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The Impact of Channel EstimationErrors on Space-Time Block Codes

Presentation for Virginia TechSymposium on Wireless Personal

Communications

M. C. Valenti

D. A. Baker

Wireless Communications Research Lab

West Virginia University

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Benefits of Space-Time

Block Codes Space-time block coding utilizes multiple transmit

antennas to create spatial diversity.

This allows a system to have better performance in afading environment.

Benefits:

Good performance with minimal decoding complexity.

Can achieve maximum diversity gain equivalent tospace-time trellis codes.

Receivers that use only linear processing.

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Diagram of Block STC

Transmission

X1 X2

0 T 2T

X1 -X2*

X2 X1

*

0 T 2T

Ant 1

Ant 2

Data

STC encoder

Data STCencoder

Fadingi

AWGN n

STC decoderx rModulation

Encoder matrix:

*

1

*

2

21

2xx

xxG

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Wireless Channel Model:

Rayleigh Fading The channel between the ith transmit antenna and

the receive antenna undergoes flat-fading:

We assume quasi-static fading:

Quasi-staticmeans that the path gains from one transmit

antenna to the receive antenna is constant over a frame.

i i i i iX jY a j exp{ }

Rayleigh UniformGaussian

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Block STC decoder

Each symbol in a block is decoded separately by

minimizing the metric

The decoder outputs the hard-decisions on the

data.

The more TXs and RXs the system has, the

better performance the system can achieve.

2

1 1

l

t

n

i

i

tit cr

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Decoding Block STC

2

12

*

212

2

22111 xxrxxr

*

2

1

2

1

*

1

*

2

21

*

2

1

x

x

r

rr

2

1 1

l

t

n

i

i

tit cr

2

2

2

2

2

1

2

21

*

2

*

21

2

1

2

2

2

1

2

12

*

2

*

11 11 xxrrxxrr

Since |x1|=|x2| (PSK), we can get:2

12

*

2

*

11 xrr

2

21

*

2

*

21 xrr

The received signals are:

In order to minimize

it is equivalent to minimize

By using: *2 ccc

we have:

and

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Simulation of STBC

Channel fading coefficients were modeled as

samples of Gaussian random variables with

variance 0.5 per dimension.

The channel was assumed to be static over the

length of a frame, and varies from frame to frame.

Noise was modeled as Gaussian with zero meanand variance n/(2*SNR). Where n is the number of

transmit antennas.

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STBC With Channel

Estimation Errors The fading coefficient between the ith transmit

antenna and the receive antenna is given as

i i ia j exp

exp i i i ia j

exp i i i iK a j

A channel estimate with phase error is of the form

A channel estimate with gain error is of the form

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QPSK With Perfect CSI

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Received SNR

BER

uncoded QPSK

STBC using QPSK

2 TX antennas

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0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

10-2

10-1

Phase error in channel 1

SNR fixed at 10 dB


BER

Simulation Results:

Phase Errors @ Low SNR

The SNR at the receiver is

fixed at 10 dB.

This shows a rapid decline

in BER performance for

small errors in the phase

of either channel estimate.

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Simulation Results:

Phase Errors @ Medium SNR

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.810

-4

10-3

10-2

10-1


SNR fixed at 20 dB


BER

The signal to noise ratio(SNR) at the receiver is

fixed at 20dB Even with the increased

SNR a rapid decline inbit error rateperformance stilloccurs.

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0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.810

-5

10-4

10-3

10-2

10-1


SNR fixed at 25 dB

Phase error in chanel 2

BER

Simulation Results:

Phase Errors @ High SNR The signal to noise ratio

(SNR) at the receiver isnow fixed at 25dB

Increasing SNR onlyresults in a steepercurve as the

performance is quicklydegraded.

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0 5 10 15 20 2510

-5

10-4

10-3

10-2

10-1

10

Simulation Results: Average

Phase Error Per Channel

0

101

Received SNR

BER

avg. phase error/channel = 0.2 radavg. phase error/channel = 0.4 radavg. phase error/channel = 0.6 radavg. phase error/channel = 0.8 rad As the average phase

error in each channel

approaches 0.5

radians, the

performance is

completely degraded

even with increasing

values of SNR at the

receiver.

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0

0.5

1

1.5

0

0.5

1

1.5

10-2

10-1

Gain error in channel one

SNR fixed at 10dB

Gain error in channel two

BER

Simulation Results: Gain Errors

The SNR is fixed at10dB.

The curve has a valley-like shape.

This shows that if theerror in both channelestimates is roughlyequal, then only a small

performance penalty isincurred.

However, if the errors ineach estimate are verydifferent, performance

can suffer.

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Normalized Gain Error

Since the performance of the system is not adversely

affected by errors in the gain of the estimates if the estimates

are the same in each channel, the concept of normalized gainerror is introduced.

Normalized Gain Error =K

K

1

2

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Simulation Results: Normalized

Gain Error SNR fixed at 10dB.

SNR fixed at 20dB.

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

SNR fixed at 20 dB

Normalized Gain error K1/K2

BER

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Simulation Results: Normalized

Gain Error The performance loss

is negligible when the

normalized gain error

is unity.

When the difference

between the gain

errors in the two

channels is nearly

double the loss

approaches 7dB at a

BER of 10-3.

0 5 10 15 20 2510

-5

10-4

10-3

10-2

10-1

100

Received SNR

BER

normalized gain error = 0.067



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Simulation Results: Combined

Gain and Phase Errors The shape of the

curves remainsimilar to the curves

generated when onlyconsidering theerrors in the gain.

However, the curvesget flattened as theaverage phase error

in each channel isincreased.

The phase errors areobviously theprimary source ofperformance loss.10-2 10-1 100 101 102

10-4

10-3

10-2

10-1

100

101

SNR fixed at 20 dB

Normalized Gain error K1/K2

BER

phase error/channel = 0.1 radians




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Pilot Sequence Estimation

A pilot sequence is a series of symbols thatare known to the receiver in advance.

By comparing what was transmitted withwhat was received, the receiver can estimatethe effects of the channel.

However, since the AWGN noise samples atthe receiver are not known, the channelestimates will be imperfect, or noisy.

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STBC Estimation Scheme:

How It Works If we have only one receive antenna then the

received signal at time t can be expressed as

follows:

r HG KJ

HG KJHGKJHG

r

r

s

s

n

n

1

2

1 2

2 1

1

2

1

2ch ch

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How It Works The received signal can also be expressed using a

matrix of transmitted signals instead of a matrix of

channel gains as shown in the following:

r

r S n

HGKJ HG KJHGKJHG

r

r

s s

s s

n

n

1

2

1 2

2 1

1

2

1

2

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How It Works If the receiver knows the signals that were

transmitted then an estimate of the channel fades

can be derived from the received signal.

FHG IKJFHG IKJFHGIKJ

FHG IKJFHGIKJLNMM

FHG

IKJFHGIKJ

FHG

IJ

LN

1

1

1 0

0

1

2

2

2

1

2

2

21 2

2 1

1 2

2 1

1

2

1 2

2 1

1

2

1

2

2

2

1

2

2

2

1

2

2

2

1

2

1 1 2 2

2 1 1 2

s s

s s

s s

s s

s s

s s

s s

s s

n

n

s s

s s

s s

s n s n

s n s n

HS r

MM

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How It Works The channel estimate can now be shown.

HGKJ HG KJ

1

2

1 1

2 2

11 1 2 2

1

2

2

2 22 1 1 2

1

2

2

2

where

ands n s n

s s

s n s n

s s

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QPSK Using Pilot Sequence

Estimation

0 5 10 15 20 25 3010

-5

10-4

10-3

10-2

10-1

100

QPSK with running average estimation

Received SNR

BER

The equation from the previous

slides was used to implement a

pilot symbol estimation scheme.

The frame size for each examplewas 60 bits.

The channel was assumed to be

quasi-static, or constant fading

over a frame.r=1/2

r=2/3

r=3/4

r=4/5

perfect CSI

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Results of Pilot Estimation

Simulations The rate 1/2 and rate 2/3 schemes perform at

a loss of only 2dB as compared to the case of

perfect CSI.

The rate 3/4 and rate 4/5 schemes perform at

a loss of approximately 3 dB as compared to

the case of perfect CSI.

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Conclusions

and Future Work Conclusions:

Block space time codes are sensitive to channel estimation errors.

The impact of phase and amplitude errors were studied separately andjointly.

Pilot symbol techniques can be used to assist estimation.

Future Work: Other modulation types, such as QAM, FSK, and DPSK, will be tested.

Correlated fading between transmit and receive pairs and variable fadingrates should be taken into account.

Turbo principles can be used to facilitate the implementation of iterativechannel estimation and decoding techniques.

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