Multi-Carrier Transmission over Mobile Radio Channels

Jean-Paul M.G. LinnartzPhilips Research and TU/e

Outline

• Introduction to OFDM• Discussion of receivers for OFDM and MC-CDMA• 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

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 (?)

I-FFT: OFDM Transmission

Transmission of QAM symbols on parallel subcarriersOverlapping, yet orthogonal subcarriers

cos(ωct+ ωst)

cos(ωct)

cos(ωct+ iωst)

cos(ωct+ (N-1)ωst)

User symbols

Seria

l-to-

para

llel = Se

rial-t

o-Pa

ralle

l

I-FFT

Para

llel-t

o-Se

rial

I-FFT: OFDM Transmission

Transmission of QAM symbols on parallel subcarriersOverlapping, yet orthogonal subcarriers

[ ]

∈=

otherwise 0

,0 1)( s

tj

sk

TteTt

kω

ψ

1,...,1,0 ; with 0 −=+= csk Nkkωωω

Although the subchannels overlap, they do not interfere with each other at f = fk; (k = 0, 1,...,Nc-1). Indeed, they are orthogonal:

∫ −=sT

lk lkdttt0

* )()()( δψψ

OFDM Subcarrier Spectra

Symbol duration : inverse of subcarrier spacing plus cyclic prefix

Sampling rate : inverse of transmit bandwidth

Frequency

Pulse shape in time domain: rectangle Π(t / NTs)

( ) ( )( ) ( )( )( )TsNNf

TsNNfTsNNfa

TNNtats

cp

cpcp

N

nn

F

scp

N

nn +

+=+=→←

+Π= ∑∑

−

=

−

=

sinsinc S(f))(

1

0

1

0

OFDM Subcarrier Spectra

OFDM signal strength versus frequency.

Rectangle <- FFT -> Sinc

before channel

after channel

Frequency

Cyclic Prefic / Cyclix postfix

The length of the cyclic prefix should be made longer than the experienced impulse response to avoid ISI and ICI. However, the transmitted energy increases with the length of the cyclic prefix. The SNR loss due to the insertion of the CP is given by

where Tcp denotes the length of the cyclic prefix and T=Tcp+Ts is the length of the transmitted symbol.

CP OFDM SYMBOL

Tcp TsT

−−=TT

SNR cploss 1log10 10

Coded OFDMReceived signal at a subcarrier is not affected by transmitted symbols in

any other subcarrier

Symbol data can be recovered using simple single tap data estimation

Viterbi decoder helps recover bits from subcarriers in deep fade

+

=

−−−− 1

0

1

0

1

0

1

0

0

0

NNNN n

n

a

a

H

H

y

yMMOM

=

−−

−

− − 1

0

1

1

1

0

1

0

0

0

ˆ

ˆ

NN y

y

H

H

a

a

N

MOM

Static Environment

Ave BER curves:slope ~ degree of diversity on fading channel

∑ −=l

mfjlm

lsehH τπ )(2

Single Frequency Networks

OFDM is robust against delay spread

We can “mis”use this by transmitting a synchronous signal from two transmit sites

The Wireless Multipath Channel

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

Doppler spread is the Fourier-dual of a delay spread

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 spectrumDoppler Spectrum

Crosstalk β caused by Doppler

For uniform angles of arrivals of waves, ICI power spilled from transmitsubcarrier n into received subcarrier m = n + ∆ equals

where f∆ is the maximum Doppler shift, and PT the local mean received power, per subcarrier

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

Pow

er o

r var

ianc

e of

ICI

Doppler spread / Subcarrier Spacing

Neighboring subcarrier2nd tier subcarrier

3rd tier subcarrier

)}(ω)ωωω(exp{

ωωsinc

2

21

1

0,

mnjTjTnj

mnD

siisic

I

i s

iinm

w

−+−++−

+−= ∑

−

=

π

β

( )∫−

∆

∆+∆+∆−

+∆

==1

1 2

2

,*

,1

sinc

8E

x

dxxff

PP sTnnnnch πββ

Y = [y0, y1, .., yN−1]T, with ym = Σnanβm,nTs

βm,n is the ‘transfer’ for a signal transmitted at subcarrier nand received at subcarrier m,

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

Pow

er o

r var

ianc

e of

ICI

Doppler spread / Subcarrier Spacing

Neighboring subcarrier2nd tier subcarrier

3rd tier subcarrier

Maximum attainable speed

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)

Loca

l-Mea

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= 896µs• FFT size: N = 8192. •Sub. spacing fs = 1.17 kHz•Data rate 9.14 Msymbol/s.

Antenna Speed [m/s]

Mobile OFDM weakness: InterCarrier Interference

An effective description of ICI

y = [H + Ξ H’] a + n

where:• H is a diagonal matrix with the complex transfer function per subcarrier• H’ is the temporal derivative of H: H’ = dH/dt

• Ξ is the ICI spreading matrix: a system value, fixed!• a is a vector of transmitted data• n is vector of white Gaussian noise

self-interferencedesired

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

mpn Tmnj

jqpqpfHH

ωπ

)(1)1(

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

−+

−

+−+

=+

+

Channel estimation DVB-T

H estimation based on pilots in frequency domain

frequency

time

pilotsempty carriers

OFDM symbol

data carriers

Estimation of H

DVB-T pilots

In every OFDM symbol, initial estimates of H at every pilot subcarriers can be obtained.

In every OFDM symbol, estimate of H at each subcarrier is obtained:– Spectral Wiener Filtering – Input: Initial estimates of H at several (scattered) pilot positions .

frequency

time

pilotsempty carriers

OFDM symbol

data carriers

Wiener Filtering for OFDM

Achievable car speeds with BB signal processing

Single antenna CoSyS result

On the market

50

120

Receiver 1: MMSE Matrix Inversion

Receiver sees Y = Q A + N, with Q=DIAG(H)+ Ξ DIAG(H(1))

• Calculate matrix Q = DIAG(H)+ Ξ DIAG(H(1))• Compute MMSE filter W = QH [Q QH + σn

2 IN]-1.

Performance evaluation:

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

ICI Handling: Brute force 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

Out

put S

INR

Conventional OFDM MMSE equalization

MMSE ICI canceller

Conventional OFDM

Doppler Diversity

Performance of (Blocked Band) Matrix Inversion

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

MMSEO

utp

ut

SIN

R

ICI cancellation

If H’n and an are known for all n, ICI can be removed almost completely from any subcarrier.

Implement Ξ as “FFT - Ramp - FFT”

kmmN

nnnnmmm naHaHyy +=Ξ−= ∑

−

=

1

0, ''

ICI Cancellation

+

Cancel Doppler

weigh

ICI

-+Y0

FFT

FFT

H’A

Ξ

ICI cancellation

However, H’ and a are not known to the receiver: They have to be estimated

– Full cancellation

– Partial cancellation: ân is used only to cancel interference it caused to p closest subcarriers

∑+

−=Ξ−=

2/

2/, ˆ'ˆ'

pm

pmnnnnmmm aHyy

∑−

=Ξ−=

1

0, ˆ'ˆ'

N

nnnnmmm aHyy

ICI Cancellation

On average:Canceling ICI originating from 4 closest subcarriers reduced the ICI power by 6 dBCanceling ICI originating from 10 closest subcarriers reduced the ICI power by 10 dB

Receiver 1: Matrix Inversion

Simulation of channel for N = 64, v = 200 km/h fc = 17 GHz, TRMS = 1 µs, sampling at T = 1µs. 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

Mag

nitu

de i

n 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

Out

put S

INR

Conventional OFDM MMSE equalization

MMSE ICI canceller

Conventional OFDM

Simplified Matrix Inversion

Rationale• ICI diminishes with increasing subcarrier difference• Approximate Ξ by band matrix with 2k+1 non-zero diagonals• Matrix Q is approximately Q = [I + ∆] Λ

– ∆ small, ∆ ~ Ξ diag(V(1) ./ V)– Λ diagonal of amplitudes V

• Approximate Q-1 = [I - ∆] Λ−1

Complexity ~2kN

Performance of (Simplified) Matrix Inversion

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

MMSEO

utp

ut

SIN

R

Multi-Carrier 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-FFTNS/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 subcarrierC-1 := Error correction decoder (non-linear operation)

S/P P/SI-CodeMatrix

C-1FFT

N N N

Y

WeighMatrix

W

N

A

Synchronous MC-CDM receiver

Receiver strategies (How to pick W ?)• equalization (MUI reduction) w = 1/β• maximum ratio combining (noise reduction) w = β• Wiener Filtering (joint optimization) w = β/(β2 + c)

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

S/P P/SI-CodeMatrix

C-1FFT

N 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-1FFT

N N N

Y

WeighMatrix

W

N

A B

s

*

*

TNββ

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 ICIOrthogonal 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 mncncwwNTbx ββ

= ∑∑

−

=

−

=][][0,

1

0,

1

1ncncwbTx knn

N

nnn

N

kksMUI β

∑∑≠∆

∆+∆+∆+

−

=∆+=

00n,n,n

1

0n ][ncwaTx n

N

nsICI β

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

Nβσ

2

,,,,ch1

1

22

2*

ch2 EEEE

−== ∑∑∑

−∈+∈

−

=nn

Annnnn

Annn

N

kk

sMUIMUIMUI wwb

NTxx ββσ

nnN

nnn

s wβNTbx ,

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 spread10-5

10-4

10-3

10-2

10-1

5 10 15Local-mean En/N0

(1)

(2)

(3)

(4)

(5)

Avg. BER

AWGN

OFDM

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 + ςP0Ts/N0).

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

( )∫∞

+

−=

022

1

0

0

0

0 21logexp2 dxxxTPN

TPNC

ssOFDM

=

ssOFDM TP

NETP

NC0

01

0

022

exp2ln

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

: Bits

per

Sub

carr

ier

-* : Rayleigh

* : MC-CDMA

- : LTI

Non-fading, LTI

Rayleigh

MC-CDM

Advantages

Simpler user separation than with DS-CDMA

Higher capacity than DS-CDMA in downlink: elegant frame work for doing simultaneous anti-multipath and interference rejection

FFT instead of rake: simpler training of receiver

MC-CDMA in uplink

In the ‘reverse’ or uplink (mobile-to-base), it is technically difficult to ensure that all signals arrive with perfect time alignment at the base station.

Frame mis-alignments cause severe interferenceDifferent Doppler spectra for each signalDifferent channels for different signalsPower control needed

BS

MS 2MS 1