ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print)
IJECT Vol. 4, Issue Spl - 1, Jan - March 2013
The Characteristic Function of a Lognormal Process
Derived Using the Saddle Point Method
Nirmal B. Chakrabarti
Dept. of E&ECE, Indian Institute of Technology, Kharagpur, India
The characteristic function (chf) of a lognormal process is derived
using saddle point method . Relations are established between
the polar coordinates of the saddle point and variance and mean
parameters. The characteristic function and cumulants are then
decomposed into a primary and a secondary component. The sum
rules for the variance and means of a sum of lognormals and
the computation of the estimates of the local variance and mean
parameters for lognormal sum are discussed.
I. Introduction
The lognormal distribution is widely used in various branches of
science and engineering and in wireless communications. Early
work on sums of lognormals in communication dates back to the
sixties [1-2]. Recent work described in [4-5] and references therein
represents a small fraction of the vast literature on lognormal. The
computation of the characteristic function of a lognormal process
has also been studied. Holgate [3] described an asymptotic
technique based on the saddle point method to derive a closed
form expression. A numerical technique which uses saddle point
method concepts has been described in [5].
This letter is concerned with the computation of the chf of a
lognormal, the estimation of the parameters of a lognormal from
the cumulant and determination of parameters of the equivalent
lognormal of a sum of lognormals. Universal (r, θ) trajectory
of saddle point, spectral equivalence, representation of z: The
probability density function of a log-normal distribution is:
, and is termed as spectral parameter.
where
On replacing
by
, one gets
(4c)
(4d)
It is seen that z lies in the second quadrant and the magnitudes of
both the real and imaginary parts increase with .. Lognormals
with the same value of the range parameter st have an identical
trajectory of z and are spectrally equivalent.. The direct relation
between
and the polar coordinates can be found based
on a two step preliminary reduction using
and
. A safe way is to use table look-up.
Primary and secondary chf and cumulant: The exponent in (2)
may be written as an expansion around z as
(2b)
Using Eq. (2b) the integral for the characteristic function may be
written as a product of a primary function P(z) and a secondary
function S(z). The primary function is
and the secondary function is
(5a)
(1)
The integral for the characteristic function
distribution may be written as
of the lognormal
(2)
One may write the exponent in Eq. (2) equivalently as
It is easily verified that
point z is defined by
(2a)
. The saddle
(3)
The trajectory of the saddle point in polar form i.e.
is defined by
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(4a)
(4b)
International Journal of Electronics & Communication Technology
(5b)
where.
. S (z ) has a strong σ
dependence. When σ is small, we have the result quoted by
Holgate, viz., S (z )= 1 1 − z . Otherwise, S (z ) has to be computed
numerically using e.g Gauss Hermite integration.
Otherwise, S (z ) has to be computed numerically using ,e..g ,Gauss
Hermite integration.
Cumulant: The cumulant χ with components χ 1 and χ2 of the
chf of a lognormal defined as log φ is given by
(6)
where φs is the secondary component of chf. The real and imaginary
parts of chf are given respectively by
and
.
The real and imaginary parts of the primary cumulant are
respectively
(7a)
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IJECT Vol. 4, Issue Spl - 1, Jan - March 2013
ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print)
(7b)
Both
and
increase with st and the polar plot is open.
The secondary cumulant for small sigma given by −0.5 ⋅ log(1 − z )
is
(8)
The imaginary part of
has a peak of about 0.22 at a low
frequency corresponding to a phase of about
while the
real part decreases continuously. Eq. (8) is an overestimation
except for the case of low variance.
The primary is a quadratic in (1-z) and is scaled down by σ2. The
secondary decreases with variance and its effect is limited to low
frequencies.but a single expression valid for the entire range is hard
to find. Fig. 1 shows the polar plots of the primary and secondary
cumulants for values of σ2 of 4,6 and 8. The relative significance
of the primary and secondary components is evident.
There has been considerable work on the determination of an
equivalent lognormal of a sum of random probability density
functions, in particular of lognormals.because of the simplicity of
the lognormal CDF The behavior of such sums with a distribution
of σ and µ is known to be complex. One may however estimate
the variance and mean parameters locally using a knowledge of
the ratio
between the real and imaginary parts of the
primary cumulant to solve for z at frequencies removed from
zero. This yields the value of
. The derived values of the
radius vector or angle enable computation of of 2*ψ(1+tan(ψ))
and hence a division by
yields 2σ2...When an estimate of σ
is available z can be found from the relation
(9)
One can compute the parameter st using Eq. (4) and hence the
value of exp(µ). The secondary component causes an increase at
low frequencies.
0. 5
Primary, Var = 4
Secondary, Var = 4
Primary, Var = 2
Secondary, Var = 2
Primary, Var = 1
Secondary, Var = 1
0. 45
0. 4
Imaginary part
0. 35
0. 3
0. 25
0. 2
0. 15
Simple relations for the mean and variance of the sum exist when
the components have identical spectral parameters. This obviously
includes i.i.d with equal means and variances. The sum of N
primary cumulants with same phase may be written in terms of
as
parameters zk and
(10)
The condition of equality of primary cumulant of the sum of
lognormals to X n is that z-parameter of the sum is the same, i.e,
where ze and
refer to the parameters
of the effective lognormal. Thus
Equating the amplitudes one derives the sum rule for equivalent
lognormal as;:
(11a)
(11b)
Where summation is over 1 to N. There is seemingly no limitation
on the number N Using,
and µe derived above, the secondary
component χse of the equivalent lognormal cumulant has now
to be found. This cumulant is not generally equal to the sum of
secondary cumulants and an error term given by
must
be taken care of.. The estimates of the mean and the variance
provided by the error cumulant near the origin can be used for
correction.
When spectral parameters are not equal but the differences are
not very large, the equivalent spectral parameter of the lognormal
may be approximated by an average of the spectral parameters of
the components. The difference between the cumulant of the sum
and that of the equivalent lognormal. can be corrected noting that
an increase of the spectral parameter causes the polar I,Q plot to
expand while an increase of the variance causes it to shrink.
Fig 3 shows the polar plots of chf of the sum of lognormal
components with different variances and means, the chf of sum
and that of equivalent lognormal. Considering that both spectral
parameters and variances have significant variations, the agreement
between the chf of the sum and the equivalent is encouraging.
When the variance and spectral parameters vary very widely it is
appropriate to combine components with comparable values and
then estimate a global equivalent .
0. 1
0. 05
0
-1
-0. 8
-0. 6
-0. 4
-0. 2
0
R eal part
Fig. 1: Polar Plots of Primary and Secondary Cumulants
Lognormal Sum
An objective of the studies on lognormal sum has been to find
the mean and variance parameters of an equivalent lognormal
by matching of moments and inverse moments [1-2, 4]. The
charcatersitic function provides an alternative approach. The
cumulant of a sum of independent lognormals is equal to the sum
of the cumulants of the components which may be derived using
Eq (6). It is to be noted that the primary and secondary components
of the cumulant have different phase behaviour.
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II. Conclusion
The cumulant of the charatcteristic function of a lognormal is a
sum of an easily computable primary component and a secondary
component which decays with variance and is limited to low
frequencies.. A knowledge of the cumulant enables estimation of
the variance and the mean of a process belonging to the lognormal
family. For spectrally equivalent lognormals a sum rule applies
for the inverse of the variance and exponential of the mean. Polar
plots of the cumulant and the I, Q components of lognormals with
different variances and their sums are presented.. The procedure
outlined may find applications when combination of lognormal
and other distributions occur,
International Journal of Electronics & Communication Technology 13
ISSN : 2230-7109 (Online) | ISSN : 2230-9543 (Print)
IJECT Vol. 4, Issue Spl - 1, Jan - March 2013
0.45
estimated from the maximum value of the imaginary component
of the chf which decreases continuously with s or otherwise,
0.4
0.35
Imag part of CHF
0.3
c
The polar plot of s shrinks as the variance increases because an
enhanced value of ψ causes the peak to shift to a lower frequency
and the attenuation to be higher due to the increased negative
real part of z.
The primary and secondary cumulants depend on st and s but
in different ways.
This implies that the low frequency behavior of the equivalent
lognormal will in general be erroneous
0.25
0.2
0.15
var=4 & mean=exp(0.23)
var=6 & mean=exp(0.12)
var=8 & mean=exp(0.01)
chf of s um
chf of equivalent
0.1
0.05
0
-0.2
0
0.2
0.4
0.6
R eal part of C HF
0.8
1
1.2
Fig. 2: Polar Plots of chf of Lognormal Components, the Sum,
and the Equivalent Lognormal
III. Acknowledgment
I am grateful to Prof Saswat Chakrabarti for fruitful suggestions
and to his students, Praful Mankar and Ms Parul Goswami for
many assistances.
Reference
[1] Fenton, L.,"The Sum of Log-Normal Probability Distributions
in Scatter Transmission Systems", Communications Systems,
IRE Transactions on, Vol. 8, No. 1, pp. 57-67, March
1960.
[2] S. C. Schwartz, Y. S. Yeh,“On the distribution function and
moments of power sums with lognormal components”, Bell
Syst. Tech. J., Vol. 61, No. 7, pp. 1441–1462, Sep. 1982.
[3] P. Holgate,“The lognormal characteristic function,
Communications”, in Statistics - Theory and Methods, Dec.
1989.
[4] Filho, J.C.S.S., Cardieri, P., Yacoub, M.D.,"Simple accurate
lognormal approximation to lognormal sums", Electronics
Letters, Vol. 41, No. 18, pp. 1016- 1017, 1 Sept. 2005
[5] Tellambura, C., Senaratne, D.,“Accurate computation of the
MGF of the lognormal distribution and its application to
sum of lognormals”, Communications, IEEE Transactions
on, Vol. 58, No. 5, pp. 1568-1577, May 2010.
Nirmal B. Chakrabrati (Department. of E&ECE., IIT Kharagpur,
India.
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