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Usage of OFDM in awideband fading channel

OFDM signal structure

Subcarrier modulation and coding

Signals in frequency and time domain

Inter-carrier interference

Purpose of pilot subcarriers

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OFDM example 1: IEEE 802.11a&g (WLAN)

48 data subcarriers + 4 pilot subcarriers. There is a null atthe center carrier. Around each data subcarrier is centered asubchannel carrying a low bitrate data signal (low bitrate =>no intersymbol interference).

52subcarriers

Frequency16.25 MHz

Subcarriers thatcontain user data

Pilot subcarrier

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OFDM example 2: IEEE 802.16a (WiMAX)

Only 200 of 256 subcarriers are used: 192 data subcarriers +8 pilot subcarriers. There are 56 nulls (center carrier, 28lower frequency and 27 higher frequency guard carriers).

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Usage of OFDM

OFDM is used (among others) in the following systems:

IEEE 802.11a&g (WLAN) systems

IEEE 802.16a (WiMAX) systems

ADSL (DMT = Discrete MultiTone) systems DAB (Digital Audio Broadcasting)

DVB-T (Digital Video Broadcasting)

OFDM is spectral efficient, but not power efficient

(due to linearity requirements of power amplifier).OFDM is primarily a modulation method; OFDMA isthe corresponding multiple access scheme.

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OFDM system block diagram

IFFTCoding

&Interl.

Bit-to-symbol

mappingS/P

AddCP

FFT P/SSync

Modu-lation

Demod.Deinterl.

&Decoding

Channel

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Why (for instance) 54 Mbit/s ?

Symbol duration = 4 ms

Data-carrying subcarriers = 48

Bits / subchannel = 6 (64-QAM)

Bits / OFDM symbol = 6 x 48 = 288

Channel coding: number reduced to 3/4 x 288

= 216 bits/symbol=> Bit rate = 216 bits / 4 ms = 54 Mbit/s

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Subcarrier modulation and coding

Ndata subcarriers or subchannels carry Ndata symbolsin parallel (= transmitted at the same time). A symbolcarries 1 bit (BPSK), 2 bits (4-PSK), 4 bits (16-QAM), or6 bits of user data (64-QAM). Ndata symbols in parallel

form one OFDM symbol.

For each modulation method, there are several codingoptions for FEC (Forward Error Control). They must betaken into account when calculating user data rates, as

shown on the previous slide. Typical coding options: 1/2(convolutional encoding), 2/3 and 3/4 (puncturing)coding rates.

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Gray bit-to-symbol mapping in QAM

Gray bit-to-symbolmapping is usuallyused in QAM systems.

The reason: it isoptimal in the sensethat a symbol error(involving twoadjacent symbols in

the QAM signalconstellation) resultsin a single bit error. 0000 0100 1100 1000

0001 0101 1101 1001

0011 0111 1111 1011

0010 0110 1110 1010

Example for 16-QAM

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BER performance of QAM (1)

2

1 1 2M M MP P P

Probability of correct symbol decision for M-ary QAM:

2

1c MP P

Probability of symbol error for M-ary QAM:

A rectangular M-ary QAM constellation, where 2kM and k is even, is equivalent to two PAM (Pulse Amplitude

Modulation) signals on quadrature carriers, each having

M signal points and symbol error probability .M

P

Proakis,

3rd Ed.5-2-9

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BER performance of QAM (2)

0

1 32 1

1

av

M

EP Q

M NM

where is the average SNR per symbol.0avE N

Proakis,3rd Ed.

5-2-6

Finally, the bit error probability for M-ary QAM:

2log

Mb M

PP P k

M

Probability of symbol error for -ary PAM:M

(Gray mapping assumed)

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Subcarrier signal in time domain

Time

Guard time for preventingintersymbol interference

In the receiver, FFT is calculatedonly over this time period

Symbol duration

Next symbol

TG

IEEE 802.11a&g: TG = 0.8 ms, TFFT = 3.2 ms

IEEE 802.16a offers flexible bandwidth allocation (i.e. differentsymbol lengths) and TG choice: TG/TFFT = 1/4, 1/8, 1/16 or 1/32

TFFT

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Orthogonality between subcarriers (1)

Guardtime

Symbol part that is used forFFT calculation at receiver

Subcarrier n

Subcarrier n+1

Previoussymbol

Nextsymbol

Orthogonality over this interval

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Guardtime


Subcarrier n

Subcarrier n+1

Previoussymbol

Nextsymbol

Orthogonality over this interval

Each subcarrier has an integer number ofcycles in the FFT calculation interval (inour case 3 and 4 cycles).

If this condition is valid, the spectrum of asubchannel contains spectral nulls at all

other subcarrier frequencies.

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0

2cos 2 cos 2

0

FFTT

FFT

FFT FFT

T m nmt T nt T dt

m n

Orthogonality over the FFT interval:

Phase shift in either subcarrier - orthogonality over theFFT interval is still retained:

0

cos 2 cos 2 0FFTT

FFT FFTmt T nt T dt m n

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Time vs. frequency domain

TG TFFT

Square-windowed sinusoid in time domain

=>

"sinc" shaped subchannel spectrum in frequency domain

sinc sinFFT FFT FFTfT fT fT

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Subchannels in frequency domain

Single subchannel OFDM spectrum

Spectral nulls atother subcarrierfrequencies

Subcarrier spacing

= 1/TFFT

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Presentation of OFDM signal

2

,

20

exp 2N

k n k

n N FFTn

ng t a j t

T

1k T t kT

kk

s t g t kT

Sequence of OFDM symbols

Thek

:th OFDM symbol (in complex LPE form) is

where N= number of subcarriers, T=TG + TFFT= symbol

period, and an,k is the complex data symbol modulating the

n:th subcarrier during the k:th symbol period.

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Multipath effect on subcarrier n (1)

Guardtime


Subcarrier n

Previoussymbol

Nextsymbol

Delayed replicas of subcarrier n

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Guardtime


Subcarrier n

Previoussymbol

Nextsymbol


Guard time not exceeded:

Delayed multipath replicas do notaffect the orthogonality behavior ofthe subcarrier in frequency domain.

There are still spectral nulls at othersubcarrier frequencies.

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Guardtime


Subcarrier n

Previoussymbol

Nextsymbol


Mathematical explanation:

Sum of sinusoids (with the samefrequency but with differentmagnitudes and phases) = still apure sinusoid with the samefrequency (and with resultantmagnitude and phase).

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Guardtime


Subcarrier n

Previoussymbol

Nextsymbol

Replicas with large delay

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Guardtime


Subcarrier n

Previoussymbol

Nextsymbol


Guard time exceeded:

Delayed multipath replicas affectthe orthogonality behavior of thesubchannels in frequency domain.There are no more spectral nulls atother subcarrier frequencies => thiscauses inter-carrier interference.

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Guardtime


Subcarrier n

Previoussymbol

Nextsymbol


Mathematical explanation:

Strongly delayed multipath replicasare no longer pure sinusoids!

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Discrete multitone (DMT) modulation

DMT is a special case of OFDM where the different signal-to-noise ratio values of different subcarriers are utilisedconstructively in the following way:

Note the requirement of a feedback channel.

Subcarriers with high S/N carry more bits (forinstance by using a modulation scheme with morebits/symbol or by using a less heavy FEC scheme)

Subcarriers with low S/N (due to frequency selective

fading) carry less bits.

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Task of pilot subcarriers

Pilot subcarriers containsignal values that are knownin the receiver.

These pilot signals are usedin the receiver for correctingthe magnitude (important inQAM) and phase shift offsetsof the received symbols (see

signal constellation exampleon the right).

Re

Im

Received symbol

Transmitted symbol

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Transmitted and received subcarrier n

Guardtime


Transmitted subcarrier n

Previoussymbol

Nextsymbol

Received subcarrier nMagnitude

error

Phase error

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Frequency offset at receiver

Frequency offset causes inter-carrier interference (ICI)

Magnitude

Frequency

Frequency offset

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Summary: Inter-carrier interference

Inter-carrier interference (ICI) means that theorthogonality between different subchannels in theOFDM signal is destroyed.

There are two causes of inter-carrier interference:

Delay spread of radio channelexceeds guard interval

Frequency offset at the receiver

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Pilot allocation example 1 (1)

To be able to equalize the frequency response of afrequency selective channel, pilot subcarriers must beinserted at certain frequencies:

Between pilotsubcarriers, someform of interpolationis necessary!

Frequency

TimePilot subcarriers

Subcarrier of an OFDM symbol

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The Shannon sampling theorem must be satisfied,otherwise error-free interpolation is not possible:

Frequency

Time

m1 2f mD T T

fD

maximum delay spread

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An alternative pilot scheme for equalizing the frequencyresponse of a frequency selective channel:

Frequency

Time

PilotOFDM

symbols

Subcarrier of an OFDM symbol

Between pilot symbols, some formof interpolation is necessary!


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Frequency

Time

The Shannon sampling theorem must again be satisfied,otherwise error-free interpolation is not possible:

max max

1

1 2t D DD B B

maximum p-p Doppler spread

tD

maximum Doppler frequency


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An efficient pilot scheme (used in DVB-T) makes use ofinterpolation both in frequency and time domain:

Frequency

Time

Black circles= Pilot

subcarriers

Interpolation necessary both infrequency and time domain!

Pilot allocation example 3

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Summary: OFDM features

In summary, OFDM offers the following features:

Multipath propagation (fading) does not cause intersymbolor intercarrier interference if the guard interval is sufficientlylarge and there is no frequency offset at the receiver.

Multipath fading, however, causes frequency selectivity inthe transmission bandwidth. Pilot signals are employed forcorrecting (equalizing) the magnitude and phase of thereceived subcarriers at the pilot subcarrier frequencies.

Some form ofinterpolation is necessary for equalization atother than pilot subcarrier frequencies. Many pilot allocationschemes have been proposed in the literature, see e.g.

www.s3.kth.se/signal/grad/OFDM/URSIOFDM9808.htm

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