EECE5984CE5984Orthogonal Frequency Division Multiplexing and Related Orthogonal Frequency Division Multiplexing and Related

TechnologiesTechnologiesFall 2007Fall 2007

Mohamed Essam Khedr

Modulation (Mapping) in OFDM

SyllabusSyllabus• Wireless channels characteristics (7.5%) 1

• OFDM Basics (10%) 1

• Modulation and Coding (10%) 2– Linear and nonlinear modulation – Interleavin\g and channel coding – Optimal bit and power allocation – Adaptive modulation

P/SQAM

decoder

invert channel

=frequencydomain

equalizer

S/P

quadratureamplitude

modulation (QAM)

encoder

N-IFFTadd

cyclic prefix

P/SD/A +

transmit filter

N-FFT S/Premove cyclic prefix

TRANSMITTER

RECEIVER

N subchannels N complex samples

N complex samplesN subchannels

Receive filter

+A/D

multipath channel

An OFDM Modem An OFDM Modem

Bits

00110

OFDM Mathematics1

2

0

( ) k

Nj f t

kk

s t X e π−

=

= t ≡ [ 0,Τos]

Orthogonality Condition

*1 2

0

( ). ( ) 0T

g t g t dt =In our case

2 2

0

. 0p q

Tj f t j f te e dtπ π− =

For p ≠ q Where fk=k/T

os

os

os

Spectrum ShapingSpectrum Shaping

• FCC manages spectrum• Specifies power spectral

density mask– Adjacent channel interference– Roll-off requirements

• Implications to OFDM– Zero tones on edge of band– Time domain windowing

‘smoothes’ adjacent symbols

IEEE 802.11a

Inband

Adjacent channel

Reference: Std 802.11a

frequency

Zero tones

Ofdm symbol

Mitigating Mitigating MultipathMultipath effectseffects• Channel estimation required • Training based methods

– Tradeoffs in overhead, complexity, and delays

• Linear equalizers can completely eliminate ISI, but this may enhance noise.

• Decision feedback (nonlinear) equalizers can improve performance.

EqualizationEqualization• Digital Equalizer

• Criterion for coefficient choice– Minimize BER (Hard to solve for ws)– Eliminate ISI (Zero forcing, enhances noise)– Minimize MSE between dn and dn

nneq zwzwwzH −− +++= ...)( 1

10

Discrete Random Variables and ProbabilityDiscrete Random Variables and ProbabilityQuick RevisionQuick Revision

Random variable X assumes a value as a function from outcomes of a process which can not be determined in advance.

Sample space S of a random variable is the set of all possible values of thevariable X.

ΩΩΩΩ: set of all outcomes and divide it into elementary events, or states

1)(

=x

xp 0)(1 ≥≥ xp

a5

a5

1

Continuous Random VariablesContinuous Random Variables

1

)()( αα

α XX Ff∂∂=

)Pr()( αα ≤= XFX

Cummulative distribution function(CDF)

Probability density function (PDF)

1)( =∂∞

∞−

ααXf

?)(

?)(

=−∞=∞

X

X

F

F

Ensemble Average Ensemble Average

• Mean:– Continuous

– Discrete

• Variance

∞

∞−

∂== ααα )( E XfXX

( )222 XX −=σ

==k

kk XX )Pr( αα

( ) ( ) ( ) ααασ ∂−=−= ∞

∞−Xx fXXX

222 E

Correlation & CovarianceCorrelation & Covariance• Crosscorrelation • Covariance

• If <X> or <Y> equal zero, correlation equals covariance

• Note that X* means the complex conjugate of X

*XYE=XYr

( )( ) ***XY YXXYYYXX-c −=−= EE

irir jXXjXXX −=+= ** )(

1−=j

Random ProcessRandom Process

• X, Y need not be separate events• X, Y can be samples of process observed at different instants t1, t2

( ) ( ) 2*

121 E),( tZtZttRZ =

( )( ) *2211

*21 )()()(t)(E),( tZtZZtZttCZ −−=

( )1tZX =

( )2tZY =

*XY XYr E=

( )( ) *XY YYXX-c −= E

Independence vs. UncorrelatednessIndependence vs. Uncorrelatedness

• R.V.s independent

• Uncorrelated (weaker condition), when

• R.V. X, Y uncorrelated if covariance is zero.

• Independent R.V. always uncorrelated.• Uncorrelated R.V. may not be independent!

)()(),( βαβα YXXY fff =

*** EEE YXYXXYrXY ===

( )( ) *EE YXXYYYXX-c **XY −=−=

Major Channel EffectsMajor Channel Effects• Propagation Loss: attenuation, also called path loss

• Gaussian Noise and Interference: Time variant nature due to mobility of objects in an environment

• Time Dispersion: multiple reflections due to obstacles leading to multipaths

• Doppler Effects: Time variant nature due to mobility of objects in an environment

2

4

=d

GGPP

RTT

R

πλ

TransmittersTransmitters

Multicarrier System –Wireless (Complex Transmission)

Map Block into N complex

IFFT BandpassFilterRate 1/T

N symbols

Rate N/T

Rate N/T

cos(wct)

sin(wct)

ReceiversReceiversMulticarrier System –Wireless (Complex Transmission)

Demap

Block

FFT

BandpassFilter

cos(wct)

j sin(wct)

SampleRate N/T

Sync

Frequency Domain EqualizationFrequency Domain Equalization

• For the kth carrier:xk = Hk sk + vk

where Hk = n hk(nTs) exp(j2π k n / N) where n = 0, …,. N-1

• Frequency domain equalizer xk

Hk-1

ssk

• Noise enhancement factor

k

|Hk|2

|Hk-1|2

kgood

bad

Example: IEEE 802.11aExample: IEEE 802.11a

• IEEE 802.11 employs adaptive modulation– Code rate & modulation depends on distance from base station– Overall data rate varies from 6 Mbps to 54 Mbps

Reference: IEEE Std 802.11a-1999

Ideal Channel EstimationIdeal Channel Estimation• Wireless channels change frequently ~ 10 ms• Require frequent channel estimation• Many systems use pilot tones – known symbols

– Given sk, for k = k1, k2, k3, … solve xk = l=0L hl e-j2π k l/N sk for hl

– Find Hk = l=0L hl e-j2π k l / N (significant computation)

• More pilot tones– Better noise resilience– Lower throughput (pilots are not informative)

frequency

mag

nitu

de Pilot tones

Channel Estimation Via InterpolationChannel Estimation Via Interpolation

• More efficient approach is interpolation• Algorithm

– For each pilot ki find Hki = xki / ski

– Interpolate unknown values using interpolation filter– Hm = αm,1 Hk1 + αm,2 Hk2 + …

• Comments– Longer interpolation filter: more computation, timing sensitivity– Typical 1dB loss in performance in practical implementation

frequency

mag

nitu

de

Vector modulator & complex baseband

• Independently modulate cos(2πfCt) & sin(2πfCt) and sum.• Coherent demodulator for ‘cos’ transmission blind to ‘sin’ trans. and vice-versa.

Mult

Mult

ADD

Cos(2ππππfCt)

Sin(2ππππfCt)

bR(t)

bI(t)

• “2 channels for price of 1”

• Still single carrier

• Complex baseband:

b(t) = bI(t) + jbR(t)

• More about this later

Vector demodulator

Mult

Mult

Cos(2ππππfCt)

Sin(2ππππfCt)

bR(t)

bI(t)

Derive local carrier(cos & sin)

Lowpassfilter

Lowpassfilter

bR(t)cos(2ππππfCt)+

bI(t)sin(2ππππfCt)

Mapping bit stream to baseMapping bit stream to base--bandband

Pulse-shaping filter

..1 1 0 1 0 ... Generate impulses

t

V V

t

‘Map to base-band’

• Stream of impulses produced according to bits & approach

e.g. for unipolar: unit impulse for ‘1’ & zero for ‘0’.

• Pass impulse stream through pulse shaping filter.

• Impulses & filter may be analogue or digital (generally digital)

Techniques for digital transmission

• Can modulate amplitude, frequency &/or phase of cos(2πƒπƒπƒπƒCt).

• These 3 forms of modulation when used independently give us

(a) amplitude shift keying (ASK) (b) frequency shift keying (FSK)(c) phase shift keying (PSK).

• There are many versions of each of these.

• Possible to use a combination of more than one form.

• Consider simplest binary forms first.

Binary frequency shift keying (B-FSK)

Modulatecarrier

Map to base-band

10110

tvolts

t

Binary amplitude shift keying (B-ASK)

Map to base-band

10110

tvolts

Multiply

Binary phase shift keying (B-PSK)

Map to base-band

10110

tvolts

Multiply

t

4-ary amplitude shift keying (ASK)

Map to base-band

10110

t

volts

Multiply

tvolts

volts

Combined multi-level ASK & PSK

Map to base-band

10110

tvolts

Multiply

Constellation diagramsShow “in phase” and “quadrature” components as a graph as illustrated below for two examples:

Binary ASK with symbols 0 & Acos(..)

In phase with carrier

Quadrature to carrier

Q

I

4-ary ASK with symbols 0, Acos(..), 2Acos(..), 3Acos(..)

0 A 2A 3A

Complex baseband & vector-modulator/demodulator

Vector modulator:

..11010.. Map

sin(2ππππfCt)

cos(2ππππfCt)

bI(t)

bR(t)

bR(t)cos(2ππππfCt)+

bI(t)sin(2ππππfCt)Map..10010..

Vector demodulator

Mult

Mult ThresholdDetector

ThresholdDetector

Cos(2ππππfCt)

Sin(2ππππfCt)

bR(t)

bI(t)Lowpass

Lowpass

..11010..

Derive localcarrier(cos & sin)

Receivedsignal r(t)

..10010..

Constellation diags for ASK with complex baseband

In phase with carrier

Quadrature to

carrier

0 A 2A 3A

Binary ASK for bR(t) & bI(t) 4-ary ASK

for bR(t) & bI(t)

In quadrature

Inphas

A

A 3A

A

0

ReReal pt

8-PSK16-PSK

Imag pt

QPSK is 4-PSK. What about 8-PSK & 16-PSK?

Can have 8-PSK (3 bits/symbol) & 16-PSK (4 bits/symbol). Constellation diagrams for shown below.

Differential forms of QPSK & M-PSK often used where changes in phase signify the data. Principle similar to DPSK .

‘Single carrier’ receiver

• Receiver must demodulate to obtain base-band b(t) .• Pulse shapes distorted & affected by noise. • Sample & detect for rectangular pulses discussed in last lecture.• May work for low bit-rates over channels with little distortion or noise• Performance can be improved by introduction of

– a matched filter optimally tuned to shape of transmitted pulses to minimize effect of noise (AWGN).

– a channel equalizer to cancel out distortion introduced by channel.

..1100..Matchedfilter

Demodulator Channelequaliser

Sample&

detectb(t)

Channelsignal + AWGN

Recall: Attenuation, Dispersion Effects: ISI!Recall: Attenuation, Dispersion Effects: ISI!

Source: Prof. Raj Jain, WUSTL

Inter-symbol interference (ISI)

Channel equaliser

• Channel equaliser’ is an ‘adaptive filter’ • Programmed to correct any differences between pulses seen at output

of matched filter & ideal RC pulses required by detector. • Aims to cancel out effect of the channel, • In particular the effects of frequency selective fading. • Received amplitude reduced at some frequencies & reinforced at

others.• Equalizer must do opposite of this. • Must adapt to changes in fading channel characteristics.• A demanding filtering task, and it cannot always be successful.• If there is a very deep fade, it will just not be possible to reverse it.• Trying to do so will just emphasize noise at frequency of deep fade.• Single carrier sine-wave modulation still widely used.

Spectra of Spectra of rectrect & & sincsinc pulsespulses

T.sinc1/T(f)

t

T/2-T/2

1

rectT(t)

f

T

1/T-1/T

2/T-2/T

3/T-3/T

4/T-4/T

Fourier transform

Real part shownImag part = 0

f

1/(2T)-1/(2T)

T

T.rect1/T(f)sincT(t)

t

1

T-T

2T-2T

3T-3T

4T-4T

Fourier transform

Real pt shownImag pt = 0

Spectra of 50% RC pulses & spectraSpectra of 50% RC pulses & spectra

RC(f)

tT/2-T/2

1

rc(t)

f

T

1/T-1/T

2/T-2/T

3/T-3/T

4/T-4/T

Fourier transform

Real part shownImag part = 0

-3T/43T/4

f

1/(2T)-1/(2T)

T

RC(f)rc(t)

t

1

T-T

2T-2T

3T-3T

4T-4T

Fourier transform

Real pt shownImag pt = 0

3/(4T)

-3/(4T)

Modulation of sub-carriers

• With IEEE802.11, each OFDM sub-carrier modulated by choice of:– binary-PSK, (1 bit per pulse)– QPSK, (2 bits per pulse)– 16-QAM (4 bits per pulse)– 64-QAM (6 bits per pulse)

• 16-QAM & 64-QAM are multi-level schemes.• Implement by vector-modulator according to ‘constellations’.• Illustrate for QPSK & 16-QAM• ‘Gray coding’ for 16-QAM makes nearest dots differ in just 1 bit.• Differential PSK, QPSK & QAM used where the difference between the current &

previous pulse specifies the bit pattern.

Constellation for QPSK

modulating cos

0,0

0,1

1,0

1,1

Bit1 Bit2 bR bI0 0 A A0 1 A -A1 0 -A A1 1 -A -A

Modulating sin

‘16_QAM’ constellation

A

3A

-A

-3A

A 3AReal

Imag

(0000)

-A

(0001)

(0010)

(0011)

(0100)(1000)

(1001)

(1010)

(1011)

(1100)

(1101)

(1110)

(1111)

(0110)

(0101)

(0111)

(modulates cos)

(modulates sin)

MulticarrierMulticarrier vsvs EqualizersEqualizers

• Equalizers use signal processing in receiver to eliminate ISI. • Linear equalizers can completely eliminate ISI (ZF), but this may enhance

noise. MMSE better tradeoff.

• Equalizer design involves tradeoffs in complexity, overhead, andperformance (ISI vs. noise).– Number of filter taps, linear versus nonlinear, complexity and overhead of training

and tracking

• Multicarrier is an alternative to equalization– Divides signal bandwidth to create flat-fading subchannels.

Introduction

-60 -40 -20 0 20 40 60-50

-40

-30

-20

-10

0

10

f [MHz]

pow

er s

pect

rum

mag

nitu

de [

dB] OFDM spectrum for N

FFT = 128, N

w in = 12, N

guard = 24, oversampling = 1

0 20 40 60 80 100 120 140 160 180 200-0.2

-0.1

0

0.1

0.2time domain signal (baseband)

sample nr.

imaginaryreal

OFDM Block DiagramOFDM Block Diagram

OFDM modulation

(IFFT)

Channel coding /

interleaving

Guard interval

I/Q I/QSymbol mapping

(modulation)

Transmitter

N symbols

OFDM demod. (FFT)

Decoding / deinter-leaving

Guard interval removal

Time sync.

I/Q I/Q

symbol de-mapping

(detection)

Channel est.

ReceiverFFT-part

time

1 OFDM symbol

Channel impulse response:

0101010010110

P/S

QAM demod

decoder

invert channel

=frequencydomain

equalizer

S/P

quadratureamplitude

modulation (QAM)

encoder

N-IFFTadd

cyclic prefix

P/SD/A +

transmit filter

N-FFT S/Premove cyclic prefix

TRANSMITTER

RECEIVER

N subchannels 2N real samples

2N real samplesN subchannels

Receive filter

+A/D

multipath channel

Summary: An OFDM Modem Summary: An OFDM Modem

Bits

00110