Error Calculation of Codes

7/29/2019 Error Calculation of Codes

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Space-Time Coding

Space-time coding (STC) systems make use of the MIMO channel gener-ated by a multiple-antenna transmit/receive setup:

+

+

+xNtr

x1r

x2r

h11

hNtNr

hij

yNrr

y2r

y1r

nNrr

n2r

n1rTransmit

AntennaArray

Receive Array

Each antenna transmits a DSB-SC signal:

yj(t) =

Nt

i=1

EsNt

hijxi(t) + ni(t)

and y(t) =

EsNt

Hx(t) + n(t),

where y = (y1, , yNr) and x = (x1, , xNt) (EE5520 Lecture Notes).Channel Gains: These are modeled as independent complex fadingcoefficients, i.e.,

p(h) =1

2exp

|h|22 ; E[hihj] = 0

which leads to a Rayleigh distributed amplitude

p(a = |h|) = a expa

2

2

p(a = |h|2) = 1

2aexp(a/2)


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EE 7950: Statistical Communication Theory 2

MIMO channels Transmit Diversity

The Alamouti scheme [1] uses two transmit antennas and one or tworeceive antennas:

+

[x1, x0]

[x0,x1]

h1

h0

[x0,x1][x0, x1]

[h0, h1]

Transmit

Antenna

Array

Combiner

Estimate

The transmitted 2 2 STC codeword is

X =

x0 x1x1 x

0

where the symbols xi can be any quadrature modulated symbols.

The received signal is now

r = [r0, r1] = [h0x0 + h1x1,h0x1 + h1x0] + [n0, n1]= [h0, h1]X + n

The demodulator calculates

[x0, x1] =

h0 h1h1 h0

r0r1

= [(|h0|2 + |h1|2)x0 + h0n0 + h1n1 n0

, (|h0|2 + |h1|2)x1 + h0n1 + h1n0 n1

]

If the channel path h0 and h1 are uncorrelated, the noise sources nihave twice the variance of the original noise sources. The system provides dual diversitydue to the factor (|h0|2+|h1|2) whichexhibits a -square distribution of fourth order.


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Multiple Receive Antennas

The Alamouti scheme can be extended to multiple receive antennas. Inthis case

R=

r0 r1r2 r3

=

h0 h1h2 h3

x0 x1x1 x

0

+N

Multiplying the received signalsRwith the channel estimateHwe obtain

[x0, x1] = h0 h1

h1 h0 r0

r1 + h2 h3h3 h2

r2

r3= (|h0|2 + |h1|2 + |h2|2 + |h3|2)x0 + h0n0 + h1n1 + h2n2 + h3n3

n0

,

+ (|h0|2 + |h1|2 + |h2|2 + |h3|2)x1 + h0n1 + h1n0 + h2n2 + h3n3 n1

]

Note that this system provides 4-fold diversity as expressed by theamplitude factor

A = (|h0|2 + |h1|2 + |h2|2 + |h3|2)This is possible due to the fact that the transmitted rows ofXare orthog-onal. In this sense the Alamouti scheme is the most basic representativeof what are known as orthogonal designs.

Example: The following is an example of a 4 4 orthogonal design

X= x1 x2 x3 x4

x2 x1 x4 x3x3 x4 x1 x2x4 x3 x2 x1

The 4 rows ofX are orthogonal for any real xi.


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Space-Time Codes

A space-time codeword is an Nt Nr matrix (array) of complex signalpoints, which is transmitted at time t:

X(t) = [x1(t), ,xNc(t)] = x11 x12 x1Nc... ...

xNt1 xNt2 xNtNc

Each row ofX is a space-time symbol, and the received STC word is given

by

Y(t) =

EsNtH(t)X(t) + N(t)

where N is a matrix of complex noise samples with variance N0, that isvariance 2 = N0/2 in each of the two dimensions.The signal energy per space-time symbol is

Es, which means that the

signal energy per constellation point is

EsNt

, and the each constellation

point xij is energy normalized to unity.

Error Calculation (Gaussian)

At any given time-instant, the channel is fixed and the impairment isGaussian noise. The probability of error between two space-time codewords, and the conditional error probability depends only on the squaredEuclidean distance between code words

P(XX) = QEsNt d2

(X

,X

)2N0

and d2(X,X) is the squared Euclidean distancebetween these two points:

d2(X,X) =EsNt

Ncn=1

Nrj=1

Nti=1

hij(xin xin)2


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Squared Euclidean Distance

We proceed to express d2(X,X) in terms of the linear algebraic proper-ties ofH,X and X.

d2(X,X) =EsNt

Nrj=1

Nti=1

Nti=1

hijhij

Ncn=1

(xin xin)(xin xin) Kii

The matrixK is a kernel matrix with entries Kii. Define hj = [h1j,

, hNtj]T

as the signature vector of receive antenna j. Then d2(X,X) can be ex-pressed as a quadratic form:

d2(X,X) =EsNt

Nrj=1

hjKhj

Since K is hermitian (K= K+), we can spectrally decompose d2(X,X)

d2(X,X) = EsNt

Nrj=1

hjV DV+hj

=EsNt

Nrj=1

vjDvj

The components of these equations are:

V is a unitary matrix

D is a diagonal matrix with the eigenvalues ofK. It can be shownthat all these eigenvalues are nonnegative real

h is a vector of complex Gaussian gains v = V+h is a vector of rotated complex channel gains


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STC Error Probability

The squared Euclidean distance can now be expressed in terms of theeigenvalues of the STC matrix K

d2(X,X) =EsNt

Nrj=1

Nti=1

di|vij|2

and, given a fixed channel H

P(XX) = QEsNrj=1Nti=1 di|vij|2

2N0Nt

Often one works with the Chernoff Bound on the error probability, whichis easier to manipulate:

P(XX) exp Es4N0NtNrj=1

Nti=1

di|vij|2

=

Nrj=1

exp

Es4N0Nt

Nti=1

di|vij|2 Dj

Fading channels If independent fading is assumed then

h is a vector of independent complex Gaussian random variables v = V+h is also a vector of independent complex Gaussian random

variables, because

Evv+

= V+E

hh+

V= I


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Fading Error Analysis

The components vij are unit-variance complex Gaussian random vari-ables, and their absolute value squared is therefore -square distributedwith two degrees of freedom:

p(a = |v|2) = exp(a); p(da = d|v|2) = 1d

exp(a/d)

d2(X,X) is now the weighted sum of -square random variables. The

PDF of the sum of independent random variables is best evaluated viathe characteristic function:

() =

fX(x)exp(jx) dx; d|v|2() =1

1 jdiWith this we have

Dj() =

Nt

i=11

1

jdi

= d2() =Nr

j=1Nt

i=11

1

j EsdiNt

Probability Density Function (PDF) The PDF is found via a partialfraction expansion and a backtransform of the individual terms:

p(x = d2) =

Nti=1

pi

Nrj=1

xm1

dmi (m 1)exp

x

di

This PDF can be integrated in closed form to

Pe =1

2

Nti=1

piNrj=1

1 11 + 4N0NtEsdi

m1n=0

2n

22nn21

1+Esdi4N0Nt

n


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Chernoff Error Bounds

The application of the Chernoff error bounding technique avoids many ofthese algebraic technicalities. The Chernoff bound is a calculated fromthe characteristic function as:

P(XX) E

exp

Es

4N0Nt

Nrj=1

Nti=1

di|vij|2

= d2()

j= Es

4N0Nt

=

Nrj=1

Nti=1

11 + Esdi

4N0Nt

The final form of the bound is:

P(X X)

1

Nti=1(1 +

Esdi4N0Nt

)

Nr

Nti=1

diNr Es

4N0Nt

NrNt

Design Criteria:Maximum diversity of NtNr can be achieved the following criteria areoptimized.

The Rank Criterion: To achieve maximum performance the ma-

trix K has to have full rank.

The Determinant Criterion: The product Nti=1 di = det(K)needs to be maximized to give maximum coding advantage.


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Rank and Rate

Rate: Each of the signals xij is drawn from a signal constellation A with2b signalpoints. We define two rates, the symbol rate Rs and the bitrate Rb:

Rs =k

Nc; Rb =

kb

Nc

Now consider the space-time codeword

X(t) = [x1(t), ,xNc(t)] = x11 x12 x1Nc... ...

xNt1 xNt2 xNtNc

Consider the rows ofX as the symbols Xi in a code of lenght Nt. Thena rank-r matrix X corresponds to a Hamming weight-r codeword in thiscode. The maximum rate is therefore

Rb,max =log(A(Nt, dH))

Nt

where A(Nt, dH) is the number of codewords in the code of length Nt with

minimum Hamming distance dH.If we want full-rank codes we require dH = Nt, and

Rb,max =log(A(Nt, Nt))

Nt= b

The maximum symbol rate of a full rank code is therefore

Rs,max = 1


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Code Construction

Following these criteria Tarokh et. al. [2] have constructed trellis codeswhich provide maximal diversity. Two examples are shown below:

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

22 23 20 21

32 33 30 31

02 03 00 01

12 13 10 11

These codes achieve full diversity on two-antenna systems. The transmis-sion rate is 2 bits per symbol using two QPSK signals over two antennas,because there are four choices at each state.

Decoding:Decoding follows the trellis using a sequence metric calculator Viterbi.

Branch Metrics:The Viterbi algorithm works by using metrics m(r) along their branchesand accumulates them to find the global minimum, where

m(r) =

Nrj=1

yjr Nti=1

hijxir

2

= need channel estimate|

The metric is simply the squared Euclidean distance between hypothesisand received signal.


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Error Performance of STCs

4 states

8states

16states

32states

64 states

4 5 6 7 8 9 10 11 12 13 14104

103

102

101

1

Frame Error Probability (BER)

SNR

Performance results copied from [2].


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Orthogonal Designs

Example: The following is an example of a 4 4 orthogonal design

X=

x1 x2 x3 x4x2 x1 x4 x3x3 x4 x1 x2x4 x3 x2 x1

Properties:

Orthogonal designs provide full diversity. This condition is equiv-alent to requiring that XX is non -singular for any X = X.Proof: The determinant ofX is

det(X) =

det(XXT)

=

det diag

i

x2i , ,i

x2i

=

i

x2i

Nt/2

and therefore

det(XX) =

i

|xi xi|2Nt/2

= 0

Therefore the maximum diversity NtNr is achieved with orthogonaldesigns.

Real orthogonal designs exist only for n = 2, 4 and n = 8. There exist very simple maximum-likelihood decoding rules (see home-

work).

Application of real orthogonal designs are in single-sideband (SSB)modulated signals.

Complex Orthogonal DesignsThere exist no complex orthogonal designs for n > 2.


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Generalized Orthogonal Designs

The generalized designs relax the tight conditions of the orthogonal de-signs.

+

+

[x1, , xk]x1 x2 x3 x4x2 x1 x4 x3

x3 x4 x1 x2

k real signals are packed into an array X of size Nt Nc; Nt Nc. Thesymbol rate of this transmission system is

R = k/Nc [real symbols/use]

A generalized orthogonal design has XXT = I, i.e., all rows are or-thogonal. The design goal is to minimize Nc for a given R and Nt.

Full rate (R = 1) real orthogonal designs exist for n 8.Theory: Using complex number theory the Hurwitz-Radon problem it is known that [2]

For any rate R 1 real generalized designs exist For any rate R 0.5 complex generalized designs exist

Example:: A rate R = 0.5 complex design is

X=

x1 x2 x3 x4 x1 x2 x3 x4x2 x1 x4 x3 x2 x1 x4 x3x3 x4 x1 x2 x3 x4 x1 x2


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References

[1] S.M. Alamouti, A simple transmit diversity technique for wirelesscommunications, IEEE J. Select. Areas Commun., Vol. 16, No. 8,October 1998.

[2] V. Tarokh, N. Seshadri, and A.R. Calderbank, Space-time codingfor high data rate wireless communication: Performance criterionand code construction, IEEE Trans. Inform. Theory, pp. 744-765,Mar. 1998.

[3] V. Tarokh, H. Jafarkhani, and A.R. Calderbank, Space-time blockcodes from orthogonal designs, IEEE Trans. Inform. Theory, vol.45, no. 5, July 1999.

[4] C. Schlegel, Error probability calculation for multibeam Rayleighchannels, IEEE Trans. Commun., Vol. 44, No. 3, March 1996.

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