Equalization of OFDM under Time-Varying Channel

Conditions

Michael D. ZoltowskiPurdue UniversityWest Lafayette, INmikedz@ecn.purdue.edu

David A. HaessigBAE SystemsWayne, NJdavid.haessig@baesystems.com

14th Annual Workshop on Adaptive Sensor Array Processing (ASAP)

MIT Lincoln Laboratory6 June 2006

2

Outline

• Introduction– OFDM

TransmitterReceiver – delay-only

– Problem Definition Delay-Doppler Channel ModelPrior Work

• Extension of OFDM for large Doppler spread– MMSE-based Receiver – Simulated Performance for complete channel information

• Parameter Estimation– Channel Model– Uncoupled estimation of Doppler, Delay, and Gain

• Conclusions & Future Work

3

Orthogonal Frequency Division Modulation (OFDM) – a waveform, TDMA in nature, designed to accommodate multi-path time dispersion

• Cyclic extensions avoid ISI due to delay spread & aid or enable synchronization• Cyclic Prefix generated by copying end of x(n) to beginning

–Suffix – add beginning to end• Windowing reduces spectral growth

timeOFDM Frame = T + TCE

OFDM Symbol = x(n) = T msecCycle suffix & window

Cycle prefix and window

Windowing

time

… …

time

… …

4

The OFDM Transceiver

Received Power

Delay Spread t

Received Power

Delay Spread t

Tx bit stream

Mod-ulator

S/P IFFTCyclic extension & windowing

P/S

Time dispersive channel

Rx bit stream

De-mod

P/S FFTRemove Cyclic extensions

S/P

noise

Channel Equalization

Extract Pilots

Add Pilots

+

Sync

x(n)x

• A transmit time-domain sequence x(n) is produced from transmit symbol vector s:

where W* is the IFFT matrix, Cp the cyclic prefix operator, N the FFT length and L the number of samples in Cp

sWCx p*=

Cp =

IL 0

IN

s

Estimate Channel

r y(n)

5

The OFDM Receiver -- handles channel time dispersion in a computationally efficient manner

sWCDCr ppr*

τ=

prC =

IN0

Rx bit stream

De-mod

P/S FFTRemove Cyclic extensions

S/P

noise

Channel Equalization

Extract Pilots

+

Sync

• Representing the channel , having K paths, in delay-operator matrix form∑=

−=K

iii nnh

1

)()( τδα

Estimate Channel

( )

W

e

ee

WD

Nj

j

j

i

i

i

i

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

−−

−

−

1

1

0

*

0000

0000

τ

τ

τ

τ ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

=

K

ii iDD

1ττ α

r

• The received signal r(n) is then given by:

where Cpr removes cyclic prefix• Channel does not destroy orthogonality of signaling basis functions

where is a diagonal matrix containing the channel frequency response, enabling coherent detection

s~s

rWs =~

H

rWHs *ˆˆ =H

6

• The signal y(n) at channel output given transmit signal x(n) at channel input:

having independent Doppler and Delays on K separate “rays” and assumed to be constant over the OFDM Frame

• The channel is given in delay-Doppler operator form by summing over each of the K paths:

• Converting from serial to parallel vector form:

• An issue, the transfer matrix (in brackets) is no longer diagonal, but diagonally dominant

Consider the general delay-Doppler channel model

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

=ΩΩ

K

ii ii

DDD1

ττ α

)()()(1

)( nnxenyK

ii

nji

ii ητα τ +−=∑=

−Ω

( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

−Ω

Ω

Ω

Ω

1

1

0

0000

0000

Nj

j

j

i

i

i

i

e

ee

D

iΩ iτ

ητ += Ω sWCDy p*

[ ] ητ += Ω sWCDWCs ppr*~

7

The Problem – a time & frequency dispersive channel with a particularly large delay-Doppler product

• In typical wireless applications frequency dispersion is a fraction of a tone spacing.

• In the problem considered here, frequency dispersion greatly exceeds the tone spacing

1/T

time

frequency

tone spacing

OFDM Symbol timeT

Typically Doppler spread << 1/T

Typical Delay spread < T/5

atypically Doppler spread >>1/T

8

Prior work related to the use of OFDM with frequency selective time-varying fading channels:

• Wang, Proakis, Zeidler, Techniques for Suppression of Intercarrier Interference in OFDM Systems, IEEE WCNC, Feb 2005:

– Propose a MMSE-based OFDM receiver structure– Estimate channel by curve fitting over time-frequency using pilots, based on

statistical model of time-varying impulse response

• Linnartz, Gorhkov, Doppler-resistant OFDM receivers for Mobile Multimedia Communications, PIMRC 2000, London, Sept 2000, and

• Mostofi, Cox, ICI Mitigation for Pilot-Aided OFDM Mobile Systems:– MMSE ICI receiver based on piece-wise linear approximation to channel

response at each toneSuitable for limited Doppler spread

• Y. Li, L.J. Cimini, and N.R. Sollenberger, Robust channel estimation for OFDM systems with rapid dispersive fading channels, IEEE, Tr. on Comm. vol. 46, pp. 902-915, July 1998:

• Many others

9

• The received signal vector through the delay-Doppler channel :

• the MMSE solution is given by:

requiring estimate of delay-Doppler channel

• Comparing delay-Doppler MMSE receiver to delay-only OFDM receiver:– OFDM - NxN FFT– MMSE - NsxNs inverse – where Ns – number of symbols; N – FFT size

An MMSE-based OFDM receiver structure is developed for the general, deterministically modeled delay-Doppler channel

ητ += Ω sWCDCr ppr*

rFIFFs n ′+′= − ˆ)ˆˆ(ˆ 12σ

r

*WCDCF ppr Ω= τ

η+= Fsr

*ˆˆ WCDCF ppr Ω= τ

ΩτD

rWHs *ˆˆ =rFIFFs n ′+′= − ˆ)ˆˆ(ˆ 12σ

Simulation results

Comparison of OFDM performance in delay-only case to that of MMSE in delay-Doppler case

11

Simulations comparing the OFDM receiver in the delay-only case to that of the MMSE receiver through the same channel but with arandom Doppler added to each channel tap

Tx packet Add 20% pilots & zero fill

1K IFFT

Inter-leave over 5 packets

BPSK

Rate 1/5 Turbo Encoder

Cyclic extension & windowing

Mod-ulator

S/PExtract pilots

& remove zero fill

1K FFT

De-Inter-leaver

Rate 1/5 Turbo Decoder

Remove Cyclic extensions

BPSK soft Demod

pilots

Channel Equalization

)(ˆ kH

Rx packet

Delay-Doppler Channel

Delay-only Channel

True Channel

1K MMSE

F Matrix Computation

α τ Ω F

Channel Response h(n)

S/PRemove Cyclic extensions

Compute Freq Resp

BPSK soft Demod

De-Inter-leaver

Rate 1/5 Turbo Decoder

Rx packet

Channel Gain, tap location and Doppler 1024FFT Size = NFFT

99

6 samples

160 samples

.0061

Value

No. of bits per packet

Cyclic Suffix

Cyclic Prefix

Tone Spacing -- (rad) -- 2π/NFFT

Parameter Name

12

Channels evaluated -- 11 tap channel with Doppler spread of a few tones, and an several hundred tap channel spanning the entire 160 sample cyclic extension with a Doppler spread of over 500 tones

0 50 100 150 200 250-0.01

-0.005

0

0.005

0.01Tap weights

50 100 150 200-0.4

-0.2

0

0.2

Tap Dopplers (rad/sam ple)

x 104

0.5 1 1.5 2

x 104

-60

-50

-40

-30

-20

Frequency (Hz)

Mag

nitu

de (d

B)

0 0.5 1 1.5 2

x 104

-60

-40

-20

0

Frequency (Hz)

Mag

nitu

de (d

B)

0 50 100 150 200 250

-0.2

-0.1

0

0.1

0.2

Tap weights

0 50 100 150 200 250

-0.2

-0.1

0

0.1

0.2

Tap Dopplers (rad/sample)

• Note: Frequency responses computed for the case in which Doppler = 0

11 tap channel many tap channel

13

-5 0 5 1010

-6

10-5

10-4

10-3

10-2

10-1

100

E b/No (dB )

Pe

B it E rror Rate vs E b/No - Des ign 3 in A W G N, Delay -only , and Delay -Doppler Channels

Performance of OFDM and MMSE Receivers with perfect channel knowledge

• Performance of MMSE receiver in delay-Doppler channel closely matches that of OFDM in delay-only channel

• Performance lost to Doppler recovered with MMSE receiver, but at a significant increase to computational expense

AWGN

Many-tap delay-only & delay-

Doppler channels

11 tap delay-only & delay-Doppler

channels

Parameter Estimation

An estimation approach for a sensor array channel structure

15

We consider the time-varying multi-path channel involving K rays from one transmitter to a P-node array

[ ]∑=

− −M

mm

Mnjm Mnye mm

111

)(1 )(11ωβ

njeNnxns 1)()( 111θα −=

Transmitter

[ ]∑=

− −M

mmPP

MnjmP Mnye mPmP

1

)( )(ωβ

njeNnxns 2)()( 222θα −=

njeNnxns 3)()( 333θα −=

njKKK

KeNnxns θα )()( −=

+)(* nxsWC p =

)(nr

• Each ray involves an unknown gain , fractional delay , and Doppler shift to be estimated

• Array signal processing parameters, , are known

• For this channel we construct the F matrix relating to

kα kNkθ

mpmpmp M ωβ ,,

s r

)(1 nz

)(nzK

)(1 ny

)(nyP

16

Where the channel Dopper & delay matrix operators are:

• Note that F is a function of the known array processing parameters, and the unknown ray parameters

• Requiring estimation of 4 real parameters per ray

The F matrix for the case of K rays

mpmpmp M ωβ ,,

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Ω

−Ω

Ω

Ω

1

)1(

0000

00001

)(

Nj

j

e

eD

( )

W

e

eWD

Nj

j

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−−

−

1

1*

0000

00001

)(

τ

τ

τ τ

( ) ( ) ( )[ ] *

1

),,,,( WCDMNDeCMNF p

M

mmkkmkk

Mjmkprmkmkmkkkk

mkk∑=

ΩΩ− +Ω+=Ω ωβωβ τ

∑=

=K

kkkkk FFF

1

),( αα

kα kNkθ

η+= sFr

17

• Training symbol vectors made entirely of pilots are transmitted– Series of sample-domain vectors used for parameter estimation

• Channel Doppler, delay, and gain estimates generated sequentially, not jointly– Estimates of Doppler generated via ESPRIT, independent and without knowledge of channel

delay and gain estimates– Delay estimates generated via series of 1-D searches, also pairing Dopplers to particular rays– matrices computed, and channel gains estimated as final step

K – coding rate

De-Inter-leaver

Rate 1/K Turbo Decoder

DeMod-ulator

Remove cyclic ext

MMSES/P

ωβ ,, m

r

MMSE receiver and parameter estimation architecture showing separate estimation of Doppler, delay, and channel gain

Rx packetof B bits

F

ESPRIT Doppler Estimation -

Residual Delay Estimation -

Compute FkMatrices

Total F Matrix Computation

Channel Gain Estimation

Array Processing

kF,α

kF

Ω,NΩ

r

18

Use of ESPRIT with Multiple Successive Training OFDM Symbol Blocks to Achieve Separable Estimation of Residual Dopplers and Delays Associated with Multiple MBAJ Channels input to MBAJ

• Estimation Doppler via ESPRIT: – train with K+1 repeated OFDM training symbols – estimates are the eigenvalues of a matrix

avoids a multi-dimensional search and is invariant to unknown gains and time delays

• Subsequent estimation of residual time delays – using Doppler shift estimates– uses eigenvector information from ESPRIT to decouple superimposed rays

allowing for separable estimation of residual time delays (pairing problem addressed)

• Unknown channel gains subsequently estimated via Least Square Error solution

• Total F matrix compute given estimates and used over subsequent OFDM Frames for symbol data detection

19

Send the same OFDM training symbol for 4 successive blocks to generate the following output vectors:

y1 = e j (N + L )ω1 z1 + e j (N + L )ω2 z2 + e j (N + L )ω3 z3

y2 = e j 2(N + L )ω1 z1 + e j 2(N + L )ω2 z2 + e j 2(N + L )ω 3z3

y0 = z1 + z2 + z3

z1 = α1 Cpr β1i Dτ ,1i Dω ,1i1,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ1

residual Dω1

residual CpW*s( )

z2 = α2 Cpr β2i Dτ ,2i Dω ,2i2,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ2

residual Dω2

residual CpW*s( )

z3 = α 3 Cpr β3iDτ ,3iDω ,3i3,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ 3

residual Dω3

residual CpW*s( )

y3 = e j 3(N + L )ω1 z1 + e j 3(N + L )ω2 z2 + e j 3(N + L )ω3 z3

ESPRIT Based Estimation of Delay – consider this example involving 3 rays:

20

Cast these equations as an eigenvalue problem, thereby avoiding a multi-dimensional search to find Doppler values

φ1 = e j (N + L )ω1

φ2 = e j (N + L )ω2 Δ =1 φ1 φ1

2

1 φ2 φ22

1 φ3 φ32

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

G1Ψ = G2

Φ =φ1 0 00 φ2 00 0 φ3

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

• With quantities defined as on previous slide, it follows that:

where

G1 = y0 | y1 | y2[ ]= z1 | z2 | z3[ ]Δ G2 = y1 | y2 | y3[ ]= z1 | z2 | z3[ ]ΦΔ

φ3 = e j (N + L )ω3

& are eigenvalues of

Ψ = Δ−1ΦΔ

Thus we can write: allowing to be determined numerically

The solution is implying that

φ1 = e j (N + L )ω1 φ2 = e j (N + L )ω2 φ3 = e j (N + L )ω3 Ψ

Ψ

21

Here is an example involving a 3 ray channel with unknown delays, Doppler, and gains -- the 3 Doppler values are estimated correctly

φ3 = e j (N + L )ω3

φ1 = e j (N + L )ω1

φ2 = e j (N + L )ω2

22

Use eigenvector information from ESPRIT to separate the three channel contributions summed at the array output:

φ1 = e j (N + L )ω1

φ2 = e j (N + L )ω2

G1 = y0 | y1 | y3[ ]= z1 | z2 | z3[ ]Δ

G2 = y1 | y2 | y3[ ]= z1 | z2 | z3[ ]ΦΔ

Ψ = ΕΛΕ−1

THUS: eigenvectors of decouple superposition of sensor array output contributions from channel 1, channel 2 & channel 3:

Ψ

G1E = z1 | z2 | z3[ ]DP

G1Ψ = G2are eigenvalues of solution to φi = e j ( N + L )ω i , i = 1,2, 3

φ3 = e j (N + L )ω3

Φ =φ1 0 00 φ2 00 0 φ3

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

where is a permutation matrix and is a diagonalmatrix representing unknown scalings, since eigenvectors are only unique to within a multiplicative scalar

P D

Λ = Φ Ε = Δ−1DPwhere and

23

Pairing problem and residual delay estimation:

There is an unknown gain on each channel -- absorb into unknown diagonal scaling matrix

z1 = Cpr β1iDτ ,1i Dω ,1i

1,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ1

residual Dω1

residual CpW*s( )

z2 = Cpr β2iDτ ,2i Dω ,2i

2,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ2

residual Dω2

residual CpW*s( )

Combinatoric pairing problem: six possibilities to assess in caseof 3 channels feeding into sensor array

z3 = Cpr β3iDτ ,3i Dω ,3i

3,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ 3

residual Dω3

residual CpW*s( )

G1E = ζ1 |ζ2 |ζ3[ ]= z1 | z2 | z3[ ]DP

D

24

For each of the permuted pairings, search for the delays producing a match (to within a scalar) of the permuted eigenvector and the vectors

G1E = ζ (1) |ζ (2) |ζ (3)⎡⎣ ⎤⎦

z1(ω

(δ ),τ1) = Cpr β1i Dτ ,1i Dω ,1i1,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ1

residual Dω (δ )residual CpW

*s( )

z2 (ω (ε ),τ 2 ) = Cpr β2iDτ ,2i Dω ,2i

2,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ2

residual Dω (ε )residual CpW

*s( )

z3(ω (γ ),τ 3 ) = Cpr β3i Dτ ,3i Dω ,3i

3,i∑⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥Dτ 3

residual Dω (δ )residual CpW

*s( )

{ }minimum0 < τ1 < LTs

ζ (δ ) | %z1(ω (δ ),τ1)⎡⎣ ⎤⎦smallest singular value of

+Γ(δ ,ε,γ ) = smallest singular value of{ }minimum

0 < τ 2 < LTs +smallest singular value of{ }minimum

0 < τ 3 < LTs

ζ (ε ) | %z2 (ω (ε ),τ 2 )⎡⎣ ⎤⎦

ζ (γ ) | %z3(ω (γ ),τ 3)⎡⎣ ⎤⎦

(δ ,ε,γ ) ∈{(1,2, 3),(1, 3,2),(2,1, 3),(2, 3,1),(3,1,2),(3,2,1)}

)ˆ,(~1

)()( τως δδ z

)ˆ,(~2

)()( τως εε z

)ˆ,(~3

)()( τως γγ z

z~

25

ESPRIT Based Method for Estimating Residual Dopplers for 3 Channels Feeding into array with Delay-Doppler Taps: Step 2. Pairing number 1.

26

ESPRIT Based Method for Estimating Residual Dopplers for 3 Channels Feeding into array with Delay-Doppler Taps: Step 2. Pairing number 2.

27

ESPRIT Based Method for Estimating Residual Dopplers for 3 Channels Feeding into aray with Delay-Doppler Taps: Step 2. Pairing no. 3 (correct)

28

ESPRIT Based Method for Estimating Residual Dopplers for 3 Channels Feeding into array with Delay-Doppler Taps: Step 2. Pairing number 4.

29

ESPRIT Based Method for Estimating Residual Dopplers for 3 Channels Feeding into array with Delay-Doppler Taps: Step 2. Pairing number 5.

30

ESPRIT Based Method for Estimating Residual Dopplers for 3 Channels Feeding into array with Delay-Doppler Taps: Step 2. Pairing number 6.

31

Channel gain estimation is accomplished as a final step after estimation of individual Fk:

[ ] ααηα SPKPPP

K

kkk FsFsFsFsFr ==+⎥⎦

⎤⎢⎣

⎡= ∑

=

ˆˆˆˆ21

1

• Recall that array processor output:

• Transmission of a train symbol vector allows this equation to be written:

• The least-square solution for alpha:

( )[ ]rFFF SSS ′′= −1α

Ps

η+= sFr

32

Conclusions

• An MMSE-based OFDM receiver robust to large Doppler spread was developed

– No restriction on magnitude of Doppler– BER Performance approaches that of OFDM receiver for same channel

(less Doppler)

• A method for separately estimating channel gains, time delays and Doppler developed and demonstrated

– dimensionality and complexity lower than a joint estimation technique

• Current Activity:– Evaluating sensitivity of MMSE-based OFDM detection performance to

estimation errors– Evaluating sensitivity of channel estimates to noise– Investigating estimation and receiver performance with non-constant

channel coefficients associated with transmitter motion