Using several mathematical examples from three different authors in texts from different courses this paper illustrates the easier way to avoid confusions and always get the correct results with the least effort was to use the proposed Excel Gamma function explained in detail for the proper use of the Q(z) and ercf(x) functions in most communication courses. The paper serves as a tutorial and introduction for such functions
Jacobi Method, For Numerical analysis. working matlab code. numeric analysis Jacobi method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Jacobi Method
Jacobi Method, For Numerical analysis. working matlab code. numeric analysis Jacobi method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Jacobi Method
FURTHER RESULTS ON THE DIRAC DELTA APPROXIMATION AND THE MOMENT GENERATING FU...IJCNC
In this article, we employ two distinct methods to derive simple closed-form approximations for the
statistical expectations of the positive integer powers of Gaussian probability integral Eg [Qp ( bWg )]
with
respect to its fading signal-to-noise ratio (SNR) g random variable. In the first approach, we utilize the
shifting property of Dirac delta function on three tight bounds/approximations for Q(.) to circumvent the
need for integration.
Tomography is important for network design and routing optimization. Prior approaches require either
precise time synchronization or complex cooperation. Furthermore, active tomography consumes explicit
probing resulting in limited scalability. To address the first issue we propose a novel Delay Correlation
Estimation methodology named DCE with no need of synchronization and special cooperation. For the
second issue we develop a passive realization mechanism merely using regular data flow without explicit
bandwidth consumption. Extensive simulations in OMNeT++ are made to evaluate its accuracy where we
show that DCE measurement is highly identical with the true value. Also from test result we find that
mechanism of passive realization is able to achieve both regular data transmission and purpose of
tomography with excellent robustness versus different background traffic and package size.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Finite-difference modeling, accuracy, and boundary conditions- Arthur Weglein...Arthur Weglein
This short report gives a brief review on the finite difference modeling method used in MOSRP
and its boundary conditions as a preparation for the Green’s theorem RTM. The first
part gives the finite difference formulae we used and the second part describes the implemented
boundary conditions. The last part, using two examples, points out some impacts of the accuracy
of source fields on the results of modeling.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
journals like IJERST to provide a valuable resource for researchers, students, and professionals who are interested in the latest developments and insights in the field of Information Technology and Computer Engineering.
Symmetric quadratic tetration interpolation using forward and backward opera...IJECEIAES
The existed interpolation method, based on the second-order tetration polynomial, has the asymmetric property. The interpolation results, for each considering region, give individual characteristics. Although the interpolation performance has been better than the conventional methods, the symmetric property for signal interpolation is also necessary. In this paper, we propose the symmetric interpolation formulas derived from the second-order tetration polynomial. The combination of the forward and backward operations was employed to construct two types of the symmetric interpolation. Several resolutions of the fundamental signals were used to evaluate the signal reconstruction performance. The results show that the proposed interpolations can be used to reconstruct the fundamental signal and its peak signal to noise ratio (PSNR) is superior to the conventional interpolation methods, except the cubic spline interpolation for the sine wave signal. However, the visual results show that it has a small difference. Moreover, our proposed interpolations converge to the steady-state faster than the cubic spline interpolation. In addition, the option number increasing will reinforce their sensitivity.
This is the main presentation for the MBA course I teach on Project Quality Management. It is based on a combination of the Critical Chain Approach by Dr Goldratt, the PMBOK (chapter 8) plus my own experience as Sr Validation Project Manager, Lawyer & US ARMY Officer
Un breve resumen de los hechos para entender el asunto: la Lcda. Iris Meléndez Vega era directora en una sección del Departamento de Justicia allá para 1990. Su secretaria, la Sra. Martha Marrero Rivera la acusa de hostigamiento sexual pero al parecer, lo que había era una guerra entre ellas, motivada posiblemente porque la secretaria no obtuvo un ascenso y esto escala gravemente hasta llegar al pleito legal. El Vocero de PR, a través de su reportero, el Sr. José A. Purcell publica una serie de reportajes de alto contenido amarillista, bastantes parcializados a favor de la posición de la Sra. Marrero Rivera; lo que motiva a la Lcda. Meléndez Vega a demandarlos por libelo. El Vocero y la Sra. Marreo pierden la demanda pues se logra demostrar que no se cumplió con los mínimos estándares de malicia requeridos según establecidos por el Tribunal Supremo federal en New York Times Co. v. Sullivan, 376 U.S. 254 (1964).
La opinión concurrente y disidente expresada por la Honorable Juez Asociada Señora Rodríguez Rodríguez da en el clavo en cuanto al asunto que siempre me ha preocupado: un buen detalle de cómo llegar a los cálculos en materia de daños, evita (o al menos minimiza) futuras controversias . El terror a las matemáticas en nuestra profesión legal solo logra que las partes consigan arreglos malos y esto, solo lleve a pleitos futuros. Si las partes no saben cuánto le cuesta un problema, es muy humano ser más optimista (más bien, muy ingenuo) de lo que debe ser y causa problemas.
Se hace una pequeña comparación entre tres de los reglamentos de educación continuada
vigentes en Puerto rico y se concluye que la regla 3.05 del Reglamentos de Ingenieros &
Agrimensores debe ser enmendada. Se propone la enmienda correspondiente al final.
‘….el intento de obligar a educarse al que no quiere
hacerlo, es una vana ilusión.”
Voto Disidente del Juez Asociado señor FUSTER BERLINGERI
2005 TSPR 44
Si de verdad vale la pena hacer algo,
vale la pena hacerlo bien.
Richard Widman
Dime como me mides y te diré como me comportaré
Dr. Eli Goldratt, La Meta
Introducción
Dejando primeramente establecido que no soy experto en educación de
ninguna forma (ni de adultos, ni de niños, ni especial, etc.); sin embargo, si quiero
aprovechar la oportunidad para traer a su atención, lo que de seguro ya han visto
y saben pero me quitaría un sinsabor menos en esta mundo.
Dijo el Hon. Juez Fuster Berlingeri que “uno puede llevar al caballo1 al
abrevadero, pero no se le puede obligar a beber”. Me opongo en principio a
cualquier programa de educación continuada (EC) en forma obligatoria2 pero ya
que lo vamos a hacer de todas formas, pues vamos a hacerlo bien
Este es un caso de pensión alimentaria, cuya revisión ya esta debida hace algún tiempo, y
solo causa mayores problemas legales de los que resuelve. En resumen, Rivera Aponte, padre no
custodio, acude en alzada de 2 puntos que afectan la pensión que debe pagar: a) un premio de
Lotería de $120,000.00 que parece que “olvidó” poner en su planilla & b) el pago de dietas y
millaje que “olvidó” también. Tribunal Supremo decide, en mi humilde opinión en parte
correctamente y en parte no, que ambos son ingresos que deben incluir en la planilla.
El caso discute la controversia entre un padre no-custodio que alega capacidad de pago
para evitar descubrir prueba sobre su ingreso y la madre custodia de dos hijos menores al
momento de solicitar aumento de pensión, pero ya universitarios.
Para ilustrar porque entiendo que la decisión del caso está equivocada, veamos varias
ejemplitos numéricos: a) qué pasaría si las necesidades de los menores fueran, digamos, unos
$4,000.00 dólares mensuales y ambos padres descubren prueba sobre su ingreso y digamos que
ambos padres ganan unos $60 mil anuales (esto es, $5 mil mensuales). Supongamos, por
simplicidad, que el padre no pasa más tiempo con los menores que el mínimo requerido (que no
es una suposición ilógica, dada la edad de los menores).
Se trabaja un modelo sencillo pero con data real de ingeniería financiera en el mercado de
divisas internacional a modo de introducir el tema. El modelo concuerda muy bien con los
datos reales.
A simple PCA model was used to find the direction of most variability for the CEF puzzle.
Evidence that the MOM factor as detailed by Carhart (1997) explains this puzzle was
found. Data sets used are available for independent verification of results.
Me gustan las matemáticas, es como un jueguito. A veces, uno
se pone a jugar y encuentra detalles que quisiera compartir. El
cálculo de la cuota viudal usufructuaria es de interés, por las implicaciones
tal vez importantes que pudiera tener.
Supongamos a modo de limitar el estudio que el caso a estimar
es una viuda de 68 años con 2 hijos del mismo matrimonio cuyo
esposo al morir deja un caudal de $953,466.24.
Existen calculadoras muy buenas que para algunos podrían ser
un salvavidas casi irremplazable1.
Para este ejemplito, la calculadora utiliza como aproximación
la fórmula para el cálculo de valor presente:
En muy apretada síntesis de los hechos, los padres del menor GARH acuden en alzada al
Alto Foro contra una decisión de Apelaciones revocando a Instancia en cuanto al pago de los
daños permanentes hechos al menor por el médico obstetra al nacer. La tablita que sigue resume
la sentencia de Instancia:
“”El 'Ay, Bendito' es el primo hermano del …”
Dicho Jibarito
Introducción
Empecemos como la Biblia…en el principio cuando todo era confusión. Allá a
finales de 1973 el Banco Popular adquirió por $86,493.87 un contrato de arrendamiento,
que tenía una duración original de 25 años, de una finca propiedad de los Talaveras por
medio de una cesión de derechos de un tercero que tenía dicho contrato desde Junio de
1968. La propiedad arrendada lo fue por canon de $8,000.00 mensuales con una opción
de compraventa, ahora a favor del Banco Popular con precio de “strike” de $10,000.00 y
pago único final (de ejercer la misma) de $22,500.00 (total a pagar en 1993; $32,500.00)
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdf
Understanding the Differences between the erfc(x) and the Q(z) functions: A Simple Approach using Excel
1. > PAPER IDENTIFICATION NUMBER TBD< 1
Abstract—Using several mathematical examples from
three different authors in texts from different courses this
paper illustrates the easier way to avoid confusions and
always get the correct results with the least effort was to
use the proposed Excel Gamma function explained in
detail for the proper use of the Q(z) and ercf(x) functions
in most communication courses. The paper serves as a
tutorial and introduction for such functions.
Index Terms—Bit error rate, BPSK, clear sky, erfc(x), M-
ary ASK, M-ary PSK, QPSK, Q(z).
I. INTRODUCTION
THE use of two different Excel functions will be explained in
order to properly address the possible confusion of the two
main approaches to estimate error probabilities.
Excel provides an excellent way to understand the
differences between the Q(z) and the erfc(x) functions. In the
main textbooks the correct calculation of this error probability
is either dismissed as an easy task or just plainly ignored,
authors mostly referring to old tables in their appendices and
resolving to interpolation to get the estimate or just telling the
student to use the function available in any software being
used in class.
I have found this could be highly confusing for the new
communication systems student just learning a lot of technical
material. It could be also potentially disastrous or costly as the
incorrect estimation could lead to overdesign a power
requirement or incorrect selection of the proper S/N figure.
II. PAPER ORGANIZATION
The paper will first discuss the two functions mathematical
formulas: Q(z) and erfc(x). There, the two approaches using
Excel formulas will be introduced. Afterwards, several
mathematical examples form main textbooks will be detailed
to illustrate the proper use of both approaches and they will be
compared against textbook’s estimates as well as interpolation
estimates, when applicable. The use of Excel in this paper as
the main estimation tool in this paper is just because of its
wide availability and simplicity as well as its ease handling of
extended significant figures.
A. The Q(z) function
Lathi (1998) defines the Q(z) as the Cumulative Density
Submitted for review.
x
Function (CDF) upper tail of the famous standardized
Gaussian function, F(y):
𝐹(𝑧) =
1
√2𝜋
∫ 𝑒
−𝑧2
2 𝑑𝑧
𝑧
−∞
(1)
Thus, Q(z) would be calculated as:
𝑄(𝑧) = 1 − 𝐹(𝑧) = 1 −
1
√2𝜋
∫ 𝑒
−𝑧2
2 𝑑𝑧 =
𝑧
−∞
𝑃𝑒 (2)
F(*) is standardized in the sense its mean, =0 and its
standard deviation, =1 (this, of course implies its variance,
2
= 1 also). This function must be numerically evaluated to
estimate its resulting error probability (𝑃𝑒) as it does not have
an analytical closed form. Until a few decades ago, this was a
major task that was mostly tackled by interpolating tables in
the back of textbooks or mathematical handbooks, by using
statistical software or coding. Nowadays, it could be easily
estimated with the Excel functions:
𝐹(𝑧) = 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(𝑧, 1) = 𝑃′ 𝑒 (3)
and,
𝐹(𝑃′ 𝑒)−1
= 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑃′ 𝑒) = 𝑧 (4)
Where the Excel function 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(𝑧, 1) requires its
second argument to be either 0 or 1. We will not discuss the
cases where the second argument is zero here as we are only
concerned with the estimation of the CDF and, in order to
obtain such CDF estimation the second argument must be
always be 1. That is, the use of 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(𝑧, 1) will
estimate the normal standardize CDF for the value z. This is,
of course, 𝑃′ 𝑒; the cumulative probability from (-∞, z). Its
inverse function 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑃′ 𝑒) provides us with the z
value when the known quantity is 𝑃′ 𝑒. Note that 𝑃𝑒 + 𝑃′ 𝑒 = 1.
It stands to logic that we will need to use (2) in Excel by
just simple subtracting the result from 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(𝑧, 1)
from 1.
B. The erfc(x) function
Lathi (1998) defines the erfc(*) function in terms of the
Q(z) one:
𝑒𝑟𝑓𝑐(𝑥) =
2
√𝜋
∫ 𝑒−𝑦^2
𝑑𝑦 = 2 ∗ 𝑄(𝑥√2)
∞
𝑥
(5)
For reasons that will become apparent in the examples,
instead of performing the change of variables and use the
Q(*), this paper will use the Excel function to define:
Understanding the Differences between the erfc(x) and the Q(z) functions: A Simple Approach using Excel
X
2. > PAPER IDENTIFICATION NUMBER TBD< 2
𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
(6)
Where the second argument, (shape parameter) = 0.5 and
the third argument, (scale parameter) 𝛽 = 1 were chosen to
approximate the normal standard distribution and the fourth
parameter is always set to 1 to estimate the CDF. It is quite
clear that:
𝑄(𝑧) = 𝑒𝑟𝑓𝑐(𝑥) (7)
In case we have the error probability, 𝑃′ 𝑒, and need the x
value, we must use, in general:
𝑥 = 𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 − 𝑃′ 𝑒)^2,0.5,1) (8)
But some adjustments must be made to fit the particular
case, as we shall see. The second argument, (shape parameter)
= 0.5 and the third argument, (scale parameter) 𝛽 = 1 were
chosen to approximate the normal standard distribution as
before.
III. MATHEMATICAL EXAMPLES
Textbooks examples were selected from Lathi (1998),
Roddy (2006) and Pratt (2003) for their different applications
of the mathematical formulas. Lathi is more concerned with
communications systems in general but Roddy and Pratt are
more interested in satellites applications.
A. MATH EXAMPLE 1: PRATT EXAMPLE 5.4.1, PAGE 195
The author is interested in obtaining the system BER (bit
error rate) for different situations (clear sky/air, rain
attenuation) with a given C/N (carrier to noise). That is, we
need to estimate the 𝑃′ 𝑒 for a given C/N - which will be the z
or x value, depending on which Excel formula is applied.
i) BER in clear sky/air condition for BPSK modulation with
a given C/N= 14 dB implying a power ratio 10
14
10
25.11886432.
Our first step will be determining what exactly would be the
values of z or x to use.
For BPSK modulation:
𝑧 = √2 ∗
𝐶
𝑁
(9)
√2 ∗ (25.11886432)√50.2377286 7.087857831
and,
𝑥 = 𝑧/√2 (10)
7.087857831
√2
7.087857831
1.414213562
5.011872336
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(7.087857831)
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(7.087857831,1) 𝟔. 𝟖𝟏𝟎𝟏𝟎𝟖𝐄 − 𝟏𝟑
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(5.0118723362,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝟐𝟓.𝟏𝟏𝟖𝟖𝟔𝟒𝟑,0.5,1,1)
2
𝟔. 𝟖𝟏𝟎𝟏𝟎𝟖𝐄 − 𝟏𝟑
It must be noted that, in this case, the use of (6) is
straightforward as the input variable is the power ratio
calculated. It should also be noted that both approaches
provide the same result for 𝑃𝑒.
Standard linear interpolation using Pratt’s Table of Q
function (page 505) for the textbook solution of Q(7.07) gives
𝑃𝑒 8.21E-13. Text must have a typo as it states the result as
7.8E-11.
ii) BER in clean air conditions, QPSK with a given C/N= 14
dB implying a power ratio 10
14
10 25.11886432.
Our first step will be determining what exactly would be the
values of z or x to use.
For QPSK modulation:
𝑧 = √
𝐶
𝑁
(11)
√(25.11886432)5.011872336
and,
𝑥 = 𝑧/√2 (10)
5.0118723361
√2
5.011872336
1.414213562
3.543928915
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(5.011872336 )
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(5.011872336 ,1) 2.695148E-07
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(3.5439289152,0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(12.55943216,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(
𝟐𝟓.𝟏𝟏𝟖𝟖𝟔𝟒𝟑
2
,0.5,1,1)
2
2.695148E-07
It must be noted that, in this case, the use of (6) is, again,
straightforward as the input variable is the power ratio
calculated divided by 2. It should also be noted that both
approaches provide the same result for 𝑃𝑒.
Standard linear interpolation using Pratt’s Table of Q
3. > PAPER IDENTIFICATION NUMBER TBD< 3
function (page 505) for the textbook solution of Q(5) from
Pratt table page 505 = 2.872E-07.
iii) BER with rain attenuation of 3 dB, BPSK with a given
now of C/N= 11 dB implying a power ratio 10
11
10
12.58925412.
Our first step will be determining what exactly would be the
values of z or x to use.
For BPSK modulation:
𝑧 = √2 ∗
𝐶
𝑁
(9)
√2 ∗ (12.58925412)√25.17850824 5.017819072
and,
𝑥 = 𝑧/√2 (10)
5.017819072
√2
5.017819072
1.414213562
3.548133892
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(5.017819072)
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(5.017819072,1) 𝟐. 𝟔𝟏𝟑𝟎𝟔𝟖𝐄 − 𝟎𝟕
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(3.5481338922,0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(𝟏𝟐. 𝟓𝟖𝟗𝟐𝟓𝟒𝟏𝟐, 0.5,1,1)
2
𝟐. 𝟔𝟏𝟑𝟎𝟔𝟖𝐄 − 𝟎𝟕
It must be noted that, again for this case, the use of (6) is
straightforward as the input variable is the power ratio
calculated. It should also be noted that both approaches
provide the same result for 𝑃𝑒.
Standard linear interpolation using Pratt’s Table of Q
function (page 505) for the textbook solution of Q(5.02) gives
𝑃𝑒 2.8E-07.
iv) BER with rain attenuation of 3 dB, QPSK with now a
given C/N= 11 dB implying a power ratio 10
11
10 12.58925412.
Our first step will be determining what exactly would be the
values of z or x to use.
For QPSK modulation:
𝑧 = √
𝐶
𝑁
(11)
√(12.58925412)3.548133892
and,
𝑥 = 𝑧/√2 (10)
3.5481338921
√2
3.5481338926
1.414213562
2.508909536
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(3.548133892 )
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(3.548133892 ,1) 1.939855E-04
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(2.5089095362,0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(6.294627059,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(
𝟏𝟐.𝟓𝟖𝟗𝟐𝟓𝟒𝟏𝟐
2
,0.5,1,1)
2
1.939855E-04
It must be noted that, in this case, the use of (6) is, again,
straightforward as the input variable is the power ratio
calculated divided by 2. It should also be noted that both
approaches provide the same result for 𝑃𝑒.
Standard linear interpolation using Pratt’s Table of Q
function (page 505) for the textbook solution of Q(3.55) from
Pratt table page 505 = 2.2E-04.
From these simple examples it should be easily inferred that
when we are dealing with C/N quantities, the preferred
methodology should be using (6) with the proper modulation
adjustment.
B. MATH EXAMPLE 2: PRATT EXAMPLE 5.4.2, PAGE 197
The author is now interested in obtaining the system BER
(bit error rate) for different situations (clear sky/air, 0.1%
average annual worst-case condition) with a given C/N
(carrier to noise). That is, we need to estimate the 𝑃′ 𝑒 for a
given C/N - which will be the z or x value, depending on
which Excel formula is applied.
i) BER in clear sky/air condition for BPSK modulation with
a given C/N= 15.2 dB implying a power ratio 10
15.2
10
33.11311215.
Our first step will be determining what exactly would be the
values of z or x to use. This situation is the same as before
with the C/N changed.
For BPSK modulation:
𝑧 = √2 ∗
𝐶
𝑁
(9)
√2 ∗ (33.11311215)√66.2262243 8.137949637
and,
𝑥 = 𝑧/√2 (10)
8.137949637
√2
8.137949637
1.414213562
5.754399373
4. > PAPER IDENTIFICATION NUMBER TBD< 4
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(8.137949637)
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(8.137949637,1) 𝟎
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(5.7543993732,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝟑𝟑.𝟏𝟏𝟑𝟏𝟏𝟐𝟏𝟓,0.5,1,1)
2
𝟐. 𝟐𝟐𝟎𝟒𝟒𝟔𝟎𝟓𝐄 − 𝟏𝟔
It must be noted that, in this case, the use of (6) is
straightforward as the input variable is the power ratio
calculated. It should also be noted that the Q(z) approach was
unable to provide a precise estimate for 𝑃𝑒.
Standard linear interpolation using Pratt’s Table of Q
function (page 505) for the textbook solution of erfc(5.75)/2
gives 𝑃𝑒 2.5E-15.
From the example, this sat link has an 8,000,000 bits/second
symbol rate; thus, with an estimated Pe= 2.22044605E-16 an
error will occur, in average, every 5.63E+08 seconds or 17.85
years. Pratt's example states one error would occur every 1.5
years on average.
The lack of precise estimation could cause a false alarm
very often.
ii) BER 0.1% of the year fall in clear sky conditions for
BPSK modulation with a given C/N= 12.2 dB implying a
power ratio 10
12.2
10 16.59586907.
Our first step will be determining what exactly would be the
values of z or x to use. This situation is the same as before
with the C/N changed.
For BPSK modulation:
𝑧 = √2 ∗
𝐶
𝑁
(9)
√2 ∗ (16.59586907)√33.19173815 5.761227139
and,
𝑥 = 𝑧/√2 (10)
5.761227139
√2
5.761227139
1.414213562
4.073802778
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(5.761227139)
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(5.761227139,1)
𝟒. 𝟏𝟕𝟓𝟐𝟐𝟖𝟕𝟏𝟖𝟑𝐄 − 𝟎𝟗
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(4.0738027782,0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(𝟏𝟔. 𝟓𝟗𝟓𝟖𝟔𝟗𝟎𝟕, 0.5,1,1)
2
𝟒. 𝟏𝟕𝟓𝟐𝟐𝟖𝟕𝟏𝟖𝟑𝐄 − 𝟎𝟗
It must be noted that, in this case, the use of (6) is
straightforward as the input variable is the power ratio
calculated. It should also be noted that both approaches
provide the same result for Pe.
Standard linear interpolation using Pratt’s Table of erfc
function (page 506) for the textbook solution of erfc(4.75)/2
gives 𝑃𝑒 5E-09.
From the example, this sat link has an 8,000,000 bits/second
symbol rate; thus, with an estimated Pe= 4.17522872E-09 an
error will occur, in average, every 29.94 seconds. Pratt's
example states one error would occur every 25 seconds on
average.
The imprecision in the calculations are not affecting as
much this time; we can conclude the BPSK modulation is
severely affected in the worst-case situation.
iii) BER in clean air conditions, QPSK with a given C/N=
14.8 dB implying a power ratio 10
14.8
10 30.1995172.
Our first step will be determining what exactly would be the
values of z or x to use.
For QPSK modulation:
𝑧 = √
𝐶
𝑁
(11)
√(30.1995172)5.495408739
and,
𝑥 = 𝑧/√2 (10)
5.4954087391
√2
5.495408739
1.414213562
3.885840784
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(5.495408739 )
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(5.495408739 ,1) 1.949032E-08
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(3.8858407842,0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(15.0997586,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(
𝟑𝟎.𝟏𝟗𝟗𝟓𝟏𝟕𝟐
2
,0.5,1,1)
2
1.949032E-08
It must be noted that, in this case, the use of (6) is, again,
straightforward as the input variable is the power ratio
calculated divided by 2. It should also be noted that both
approaches provide the same result for 𝑃𝑒.
5. > PAPER IDENTIFICATION NUMBER TBD< 5
Standard linear interpolation using Pratt’s Table of erfc
function (page 506) for the textbook solution of erfc(3.89)/2
gives Pe = 2E-08.
From the example, this sat link has an 16,000,000
bits/second symbol rate; thus, with an estimated Pe=
1.949032E-08 an error will occur, in average, every 3.21
seconds. Pratt's example states one error would occur every
3.12 seconds on average.
The imprecision in the calculations are not affecting as
much this time; we can conclude the QPSK modulation should
not be used in this application.
iv) BER in 0.1% of the year fall in clear sky conditions,
QPSK with a given C/N= 11.8 dB implying a power ratio
10
11.8
10 15.13561248.
Our first step will be determining what exactly would be the
values of z or x to use.
For QPSK modulation:
𝑧 = √
𝐶
𝑁
(11)
√(15.13561248)3.89045145
and,
𝑥 = 𝑧/√2 (10)
3.89045145
√2
3.89045145
1.414213562
2.750964602
Now, we use (3) with (2) to estimate 𝑃𝑒:
𝑃𝑒 = 1 − 𝐹(3.89045145 )
1 − 𝑁𝑂𝑅𝑀. 𝑆. 𝐷𝐼𝑆𝑇(3.89045145 ,1) 5.002895E-05
Using (6) we obtain for 𝑃𝑒:
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(𝑥2,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(2.7509646022,0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(7.567806242,0.5,1,1)
2
1−𝐺𝐴𝑀𝑀𝐴.𝐷𝐼𝑆𝑇(
𝟏𝟓.𝟏𝟑𝟓𝟔𝟏𝟐𝟒𝟖
2
,0.5,1,1)
2
5.002895E-05
It must be noted that, in this case, the use of (6) is, again,
straightforward as the input variable is the power ratio
calculated divided by 2. It should also be noted that both
approaches provide the same result for 𝑃𝑒.
Standard linear interpolation using Pratt’s Table of erfc
function (page 506) for the textbook solution of erfc(15.14)/2
gives Pe = 5E-05.
From the example, this sat link has an 16,000,000
bits/second symbol rate; thus, with an estimated Pe=
5.00289478E-05 an error will occur, in average, every
0.00125 seconds. Pratt's example states one error would occur
every 0.00125 seconds on average.
The imprecision in the calculations are not affecting as
much this time; we can conclude the QPSK modulation should
not be used in this application. Worst-case condition there will
be 800 errors per second.
In summary, when dealing with C/N the best way to
estimate the Pe is by using (6) properly adjusted for
modulation: Use the C/N as input directly for BPSK or use
(C/N)/2 for QPSK.
We will now use Lathis’ examples to illustrate calculations
for both approaches when the input is the Pe and a z or x
values is required to be determined.
C. MATH EXAMPLE 3: LATHI EXAMPLE 13.1 page 620
The author interest is to compare power requirements for
different transmissions schemes under a Pe constraint.
Given:
a) Required transmissions rate, 𝑅 𝑏: 2.08E06 bps
b) Channel noise PSD, N: 2E-08
c) Pe ≤ 1E-06
i) Binary, polar signaling; Required Power=𝐸 𝑏 ∗ 𝑅 𝑏 ∗ 𝑁
𝑃𝑒 = 𝑄 (√
2𝐸 𝑏
𝑁
) = 1E − 06
Using (4),
𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑃𝑒) = 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(1𝐸 − 06)
−4.753424309 = √
2𝐸 𝑏
𝑁
; −4.7534243092
2𝐸 𝑏
𝑁
22.59504266
2𝐸 𝑏
𝑁
𝟏𝟏. 𝟐𝟗𝟕𝟓𝟐𝟏
𝐸 𝑏
𝑁
𝑃𝑜𝑤𝑒𝑟 𝟏𝟏. 𝟐𝟗𝟕𝟓𝟐𝟏 ∗ 2E − 08 ∗ 2.08E06 𝟎. 𝟒𝟔𝟗𝟗𝟕𝟔𝟖𝟗 𝐖
Using (8),
𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 − 𝑃′ 𝑒)^2,0.5,1) = 𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 −
1𝐸 − 6)^2,0.5,1)11.297522.
𝑃𝑜𝑤𝑒𝑟 𝟏𝟏. 𝟐𝟗𝟕𝟓𝟐𝟐 ∗ 2E − 08 ∗ 2.08E06𝟎. 𝟒𝟔𝟗𝟗𝟕𝟔𝟗𝟏𝐖
The text estimates
𝐸 𝑏
𝑁
= 11.35; thus,
𝑃𝑜𝑤𝑒𝑟 𝟏𝟏. 𝟑𝟓 ∗ 2E − 08 ∗ 2.08E06𝟎. 0.47216000 W
ii) 16-ary ASK, M=16; Required Power=𝐸 𝑏 ∗ 𝑅 𝑏 ∗ 𝑁 ∗
(𝑙𝑜𝑔2(𝑀)
First, we made the proper adjustments for this scheme,
𝑃𝑒𝑀 = 𝑙𝑜𝑔2(16) ∗ 𝑃𝑒 = 4 ∗ 1𝐸(−06)
𝑃𝑒 = 2(
𝑀 − 1)
𝑀
)𝑄 (√
6𝑙𝑜𝑔2(16)𝐸 𝑏
𝑁(𝑀2 − 1)
) = 4E − 06
= 2(
16 − 1)
16
)𝑄 (√
6 ∗ 4 ∗ 𝐸 𝑏
𝑁(162 − 1)
) = 4E − 06
6. > PAPER IDENTIFICATION NUMBER TBD< 6
= 2(
15)
16
)𝑄 (√
24𝐸 𝑏
𝑁(256 − 1)
) = 4E − 06
= (
15)
8
)𝑄 (√
24𝐸 𝑏
𝑁(255)
) = 4E − 06
= (
15)
8
)𝑄 (√
24𝐸 𝑏
𝑁(255)
) = 4E − 06
= 𝑄 (√
24𝐸 𝑏
𝑁(255)
) = 4E − 06 ∗ (
8
15
)
= 𝑄 (√
24𝐸 𝑏
𝑁(255)
) = 4E − 06 ∗ (
8
15
)
= 𝑄 (√
24𝐸 𝑏
𝑁(255)
) = 2.13𝐸 − 06
Using (4),
𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑃𝑒) = 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(2.13𝐸 − 06)
−4.597950823 = √
24𝐸 𝑏
𝑁(255)
; −4.5979508232
24𝐸 𝑏
𝑁(255)
21.14115177
24𝐸 𝑏
𝑁(255)
21.1411517 ∗ (
255
24
)
𝐸 𝑏
𝑁
224.6247
𝐸 𝑏
𝑁
𝑃𝑜𝑤𝑒𝑟224.6247 ∗ 2E − 08 ∗
2.08E06
𝑙𝑜𝑔2(16)
∗ 𝑙𝑜𝑔2(16)
224.6247 ∗ 2E − 08 ∗ 2.08E06𝟗. 𝟑𝟒𝟒𝟑𝟖𝟗 𝐖
Using (8)
𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 − 𝑃′ 𝑒)^2,0.5,1) = 𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 −
2.13𝐸 − 6)^2,0.5,1) 10.57058
𝑃𝑜𝑤𝑒𝑟10.57058 ∗ 2E − 08 ∗ 2.08E06 ∗ 2
∗ (
255
24
)𝟗. 𝟑𝟒𝟒𝟑𝟗𝟎 𝐖
The text estimates 𝐸 𝑏 = 0.499𝐸 − 05,
𝑡ℎ𝑢𝑠, 𝑔𝑖𝑣𝑒𝑠 𝑡ℎ𝑒 𝑃𝑜𝑤𝑒𝑟 𝟗. 𝟑𝟒 𝑾. Our estimates of
𝐸 𝑏 =4.49249E-06 for the Gaussian formula and 𝐸 𝑏 =
4.4925𝐸 − 06 for the Gamma methodology. Using 𝐸 𝑏 =
0.499𝐸 − 05 results in a 𝑃𝑜𝑤𝑒𝑟 𝟏𝟎𝟑. 𝟕𝟗𝟐 𝑾;therefore, the
text must have a typo. If it meant 𝐸 𝑏 = 0.499𝐸 − 06, this
result in a 𝑃𝑜𝑤𝑒𝑟 𝟏𝟎. 𝟑𝟕𝟗𝟐 𝑾 which is not the stated power
value.
Note that the power estimates are congruent until the sixth
significant figure between the two approaches. Again, the use
of the Gamma function is the most straightforward method.
iii) 16-ary PSK, M=16; Required Power=𝐸 𝑏 ∗ 𝑅 𝑏 ∗ 𝑁 ∗
(𝑙𝑜𝑔2(𝑀)
First, we made the proper adjustments for this scheme,
𝑃𝑒𝑀 = 𝑙𝑜𝑔2(16) ∗ 𝑃𝑒 = 4 ∗ 1𝐸(−06)
𝑃𝑒 2𝑄 (√
2(𝜋2)𝑙𝑜𝑔2(16)𝐸 𝑏
256𝑁
) 4E − 06
= 𝑄 (√
(𝜋2)(2)(4)𝐸 𝑏
256𝑁
) 2E − 06
= 𝑄 (√
(𝜋2)(8)𝐸 𝑏
256𝑁
) 2E − 06
= 𝑄 (√
(𝜋2)𝐸 𝑏
32𝑁
) 2E − 06
= 𝑄 (√
(𝜋2)𝐸 𝑏
32𝑁
) 2E − 06
Using (4),
𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(𝑃𝑒) = 𝑁𝑂𝑅𝑀. 𝑆. 𝐼𝑁𝑉(2. 𝐸 − 06)
−4.611382362 = √
(𝜋2)𝐸 𝑏
32𝑁
; −4.6113823622
(𝜋2)𝐸 𝑏
32𝑁
21.26484729
(𝜋2
)𝐸 𝑏
32𝑁
21.1411517 ∗ (
32
𝜋2
)
𝐸 𝑏
𝑁
68.9465
𝐸 𝑏
𝑁
𝑃𝑜𝑤𝑒𝑟68.9465 ∗ 2E − 08 ∗
2.08E06
𝑙𝑜𝑔2(16)
∗ 𝑙𝑜𝑔2(16)
68.9465 ∗ 2E − 08 ∗ 2.08E06𝟐. 𝟖𝟔𝟖𝟏𝟕𝟔𝟐 𝐖
Using (8)
𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 − 𝑃′ 𝑒)^2,0.5,1) = 𝐺𝐴𝑀𝑀𝐴. 𝐼𝑁𝑉((1 −
2. 𝐸 − 6)^2,0.5,1) 10.63242
𝑃𝑜𝑤𝑒𝑟10.63242 ∗ 2E − 08 ∗ 2.08E06 ∗ 2
∗ (
32
𝜋2
) 𝟐. 𝟖𝟔𝟖𝟏𝟕𝟔𝟓 𝐖
The text estimates 𝐸 𝑏 = 137.8𝐸 − 08 = 1.378𝐸 − 06,
𝑡ℎ𝑢𝑠, 𝑔𝑖𝑣𝑒𝑠 𝑡ℎ𝑒 𝑃𝑜𝑤𝑒𝑟 𝟐. 𝟖𝟔𝟔𝟐𝟒𝟎𝟎 𝑾. Our estimates of
𝐸 𝑏 =1.3789309E-06 for the Gaussian formula and 𝐸 𝑏 =
1.3789310𝐸 − 06 for the Gamma methodology.
Note that the power estimates are congruent until the
seventh significant figure between the two approaches. Again,
the use of the Gamma function is the most straightforward
method.
7. > PAPER IDENTIFICATION NUMBER TBD< 7
D. MATH EXAMPLE 4: RODDY EXAMPLE 10.1 page
304
The author interest is to estimate BER for binary polar
transmission; given 𝐸 𝑏 = 1𝐸 − 06 & 𝑁 = 1𝐸 − 07. This
example is surprisingly confusing as simple as it can look. As
known,
𝐸 𝑏
𝑁
=
1𝐸 − 06
1𝐸 − 07
= 10
Now, the author uses erf instead of erfc. The relationship
between these two functions is that their sum =1. That is,
erf(𝑥) + 𝑒𝑟𝑓𝑐(𝑥) = 1. (12)
Now, in order to properly solve the problem, you must
know that for polar signaling the correct formulas are the ones
we used with the above example:
𝑃𝑒 = 𝑄 (√
2𝐸 𝑏
𝑁
) (13)
Solving,
𝑃𝑒 = 𝑄 (√
2𝐸 𝑏
𝑁
) = 𝑄 (√(2 ∗ 10)) =
𝑄(√20) 𝑄(4.472135955)3.872108𝐸 − 06
It is important to understand that 𝑧 = √20 4.472135955
here; thus, 𝑥 =
𝑧
√2
=
√20
√2
= √103.16227766
Using erfc,
𝑃𝑒 = 𝑒𝑟𝑓𝑐(𝑥) =
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(𝑥2
, 0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(3.162277662
, 0.5,1,1)
2
1 − 𝐺𝐴𝑀𝑀𝐴. 𝐷𝐼𝑆𝑇(𝟏𝟎, 0.5,1,1)
2
3.872108𝐸 − 06
The author’s solution using a graphical approach is
surprisingly precise, it gives 𝑃𝑒 = 3.9𝐸 − 06.
And again, the Gamma approach is the easier and
straightforward methodology to find the result.
A student learning from different texts could get very
confused as to what value to input in which function in order
to correctly estimate the BER. This text directs the student to
look for erf(√10) erf(3.16227766). There are no details on
how to do this.
IV. CONCLUSION
It could be appreciated that using examples from three
different authors in texts with different topics the easier way to
avoid confusions and get the correct results with the least
effort was to use the proposed Excel Gamma function.
It is a mistake not to explain in detail the use of the Q(z)
and ercf(x) functions in most communication courses as it is
looked as an easy task not worthy of the time. This paper
illustrates the correct way to code with Excel in order to
always obtain the correct result.
V. APPENDIX
Appendices not needed in this work.
REFERENCES
[1] B.P. Lathi Modern Digital and Analog Communications
Systems 3rd
Ed., Oxford University Press NY, 1998.
[2] T. Pratt, C. W. Bostian & J. E. Allnut Satellite
Communications 2nd
Ed. John Wiley & Sons, MA,
2003.
[3] D. Roddy Satellite Communications 4th
Ed. Mc-Graw
Hill, NY 2006.
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