Download Error probability of digital signaling

Error probability of digital signaling
15, Jan. 2001
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CONTENTS
■
Error probability of digital signaling
Pairwise error probability
Bounds on error probability
Bit and symbol error probabilities
■
Error probability of digital signaling for an AWGN
channel
BPSK, M-PSK, M-PAM, M-QAM...
■
Error probability of digital signaling in the fading
channel
error probability in a flat fading channel
error probability over frequency selective fading channel
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Vector space representation of received signals
■
In a flat fading channel with AWGN, the received complex envelope
can be defined as:
~
r (t ) = g (t )~
si (t ) + n~ (t )
If fmT<<1
~
r (t ) = g~
si (t ) + n~ (t )
~
Si (t) is M complex low-pass waveforms
~
{ S k (t)} (k: from 0 to M-1)
g(t): a complex Gaussian random variable.
n~ (t ) : zero mean complex AWGN with a
power spectral density (psd) of N0 watts/Hz.
The vector can be expressed as:
~
~
~
r = gs i + n
r~ = (~
r0 , ~
r1 ,..., r~N −1 )
~
si = ( ~
si 0 , ~
si 1 ,..., ~
si N −1 )
n~ = (n~ , n~ ,..., n~ ).
0
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1
N −1
Probability of error
■
■
{ }
the set M signal vectors S~
m
The decision regions R = {~r :
m
■
P (e ~
sm ) = 1 − ∫ p ( ~
r g~
s m ) d~
r
M −1
m =0
~
r − g~
sm
2
2
≤ ~
r − g~
smˆ , ∀mˆ ≠ m
1
P ( e) =
M
Rm
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M −1
~
P
(
e
S
∑
m)
m =0
}
Upper bounds and lower bounds
on error probability
■
Upper bound can be obtained by computing the
minimum squared Euclidean distance between any
2
2
two-signal points d~min
= min ~
sn − ~
sm
n,m
the pair wise error probability
,
~2
α 2 d min
P (e) ≤ ( M − 1)Q (
)
4N 0
~2
α d min
~ ~
)
P(S j , S k ) ≤ Q(
4N 0
2
■
lower bound on the error probability
~2
α d min
2
P ( e) ≥
Q(
)
M
4N 0
2
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Pairwise error probability
It can be defined for each pair of signal vectors in the signal
constellation
decision
boundary
~
gS j
the squared Euclidean distance
~
gS k
~2
~ ~
d jk = S i − S k
2
So the pairwise error probability
between the message vectors
2 ~2
d jk
α
~ ~
~
~
~ and ~
P
S
S
P
e
S
P
e
S
Q
(
,
)
=
(
)
=
(
)
=
(
)
j
k
j
k
Sj
Sk
4N 0
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Bit and symbol error probabilities
The symbol error probability is PM and
the bit error probability is Pb
■
PM
≤ Pb ≤ PM
log 2 M
■
Gray codes :symbol errors correspond to single bit errors
where k= log2M.
PM
Pb ≈
k
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Error probability of BPSK
■
two signal waveforms are s1(t)=g(t) and s2(t)= -g(t).
■
s1 = Eb , s 2 = − Eb
When noise n is present, The received signal from the
demodulator is r = s1+n
p ( r s1 ) =
0
1
−( r −
e
πN 0
P (e s1 ) = ∫ p ( r s1 ) dr = Q (
−∞
P (e ) =
Eb ) 2 / N 0
_ AND _ p ( r s2 ) =
1
−( r +
e
πN 0
Eb ) 2 / N 0
2 Eb
)
N0
2 Eb
1
1
P(e s1 ) + P(e s 2 ) = Q(
)
2
2
N0
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Fig. Conditional pdfs
of two signals
Error probability of M-PSK
The probability of M-ary symbol error, is just the probability that
the received angle outsider the region [-´/M, ´ /M]
■
PM (γ s ) = 1 − ∫
π /M
−π / M
p(θ )dθ
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Error probability of M-PAM
■
the Gray coded 8PAM system signal constellation
 2α 2 E h
Pi = 2 Q 
 N0




 2α 2 E h
Po = Q
 N0



M −2
2
1  2α 2 E h
Pm =
Pi +
P0 = 2(1 − )Q
M
M
M 
N0
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



Error probability of M-QAM
■
■
Rectangular QAM signal constellations
in practice
two -PAM systems in quadrature
most frequently used
e.g.16-QAM system can be treated as two independent Gray coded 4PAM systems in quadrature, each operating with half the power of the
16-QAM system
The symbol error probability
for each-PAM system is

1
6 γs
P M (γ s ) = 2(1 −
)Q
M  M −1 2




the probability of error symbol
reception in the M-QAM system
is
PM (γ s ) = 1 − (1 − P
2
)
M
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Error probability in a flat fading channel
(1/2)
■
When the channel experiences fading, the error probability must
be averaged over the fading statistics.
■
The bit error probability of a fading channel is given by
∞
Pb = ∫ Pe (γ ) p(γ )dγ
0
and the average symbol error probability is the symbol error
∞
probability over the p (γ )
PM = ∫ PM (γ ) p (γ )dγ
0
■
If the channel is Rayleigh faded. p (γ ) = 1 exp( − γ )
Eb
2σ 2 Eb
2
Ea =
γ0 =
,
N0
N0
[ ]
γ0
γ0
is the mean value of the SNR.
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Error probability in a flat fading channel
(2/2)
■
Rayleigh fading converts an exponential dependency of the
bit error probability on the bit energy-to-noise ratio into an
inverse linear one.
Bit error probability for
BPSK and QPSK on an
AWGN channel and a
Rayleigh fading channel
with AWGN
Bit error probability for M-QAM on
an AWGN channel and a Rayleigh
fading channel with AWGN
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Doppler spread impacts
For OFDM, if the channel is time-varying, inter-channel interference
(ICI) is introduced due to a loss of
sub-channel orthogonality
■ At low ~ , additive noise dominates the
γb
performance so that the extra noise due to
ICI has little effect. However, ICI dominates
the performance and causes an error floor
at large .
■
γ~b
Figure: BER for 16-QAM OFDM on a
Rayleigh fading channel with various Doppler
frequencies
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Error probability over frequency selective
fading channel
■
The frequency selective channel will lead to ISI in the receiver,
which will cause signal distortion. The result is irreducible bit
error probability floor, which means increasing the signal
energy does not have any effect on the bit error probability.
■
This irreducible error floor can also due to the random FM
caused by the time-varying Doppler spread, which in turn is due
to the motion of the mobile.
■
The computer simulations are the main tool used for analyzing
frequency selective fading effects
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Reference
[1] Gordon L. Stuber, "Principles of Mobile
Communication (Second Edition)", Kluwer Academic
Publishers
[2] John G. Proakis, “Digital communications (Third
Edition)”, by McGraw-Hill, Inc
[3] L.Ahlin & J. Zander,
Principle of Wireless
Communications , Studentlitteratur, 1998
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