OFDM Lecture

5/23/2018 OFDM Lecture

1/128

Orthogonal Frequency

Division MultiplexingOFDM

fred harris

Cubic Signal Processing Chair

San Diego State [email protected]

Vehicular Technology Conference - 2004


2/128

Textbooks and References

Wireless OFDM Systems: How to Make Them WorkMarc Engels, Editor

OFDM Wireless LANs: A Theoretical and Practical Guide

Juha Heiskala and John Terry

OFDM for Wireless Multimedia Communications

Richard Van Nee and Ramjee Prasad Single and Multi-Carrier Quadrature Amplitude Modulation

Lajos Hanzo, William Webb, and Thomas Keller

ADSL, VDSL, and Multicarrier Modulation

John Bingham

Implementing ADSL

David Ginsburg

DSL Advances

Massimo Sorbara, John Cioffi, and Peter Silverman


3/128

OFDM

OFDM also known as

Multi-Carrier or Multi-Tone Modulation

DAB-OFDM

Digital Audio Broadcasting DVD-OFDM

Digital Video Broadcasting

ADSL-OFDM

Asynchronous Digital Subscriber Line Wireless Local Area Network

IEEE-802.11a, IEEE-802.11g

ETSI BRAN (Hyperlan/2)


4/128

OFDM Systems

System Transform

Size

Number

Carriers

Channel

Spacing

kHz

Bandwidth

MHz

Sample

Rate

MHz

Symbol

Duration

sec

Data

Rate

Mbits/s

HyperLAN/2 64 524 312.5 16.25 20 3.20.8 6-54

802.11a 64 52

4

312.5 16.56 20 3.2

0.8

6-54

DVB-T 2048

1024

1712

842

4.464 7.643 9.174 224 0.68-14.92

DAB 2048

8192

1536 1.00 1.536 2.048 24/48/96

msec

3.072

ADSL 256 (down)

64 (up)

36-127

7-28

4.3125 1.104 1.104 231.9 0.64-8.192


5/128

OFDM Advantages

Efficiently Deals With Multi-path Fading

Efficiently Deals With Channel Delay Spread

Enhanced Channel CapacityAdaptively Modifies Modulation Density

Robustness to Narrowband Interference


6/128

OFDM Disadvantages

OFDM Sensitive to

Small Carrier Frequency Offsets

OFDM ExhibitsHigh Peak to Average Power Ratio

OFDM Sensitive to

High Frequency Phase Noise OFDM Sensitive to

Sampling Clock Offsets


7/128

Single Carrier System

Sequential Transmissionof Waveforms

Waveforms are

Short Duration T

Waveforms Occupy

Full System Bandwidth 1/T


8/128

Multi-Carrier System

Parallel Transmission

of Waveforms

Waveforms areLong Duration MT

Waveforms Occupy 1/M th

Of System Bandwidth 1/T


9/128

OFDM: Dense Multichannel System

Conventional Multichannel System

Non Overlapping Adjacent Channels.

Channels separated by MoreThan Their Two Sided bandwidth

OFDM Multichannel System

50% Overlap of Adjacent ChannelsAvailable bandwidth is Used Twice

Channels separated by Half

Their Two Sided bandwidth


10/128

Standard Digital

Communication System


11/128

Bandlimited Channel

Nyquist Spectrum Nyquist Spectrum

With Cosine Taper

Infinite Duration Nyquist Pulse Finite Duration Nyquist Pulse


12/128

Continuous Time: Orthogonal Time Signal Set

T

mn

k

mnifTmnifdttt

Ttktk

Tjt

Ttkt

0

k

0)()(

0,,2,1,0,1,2,:)2exp()(

0,,2,1,0,1,2,:)(


13/128

Discrete Time: Orthogonal Time Signal Set

k

1

0

..., 2, 1, 0 ,1, 2, ....,( ) :

0 1

0 ,1, 2, ...., 12( ) exp( ) :

0

0 ,1, 2, ...., 12exp( ) :0

0( ) ( )

: ( ) ( ) ( ) ( )

k

N

n m

n

k N k k N k

kn

n N

k Nn j k nT

NT nT NT

k Nj k nN n N

if n mn n

N if n m

NOTE n n n n


14/128

OFDM Modulator


15/128

OFDM Demodulator


16/128

OFDM is a Block Process


17/128

Adjacent Symbol Interference (ASI)

Symbol Smearing Due to Channel


18/128

Guard Interval Inserted Between Adjacent

Symbols to Suppress ASI


19/128

Cyclic Prefix Inserted in Guard Interval to

Suppress Adjacent Channel Interference (ACI)


20/128

Data Length Defines Sinc Width:

Spectral Spacing Matches Width


21/128

Extended Data Length Reduces Sinc

Width: Spectral Spacing Preserved


22/128

OFDM Symbol: Time and Spectra

Channel Input and Output

20 40 60 80 100 120 140 160 180

-0.4

-0.2

0

0.2

0.4

0.6Real Part of Time Series, Input to Channel

20 40 60 80 100 120 140 160 180

-0.4

-0.2

0

0.2

0.4

0.6Real Part of Time Series, Output of Channel

-0.5 0 0.5-30

-25

-20

-15

-10

-5

0

5

10Spectrum

-0.5 0 0.5-30

-25

-20

-15

-10

-5

0

5

10Spectrum


23/128

OFDM Spectra

Without and with Cyclic Prefix

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-1.5

-1

-0.5

0

0.5

1

1.5OFDM spectral lines With Channel Without Cyclic Prefix

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-1.5

-1

-0.5

0

0.5

1

1.5OFDM spectral lines With Channel With Cyclic Prefix


24/128

Overlaid Constellations , All Frequencies,

Without and With Cyclic Prefix

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5OFDM Constellations With Channel Without Cyclic Prefix

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5OFDM Constellations With Channel With Cyclic Prefix


25/128

Constellations: Different OFDM Bins

Without Cyclic Prefix

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1


26/128

Constellations: Different OFDM Bins

With Cyclic Prefix

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1


27/128

Channel Estimate with Pilots

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

Channel, Bandwidth, and Samples

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1Zero Packed

Spectral Samplesand

Extended Reflection

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1Interpolated

Spectral Points

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.005

0.01

Magnitude of Interpolation Error For In-Band Frequencies

Normalized Frequency


28/128

DFT (FFT) as Signal Generator

for Complex Sinusoids


29/128

DFT (FFT) As Signal Analyzer

for Complex Sinusoids

1,...,2,1,0:)()(1

0

2

NkenhkH

N

n

nkN

j


30/128

Radix-2 FFT Flow Diagrams


31/128

Input Vector FFT Mapped to Output Time Series,

Up-Sampled, Converted Via DAC to Waveform,

and I-Q Up-Converted


32/128

The FFT as Signal Generator

and Interpolator


33/128

OFDM Modulation With IFFT

and Interpolator


34/128

OFDM Demodulation With FFT


35/128

OFDM Transceiver


36/128

Time and Spectra of Sparse OFDM Symbol

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1Real Part OFDM Time Series

Normalized Time

Amplitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-50

-40

-30

-20

-10

0

10Spectrum


LogMagnitude(dB)


37/128

Time and Spectra With Frequency Offset = 0.1 Bin

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1Real Part OFDM Time Series with Offset Frequency = 0.1 Bin Width

Normalized Time

Amplitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-50

-40

-30

-20

-10

0

10Spectrum With Frequency Offset = 0.1 Bin Width


LogMagnitude(d

B)


38/128

Time and Spectra With Sample Clock Offset = 1.02 fs

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

Real Part OFDM Time Series with Sampling Clock = 1.02 fs

Normalized Time

Amplitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-50

-40

-30

-20

-10

0

10

Spectrum With Sampling Clock = 1.02 fs


LogMagnitude(dB)


39/128

Time and Spectra With Sample Clock Offset = 0.98 fs

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

Real Part OFDM Time Series with Sampling Clock = 0.98 fs

Normalized Time

Amplitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-50

-40

-30

-20

-10

0

10

Spectrum With Sampling Clock = 0.98 fs


LogMagnitude(d

B)


40/128

Ideal I-Q Up and Down Conversion

Shape

Shape

Match

Match

CHANNEL

cos( t) cos( t)

-sin( t) -sin( t)

n(t)

I(t) I(t)

Q(t)Q(t)

^

^


41/128

Spectral and Time Description of

Real Sinusoids


42/128

Complex Sinusoids-I


43/128

Complex Sinusoids-II

C l B b d d C l


44/128

Complex Baseband and Complex

Band-Centered Spectra


45/128

Complex Baseband and Real

Band-Centered Spectra


46/128

Complex Down Conversion


47/128

Gain and Phase Imbalance in

I-Q Mixers


48/128

Spectral Image Due to Gain Imbalance


49/128

Spectral Image Due to Phase Imbalance

Li S t l I D t I Q Mi t h


50/128

Line Spectral Images Due to I-Q Mismatch


51/128

Coupling Between Positive and Negative FFT Indices Due to

I-Q Imbalance and First Order Correction Mechanism

(1 ) ( )( ) ( )2 2 2

( ) ( )( ) (1 )

2 2 2

(1 ) ( )( ) ( )2 2 2

( ) ( )( ) (1 )2 2 2

j jG k H k

G k H k j j

j jH k G k

H k G k j j

Test Bench: Demonstration of Receiver I-Q


52/128

Test Bench: Demonstration of Receiver I-Q

Imbalances, Carrier Offset, and Timing Offset


53/128

Carrier Offset: 4% of FFT Bin Width


54/128

Timing Offset: 10% of Sampling Time Period


55/128

Timing Clock Offset: 5% of Sampling Time Period per Frame

Gain Imbalance: 10% Error


56/128

Gain Imbalance: 10% Error

Ph I b l 0 1 R di E


57/128

Phase Imbalance: 0.1 Radian Error

I Q Mixer Imbalance; 20% Gain 0 2 Radians


58/128

I-Q Mixer Imbalance; 20% Gain, 0.2 Radians

Differential Delay to I/Q Mixers,

10% f S l I t l


59/128

10% of Sample Interval

Periodic Time Segments in OFDM Frame


60/128

Periodic Time Segments in OFDM Frame

Obtained by Zero Packing Spectrum


61/128

Probe Mismatch During Short Repeated Preamble


62/128

Power Amplifier Non-Linearity

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4Nonlinear Transfer Function of Amplifier

1-dB Compression Point

0 2 4 6 8 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Input and Output of Non-Linear Amplifier

-0.5 0 0.5-60

-50

-40

-30

-20

-10

0

10Spectrum of Two Input Sinusoids


-0.5 0 0.5-60

-50

-40

-30

-20

-10

0

10Spectrum of Two Output Sinusoids



63/128

16-QAM Input and Output Envelopes.

Saturation and 1-dB Compression Circles

-2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Envelope at Output of Amplifier

-2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Envelope at Input to Amplifier

Saturation at 2-Times RMS Signal Level

Saturation Saturation

1-dB

Compression

1-dB

Compression


64/128

Limiting Amplifier Effect on Received QAM Constellation

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Matched Filter Applied to Input of Amplifier

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Matched Filter Applied to Output of Amplifier


65/128

Limiting Amplifier Effect on Signal Spectra

-4 -3 -2 -1 0 1 2 3 4

-60

-50

-40

-30

-20

-10

0

10Spectrum at Input to Amplifier

Normalized Frequency (f/fsym

)

LogMagnitude(dB)

-4 -3 -2 -1 0 1 2 3 4

-60

-50

-40

-30

-20

-10

0

10Spectrum at Output of Amplifier

Normalized Frequency (f/fsym

)

LogMagnitude

(dB)

SPECTRAL REGROWTH

16 QAM ( 0 2) E l St ti ti


66/128

16-QAM (=0.2) Envelope Statistics

0 0.5 1 1.5 2 2.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

16-QAM Histogram at Amplifier Input

Normalized Amplitude (x/x)

0 0.5 1 1.5 2 2.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016 std dev =1.03 clip level

16-QAM Histogram at Amplifier Output

Normalized Amplitude (x/x

)

0 0.5 1 1.5 2 2.5 310

-6

10-5

10-4

10-3

10-2

10-1

100

Probabilty of Level Crossing

Normalized Amplitude (x/x

)

OFDM I t d O t t E l


67/128

OFDM Input and Output Envelopes:

Saturation and 1-dB Compression Circles

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Envelope at Input to Amplifier

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Envelope at Output of Amplifier

Saturation at 2-Times RMS Signal Level

SaturationSaturation

1-dB

Compression

1-dB

Compression


68/128

Limiting Amplifier Effect on OFDM Constellation

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

OFDM Constellation at Input to Amplifier

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

OFDM Constellation at Output of Amplifier


69/128

OFDM Envelope Statistics

0 1 2 3 40

0.005

0.01

0.015OFDM Histogram at Amplifier Input


0 1 2 3 40

0.005

0.01

0.015

std dev =1.06 clip level

OFDM Histogram at Amplifier Output


0 1 2 3 410

-6

10-5

10-4

10-3

10-2

10-1

100



OFDM Envelope Statistics with


70/128

OFDM Envelope Statistics with

Selected Alternate Mapping

0 1 2 3 40

0.005

0.01

0.015

0.02OFDM Histogram: One FFT

0 1 2 3 40

0.005

0.01

0.015

0.02OFDM Histogram: Two FFTs

0 1 2 3 40

0.005

0.01

0.015

0.02

OFDM Histogram: Four FFTs


0 1 2 3 410

-6

10-5

10-4

10-3

10-2

10-1

100



One FFTTwo FFTs

Four FFTs


71/128

Clipping


72/128

Smart Clipping


73/128

Reserve Frequency Bins Form Clipping

Pulses


74/128

Selecting Reserve Frequency Bins

-60 -40 -20 0 20 40 600

0.2

0.4

0.6

0.8

1

Spectrum 11-Adjacent Frequencies

-0.5 0 0.50

0.2

0.4

0.6

0.8

1

Time Series for 11-Adjacent Frequencies

-60 -40 -20 0 20 40 600

0.2

0.4

0.6

0.8

1

Spectrum 11-Equally Spaced Frequencies

-0.5 0 0.50

0.2

0.4

0.6

0.8

1

Time Series for 11-Equally Spaced Frequencies

-60 -40 -20 0 20 40 600

0.2

0.4

0.6

0.8

1

Spectrum 11-Randomly Spaced Frequencies

-0.5 0 0.50

0.2

0.4

0.6

0.8

1

Time Series for 11-Randomly Spaced Frequencies


75/128

Reserve Bin Canceller Clipping at 2.5 (8 dB)

0 50 100 150 200 2500

1

2

3

4

5Peak envelope, input to PAR control

0 50 100 150 200 2500

1

2

3

4

5Peak envelope, output of first pass PAR control

data

clip level

data std devaverage peak

0 50 100 150 200 2500

1

2

3

4

5Peak envelope, output of second pass PAR control

data

clip level

data std dev

average peak

0 50 100 150 200 2500

1

2

3

4

5Peak envelope, output of third pass PAR control

data

clip level

data std dev

average peak

St ti ti f Cli t 2 5 (8 dB)


76/128

Statistics for Clip at 2.5 (8 dB)

0 1 2 3 40

0.005

0.01

0.015

0.02

0.025input histogram

0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

std dev =0.928

clip level

output histogram

-5 0 5 1010

-6

10-5

10-4

10-3

10-2

10-1

100

average =-0.648 dB

prob of level crossing

PAR (dB)

inputpass-1

pass-2pass-3

R Bi C ll Cli i t 2 2 (6 9 dB)


77/128

Reserve Bin Canceller Clipping at 2.2 (6.9 dB)

0 50 100 150 200 2500

1

2

3

4


0 50 100 150 200 2500

1

2

3

4


data

clip level

data std dev

average peak

0 50 100 150 200 2500

1

2

3

4


data

clip level

data std dev

average peak

0 50 100 150 200 2500

1

2

3

4


data

clip level

data std dev

average peak

St ti ti f Cli t 2 2 (6 9 dB)


78/128

Statistics for Clip at 2.2 (6.9 dB)

0 1 2 3 40

0.005

0.01

0.015

0.02


0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

std dev =0.928

clip level

output histogram

-5 0 5 1010

-6

10-5

10-4

10-3

10-2

10-1

100

average =-0.653 dB


PAR (dB)

input

pass-1

pass-2pass-3


79/128

Reserve Bin Canceller Clipping at 2.0 (6 dB)

0 50 100 150 200 2500

1

2

3

4


0 50 100 150 200 2500

1

2

3

4


data

clip level

data std devaverage peak

0 50 100 150 200 2500

1

2

3

4


data

clip level

data std dev

average peak

0 50 100 150 200 2500

1

2

3

4


data

clip level

data std dev

average peak


80/128

Statistics for Clip at 2.0 (6 dB)

0 1 2 3 40

0.005

0.01

0.015

0.02


0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

std dev =0.927

clip level

output histogram

-5 0 5 1010

-6

10-5

10-4

10-3

10-2

10-1

100

average =-0.659 dB


PAR (dB)

input

pass-1

pass-2pass-3

OFDM 802 11


81/128

OFDM 802.11a

Time Frequency Profile of 802 11a Tones


82/128

Time-Frequency Profile of 802.11a Tones

Pilot Tones Shown in Yellow

Preamble and Pilot Str ct re


83/128

Preamble and Pilot Structure

Short Symbols

Start of Frame Detection

Signal Strength Indication

Frequency Offset Resolution

Long Symbols

Channel Estimate

Fine Time Resolution

Distributed Pilots Carrier Tracking

Sample Clock Tracking


84/128

Preamble Time Structure

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

Magnitude IEEE 802.11a Preamble

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.5

0

0.5

1

Real Part

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5

0

0.5

1Imaginary Part

Detecting Frame Start with


85/128

Detecting Frame Start with

Repeated Short Symbols

Si l i P bl D t t


86/128

Signals in Preamble Detector

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8Envelope of Input Signal

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8Delayed Envelope of Input Signal

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8cross and auto correlations of Input Signal

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

Ratio of Cross to Auto Correlation

Detection

Threshold

Cross Correlation

Detail of Signal in Preamble Detector


87/128

Detail of Signal in Preamble Detector

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.2

0.4

0.6

0.8Envelope of Input Signal

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.2

0.4

0.6

0.8Delayed Envelope of Input Signal

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.2

0.4

0.6

0.8cross and auto correlations of Input Signal

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.5

1

Ratio of Cross to Auto Correlation

Detection

Threshold

Cross Correlation

Auto Correlation

Maximum Likelihood Estimator for


88/128

Frequency Offset

F d Si l St th E ti t


89/128

Frequency and Signal Strength Estimates

0 0.5 1 1.5 2 2.5 3 3.5 4-10

-5

0

5

10Estimate of Frequency Offset

Sample Time

FrequencyOffsetinFFTBins

0 0.5 1 1.5 2 2.5 3 3.5 40

0.02

0.04

0.06

0.08

0.1Estimate of Signal Strength

Sample Time

MagSquare

Known Offset3.3 FFT Bins


90/128

Cross Correlation of Long Preamble

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

Cross Correlation of Input Signal With Long Preamble Section

Time Samples

Amplitude

4.3 4.4 4.5 4.6 4.7 4.80

5

10

15Zoom to First Correlation Peak

Time Samples

Amplitude

5.3 5.4 5.5 5.6 5.7 5.80

5

10

15Zoom to Second Correlation Peak

Time Samples

Amplitude

Expected Peak Position Expected Peak Position

Clipped Cross Correlation of Long Preamble


91/128

Clipped Cross Correlation of Long Preamble

0 100 200 300 400 500 600 700 800 9000

10

20

30

40

50Clipped Cross Correlation of Input Signal With Long Preamble Section

Time Samples

Amplitude

4.3 4.4 4.5 4.6 4.7 4.80

10

20

30

40

50Zoom to First Correlation Peak

Time Samples

Amplitude

5.3 5.4 5.5 5.6 5.7 5.80

10

20

30

40

50Zoom to Second Correlation Peak

Time Samples

Amplitude

Clipped CorrelatorReplica Signal Clipped Version of Template SignalSign[Real(Template)]+j*sign[Imag(Template)]

Channel Probe With Long Preamble


92/128

Channel Probe With Long Preamble

-0.5 0 0.50

1

2

3

4

5

6

7Channel Probe, Long Segment of Preamble


Amplitude

-0.5 0 0.50

1

2

3

4

5

6

7Response of Channel Probe, One Look


Amplitude

-0.5 0 0.50

1

2

3

4

5

6

7Response of Channel Probe, No Noise


Amplitude

-0.5 0 0.50

1

2

3

4

5

6

7Response of Channel Probe, Average of Two Looks


Amplitude

Constellation with Residual Carrier Offset


93/128

Constellation with Residual Carrier Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.55000 Constellations, Zero Carrier Frequency Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5400 pilot, Zero Carrier Frequency Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.55000 Constellations, 5 ppm Carrier Frequency Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5400 pilot, 5 ppm Carrier Frequency Offset

Frequency Domain Residual Carrier Offset


94/128

Frequency Domain Residual Carrier Offset

-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.560 Frames Real Part FFT: Zero Carrier Frequency Offset

Frequency Index

Amplitude

-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.560 Frames Real Part FFT: 5 ppm Carrier Frequency Offset

Frequency Index

Amplitude

Constellations with Sample Clock Offset


95/128

Constellations with Sample Clock Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.55000 Constellations, Zero Clock Frequency Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5400 pilot, Zero Clock Frequency Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.55000 Constellations, 300 ppm Clock Frequency Offset

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5400 pilot, 300 ppm Clock Frequency Offset

Frequency Domain With Sample Clock Offset


96/128

Frequency Domain With Sample Clock Offset

-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.580 Frames Real Part FFT: Zero Clock Frequency Offset

Frequency Index

Amplitude

-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.580 Frames Real Part FFT: 300 ppm Clock Frequency Offset

Frequency Index

Amplitude

Other Variants of OFDM


97/128

Other Variants of OFDMAmplitude and Phase Overlays

Shaped OFDM

OQAM OFDM Coded OFDM

CI OFDM

Shape to Control Spectral Side Lobes


98/128

Shape to Control Spectral Side Lobes


99/128

Overlapped OFDM Frames

Polyphase Filter For Shaped OFDM


100/128

Shaping and Matched Filter

Impulse Response of Shaped OFDM


101/128

p p p

Modulator and Demodulator

0 1 2 3 4 5 6 7 8

0

0.5

1

Impulse at Input to IFFT (DC-bin)

0 1 2 3 4 5 6 7 8

0

0.5

1

Impulse Response at Output of IFFT

0 1 2 3 4 5 6 7 8

0

0.5

1

Impulse Response at Output of Polyphase Shaping Filter

0 1 2 3 4 5 6 7 8

0

0.5

1

1.5Impulse Response at Output of Polyphase Matched Filter

0 1 2 3 4 5 6 7 8

0

0.5

1

Impulse Response at Output of FFT

Orthogonal: Adjacent Time Slots Non


102/128

Adjacent Frequency Bins

0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

Impulse Response Shaping Filter

0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1

Auto Correlation Response

-4 -3 -2 -1 0 1 2 3 4

-60

-40

-20

0Adjacent Spectral Bins Correlated

Alternate Spectral Bins Not Correlated

Spectrum: Shaping Filter Centered on IFFT Spectral Bins

Impulse Response Time Frequency Profile


103/128

Impulse Response Time-Frequency Profile

-10

-5

0

5

10

0

2

4

6

8

0

0.2

0.4

0.6

0.8

1

frequencybin

OFDMFrameNumber

Orthogonality Between Real and Imaginary


104/128

g y g y

Part of Shaped OFDM Frequency Bins

Even and Odd Symmetric Wave Shapes from

Adjacent Bins are Orthogonal in Shaped OFDM


105/128

Adjacent Bins are Orthogonal in Shaped OFDM

Symmetry Considerations in Real and Imaginary

C t f Off t Sh d OFDM F


106/128

Components of Offset Shaped OFDM Frames

Off O


107/128

Offset OFDM

Compare Spectra


108/128

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-60

-40

-20

0

Spectrum of Standard OFDM With Cyclic Prefix


LogMagnitude(dB

)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-60

-40

-20

0

Spectrum of OFDM/OQAM Without Cyclic Prefix


LogMagnitude(dB)

OFDM and Shaped OFDM PAR


109/128

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Histogram: Standard OFDM


0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Histogram: Shaped OFDM/OQAM


0 1 2 3 4

10-4

10-3

10-2

10-1

100



Standard OFDM

Shaped OFDM/OQAM

Complementary Codes


110/128

Complementary Codes

Canceling Correlation Side Lobes


111/128

g

Inserting CC in OFDM


112/128

2 2

2

( ) ( ) ( ) ( ) 2 ( )

( ) ( ) 2 (A Constant Power Level)

Since Sample Values ( ) are equal to 1

Average Power in ( ) = N

2NThus Peak to Average Power Ratio 2

N

N N N N

N N

N

N

A n A n A n A n N n

A B N

A n

A

Now Reverse Domains

Use Complementary Code Sequence

as amplitude of Carriers in Frequency Domain

Then time series has

Peak Squared Magnitude = 2N

Average Magnitude = N

for Peak to Average Power Ratio = 2


113/128

CC and Digital Filters

Equivalent Phase Coding


114/128

q g

PAR in CCK OFDM and Standard OFDM


115/128

0 500 1000 1500 2000 2500 3000 35000

0.5

1

1.5

2

2.5Mag Square of CC Rate 1/2 OFDM

0 500 1000 1500 2000 2500 3000 35000

2

4

6

8Mag Square of Coded Rate 1/2 OFDM

Mean = 1

Mean = 1

Peak = 2

Peak = 6.26

OFDM and CC-OFDM PAR


116/128

C-I OFDM

C i I t f t


117/128

Carrier Interferometry

OFDM with Phase Overlay

In Conventional OFDM

Rectangle Envelope in Time

Dirichlet Kernel in Frequency In CI-OFDM

Rectangle Envelope in Frequency

Dirichlet Kernel in Time

Sin(x)/x in Time Domain Without Excess Bandwidth,

No Square-Root Nyquist Shaping Filter

Frequency Domain Phase Slope in


118/128

Continuous and in Sampled Data Domains

Circularly Shifted Time Domain Dirichlet Kernels


119/128

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

-10 0 10-0.5

0

0.5

1

exp(-j 2k 0/N) exp(-j 2k 1/N) exp(-j 2k 2/N) exp(-j 2k 3/N) exp(-j 2k 4/N)

exp(-j 2k 9/N)exp(-j 2k 8/N)exp(-j 2k 7/N)exp(-j 2k 6/N)exp(-j 2k 5/N)

exp(-j 2k 10/N) exp(-j 2k 11/N) exp(-j 2k 12/N) exp(-j 2k 13/N) exp(-j 2k 14/N)

Linear Versus Circular Convolution


120/128


121/128

Fast Circular Convolution with the FFT

Single Symbol in CI-OFDM


122/128M-Symbols in CI-OFDM


123/128

1-to-2 Interpolated Time Domain Data PointsCI-OFDM Real Time Series and 1-Modulation Sample


124/128

-20 -15 -10 -5 0 5 10 15 20

-0.2

0

0.2

0.4

0.6

0.8

1

CI OFDM Real Time Series and 1 Modulation Sample

Time

Amplitude

-20 -15 -10 -5 0 5 10 15 20

-0.2

0

0.2

0.4

0.6

0.8

1

CI-OFDM Real Time Series and 4-Modulation Samples

Time

Amplitude

CI-OFDM Data Frame


125/128

-25 -20 -15 -10 -5 0 5 10 15 20

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

CI-OFDM Real Time Series and 32-Modulation Samples

Time

Amplitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-50

-40

-30

-20

-10

0

10Spectrum


LogMagnitude

(dB)

Cyclic Prefix

CI-OFDM Statistics


126/128

0 1 2 3 40

0.005

0.01

0.015

Histogram: Standard OFDM (QPSK)


0 1 2 3 40

0.05

0.1

Histogram: CI-OFDM (QPSK)


0 1 2 3 4

10-4

10-3

10-2

10-1

100



CI-OFDM Statistics


127/128

0 1 2 3 40

0.005

0.01

0.015

Histogram: Standard OFDM


0 1 2 3 40

0.02

0.04

0.06

0.08Histogram: CI-OFDM (16-QAM)


0 1 2 3 4

10-4

10-3

10-2

10-1

100



Thats all Folks


128/128

OFDM Lecture

Documents

Transcript of OFDM Lecture