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An Introduction To OFDM

John Wiss

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OFDM Origins• OFDM is a modulation type whereby the information coming from a

single source is intentionally split up into many carriers (subcarriers) to combat multipath effects and to add a dimension of “diversity” to the transmit signal

• Initial work in 1960s for military applications• OFDM stands for “Orthogonal Frequency Division Multiplexing”

– “Orthogonal” in the sense that each information-bearing subcarrier may be demodulated without interference from adjacent subcarriers

– “Frequency Division” in the sense that the subcarriers are generated as frequency-disjoint coupled carriers with fixed spacing and modulation type

– “Multiplexed” in the sense they are synthesized into a single channel

• Error-correction coding essential because some subcarriers may be in deep fades so encoding of all information bits with interleaving necessary

3

OFDM-BASED SYSTEMS• IEEE 802.11a was the first wireless networking standard with

widespread usage in unlicensed 5 GHz band providing up to 54 Mbps 16.67 MHz wide channel containing 52 subcarriers with BPSK,

QPSK, 16-QAM, 64-QAM modulation on each subcarrier OFDM “symbol rate” of 250 ksym/sec and up to 288 bits/symbol of

encoded information• IEEE 802.11g is a dual mode system fusing 802.11b WiFi and 802.11a

except at a lower frequency band (unlicensed 2.4 GHz)• UWB- Multiband OFDM is the leading contender for ultra-wideband

communications (>100 Mbps) Hopped OFDM over three bands to spread energy for short range

high-speed communications

• Spatially diverse Vector OFDM (VOFDM) Multiple Input/Multiple Output (MIMO) diversity to combat flat fading

• Wideband Networking Waveform for military communications • 4-G Cellular• IEEE 802.16 Metropolitan Area Networking--OFDMA

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• One QAM symbol per subcarrier

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Why Use OFDM?• There are a few ways to cope with multipath channels:

Use spread spectrum techniques to allow separation of paths Direct Sequence Spreading (CDMA/DSSS)—Low bps/Hz Use Frequency Hopping (Bluetooth, GSM)—MAC inefficiencies/complex

Use high bandwidth single-carrier signal —Large delay spread vs Tsym Use lower data rates to allow channel to be flat over signal bandwidth

• The problem is that the signal bandwidth needs to be as narrow as possible to mitigate multipath (the symbol period needs to be as long as possible), Unless…

The signal is made bandwidth-inefficient in order to perform combining of dispersed rays (e.g., RAKE combining) The optimal signal would be one which has high bandwidth efficiency and immunity to channel variations due to multipath

• Lower complexity equalization for given delay spread than single-carrier modulation

OFDM allows tightly packed carriers to convey information orthogonally and with high bandwidth efficiency

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Equalization Complexity• Let’s compare equalization complexity of a single carrier (SC) system with the same total bandwidth as an OFDM system subject to similar channel conditions

Frequency domain equalization (via FFT) for OFDM Rx Decision feedback equalization for SC Rx

• OFDM complexity grows slightly greater than linear with BW*Delay Spread product• Single carrier complexity grows quadratically with BW*Delay Spread product

+XX

Adj Proc.

CmplexData In

D D D D

Tap 0 Tap 1 Tap 2 Tap 3

[ . ]*

X

+

From other Taps FFF

+

D(k) {Training, Demod Decisions}

-

+

+Error

Slicer

D

1 SPS

Tap 4

Decisions

D D D

X + X X

LMS

Error

[ . ]*

From FBF Taps

Tap FB0Tap FB1Tap FB2

Tap FB3

D

+

FEEDFORWARD FILTER (FFF)

FEEDBACK FILTER (FBF)

Eq. Output

Peamble Sequence

•Single Carrier Case GMSK or OQPSK 24 Mbps Delay Spread = 250 nsec

Requires 20 FF, 20 FB taps•OFDM Case

QAM 52-OFDM 24 Mbps (QAM 16) Delay Spread = 250 nsec

64 Point FFT 52 Subcarriers Guard Interval = 800 nsec Rsym = 250 kHz

• GMSK Eq (LMS) = 2*20*24*106 (960E6) rmult/sec• OFDM (radix 4)= 96*106 = (96E6) cmult/4sec = 96E6 rmult/sec

Single Carrier DFE is 10 times as complex!!!

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Indoor Channels ETSI/BRAN Low Mobility Indoor

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Rayleighk ,Gaussian0k

k is the ratio of power of the direct path to the scattered paths

Small Room Fading (amb. Motion) Tunnel Fading (amb. Motion)

Indoor Channels (Cont)

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Tx Height=15 m, 3.5 GHz/Outdoor Vs Antenna Height—12 &15dBi antennas 3km Separation

Outdoor Channels

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11

-- Channel can be inverted (Equalized) by spectrum division

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13

14

15

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OFDM Channel Estimation• Each subcarrier corrected by a single tap (complex) equalizer• Typically have training symbols with to train frequency domain equalizer• Sometimes have embedded pilot subcarriers that are used to estimate channel

Uniformly spaced throughout symbol to allow interpolation between pilots to correct data-bearing subcarriers

• Data subcarriers corrected post-FFT in frequency domain• Additional performance by weighting soft decisions based on channel estimates

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18

Conv.Encode

InterleavePN Data QAM MapSubcarrier

Mapper

I

Q

PilotCarriers

DACCompensate

InsertShort Train

InsertLong Train

IFFTAdd Cyclic

PrefixSpectralWindow

Polyphase Interpolate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-70

-60

-50

-40

-30

-20

-10

0

10

-80-70-60-50-40-30-20-10

010

DigitalTo

Analog

OscillatorPhase Noise

Effects

PA Nonlinearity

Effects

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Pin (Normalized : 1dB compression Pt=1.0) --Real Units

Normalized Pout Vs Pin GaAsTEK ITT6401FM

Y = 0.0781X^3-0.5285X^2+1.245X for (0<X<2)

Y=1 for (X>2.0)

Freq.

A0cos(2pf1t+F1(t))

f1 f2 f3-f3 -f2 -f1

A1cos(2pf2t+F

2(t))

A3cos(2pf3t+F

3(t))

Phase PSDSxx(f)

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-2.0E+7 -1.5E+7 -1.0E+7 -5.0E+6 0.0E+0 5.0E+6 1.0E+7 1.5E+7 2.0E+7

Frequency (MHz)

Standard (52-Carrier) Mode Spectrum For Various Output Backoffs

Linear PA

12 dB OBO

10 dB OBO

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

I

Q

Subc

arrie

r #

0

1

Up to 64 QAMPer Carrier

To Channel

Analog ToDigital

Fs = 40,80M

Channel

DQM Decimate

Filter

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10Decimate By 2 Filter Frequency Response

Frequency (1.0 = Nyquist)

Am

plitu

de S

pect

rum

(dB

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-80

-60

-40

-20

0

20Decimate By 4 Filter Frequency Response

Frequency (1.0 = Nyquist)

Am

plitu

de S

pect

rum

(dB

)

By 2

By 4

I

Q

ShortTrainCorrel

NoiseAverage

Mag

Mag

Gen TestStatistic

MultipathResistantDetection

Orthog-onalizer

Programmable FFT

Phase NoiseReductionAlgorithm

Equalization

De-InterleaveAnd Decode

PN DataOut

ChannelInverter

AdaptiveFilters

TapUpdate

Slicer

AGC

000 tSjexp

111 tSjexp

222 tSjexp

-60

-50

-40

-30

-20

-10

0

10

-512 -312 -112 88 288 488

-100

-50

0

50

100

150

200

250

0 500 1000 1500 2000 2500 3000 3500 4000

Time (Samples)

-600

-400

-200

0

200

400

600

800

1000

1200

7400 7450 7500 7550 7600 7650 7700 7750 7800

Sample Output Of H0-H15 Under Severe Multipath

Peak

Left Adjacent Right Adjacent

MultipathPeaks

Multipath peak > 0

Burst Detector/Orthogonalizer/Synchronizer+ Additive Noise

RADIX2

STAGE128-Points

RADIX8

FFT ASTAGE 1

{64-Points}

RADIX8

FFT BSTAGE 1

{64-Points}

RADIX8

FFT ASTAGE 2

{64-Points}

RADIX8

FFT BSTAGE 2

{64-Points}

UNSCRAMBLE128-pt

Transform

Unscramble64-pt

Transform

AGC

1.0E-8

1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

20 21 22 23 24 25 26 27 28

Ecarr/No (dB)

(1) Theory (2) Simulation

QAM 64 Uncoded BER Performance {Enhanced 105-Carrier Mode}

Real System

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OFDM Waveform/Receiver• 802.11a/g uses training symbols that can be used to estimate/correct gain frequency offsets and to equalize the channel and Pilot carriers to correct phase offsets during the data symbols

• VOFDM uses dense training structure in the data symbols to allow for interpolation between pilots to equalize data-bearing carriers

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The Peak to Average Power (PAPR) Problem

0E+0

5E-3

1E-2

2E-2

2E-2

3E-2

3E-2

4E-2

4E-2

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

Amplitude Relative To Mean Amplitude (dB) (mean=0 dB)

QAM 64 52-Carrier OFDM W/Shaping Crest Factor PDF

99.9 % =9.5 dB

• 802.11a/g consists of 52 subcarriers which are modulated using Digital rather than analog techniques forcing output PA backoffs of ~5-9 dB from P1• The peak signal envelope is the vector sum of the instantaneous amplitude and phases of the 52 subcarriers rotating at nF where n ranges from –26 to +26 (in the case of 802.11a)• The PAPR is more dependent on the number of subcarriers than the modulation on each subcarrier when the #subcarriers grow

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

RMS Phase Jitter (Degrees)

QAM 64 -OFDM Loss Vs Jitter

The Phase Noise and AFC Problem• Since the subcarriers are packed closely (at the 1st Null of Sinx/x) frequency errors in the Rx will cause signal leakage and Inter-carrier interference• Phase Jitter will cause the carriers to leak as well • However, for 802.11a/g 64 QAM-OFDM the single carrier phase noise spec. is adequate to supress LOT (loss of orthogonality)• Low OFDM symbol rate (250 Ksym/sec) implies low loop bandwidth which may allow a lot of Flicker Noise through

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OFDM Waveform Design

• The parameters to design are function of channel characteristics Number of subcarriers

Efficiency, delay spread of channel Subcarrier bandwidth

Mobility/Time variation of channel, coherence BW of channel Guard period

Delay spread of channel, modulation order• Fundamental Design criterion of OFDM is that the channel must appear flat across one subcarrier bandwidth• Coherence BW determines subcarrier bandwidth to ensure flatness• Cyclic prefix period must be > delay spread of channel

For BPSK/QPSK-OFDM cyclic prefix should be > 1 x Delay Spread For QAM 16 cyclic prefix should be > 2 x Delay Spread For QAM 64 cyclic prefix should be > 3.5 x Delay Spread

• For efficiency want guard period to be less than 50% or so of total symbol period—determines # subcarriers/symbol

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Non-Idealities / Mitigation /Modem Design• Peak-Average Power Problem

The signal band consists of many closely-spaced subcarriers (in fact, overlapped by 50%)

With large modulation orders, the signal behaves like bandlimited Gaussian noiseThe signal envelope exhibits a Rayleigh density—different than SC modulations

Mitigate by either clipping signal or by modifying data or adding sacrificial subcarriers to reduce PAPR

Low order modulations can be improved by phase whitening the BPSK or QPSK subcarriers Data symbols may be multiplied by one of N stored sequences—Sequence with lowest PAPR is transmitted along with best sequence indexPeak Cancellation Coding Techniques—Golay complementary codes etc. Sacrifice subcarriers to cancel biggest peaks (need about 50% for canceling to limit PAPR < 3 dB)

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Non-Idealities / Mitigation /Modem Design Cont.{PAPR}

0E+0

1E-1

2E-1

3E-1

4E-1

5E-1

6E-1

7E-1

8E-1

-1 0 1 2 3 4 5

Normalized Amplitude

(1) Simul. Signal

(2) Quadrature Noise Model

Shaped Amplitude PDF Vs Quadrature Gaussian Noise Model

2n

2

2n

m 2

mexp

mmf

2n

2m 2

2

p

2

2n

m

p

2

E811.0)equiv()equiv( S2

nQ2nI

• The signal envelope has nearly identical statistics to BL Gaussian Noise (52 Subcarriers)

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Non-Idealities / Mitigation /Modem Design Cont.

• Phase Noise & Frequency Errors The symbol rate of OFDM is slow (equal to the BW of a single subcarrier + 1/Guard Period thus close-in phase noise more critical Zero IF (Direct Conversion) designs will admit a Flicker noise component Frequency control very important to keep the subcarriers orthogonal relative to FFT Rx Bin reference. SinX/X leakage will cause Inter-Carrier Interference Phase noise will jitter subcarriers (SCs) into each other and a given SC will interfere with all other SCs resulting in a maximum attainable SNR Phase tracking via pilots or decision-directed techniques allow tracking of so-called “Common Phase Error”

1N

0k

Sig2eee mkS

N

E0RError Power =

Where S( ) = Phase noise PSD, N = # subcarriers

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0

5

10

15

20

25

30

35

40

45

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Phase Jitter Standard Deviation (Radians)

Maximum Attainable SNR Vs Phase Jitter St Dev

Non-Idealities / Mitigation /Modem Design Cont.Phase Jitter Max SNR (52 Subcarriers)

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Non-Idealities / Mitigation /Modem Design Cont.{Phase, Frequency Recovery}

Phase may be recovered by Noting common phase shift of pilot symbols and constructing a loop With this information

Frequency information may exploit cyclic prefix since same time waveform is at start & endOf OFDM symbols—Use Lag Product Correlation

Ttntnd-ts hTf2jexptrTtr *2T

0

2

d*

CP

p

T2

trTtrARGf

*

d p

+

Conjugate

X

FIFO

Symbol Length Less Cyc prefix

0

Z-1ARG ( . )

Freq Est

Select

Switch

To Phase Rotator/NCO/FFT

I In

Q In

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Non-Idealities / Mitigation /Modem Design Cont.{Fine Timing Adjust & Tracking}

aF

a

1atf

Because each subcarrier is narrowband and typically the OFDM access scheme is TDMA, We can correct spectrally any time offset or residual sampling errors

Typically the RF reference is used as the sampling reference so a frequency error may be translated into a ppm error for the sampling frequency error as well –bursts must be short enough so that the total timing slew is << cyclic prefix or guard period

Can correct timing drift using FT time scaling property

SCs will rotate at a speed linearly related to their distance from DC subcarrier—rotation in oppositedirections for positive vs negative subcarriers—delta phase between subcarrier and its negative frequency complement provides timing frequency error signal—can rotate each subcarrier indirection and rate to stop spinning

DC

slow fast

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Non-Idealities / Mitigation /Modem Design Cont.{DC Offsets, I/Q Imbalance}

• DC Offsets Zero-IF designs may suffer from large DC offsets

Analog cancellation on I/Q arms Digital DC offset detection and cancellation feedback to analog Progressive notch filter near DC Low-IF improves Zero (DC) subcarrier not used for data in waveform design

• I/Q imbalances I/Q Imbalance causes positive/negative frequency aliasing

I/Q mismatch will introduce crosstalk between I, Q branches May be mitigated digitally in Rx to eliminate crosstalk Isolation of imbalance to one side of link desirable due to multi-users each of which may have different Tx imbalances Low-IF rather than Zero-IF eliminates problem to a large extentDouble complex-mixing mitigates phantom image products

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Soft Symbol/LLR Generation• Typically every bit is encoded in OFDM to provide full time/frequency diversity—for QAM constellations the optimum approach is “Bit/Slice” decisioning—based on bit qualities of symbol.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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

I or Q Input (To Slicer)

QAM 16 LSB Unweighted Soft Decision Mapping

0

2

4

6

8

10

12

14

16

-8 -6 -4 -2 0 2 4 6 8

I or Q Input (To Slicer)

QAM 16 MSB Unweighted Soft Decision Mapping

Gray-Coded Square QAM

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MIMO (Multiple-Antenna OFDM)• OFDM is very amenable to space-time diversity

Each subcarrier easy to combine from different antenna Channel flat over a single subcarrier

Huge BER gains possible with just dual Tx/Rx diversity Transmitter signal sent to two antennas/Tx chains Receiver uses two antennas/Rx chains

• Strict Line-of-Sight (LOS) not required

Channel

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Antenna#2 ChannelEstimatesVia FFT

(Training)

MIMO System (2 Antenna Rx, 2 Antenna Tx—Delay Diversity)

Antenna#1 ChannelEstimatesVia FFT

(Training)

FFT Out #1

FFT Out #2

Chan Est #1(Data)

Chan Est #1(Data)

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Cross Correlation From Rx Antennas

• WT1 & WT2 are the training tone channel estimates HT1, HT2 with delay compensation for antenna 1 and 2 respectively to compute Covariance Matrix R

• Channel Estimates are then used with R to form a weighted channel estimate

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Combiner

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Diversity Advantage

• Dual Tx/Rx up to 23 dB advantage for 1E-3 Uncoded for flat fading channel over single antenna OFDM

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OFDM for multiple access (OFDMA)

• IEEE 802.16.3 uses OFDM as a multiple access scheme for fixed wireless access

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Multicarrier CDMA•Multi-Carrier CDMA is a combination of OFDM & CDMA:

First an OFDM system is used to provide orthogonal carriers, free from ISI. Second each carrier is modulated by an individual code chip, to provide a spread spectrum system

• The main advantage of doing this is that when the multiple-access interference becomes a problem, the resulting linear detectors are much simpler to implement as only a single tap equalizer is required for each channel. • Rake reception can also be employed to exploit the channel diversity by channel matching in the frequency domain allowing optimal reception for a single user. • In the uplink another advantage of MC-CDMA can be exploited. If the signals can be synchronized to arrive within a small fraction of the symbol time ( e.g. indoor, or very small cell environment) then this asynchronism can be overcome by cyclically-extending the signal further allowing synchronous reception of all signals, with no ISI from other users• Makes multiple-access easier with ability to correlate out interfering users

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• Transmitter Multiplies symbols by spreading sequence (Walsh etc.) prior to IFFT

• Receiver Multiplies symbols by same seq. as was used to transmit after FFT

Spread

De-Spread

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Further Reading

• OFDM for Wireless Multimedia Communications, Van Nee & Prasad, Artech House Publishers, 2000, Boston, MA

• OFDM Wireless LANs: A Theoretical and Practical Guide, Heiskala & Terry, SAMS Publishers, 2002, Indianapolis, IN

• “Supplement To Standard For Information Technology-Telecommunications and Information Exchange Between Systems-Local And Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control(MAC) and Physical Layer (PHY) Specifications: High Speed Physical Layer In The 5 GHz Band {IEEE 802.11a}”