ofdm

Multi-Carrier Transmission over Mobile Radio Channels

Jean-Paul M.G. Linnartz

Nat.Lab., Philips Research

Outline

• Introduction to OFDM• Introduction to multipath reception • Discussion of receivers for OFDM and MC-CDMA• Introduction to Doppler channels• Intercarrier Interference, FFT Leakage• New receiver designs• Simulation of Performance• Conclusions

OFDM

OFDM: a form of MultiCarrier Modulation. • Different symbols are transmitted over different subcarriers• Spectra overlap, but signals are orthogonal.• Example: Rectangular waveform -> Sinc spectrum

I-FFT: OFDM Transmission

Transmission of QAM symbols on parallel subcarriers

Overlapping, yet orthogonal subcarriers

cos(ct+ st)

cos(ct)

cos(ct+ ist)

cos(ct+ (N-1)st)

User symbols

Ser

ial-

to-

par

all

el = Ser

ial-

to-

Par

alle

l

I-F

FT

Par

alle

l-to

-S

eria

l

OFDM Subcarrier Spectra

OFDM signal strength versus frequency.

Rectangle <- FFT -> Sinc

before channel

after channel

Frequency

Applications

Fixed / Wireline:• ADSL Asymmetric Digital Subscriber Line

Mobile / Radio:• Digital Audio Broadcasting (DAB)• Digital Video Broadcasting - Terrestrial (DVB-T)• Hiperlan II• Wireless 1394• 4G (?)

The Wireless Multipath Channel

The Mobile Multipath Channel

Delay spread Doppler spread

Frequency Time

FT

Frequency

FT

Frequency

Time

Effects of Multipath Delay and Doppler

Frequency

Tim

e

Narrowband

Frequency

Tim

e

OFDMWideband QAM

Frequency

Tim

e

Effects of Multipath (II)

Frequency

Tim

e

+-+--+-+

DS-CDMA

Frequency

Tim

e +

-

-

FrequencyHopping

Frequency

Tim

e + - + -

+ - +-

+ - +-

MC-CDMA

Multi-Carrier CDMA

Various different proposals.• (1) DS-CDMA followed by OFDM • (2) OFDM followed by DS-CDMA• (3) DS-CDMA on multiple parallel carriers

First research papers on system (1) in 1993:– Fettweis, Linnartz, Yee (U.C. Berkeley)– Fazel (Germany) – Chouly (Philips LEP)

System (2): Vandendorpe (LLN)

System (3): Milstein (UCSD); Sourour and Nakagawa

Multi-Carrier CDM Transmitter

What is MC-CDMA (System 1)?

• a form of Direct Sequence CDMA, but after spreading a Fourier Transform (FFT) is performed.

• a form of Orthogonal Frequency Division Multiplexing (OFDM), but with an orthogonal matrix operation on the bits.

• a form of Direct Sequence CDMA, but the code sequence is the Fourier Transform of the code.

• a form of frequency diversity. Each bit is transmitted simultaneously (in parallel) on many different subcarriers.

P/SI-FFT

NS/P N

B

Code Matrix

C

N

A

MC-CDM (Code Division Multiplexing) in Downlink

In the ‘forward’ or downlink (base-to-mobile): all signals originate at the base station and travel over the same path.

One can easily exploit orthogonality of user signals. It is fairly simple to reduce mutual interference from users within the same cell, by assigning orthogonal Walsh-Hadamard codes.

BS

MS 2MS 1

Synchronous MC-CDM receiver

The MC-CDM receiver • separates the various subcarrier signals (FFT)• weights these subcarriers in W, and

• does a code despreading in C-1:(linear matrix over the complex numbers)

Compare to C-OFDM:

W := equalization or AGC per subcarrier

C-1 := Error correction decoder (non-linear operation)

S/P P/SI-CodeMatrix

C-1

FFTN N N

Y

WeighMatrix

W

N

A

Synchronous MC-CDM receiver

Receiver strategies (How to pick W ?)• equalization (MUI reduction) w = 1/• maximum ratio combining (noise reduction) w = • Wiener Filtering (joint optimization) w = /( c

Next step: W can be reduced to an automatic gain control, per subcarrier, if no ICI occurs

S/P P/SI-CodeMatrix

C-1

FFTN N N

Y

WeighMatrix

W

N

A

Synchronous MC-CDM receiver

• Optimum estimate per symbol B is obtained from B = EB|Y = C-1EA|Y = C-1A.

• Thus: optimum linear receiver can implement FFT - W - C-1

• Orthogonality Principle: E(A-A)YH = 0N, where A = WYH

• Wiener Filtering: W = E AYH (EYYH)-1 • EAYH diagonal matrix of signal power• EYYH diagonal matrix of signal plus noise power• W can be reduced to an AGC, per subcarrier

S/P P/SI-CodeMatrix

C-1

FFTN N N

Y

WeighMatrix

W

N

A B

s

*

*

T

Nββ

βw

0

MC-CDM BER analysis

Rayleigh fading channel– Exponential delay spread– Doppler spread with uniform angle of arrival

Perfect synchronisation

Perfect channel estimation, no estimation of ICI

Orthogonal codes

Pseudo MMSE (no cancellation of ICI)

Composite received signal

Wanted signal

Multi-user Interference (MUI)

Intercarrier interference (ICI)

000,

1

0,,

1

0,00 ][][

mnn

N

nnmnn

N

nnn

s mncncwwN

Tbx

][][0,

1

0,

1

1ncncwbTx knn

N

nnn

N

kksMUI

00n,n,n

1

0n ][ncwaTx n

N

nsICI

Composite received signal

Wanted signal

Multi-User Interference (MUI)

Intercarrier interference (ICI)

1

0

2,ch

1

0

2,ch

0

20

1

1

22 EE)()(EE1 N

nnn

N

nnmk

N

kkICI wmncncb

N

2

,,,,ch

1

1

22

2*

ch2 EEEE

nn

Annnnn

Annn

N

kk

sMUIMUIMUI wwb

N

Txx

nn

N

nnn

s wβN

Tbx ,

1

0,00

BER for MC-CDMA

BER for BPSK versus Eb/N0

(1) 8 subcarriers

(2) 64 subcarriers

(3) infinitely many subcarriers

(4) 8 subc., short delay spread

(5) 8 subc., typical delay spread1 0 -5

1 0 -4

1 0 -3

1 0 -2

1 0 -1

5 1 0 1 5L o cal-m e an E n /N 0

E b /N 0 E b /N o (d B )

(1 )

(2 )

(3 )

(4 )

(5 )

A v g. B E R

A W G N

O F D M

Local-mean Eb/N0

Capacity relative to non-fading channel

Coded-OFDM

same as N fading channels

For large P0Ts/N0 on a Rayleigh fading channel, OFDM has 0.4 bit less capacity per dimension than a non-fading channel.

MC-CDMData Processing Theorem:

COFDM = CMC-CDM

In practise, we loose a little.

In fact, for infinitely many subcarriers,

CMC-CDM = ½ log2(1 + P0Ts/N0).

where is MC-CDM figure of merit, typically -4 .. -6 dB.

022

1

0

0

0

0 21logexp2 dxxxTP

N

TP

NC

ssOFDM

ssOFDM TP

NE

TP

NC

0

01

0

0

22exp

2ln

1

Capacity

Capacity per dimension versus local-mean EN/N0,

no Doppler.

-5 0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

Local-mean En/N0 (dB)

Cap

acity

: B

its p

er S

ubca

rrie

r

-* : Rayleigh

* : MC-CDMA

- : LTI

Non-fading, LTI

Rayleigh

MC-CDM

OFDM and MC-CDMA in a rapidly time-varying channel

Doppler spread is the Fourier-dual of a delay spread

Doppler Multipath Channel

Describe the received signal with all its delayed and Doppler-shifted components

Compact this model into a convenient form, based on time-varying amplitudes.

Make a (discrete-frequency) vector channel representation

Exploit this to design better receivers

Mobile Multipath Channel

Collection of reflected waves, each with • random angle of arrival• random delay

Angle of arrival is uniform

Doppler shift is cos(angle)

U-shaped power density

spectrum

Doppler Spectrum

ICI caused by Doppler

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10

-4

10-3

10-2

10-1

100

Normalized Doppler [fm/fsub]

Pow

er, V

aria

nce

of IC

I

P0

P1 P2 P3

Po

we

r o

r va

ria

nce

of I

CI

Doppler spread / Subcarrier Spacing

Neighboring subcarrier2nd tier subcarrier

3rd tier subcarrier

BER in a mobile channel

0 5 10 15 20 25 30 35 4010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Antenna Speed (m/s)

Lo

cal-M

ea

n B

ER

for

BP

SK

OFDM, 10 dB

MC-CDMA, 20 dB 30 dB

MC-CDMA, 10 dB

OFDM, 20 dB

OFDM, 30 dB

• Local-mean BER for BPSK, versus antenna speed.

• Local mean SNR of 10, 20 and 30 dB.

• Comparison between MC-CDMA and uncoded OFDM for fc = 4 GHz

• Frame durationTs= 896s

• FFT size: N = 8192.

•Sub. spacing fs = 1.17 kHz

•Data rate 9.14 Msymbol/s.

Antenna Speed [m/s]

Doppler Multipath Channel

Received signal r(t)

Channel model:

Iw reflected waves have

the following properties:

• Di is the amplitude

I is the Doppler shift

• Ti is the delay

OFDM parameters:•N is the number of subcarriers•Ts is the frame duration•an is the code-multiplexed datac is the carrier frequencys is the subcarrier spacing

)(}))((exp{)(1

0

1

0tntjTtnjDatr

wI

iiiscin

N

n

Taylor Expansion of Amplitude

Rewrite the Channel Model as follows

Tayler expansion of the amplitude

Vn(t) = vn(0)+ vn

(1) (t-t) + vn(2) (t-t)2/2 + .. .

vn(q) : the q-th derivative of amplitude wrt time, at instant t = t.

vn(p) is a complex Gaussian random variable.

)(})(exp{)()(1

0tntnjtVatr sc

N

nnn

1

0

)( })(exp{wI

itiisci

qi

qn jTnjDjv

)(}))((exp{)(1

0

1

0tntjTtnjDatr

wI

iiiscin

N

n

Random Complex-Gaussian Amplitude

It can be shown that for p + q is even

and 0 for p + q is odd.

• This defines the covariance matrix of subcarrier amplitudes and derivatives,

• allows system modeling and simulation between the input of the transmit I-FFT and output of the receive FFT.

srms

qpqqp

Dq

mp

n Tmnj

j

qp

qpfvv

)(1

)1(

!)!(

!)!1(2E )*()(

DF Vector Channel Model

Received signal Y = [y0, y1, … yN-1 ],

Let’s ignore

f : frequency offset

t : timing offset

We will denote = (0) and = (1)

• For integer , :: 0 (orthogonal subcarriers)

models ICI following from derivatives of amplitudes

0 does not carry ICI but the wanted signal

m

N

n q

qqmn

qn

ftsnm nq

Tvnjay f

1

0 0

)()(

!exp

Complex amplitudes and derivatives

System constants (eg sinc) determined by waveform

DF-Domain Simulation

Simulation of complex-fading amplitudes of a Rayleigh channel with Doppler and delay spread

• Pre-compute an N-by-N matrix U, such that UUH is the channel covariance matrix with elements n,m = Evn

(0)vm*(0)

– Simply use an I-FFT, multiply by exponential delay profile and FFT

• Generate two i.i.d vectors of complex Gaussian random

variables, G and G’, with unity variance and length N.

• Calculate V = U G.

• Calculate V(1) = 2fT U G’.

DF Vector Channel Model

Received signal Y = [y0, y1, … yN-1 ],

models ICI following from derivatives of amplitudes

0 does not carry ICI but the wanted signal

NTAV

AVIχY N

*'.

*.30

021

201

110

3

..

........

..

..

NN

N

N

FFT leakage

Amplitudes & Derivatives

User data

Possible Receiver Approaches

Receiver

1) Try to invert adaptive matrix (Alexei Gorokhov)

2) See it as Multi-user detection: (J.P. Linnartz, Ton Kalker)

– try to separate V .* A and V(1).* A

3) Decision Feedback (Jan Bergmans)– estimate iteratively V, V (1) and A

)DIAG()DIAG( )1(0 VVχ

NATV

AVIχY N

*.'

*.0

Receiver 1: Matrix Inversion

• Estimate amplitudes V and complex derivatives V (1)

• create the matrix Q1 = DIAG(V)+ T DIAG(V(1))

• Invert Q1 to get Q1-1 (channel dependent)

• Compute Q1-1Y

Zero-forcing: For perfect estimates V and V (1), Q1

-1Y = A + Q1-1N,

i.e., you get enhanced noise.

MMSE Wiener filtering inversion W

ChannelEstimator

SlicerQ-1Y

3

X1 X2

X3

+x

x

V

V’

A

N

V’

A

V

Receiver 1: MMSE Matrix Inversion

Receiver sees Y = Q A + N, with Q=DIAG(V)+ T DIAG(V(1))

Calculate matrix Q = DIAG(V)+ T DIAG(V(1))

Compute MMSE filter W = QH [Q QH + n

2 IN]-1.

Performance evaluation:

• Signal power per subcarrier• Residual ICI and Noise enhancement from W

Receiver 1: Matrix Inversion

Simulation of channel for N = 64, v = 200 km/h fc = 17 GHz, TRMS = 1 s, sampling at T = 1s. fDoppler = 3.14 kHz, Subcarrier spacing fsr = 31.25 kHz, signal-to-ICI = 18 dB

0 10 20 30 40 50 60 70-80

-70

-60

-50

-40

-30

-20

-10

0

10

Subcarrier number

Ma

gn

itud

e in

d

B

Amplitudes Derivatives

Amplitudes

First derivatives

Determined by speed of antenna, and carrier frequency

Receiver 1: Matrix Inversion

SNR of decision variable. Simulation for N = 64, MMSE Wiener filtering to cancel ICI

0 10 20 30 40 50 60 700

5

10

15

20

25

30

Subcarrier number

Ou

tpu

t S

INR

Conventional OFDM MMSE equalization

MMSE ICI canceller

Conventional OFDM

Performance of (Simplified) Matrix Inversion

N = 64, v = 200 km/h, fc = 17 GHz, TRMS = 1 s, sampling at T = 1s.

fDoppler = 3.15 kHz, Subc. spacing fsr = 31.25 kHz:

Compare to DVB-T: v = 140 km/h, fc = 800MHz: fdoppler = 100 Hz while fsr = 1.17 kHz

5 10 15 20 25 300

5

10

15

20

25

30

Input SNR

Conventional OFDM MMSE equalization simplified MMSE

k = 4

ConvOFDM

MMSE

Ou

tpu

t S

INR

Receiver 1: Subconclusion

• Performance improvement of 4 .. 7

• Complexity can be reduced to ~2kN, k ~ 5 .. 10.

• Estimation of V(1) to be developed, V is already being estimated

Receiver 3: Decision Feedback

Estimate

• data, • amplitudes and • derivatives

iteratively

Receiver 3: Decision Feedback

Iteratively do the following:• Compare the signal before and after the slicer• Difference = noise + ICI + decision errors• Invert to retrieve modulated derivatives from ICI

– V(1).*A = -1 ICI– MMSE to minimize noise enhancements

• Remove modulation 1/A• Smooth to exploit correlation in V(1)

• Modulate with A• Feed through to estimate ICI• Subtract estimated ICI

Receiver 3: DFE

Estimate V(1) in side chain

Pilot

Slicer

M6

+

x

x

+

Cancel Doppler

Estimated Amplitudes

3

ICI

-+

M7

Z10

Z8

Z7

Z6

Y2Y0

3

X1 X2

X3

+x

x

V

V’

A

N

Z9 =V’

A

1/A

A.*V

V

-

A

INT

Channel Model

Performance of Receiver 3: DFE

Variance of decision variable after iterative ICI cancellation versus variance in conventional receiver

10-2

10-1

100

101

102

10-3

10-2

10-1

100

101

102

Variance Conventional

Va

ria

nce

Ne

w S

yste

m 3

Variance decision variable in conventional receiver

Var

ianc

e of

dec

isio

n va

riabl

e in

DF

E r

ecei

ver

afte

r IC

I ca

ncel

ling

Error Count

Receiver 3: DFE

N = 64 out of 8192 subcarriers, v = 30 m/s, fc = 600 MHz TRMS / NT= 0.03, fDoppler = 60 Hz, Subcarrier spacing fsr = 1.17 kHz

0 10 20 30 40 50 60 70-30

-25

-20

-15

-10

-5

0

5

10

Decision FeedbackSample run N=64 9 errors -> 4 errors

Subcarrier Number

Am

plit

ud

e

Amplitudes

Derivatives

Conclusions

Modeling the Doppler channel as a set of time-varying subcarrier

amplitudes leads to useful receiver designs.

Estimation of V(1) is to be added, V is already being estimated

Basic principle demonstrated by simulation

Gain about – 3 .. 6dB, – factor of 2 or more in uncoded BER, – factor 2 or more in velocity.

Promising methods to cancel FFT leakage (DVB-T, 4G)

More at http://wireless.per.nl

Further Research Work

Optimise the receiver design and estimation of derivatives

Can we play with the waveform (or window) to make the tails of the filter steeper?

Can we interpret the derivatives as a diversity channel?

Can estimation of derivatives be combined with synchronisation?

Isn’t this even more promising with MC-CDMA?

Apply it to system design.