reliable symbols: decision-directed learning with quality control - winlab · verticalband...

VERTICALBAND TECHNOLOGY

Reliable Symbols:Decision-Directed Learning with

Quality ControlEilon Riess and Stuart SchwartzVerticalband Ltd. and Princeton

University


Higher Data Rates

• Higher data rates can be achieved by:– Shorter symbol length - more bandwidth,

more ISI– MIMO– More bits per symbol (higher order

constellation)

• Problem with HOC -- increases the ISI


Proposed Solution

• Use HOC to increase the data rate in conjunction with ‘Reliable Symbols’ to do blind equalization of HOC

• ‘Reliable Symbols’ greatly reduce (maybe eliminate) the need for training samples

• Can then track dynamic channels


HOC and Equalization

Observation: There is no known blind equalizer algorithm that works well with HOC!

J. J. Werner et al, “Blind Equalization for Broadband Access”, IEEECommunication Magazine, pp. 87-93, April 1997

J. R. Treichler et al, “Practical Blind Demodulators for High-Order QAM Signals,” Proceeedings of the IEEE, vol. 86, No. 10, pp. 1907-1926, October 1998


ISI or Multipath ChannelsIn a typical digital communication system, the received signal is:

x --received signal

d --transmitted data point

w --additive white Gaussian noise (AWGN)

a -- channel coefficients, normalized to

“noise” due to the ISI (or multipath) given by the

summation term

∑ ++= −i

nininn wdadx

10 =a


HOC is an ISI Amplifier

222222 )()( ISIadaEnE didiniISI σσ === ∑ ∑−

Look at the summation term:


Std of the ISI noise as a function of the constellation sizeISI= 1

1

3

5

7

9

11

13

15

17

19

2 4 8 16 32Constellation size [PAM]

STD


HOC and ISI

QPSK

16 QAM

64 QAM

256 QAM


Verticalband Equalizer

QPSK 16 QAM

64 QAM256 QAM


Nature of the ISI and HOC

∑ −=i

iniISI dan

The ISI term

is well approximated by a Gaussian random variable


The Gaussian Approximation

0

0.02

0.04

0.06

0.08

0.1

0.12

-6 -4 -2 0 2 4 6

ISI length 3Gaussian

1-1


True Symbols‘True Symbol’: is a received data point that is still located within its correct decision region despite the effects of ISI and the AWGN

nnn dx υ+=2222wd ISI σσσ

υ+=

Received signal and ‘noise’ variance is:


True Symbols‘True Symbol’: is a received data point that is still located within its correct decision region despite the effects of ISI and the AWGN

nnn dx υ+=2222wd ISI σσσ

υ+=

Received signal and ‘noise” variance is:

{ } ))2/(1(11Pr υσυ erfP nRS =<<−=

Probability of having a TS is:


The Gaussian Approximation

0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1

0 .1 2

-6 -4 -2 0 2 4 6

IS I le n g th 3G a u ssia n

1-1


• Sometimes the interference is low and the received data point is located within its correct decision region. In this case the received symbol is actually a True Symbol.

Decision region for symbol

Noise effecting the transmitted symbolReceived data point

Problem: The receiver sees onlyxn = dn + noise

Question: Is it possible to determine if a received data point is actually a True Symbol

Answer: YES, with a high probability

Solution: Reliable Symbols

x


Reliable Symbols

‘Reliable Symbol’: is a received data point whose surrounding symbols have energies that are below a pre-defined threshold and are therefore unlikely to contribute a great deal of ISI.

RELIABLE SYMBOLS ARE , THEREFORE, HIGHLY LIKELY TO BE ‘TRUE SYMBOLS’.


Finding Reliable Symbols - 1∑ ++= −

inininn wdadx

∑∑∑ ≤≤= −− iiniiniISI addadan max

We can bound the ISI:

For initial ISI < 1, easy to define low energy signals


Finding Reliable Symbols - 2

KRS MP )/2(=

PAM constellation of order M,with data at +-1, +-3, +-5, ... Define a low energy signal as having magnitude = 1.

With K surrounding symbols (pre-cursors and post cursors) of low energy, probability of such a sequence is:


Finding Reliable Symbols - 3

Define a reliable symbol weight (RSW) which is monotonically decreasing with magnitude of hard decision on surrounding symbols:

−= ∑=

−

N

iinn dC

CRSW

1

^1

Choose C so that 0<RSW<1


Reliable Symbol Simulation - 1

8-PAM constellation

Normalized SNR = 16dB

20,000 received points in the simulation

ISI coefficients: (1, 0.4, 0.2, 0.1)

7,5,3,1 ±±±±=nd


Soft decision error vs. RS weights


Probability of correct decision vs. RS weights


Probability of a RS weight vs. RS weights


Simulation 1--Summary8-PAM constellation


20,000 received points in the simulation


Note: 1. With a RSW > 0.8, RS can be identified with Prob. =1.

2. One RS can be detected in about every

59 received signals


Reliable Symbol Simulation - 2

32-PAM constellation


Received signal points = 30,000


31,...,3,1 ±±±=nd


Soft decision error vs. RS weights


Probability of correct decision vs. RS weights


Probability of a RS weight vs. RS weights


Simulation 2 -- Summary32-PAM constellation

SNR = 16dB

Received signal points = 30,000


Note: 1. With a RSW >0.95, a RS can be identified with Prob. = 1.

2. For RSW > 0.95, Prob. 0.00035

One RS in about 3000 received signals

≈


Subtractive Equalizer

Rather than work with received data, we “rescatter” by subtracting our best estimate of the ISI:

ini

inn daxy −∑−=^^

A DFE without a linear feedforward filter


RSD- EqualizerDecision

Device

Subtractive

Equalizer

Channel EstimatorRSD

nx nd^

ny

^

ia


• Express the rescattered data point, yn, in terms of the Tx symbol, estimated Txsymbols, and the estimated channel coefficients:

Error in Tx Symbol

Rescattered Data Point

Estimated Channel Coefficient

Decision errorsDecision errorsChannel estimation errorsChannel estimation errors

Tx Symbols

Channel Coefficients

WGN

∑=

− ++=N

iGininn ndadx

1iii aaa ~ˆ +=iii ddd −=

^~

∑=

−−=N

iininn daxy

1

ˆˆ

∑ ∑= =

−− +−−=N

iG

N

iiniininn ndadady

1 1

~~


RSD in the Subtractive Equalizer

M is m a tc h E q u a l is e rL S E s t im a to r

nx

∑=

−⋅−′=N

iininn dayy

1

ˆ

ny

nd̂

Na 1}ˆ{

1}ˆ{ −− Nd

M e m o r y ( s iz e L )

HW ∆

M e m o r y ( S iz e L )

S in g le e le m e n t

V e c to r

M a t r ix

{ } 1ˆ −

− Nnd

{ }nd

t i o nI n i t i a l i s a

R S D

H a r d D e c is io n

S ig n a l in


LS Channel Estimator

is nxk, rows contain k surrounding signals

W is a weighting matrix

After about 10 reliable symbols:

^

nn dx −=∆

∆= − WDDWDa TT ˆ)ˆˆ( 1^

D̂


LS Channel Estimation(Single Batch Processing)

Constellation Channel Number of surrounding symbols

Sample size

RS min

RS max

aaverage ˆ aStd ˆ

8-PAM 1 2+1 750 8 18 0.301 0.099

0.0044 0.0082

8-PAM 1 2 750 50 57 0.307 0.089

0.019 0.029

8-PAM 1 2 200 7 18 0.3114 0.0969

0.0383 0.0479

16-PAM 1 2+1 6000 7 18 0.301 0.1007

0.0034 0.003

16-PAM 1 2 6000 82 114 0.287 0.0486

0.0314 0.0306

16-PAM 1 2 2000 20 38 0.27 0.069

0.05 0.052

32-PAM 1 2+1 45000 8 18 0.2892 0.0029

0.0811 0.04

32-PAM 1 2+1 30000 5 15 0.344 0.0028

0.07 0.085

8-PAM 2 2+3 20000 18 24 0.474 0.244

0.026 0.017

Simulation results summary table for channel, 1. 1, 0.3, 0.12. 1, 0.5, 0.25


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10000 20000 30000 40000 50000 60000

Symbols

Estim

ated

taps

Adaptive Batch Estimation and Subtractive Equalization - 1

32 PAM, ISI: 1, 0.6, 0.5Initial Estimate: 0.3, 0.25


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2000 4000 6000 8000 10000 12000

Symbols

Estim

ated

taps

Adaptive Batch Estimation and Subtractive Equalization - 2

8 PAM, ISI: 1, 0.6, 0.4, 0.3Initial Estimate: 0.4, 0.2, 0.1


RS and Adaptive Estimation and Equalization

Channel estimates and the multipath coefficients, when using 1024QAM and a SNR of 8dB,

025.1 aai =∑


Bit Error Rate vs. Normalized SNR(same ISI channel)

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

2 4 6 8 10 12Normalised SNR

Bit

erro

r rat

e

64QAM256QAM1024QAM


Channel Estimation

0

8

135.1 aa

ii =∑

=

Randomly selected 8 coefficient multipath channel with closed binary eye:


Constellation Samples required

to estimate channel

64QAM 3000 symbols

256 5000

1024 12000

4096 20000


Channel estimates of multipath coefficients, for 4096QAM and SNR of 12dB, (35.57 SNR)


The error distance when using 4096QAM and the SNR equals 12 dB


Bit error rate vs. normalized SNR, ISI=[0.4, -0.1, 0.2, 0.3, -0.05, -0.15, 0.15, 0.2, 0.1, 0.05]

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

0 2 4 6 8 10 12SNR

Bit

erro

r rat

e

64 QAM256 QAM1024 QAM


Dynamic Channels

Distance between transmitted and rescattered point for 16 PAM, SNR_Norm = 14dB, Doppler Spread = 125Hz (2G carrier – about 67km/h, 1G carrier = about 134km/h), RSW >

0.6, initial error – [0.05,0.07]

I


Distance between transmitted and rescattered point for of 32PAM, SNR_Norm = 14dB,Doppler Spread = 41Hz (2G carrier – about 22km/h,

1G carrier = about 44km/h), RSW > 0.75, initial error – [0.03,0.05]

Dynamic Channels


Tracking Comparison• Channel – 76Hz Max Doppler spread (Jakes Model) +

discontinuities every 2500 Symbols.• 16 PAM.

RSW = 0 RSW = 0.65


Application Areas

• Mobile Wireless• Fixed Wireless• Digital TV• DSL


Conclusions

• Reliable symbols – the “next best thing to training samples”

• Subtractive equalizer – accelerates the convergence process

-- Get RS with increasing frequency-- Can track dynamic channels

• Combine the two concepts for a blind equalizer that converges rapidly for HOC


Future Work• Coded systems• PSK modulation• Better channel estimators using RS• Study a variety of RSW• Relationship to turbo-equalization and other

iterative methods that compute and use posterior probabilities

• MIMO and QAM – are they a good fit• Follow up on good suggestions from colleagues in

the audience


Channel B:

Complex Channel


Received Signal before and after VB equalization (Channel B)


Dynamic Channels - 1• 64 QAM – VB Tracking (SNR=15dB, Doppler Spread =

50Hz, Symbol rate = 1.2288MSym/Sec)

Channel variation and VB tracking. Four real and four imaginary channel coefficients.

I


256 QAM – VB Tracking (SNR=20dB, Doppler = 50Hz, Symbol rate = 1.2288MSym/Sec)

Channel variation and VB tracking. Four real and four imaginary channel coefficients.

I Q

Dynamic Channels -2


Dynamic Channels - 3Variation SNR Constellation Order

ISI= 1.25

0.05/90 samples 10dB 64QAM

0.05/400 10dB 256

0.05/4000 10dB 1024

ISI=.65

0.05/400 14dB 1024

0a

0a


Effect of HOC

4QAM

16QAM

64QAM

256QAM

1024QAM

1

10

100

1000

1 10 100 1000 10000Constellation size [Log]

STD

[Log

]


A generic system with a RSD and channel estimator and decoder.

M E M O R Y

C A P T U R E DS A M P L EB U F F E R

D E M O D U L A T O R

D E C O D E DS Y M B O LB U F F E R

R E L I A B L ES Y M B O L

D E T E C T O R

S Y M B O LD E C O D E R

I S IE S T I M A T O R

x n x n

d̂ n

P R O G R A MI N S T R U C T I O N S

P R O C E S S O R

D a t a o u t


High Data Rates (ISI)MobilityMobility

Today’s voice-enabled networks are transitioning

to 2.5 and 3G data-enabled networks

Today’s voice-enabled networks are transitioning

to 2.5 and 3G data-enabled networks

Broadband AccessBroadband Access

LMDS, MMDS and other broadband access

networks are providing scalable alternatives to

incumbent networks

LMDS, MMDS and other broadband access

networks are providing scalable alternatives to

incumbent networks

Digital Television promises a world of High-Definition video

coupled with High-Fidelity audio

Digital Television promises a world of High-Definition video

coupled with High-Fidelity audio

Digital TelevisionDigital Television

Cable

TerrestrialBroadcast

Satellite

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