1. A STUDY ON THE STATISTICAL MODELING OF FADING AND ITS EFFECTS ON
SYSTEM PERFORMANCE USING SIRP AND SDP METHODS
Cheng-An Yang*, Kung Yao* and Ezio Biglieri*†
Unified Fading Model Performance Evaluation Results
Spherically Invariant Process
Conclusions
Nakagami-𝑚𝑚
Rayleigh
Rician
Weibull
SIRP
Solving GMP
• Many field measured data have non-
Gaussian statistics.
• Various fading models such as Weibull,
Nakagami-𝑚𝑚 and Generalized Gamma
have been proposed.
• The Spherically Invariant Random Process
(SIRP) provides a unified theory to study
these models systematically.
• We study the performance degradation of
a wireless communication system under
the SIRP fading model.
• When the pdf of the SIRP model is not
known, we use moment bound theory to
find the performance bounds.
Many well-known fading models
are special cases of SIRP.
The Spherically Invariant
Random Process (SIRP) has an
nth order pdf of the form
• Every SIRP 𝑌𝑌 𝑡𝑡 can be decomposed as a
product of a random scalar 𝑉𝑉 and an
independent Gaussian process 𝑍𝑍 𝑡𝑡 as
• The SIRP fading envelop is given by
where 𝑌𝑌𝐼𝐼 𝑡𝑡 and 𝑌𝑌𝑄𝑄 𝑡𝑡 are independent
Gaussian processes.
• By properly selecting 𝑉𝑉, 𝑋𝑋 can have a
wide range of distributions, including
many well-known models, such as
Generalized Gamma distribution.
• The performance 𝜙𝜙(𝑋𝑋) of a communication
system is a function of the fading envelop 𝑋𝑋,
modelled as an SIRP fading envelop.
• When the pdf of 𝑋𝑋 is not exactly known, we
can not evaluate E[𝜙𝜙(𝑋𝑋)].
• Instead, we seek the sharpest upper and
lower bound of E[𝜙𝜙(𝑋𝑋)].
• This can be formulated as a Generalized
Moment Problem (GMP):
A BPSK system with SIRP fading
envelop X and Gaussian noise
n. When the distribution of X is
unknown, finding the lower
bound of the BER becomes a
GMP:
• The GMP with SIRP constraint in its original form is not solvable.
• Using the SIRP representation theorem, we may reformulate the original GMP
as the following classical moment problem:
• This GMP is known as the Stieltjes type of problem. It can be transformed into
an Semi-Definite Program (SDP) and efficiently solvable.
• An SDP is a special case of the conic optimization problem. It has the form:
• SDP can be efficiently solved using algorithms like interior point method.
• Consider a BPSK system experiencing unknown SIRP fading.
• Performance metrics: bit error rate (BER), outage probability and channel
capacity.
• The SIRP fading model characterizes a broad class of fading statistics.
• When the pdf of the SIRP model is not exactly known, we showed how to
efficiently solve the performance bound problem of the communication
system using the moment bound theory and SDP.
* †
Comparing the BER bounds of SIRP
fading model with the Nakagami-𝑚𝑚
fading model.
Comparing with the case 𝑚𝑚 = 1/4,
the bounds of ergodic capacity of
the locally SIRP model is given.
Comparing the outage probability
bounds of SIRP fading model with
the Nakagami-𝑚𝑚 fading model.
When the underlying Gaussian
process of the SIRP fading has a
non-zero mean, the lower bound of
the BER can be further reduced.
min
𝑋𝑋~𝑃𝑃
E[𝜙𝜙(𝑋𝑋)]
s. t. E[𝒈𝒈 𝑋𝑋 ] ≤ 𝒄𝒄
min
𝑋𝑋~𝑃𝑃
𝑃𝑃e = 𝐸𝐸 𝑄𝑄 2𝑋𝑋2SNR
s. t. E 𝑋𝑋2 = 1
𝑌𝑌 𝑡𝑡 = 𝑉𝑉𝑉𝑉(𝑡𝑡)
𝑋𝑋 𝑡𝑡 = 𝑉𝑉 𝑌𝑌𝐼𝐼
2
𝑡𝑡 + 𝑌𝑌𝑄𝑄
2
𝑡𝑡
𝑓𝑓𝒀𝒀 𝒚𝒚 = 𝑔𝑔 𝒚𝒚𝑇𝑇
𝑹𝑹𝑹𝑹 .
𝑠𝑠
𝑋𝑋 𝑛𝑛
𝑟𝑟 = 𝑋𝑋𝑋𝑋 + 𝑛𝑛
min
𝑉𝑉≥0
E[Φ(𝑉𝑉)]
s. t. E[𝑮𝑮 𝑉𝑉 ] ≤ 𝒄𝒄
Φ 𝑉𝑉 ≡ E[Φ 𝑋𝑋 |𝑉𝑉]
𝑮𝑮 𝑉𝑉 ≡ E[𝒈𝒈 𝑋𝑋 |𝑉𝑉]
min
𝒙𝒙
𝒂𝒂𝑇𝑇
𝒙𝒙
s. t. 𝑨𝑨0 + � 𝑥𝑥𝑘𝑘 𝑨𝑨𝑘𝑘
𝑘𝑘
≽ 0
20 22 24 26 28 30
0
0.2
0.4
0.6
0.8
1
SNR (dB)
OutageProbability
K=0 (Rayleigh)
K=6
K=16
m=1/2Lower Bounds
Upper Bounds
0 10 20 30
10
-4
10
-2
10
0
SNR (dB)
BER
K=16
Worst Case
K=0
(Rayleigh)
K=6
m=1/2
WGN
0 10 20 30
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
BER
m=1/4
m=1/2
m=3/4
m=1
Upper Bound
Lower Bound
0 5 10 15 20 25 30
0
1
2
3
4
5
6
7
8
9
10
SNR (dB)
Capacity(bits/sec/Hz)
AWGN
Rayleigh
m=1/4
Upper Bound
Lower Bound
Department
of Information and Communication
Technologies