Orthogonal Frequency Division Multiplexing and Related Technologieswebmail.aast.edu/~khedr/Courses/VT/OFDM/lecture fou… ·  · 2007-10-31Orthogonal Frequency Division Multiplexing

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