PK_MTP_PPT

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Overview Introduction ABER Analysis of MPSK -MRC Results Conclusion Future Work References
Performance of MRC Receivers in κ−µ Fading
Channels with Channel Estimation Error
Presented by:
Pawan Kumar
Roll No.: 11410248
Supervisor:
Dr. P. R. Sahu
Department of Electronics and Electrical Engineering
Indian Institute of Technology Guwahati
Guwahati, Assam

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Overview
Introduction
Fading Channels
Modeling of Fading Channels
Diversity Combining Receivers
ABER Analysis of MPSK-MRC
System Model and Channel Estimation
MGF of Effective Output SNR
Error Probability Analysis using Half-Plane Decision Method
Numerical Results
Conclusion
Future Work
References

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Introduction
Fading Channels
r = dc + n,
where d is transmitted symbol, c = αe−jθ is complex channel gain, r
is received symbol and n is AWGN
Y-axis
X-axis
Phase, θ
Amplitude, α
Centre=(px , py)
m=1 (Rayleigh/Rician)
Figure 1: Cluster of scattered components

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Effects of Fading
X-axis
Y-axis
X-axis
Y-axis
Figure 2: Phaser diagrams of transmitted symbol and phase introduced by
fading channel

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Receiver
1 Coherent receiver
Needs phase synchronization
Phase recovery techniques can be used to estimate the phase
2 Non-coherent receiver
No need of phase synchronization

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Coherent Receiver’s Performance over Fading Channels
In presence of fading, phase have different characteristics
In the presence of deep fade standard phase estimation
techniques may loose lock
It needs to use some special techniques to estimate the phase
Alternative methods
Pilot Symbol Assisted Modulation (PSAM)[1] has been attracting
researchers’ attention due to its straight forward application
P number of pilot symbols are inserted periodically into N
number of data symbols

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Modeling of Fading Channels
1 Homogeneous
Rayleigh (no line-of-sight component)
Nakagami-n (Ricean)(line-of-sight component)
Nakagami-q (Hoyt)
Nakagami-m
2 Non-homogeneous
κ-µ (line-of-sight components)
η-µ (for no line-of-sight component)
Non-homogeneous channel models can characterize
homogeneous channel models as special cases

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Table 1: κ − µ distribution related to other distributions [2]
Parameters Distribution
κ µ κ − µ
κ → 0 µ = m Nakagami-m
κ → 0 µ = m = 1 Rayleigh
κ → 0 µ = m = 0.5 One sided Gaussian
κ = K µ = 1 Rice
Table 2: η − µ distribution related to other distributions [2]
Parameters Distribution
η µ η − µ
format 1 format 2
η → 1 η → 0 µ = m/2 Nakagami-m
η → 1 η → 0 m = 1 Rayleigh
η → 1 η → 0 m = 0.5 One sided Gaussian
η = q2 1−η
1+η = q2 µ = 0.5 Hoyt (Nakagami-q)

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κ − µ model [2]
|c|2
=
n
i=1
(Xi + pi)2
+
n
i=1
(Yi + qi)2
(Xi, Yi) ∼ N(0, σ2), and pi, qi are mean

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Diversity Combining Receivers
ObstaclesTransmitting end
Receiving end
τ1
τ2
τ3
τ4
1
Figure 3: Reception using single Receiver

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ObstaclesTransmitting end
Receiving end
τ1
τ2
τ3
τ4
1
2
Figure 4: Reception using dual-Receiver

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Basically, there are three types of principle (pure) combining
techniques:
1 Maximal Ratio Combining (MRC)
2 Equal Gain Combining (EGC)
3 Selection Combining (SC)
Output
w1
Sum
and
Detection
w2
wL
Lreceivingbranches
Figure 5: Maximal Ratio Combining Receiver.

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MRC
needs knowledge of channel’s amplitude and phase
most complex among all combining receivers
better performance than EGC and SC even with imperfect
channel estimates
output SNR is maximum when wl = cl
EGC
needs knowledge of channel’s phase only
lower complexity and performance than MRC
SC
needs no knowledge of channel parameters
lower complexity but poorer performance

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Error Rate Analysis
pdf-based approach
MGF based approach
MGF based approach is easier to analyze for non-iid branches [3]

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Motivation
MRC
possess better performance than EGC and SC
MGF based approach
easier to analyze and has advantage over pdf-based approach for
non-iid branches
κ − µ fading channels
analysis with ICE has been done for homogeneous channels
it characterizes non-homogeneous channel
suitable to model fading channels with LOS components
There are several works in literature related to Rayleigh, Rician
and Nakagami-m fading channels with ICE
Half plane decision method
provides an easier way to analyze the error rate of MPSK
modulations

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System Model
For transmitted symbol d(i), {d(i) ∈ e(j2πn/M), n = 0, 1, ..., M − 1},
over slow and ﬂat fading channel the received signal over L branch
MRC receiver in the ith symbol interval
r(i) = c(i)d(i) + n(i),
where
n(i) = [nl(i), ..., nL(i)]T
is zero-mean complex Gaussian vector
for l = 1, 2, ..., L.
c(i) = [cl(i), ..., cL(i)]T
is the channel-coefﬁcient vector for L
branches and its each element is κ − µ distributed.

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Channel Estimation
Assuming MMSE channel estimation, a model for channel estimation
error was proposed in [5] as
cf,l(i) = ρlˆcf,l(i) + zf,l(i),
where
cl = cf,l + pl and f for diffuse component
the correlation coefﬁcient between c(i) and ˆc(i) is
ρl = |ρl|ej∆θl = ρl,C + jρl,S, where ∆θl = tan−1 (ρl,S/ρl,C)
denotes the phase offset (or mismatch) of ρl [5].
Due to ICE the cases |ρl| < 1 and (or) ∆θl = 0 may arise which
degrades the performance of receiver.

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Effective Output SNR
Using the estimated channel vector ˆc(i) the complex decision
variable (DV) to detect the transmitted symbol d(i), with MRC
receiver, is given as
˜D = ˆcH
(i)r(i) =
L
l=1
ˆc∗
l (i)rl(i)
The symbol is estimated as ˆd(i) = e−j2πn/M, where ˆn = arg maxn
ℜ(˜De−j2πn/M) [5].

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To facilitate a half-plane decision method the complex DV ˜D
will be rotated with a plane angle β = ±(π/2 + π/M) to obtain
a new DV as [4]
D(β) = ℜ ˜De−jβ
=
L
l=1
D(l)(β)
where, D(l)(β) = ℜ(ˆc∗
l (i)rl(i)e−jβ) is DV element at each
branch, and it can be rewritten as
D(l)(β) = ℜ{ˆc∗
l (i)[cl d(i) + nl(i)]e−jβ
}

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The effective SNR at the output of MRC receiver can be given as
[4]
γMRC
ICE =
|ρl|2 cos2 (∆θl − β) (log2 M) L
l=1 |ˆcl|2
[(1 − |ρl|2) (log2 M)(2nσ2
c + p2) + N0]
= B(β)
L
l=1
ˆγl
where
B(β) =
|ρl|2
cos2
(∆θl−β)(log2 M)
[(1−|ρl|2)¯γl(log2 M)+1] ,
ˆγl = |ˆcl|2
/N0, and ¯γl = (2nσ2
c + p2
)/N0 with p2
= p2
i + q2
i and
¯γl = ¯ˆγl,

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The instantaneous SNRs ˆγ and γ, at each branch, have identical
statistics for Rayleigh fading and for general fading channels with ρ
close to one they may be assumed to have identical statistics [4].
In same way, we assume here that ˆµ = µ and ˆκ = κ, where ˆκ and ˆµ
are the fading parameters of the estimated channel coefﬁcient ˆc.

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MGF of the Effective Output SNR
The MGF for effective SNR γMRC
ICE can be obtained as [4]
MγMRC
ICE
(s) = E exp −sγMRC
ICE
= E exp −sB(β)
L
l=1
ˆγl
=
L
l=1
Mˆγl
(sB(β))
where Mˆγl
(s) = E exp −s¯ˆγl is the MGF of ˆγl.

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The MGF Mˆγl
(s) can be given as [7]
Mˆγl
(s) =
1
exp (µκ)
∞
n=0
(µκ)n
n!Γ(n + µ)
G1,1
1,1
µ(1 + κ)
s¯ˆγl
1
n+µ
where Gm,n
p,q [·] is the Meijer’s G-function [6]. For m = n = p = q = 1
it can be converted into an elementary form as [6]
Gm,n
p,q z|
a
b
= Γ(1 − a + b)zb
(1 + z)a−b−1
which can be used to obtain MGF MγMRC
ICE
(s) as
MγMRC
ICE
(s) =
1
exp (Lµκ)
∞
n=0
(Lµκ)n
n!Γ(n + Lµ)
G1,1
1,1
µ(1 + κ)
sB(β)¯ˆγl
1
n + Lµ

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Error Probability Analysis using Half-Plane Decision Method
For transmitted phase φ = 0, or n = 0, error occurs if this DV
becomes less than zero, i.e., when it falls in the left half-plane
Pb,MP = Pr{D(β) < 0|d(i) = 1} = FD(0|β)
For Gray-coded MPSK the BEP can be calculated using [4, 5]
Pb,MP ≃
1
log2 M
β=±(π/2−π/M)
FD(0|β)

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Evaluation of FD(0|β)
Using [7]
FD(0|β) =
1
2
√
π exp (Lµκ)
∞
n=0
(Lµκ)n
n!Γ(n + Lµ)
G1,2
2,2
µ(1 + κ)
B(β)¯ˆγl
1/2, 1
n + Lµ, 0
The equation [8, Equation 2] is used for ABER analysis of different
M-levels
Pb =
1
m
M−1
k=1
¯d(k)Pk
m = log2 M,
¯d(k) is the average distance spectrum and can be calculated using [8]
for Gray-coded symbol, Si (i = 0, 1, ..., M − 1).

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1 ABER evaluation for BPSK
{¯d(k)}k=1 = 1
β = 0
Pb,BP = Ps,BP = P1 = FD(0|β)
Figure 6: Constellation space for BPSK

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2 ABER evaluation for QPSK
β = ±π/4
{¯d(k)}3
k=1 = {1, 2, 1}
Figure 7: Constellation space for QPSK

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Figure 8: Error regions for QPSK
Pb,QP =
1
2
(P1 + 2P2 + P3)
=
1
2
(FD(0|π/4) + FD(0| − π/4))

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3 ABER evaluation for 8-PSK
β = ±3π/8
{¯d(k)}7
k=1 = {1, 2, 2, 2, 2, 2, 1}
Figure 9: Decision regions for 8-PSK

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The covered error region is R1 + R2 + R3 + 2R4 + R5 + R6 + R7 with
probability P1 + P2 + P3 + 2P4 + P5 + P6 + P7.
The ABER can be obtained using [4, 5]
Pb,8P =
1
3
{FD(0|3π/8) + FD(0| − 3π/8)
+FD(0| − 5π/8)FD(0| − π/8)
+FD(0|5π/8)FD(0|π/8)}

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4 ABER evaluation for 16-PSK
The rotating plane angles β = ±7π/16
the respective CDFs are FD(0|7π/16) and FD(0| − 7π/16) with covered
error regions
(R8 + R9 + R10 + R11 + R12 + R13 + R14 + R15) and
(R1 + R2 + R3 + R4 + R5 + R6 + R7 + R8) respectively.
{¯d(k)}15
k=1 = {1, 2, 2, 2, 2.5, 3, 2.5, 2, 2.5, 3, 2.5, 2, 2, 2, 1}

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Pb,16P =
1
4
{FD(0|7π/16) + FD(0| − 7π/16)
+FD(0|9π/16)FD(0|π/16)
+FD(0| − 9π/16)FD(0| − π/16)
+FD(0|17π/16)FD(0|3π/16)
+FD(0| − 17π/16)FD(0| − 3π/16)}
It is (2P5 + 2P6) and (P5 + 2P6 + P7) less than the results in [9] and
[8], respectively

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Results I
0 2 4 6 8 10 12 14 16 18
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Average SNR per bit (dB)
ABER,P
b
increasing ∆θ
Dashed lines: ∆θ=0
Solid lines : ∆θ=π/20
µ=1, κ=0
(Rayleigh)
8−PSK
BPSK
QPSK
Figure 10: ABER versus average SNR per branch per bit for BPSK, QPSK
and 8-PSK with L = 2 for µ = 1, κ = 0 (Rayleigh fading channel) and
|ρ| = 1, ∆θ = {0, π/20} rad.
.

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Results II
0 2 4 6 8 10 12 14 16 18 20
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
ABER,P
b
π/10
8PSK
π/20
BPSK
QPSK
Figure 11: ABER versus average SNR per branch per bit for BPSK, QPSK
and 8-PSK with L = 2 for µ = 2, κ = 3 dB and |ρ| = 0.995,
∆θ = {π/20, π/10} rad.
.

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Results III
0 5 10 15 20 25 30 35 40
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
ABER,Pb
π/8
|ρ|=0.999,
∆θ={0, π/32,π/16, π/8}
Increasing ∆θ
|ρ|=1
∆θ=0
µ=1, κ=0
(Rayleigh)
Figure 12: ABER versus average SNR per branch per bit for 8-PSK with
L = 3 for µ = 1, κ = 0 (Rayleigh fading channel) and |ρ| = {1, 0.999},
∆θ = {0, π/32, π/16, π/8} rad.
.

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Results IV
0 5 10 15 20 25 30 35 40
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
ABER,Pb
increasing ∆θ
|ρ|=0.999
∆θ={0, π/48, π/32, π/16}
∆θ=π/16
|ρ|=1
∆θ=0
Figure 13: ABER versus average SNR per branch per bit for 16-PSK with
L = 4 for µ = 1.25, κ = 5 dB and |ρ| = {1, 0.999},
{∆θ = 0, π/48, π/32, π/16} rad
.

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Conclusion
The ABER performance of M-PSK modulation is analyzed with
MRC receiver over κ − µ fading channel with imperfect channel
estimates. The useful half-plane decision method is used for
ABER evaluation for different M-levels.
The numerical results show that for correlation coefﬁcient
having magnitude less than unity (|ρ| < 1) at high SNRs, there
would be irreducible error ﬂoors and thus the diversity order
approaches zero as ¯ˆγ → ∞.
Due to phase mismatch the ABER performance of M-PSK
degrades very rapidly as ∆θ increases.
The effect of phase mismatch (∆θ) become more adverse with
increase in M-level.

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Future Work
ABER analysis of MQAM - MRC
over κ − µ fading channels with ICE.
Analysis over correlated κ − µ fading channels with ICE.
Effect of ICE on the performance of Diversity receivers over
η − µ fading channels.
Effect of ICE on the performance of Diversity receivers over
κ − µ and η − µ fading channels with co-channel interference.

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References I
[1] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers New
York: Plenum Press, 1997.
[2] M.D. Yacoub, "The κ − µ distribution and the η − µ distributuion," IEEE Antennas
Propag. Mag., vol. 49, no. 1, pp. 68-81, Feb. 2007.
[3] M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 4th
edition Wiley, 2005.
[4] Y. Ma, R. Schober and S. Pasupathy, "Performance of M-PSK with GSC and EGC with
Gaussian weighting errors," IEEE Trans. Vehicular Tech., vol. 54, pp. 149-162, Jan. 2005.
[5] Y. Ma, R Schober and S Pasupathy, "Effect of channel estimation error on MRC Diversity
in Rician Fading Channels," IEEE Trans. Vehicular Tech., vol. 54,pp. 2137-2142,
Nov. 2005.
[6] http://www.wolframalpha.com/.
[7] D. B da Costa and M. D. Yacoub, "Moment Generating Functions of generalized fading
distributions and applications," IEEE Commun. Lett., vol. 10, no. 5, pp. 353-355, May
2008.

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References II
[8] J. Lassing, E.G. Strom, E. Agrell and T. Ottosson, "Computation of the exact bit-error rate
of coherent M-ary PSK with Gray code bit mapping," IEEE Trans. Commun., vol. 51, pp.
1758-1760, Nov. 2003.
[9] P. J. Lee, "Computation of the bit error rate of coherent M-ary PSK with Gray code bit
mapping," IEEE Trans. Commun., vol. COM-34, no. 5, pp. 488-491, May 1986.

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