

B B

Pulse length ~1/B Pulse length ~ N/B

– Data are transmited over only one carrier – Data are shared among several carriers

and simultaneously transmitted

Drawbacks Advantages

Furthermore

– Selective Fading – Flat Fading per carrier

– It is easy to exploit

– Very short pulses – N long pulses Frequency diversity

– ISI is compartively long – ISI is comparatively short – It allows to deploy

2D coding techniques

– EQs are then very long – N short EQs needed

– Dynamic signalling

– Poor spectral efficiency – Poor spectral efficiency

because of band guards because of band guards

To improve the spectral efficiency:

Eliminate band guards between carriers

To use orthogonal carriers (allowing overlapping)

2

Orthogonal Frequency Division Modulation

N carriers

Symbol: 2 periods of f0

Transmit

f

+

Symbol: 4 periods of f0

f

B

Symbol: 8 periods of f0

Channel frequency

Data coded in frequency domain Transformation to time domain: response

each frequency is a sine wave

in time, all added up.

Decode each frequency

bin separately

Receive

time f

B

Time-domain signal Frequency-domain signal

OFDM uses multiple carriers

to modulate the data

Time-frequency grid Data

N carriers

Frequency

B Carrier

f0

B

One OFDM symbol

T=1/f0

Features Time

– No intercarrier guard bands

Intercarrier Separation =

– Controlled overlapping of bands

– Maximum spectral efficiency (Nyquist rate) 1/(symbol duration)

– Easy implementation using IFFTs

– Very sensitive to freq. synchronization

Modulation technique

A user utilizes all carriers to transmit its data as coded quantity at each

frequency carrier, which can be quadrature-amplitude modulated (QAM).

3

OFDM Modulation and Demodulation

using FFTs

d0

b0

d1 P/S

b1 IFFT

d2 d0, d1, d2, …., dN-1

b2 Inverse fast d3 Parallel to

. Fourier transform . serial converter

.

f . . Transmit time-domain

. samples of one symbol

.

.

bN-1 time

dN-1

Data coded in

frequency domain: Data in time domain:

one symbol at a time one symbol at a time

d0’ Decode each

b0’

d0’, d1’, …., dN-1’ S/P d1’ FFT b1’

frequency bin

d2’ Fast Fourier independently

b2’

Serial to . transform .

Receive time-domain parallel converter . .

samples of one symbol . f .

. .

dN-1’ bN-1’

time

Loss of orthogonality (by frequency offset)

Transmission pulses ψ k (t) = exp( jk 2π t / T ) y ψ k +m ( t) = exp ( j2π (k + m )t / T )

ψ k+ m (t) = exp ( j2π (k + m + δ ) / T ) con δ ≤ 1 / 2

δ

Reception pulse with offset δ

T (1 − exp(− j2πδ ))

I m (δ ) = ∫ exp( jk2πt / T ) exp(− j(k + m + δ )2πt / T )dt =

T

Interference between

channels k and k+m 0

j 2π(m + δ )

N −1

T sin πδ 1 23

I m (δ) =

π m+δ

Summing up

∀m ∑ I (δ ) ≈ (Tδ) ∑ m ≈ (Tδ ) 14

2

m

2

2

2

for N >> 1 (N > 5 Is enough )

m m =1

Loss for 8 carriers Total ICI due to loss of orthogonality

0

-10

-10 m=1

Interference: Im(? )/T en dB

-15 δ =0.05

-20 -20

δ =0.02

m=3

m=5 -25

-30 m=7 -30 δ =0.01

ICI in dB

-40 -35 δ =0.005

-40

Practical limit

-50 -45 δ =0.002

Asymetric δ assumed

-50 r.v. δ =0.001

-60

Gaussian

-55 σ=δ

-70 -60

0

-0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 2 4 6 8 10 12 14 16

Frequency offset: ∂ Carrier position within the band (N=16)

4

Loss of orthogonality (time)

− T /2+ τ T/ 2 2 consecutive

Let us assume Xi = c 0 ∫ ψ k (t )ψ l (t − τ )dt + c 1 ∫ ψ k (t )ψ l (t − τ )dt

* *

a misadjustment τ −T /2 −T / 2+τ symbols

τ

senmπ

2 T T , c ≠c τ

Then Xi = mπ

0 1

Or approximately, Xi 2mπ T τ independent

≈ =2

when τ< if m=k-l 0, c0 = c1

τ

X 2 ICI ≈ 20log 2 , τ << T

τ 1 τ

2 2

In average, the interfering T

E i2 = 4 + 0 = 2

1

power in any carrier is T 2 T

T 2 Per carrier

Loss for 16 carriers ICI due to loss of orthogonaliy

0 45

-5

m=1 40 Doubling N means 3 dB more ICI

-10

Interference en dB

-15 35

τ assumed an Uniform r.v.

-20

ICI in dB

m=5 30

-25

-30 m=10 25 N=8 Max. practical limit

-35 20

-40 N=64

15

-45

-50 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Relative misadjustment τ Typical deviation for the relative misadjustment

Zone of interest

Including a “cyclic prefix”

To combat the time dispersion: including ‘special’ time guards in the symbol transitions

co p y

Furthemore it converts Linear conv. = Cyclic conv.

CP

τ T (Method: overlap-save)

Tc

Without the Cyclic Prefix Including the Cyclic Prefix

Symbol: 8 periods of fi

CP

Symbol: 8 periods of fi

Passing the channel h(n)

Passing the channel h(n)

Ψi(t)

Ψi(t)

Channel:h(n )=(1 ) – n / n n =0 , …,2 3

≠Ψ i(t)

Initial transient The inclusion of a CP Final transient

remains within maintains the orthogonality remains within

Initial transient Loss of orthogonality Decaying transient

the CP the CP

Ψ j(t) Ψj (t)

Symbol: 4 periods of fi

Symbol: 4 periods of fi

CP functions:

– It acomodates the decaying transient of the previous symbol

– It avoids the initial transient reachs the current symbol

5

Cyclic Prefix

Tg T

Multi-path components

τmax

Tx Sampling start T

802.11a System Specification

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 GI2 T1 T2 GI OFDM Symbol GI OFDM Symbol

Short training sequence: Long training sequence:

AGC and frequency offset Channel estimation

l Sampling (chip) rate: 20MHz

l Chip duration: 50ns

l Number of FFT points: 64

l FFT symbol period: 3.2µs

l Cyclic prefix period: 16 chips or 0.8µs

» Typical maximum indoor delay spread < 400ns

» OFDM frame length: 80 chips or 4µs

» FFT symbol length / OFDM frame length = 4/5

l Modulation scheme

» QPSK: 2bits/sample

» 16QAM: 4bits/sample

» 64QAM: 6bits/sample

l Coding: rate ½ convolutional code with constraint length 7

6

Frequency diversity using coding

Random errors: primarily introduced by thermal and circuit noise.

Channel-selected errors: introduced by magnitude distortion in

channel frequency response.

Data bits

Time-frequency grid

Frequency

B

Bad carriers

f0

f Time

Frequency response T=1/f0

Errors are no longer random. Interleaving is often used to scramble

the data bits so that standard error correcting codes can be applied.

Spectrum Mask

Power Spectral Density

-20 dB

-28 dB

-40 dB

-30 -20 -11 -9 9 11 20 30

f carrier

Frequency (MHz)

• Requires extremely linear power amplifier design.

7

Adjacent Channel and

Alternate Channel Rejection

Date M inimum Adjacent Channel Alternate

rate Sensibility Rejection Channel rejection

6 Mbps -82 d B m 16 dB 32 dB

12Mbps -79 d B m 13 dB 29 dB

24Mbps -74 d B m 8 dB 24 dB

36Mbps -70 d B m 4 dB 20 dB

54Mbps -65 d B m 0 dB 15 dB

32 dB blocker

16 dB blocker

Signal Frequency

• Requires joint design of the anti-aliasing filter and ADC.

OFDM Receiver Design

Yun Chiu, Dejan Markovic, Haiyun Tang,

Ning Zhang

EE225C Final Project Report, 12 December

2000

8

OFDM System Block Diagram

Synchronization

l Frame detection

Tg T

Frame start

l Frequency offset compensation

l Sampling error

» Usually less 100ppm and can be ignored

– 100ppm = off 1% of a sample every 100 samples

9

System Pilot Structure

IEEE 802.11a OFDM Txer

Short Preamble Gen.

Long Preamble Gen.

OFDM Data Path

10 x 0.8 = 8 uS 2 x 0.8 + 2 x 3.2 = 8 uS 0.8 + 3.2 = 8 uS 0.8 + 3.2 = 8 uS 0.8 + 3.2 = 8 uS

1 2 3 4 5 6 7 8 9 10 GI2 T1 T2 GI Signal GI Data GI Data

Signal Detection, AGC, Channel & Fine Freq. Rate, Length Data Data

Diversity Selection Offset Estimation

Coarse Freq. Offset

Est.,Timing Sync.

10

Short & Long Preambles

1+j

-20

-24 -12

-16 f

-1-j

Short Preamble

+1 Period = 16 Chips

-24

-26

-16 -12

f

-1

Long Preamble

Period = 64 Chips

Correlation of Short Preamble

Correlation

Fine Timing

Auto-

Correlation

Coarse Timing

11

Synchronization

From AGC

16Td Td Td Td ... T d

* * *

... * Σ

Moving Auto-

Corr. Unit

Td Td Td ... Td

From AGC

Td Td Td ... T d

* * *

... * Σ

Moving SP

Corr. Unit ...

Short Preamble (LUT)

Impairments: Multi-Path Channel

Tc

2T 0 0

0 T

T 2T

3T t T t

t 4T c

0 T

T 2T

2T 3T t t

3T t 4T

4T 5T

0 0 0

T T

2T 2T

T t 3T t 3T t

c 4T 4T

5T 5T

Auto-Correlation w/

Ch. Impulse

Multi-Path Channel

Response

Response.

12

OFDM Introduction

EE225C

Introduction to OFDM

l Basic idea

» Using a large number of parallel narrow-band sub-

carriers instead of a single wide-band carrier to

transport information

l Advantages

» Very easy and efficient in dealing with multi-path

» Robust again narrow-band interference

l Disadvantages

» Sensitive to frequency offset and phase noise

» Peak-to-average problem reduces the power

efficiency of RF amplifier at the transmitter

l Adopted for various standards

– DSL, 802.11a, DAB, DVB

1

Multipath can be described in two domains:

time and frequency

Time domain: Impulse response

time time

time

Impulse response

Frequency domain: Frequency response

time time

time f time

Sinusoidal signal as input Frequency response Sinusoidal signal as output

Modulation techniques:

monocarrier vs. multicarrier

Channel

Channelization N carriers

Similar to

Guard bands

FDM technique

B B

Pulse length ~1/B Pulse length ~ N/B

– Data are transmited over only one carrier – Data are shared among several carriers

and simultaneously transmitted

Drawbacks Advantages

Furthermore

– Selective Fading – Flat Fading per carrier

– It is easy to exploit

– Very short pulses – N long pulses Frequency diversity

– ISI is compartively long – ISI is comparatively short – It allows to deploy

2D coding techniques

– EQs are then very long – N short EQs needed

– Dynamic signalling

– Poor spectral efficiency – Poor spectral efficiency

because of band guards because of band guards

To improve the spectral efficiency:

Eliminate band guards between carriers

To use orthogonal carriers (allowing overlapping)

2

Orthogonal Frequency Division Modulation

N carriers

Symbol: 2 periods of f0

Transmit

f

+

Symbol: 4 periods of f0

f

B

Symbol: 8 periods of f0

Channel frequency

Data coded in frequency domain Transformation to time domain: response

each frequency is a sine wave

in time, all added up.

Decode each frequency

bin separately

Receive

time f

B

Time-domain signal Frequency-domain signal

OFDM uses multiple carriers

to modulate the data

Time-frequency grid Data

N carriers

Frequency

B Carrier

f0

B

One OFDM symbol

T=1/f0

Features Time

– No intercarrier guard bands

Intercarrier Separation =

– Controlled overlapping of bands

– Maximum spectral efficiency (Nyquist rate) 1/(symbol duration)

– Easy implementation using IFFTs

– Very sensitive to freq. synchronization

Modulation technique

A user utilizes all carriers to transmit its data as coded quantity at each

frequency carrier, which can be quadrature-amplitude modulated (QAM).

3

OFDM Modulation and Demodulation

using FFTs

d0

b0

d1 P/S

b1 IFFT

d2 d0, d1, d2, …., dN-1

b2 Inverse fast d3 Parallel to

. Fourier transform . serial converter

.

f . . Transmit time-domain

. samples of one symbol

.

.

bN-1 time

dN-1

Data coded in

frequency domain: Data in time domain:

one symbol at a time one symbol at a time

d0’ Decode each

b0’

d0’, d1’, …., dN-1’ S/P d1’ FFT b1’

frequency bin

d2’ Fast Fourier independently

b2’

Serial to . transform .

Receive time-domain parallel converter . .

samples of one symbol . f .

. .

dN-1’ bN-1’

time

Loss of orthogonality (by frequency offset)

Transmission pulses ψ k (t) = exp( jk 2π t / T ) y ψ k +m ( t) = exp ( j2π (k + m )t / T )

ψ k+ m (t) = exp ( j2π (k + m + δ ) / T ) con δ ≤ 1 / 2

δ

Reception pulse with offset δ

T (1 − exp(− j2πδ ))

I m (δ ) = ∫ exp( jk2πt / T ) exp(− j(k + m + δ )2πt / T )dt =

T

Interference between

channels k and k+m 0

j 2π(m + δ )

N −1

T sin πδ 1 23

I m (δ) =

π m+δ

Summing up

∀m ∑ I (δ ) ≈ (Tδ) ∑ m ≈ (Tδ ) 14

2

m

2

2

2

for N >> 1 (N > 5 Is enough )

m m =1

Loss for 8 carriers Total ICI due to loss of orthogonality

0

-10

-10 m=1

Interference: Im(? )/T en dB

-15 δ =0.05

-20 -20

δ =0.02

m=3

m=5 -25

-30 m=7 -30 δ =0.01

ICI in dB

-40 -35 δ =0.005

-40

Practical limit

-50 -45 δ =0.002

Asymetric δ assumed

-50 r.v. δ =0.001

-60

Gaussian

-55 σ=δ

-70 -60

0

-0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 2 4 6 8 10 12 14 16

Frequency offset: ∂ Carrier position within the band (N=16)

4

Loss of orthogonality (time)

− T /2+ τ T/ 2 2 consecutive

Let us assume Xi = c 0 ∫ ψ k (t )ψ l (t − τ )dt + c 1 ∫ ψ k (t )ψ l (t − τ )dt

* *

a misadjustment τ −T /2 −T / 2+τ symbols

τ

senmπ

2 T T , c ≠c τ

Then Xi = mπ

0 1

Or approximately, Xi 2mπ T τ independent

≈ =2

when τ<

τ

X 2 ICI ≈ 20log 2 , τ << T

τ 1 τ

2 2

In average, the interfering T

E i2 = 4 + 0 = 2

1

power in any carrier is T 2 T

T 2 Per carrier

Loss for 16 carriers ICI due to loss of orthogonaliy

0 45

-5

m=1 40 Doubling N means 3 dB more ICI

-10

Interference en dB

-15 35

τ assumed an Uniform r.v.

-20

ICI in dB

m=5 30

-25

-30 m=10 25 N=8 Max. practical limit

-35 20

-40 N=64

15

-45

-50 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Relative misadjustment τ Typical deviation for the relative misadjustment

Zone of interest

Including a “cyclic prefix”

To combat the time dispersion: including ‘special’ time guards in the symbol transitions

co p y

Furthemore it converts Linear conv. = Cyclic conv.

CP

τ T (Method: overlap-save)

Tc

Without the Cyclic Prefix Including the Cyclic Prefix

Symbol: 8 periods of fi

CP

Symbol: 8 periods of fi

Passing the channel h(n)

Passing the channel h(n)

Ψi(t)

Ψi(t)

Channel:h(n )=(1 ) – n / n n =0 , …,2 3

≠Ψ i(t)

Initial transient The inclusion of a CP Final transient

remains within maintains the orthogonality remains within

Initial transient Loss of orthogonality Decaying transient

the CP the CP

Ψ j(t) Ψj (t)

Symbol: 4 periods of fi

Symbol: 4 periods of fi

CP functions:

– It acomodates the decaying transient of the previous symbol

– It avoids the initial transient reachs the current symbol

5

Cyclic Prefix

Tg T

Multi-path components

τmax

Tx Sampling start T

802.11a System Specification

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 GI2 T1 T2 GI OFDM Symbol GI OFDM Symbol

Short training sequence: Long training sequence:

AGC and frequency offset Channel estimation

l Sampling (chip) rate: 20MHz

l Chip duration: 50ns

l Number of FFT points: 64

l FFT symbol period: 3.2µs

l Cyclic prefix period: 16 chips or 0.8µs

» Typical maximum indoor delay spread < 400ns

» OFDM frame length: 80 chips or 4µs

» FFT symbol length / OFDM frame length = 4/5

l Modulation scheme

» QPSK: 2bits/sample

» 16QAM: 4bits/sample

» 64QAM: 6bits/sample

l Coding: rate ½ convolutional code with constraint length 7

6

Frequency diversity using coding

Random errors: primarily introduced by thermal and circuit noise.

Channel-selected errors: introduced by magnitude distortion in

channel frequency response.

Data bits

Time-frequency grid

Frequency

B

Bad carriers

f0

f Time

Frequency response T=1/f0

Errors are no longer random. Interleaving is often used to scramble

the data bits so that standard error correcting codes can be applied.

Spectrum Mask

Power Spectral Density

-20 dB

-28 dB

-40 dB

-30 -20 -11 -9 9 11 20 30

f carrier

Frequency (MHz)

• Requires extremely linear power amplifier design.

7

Adjacent Channel and

Alternate Channel Rejection

Date M inimum Adjacent Channel Alternate

rate Sensibility Rejection Channel rejection

6 Mbps -82 d B m 16 dB 32 dB

12Mbps -79 d B m 13 dB 29 dB

24Mbps -74 d B m 8 dB 24 dB

36Mbps -70 d B m 4 dB 20 dB

54Mbps -65 d B m 0 dB 15 dB

32 dB blocker

16 dB blocker

Signal Frequency

• Requires joint design of the anti-aliasing filter and ADC.

OFDM Receiver Design

Yun Chiu, Dejan Markovic, Haiyun Tang,

Ning Zhang

EE225C Final Project Report, 12 December

2000

8

OFDM System Block Diagram

Synchronization

l Frame detection

Tg T

Frame start

l Frequency offset compensation

l Sampling error

» Usually less 100ppm and can be ignored

– 100ppm = off 1% of a sample every 100 samples

9

System Pilot Structure

IEEE 802.11a OFDM Txer

Short Preamble Gen.

Long Preamble Gen.

OFDM Data Path

10 x 0.8 = 8 uS 2 x 0.8 + 2 x 3.2 = 8 uS 0.8 + 3.2 = 8 uS 0.8 + 3.2 = 8 uS 0.8 + 3.2 = 8 uS

1 2 3 4 5 6 7 8 9 10 GI2 T1 T2 GI Signal GI Data GI Data

Signal Detection, AGC, Channel & Fine Freq. Rate, Length Data Data

Diversity Selection Offset Estimation

Coarse Freq. Offset

Est.,Timing Sync.

10

Short & Long Preambles

1+j

-20

-24 -12

-16 f

-1-j

Short Preamble

+1 Period = 16 Chips

-24

-26

-16 -12

f

-1

Long Preamble

Period = 64 Chips

Correlation of Short Preamble

Correlation

Fine Timing

Auto-

Correlation

Coarse Timing

11

Synchronization

From AGC

16Td Td Td Td ... T d

* * *

... * Σ

Moving Auto-

Corr. Unit

Td Td Td ... Td

From AGC

Td Td Td ... T d

* * *

... * Σ

Moving SP

Corr. Unit ...

Short Preamble (LUT)

Impairments: Multi-Path Channel

Tc

2T 0 0

0 T

T 2T

3T t T t

t 4T c

0 T

T 2T

2T 3T t t

3T t 4T

4T 5T

0 0 0

T T

2T 2T

T t 3T t 3T t

c 4T 4T

5T 5T

Auto-Correlation w/

Ch. Impulse

Multi-Path Channel

Response

Response.

12