Mulailah

Tanpa permulaan, anda tidak akan sampai ke mana-mana.

Semangat

Semangat yang kuat mampu mengatasi apapun cobaan yang datang.

Konsisten

Lumbung emas dalam diri kamu adalah pikiran kamu. Kamu dapat menggalinya sedalam-dalamnya dan sepuas-puas yang kamu inginkan.

Pantang Menyerah

Gagal selepas usaha adalah hikmah, anda akan mendapat sesuatu yang lebih besar daripada apa yang anda sangkakan.

Be The One

Be the one is better than be the best.

Tuesday, January 17, 2012

Mobile and Personal Communication : Past, Present and Future


PAST
From the introduction of public mobile radio in the United States in 1946 until the first analog cellular system went into operation in Chicago in 1983,mobile radio systems were based on the trunking principle. In other words,the available frequency spectrum (in the 150 or 450 MHz band) was divided into a suitable number of frequency channels. A centralized, high power antenna was used to transmit signals to mobile receivers. Large mobile receivers were installed in automobiles (in the trunks), and the telephone sets also were rather large. A call originating from or terminating on a mobile terminal had to compete for one of the limited number of channels. The quality of service in terms of call blocking probabilities was very high—in the order of 20-25%. However, the users were willing to trade off the convenience of mobility against the poor quality of service in terms of call blocking and received signal quality. These systems were also severely limited in terms of capacity and coverage.

To alleviate the high blocking problem in the early systems, efforts were made to allow call originations from the mobile telephones to wait for a free channel. In the so-called automated mobile telephone system (AMTS), the mobile telephone user would key in the called number and press the send button. The receiver system would then start scanning for an idle channel by cycling through all the channels in the system. In some systems, the number of scan cycles was restricted, so that if an idle channel was not found within the allowed number of scans, the call would be blocked. However, incoming calls to mobile terminals (mostly originating from fixed terminals in public switched telephone networks) had no mechanism for awaiting a free channel and were blocked on all-channels-busy condition. Though the improvement in the quality of service in these systems was only marginal, they did provide some interesting performance modeling problems



PRESENT
Since the initial commercial introduction of advanced mobile phone system (AMPS) service in 1983, mobile communications has seen an explosive growth worldwide. Besides the frequency reuse capabilities provided by the cellular operation, advances in technologies for wireless access, digital signal processing, integrated circuits, and increased battery life have contributed to exponential growth in mobile and personal communication services. Systems are evolving to address a range of applications and markets, which include digital cellular, cordless telephony, satellite mobile, and paging and specialized mobile radio systems. Data capabilities of these systems are also coming into focus with the increasing user requirements for mobile data communications, driven by the need for e-mail and Internet access. Whereas the analog cellular mobile systems fall in the category of first-generation mobile systems, the digital cellular, low power wireless, and personal communication systems are now perceived as second-generation mobile/PCS systems.

The first digital cellular system specification was released in 1990 by the European Telecommunications (ETSI) for the global system for mobile communication (GSM) system. The GSM, DCS 1800 (1800 MHz version of GSM), and DECT (digital enhanced cordless telecommunications) systems developed by ETSI form the basis for mobile and personal communication services not only in Europe but in many other parts of the world including North America. The number of GSM subscribers worldwide exceeds 100 million and is growing rapidly.

FUTURE
To complement the cellular and personal communication networks, whose radio coverage will be confined to populated areas of the world (less than 15% of the earth's surface), a number of global mobile satellite systems are in advanced stages of planning and implementation. These systems are generally referred as global mobile personal communications by satellites (GMPCS).GMPCS systems like Iridium, Globalstar, and ICO use constellations of low earth orbit (LEO) or medium earth orbit (MEO) satellites and operate as overlay networks for existing cellular and PCS networks. Using dual-mode terminals, they will extend the coverage of cellular and PCS networks to any and all locations on the earth's surface. On the other hand, a LEO satellite system like Teledesic aims to provide high capacity satellite links to enable delivery of high bitrate and multimedia services to every location on the earth.
International Mobile Telecommunications -2000 (IMT-2000) is the standard being developed by the ITU to set the stage for the third generation of mobile communication systems. The IMT-2000 standard not only will consolidate under a single standard different wireless environments (cellular mobile, cordless telephony, satellite mobile services), but will also ensure global mobility in terms of global seamless roaming and delivery of services. ETSI is also developing a third-generation mobile communication system called Universal Mobile Telecommunication System (UMTS), which will belong to the family of IMT-2000 systems.
Posted by 0 comments
Location: Dubai - United Arab Emirates

Monday, January 16, 2012

Matrix Operations



Before we give the formal definition of how to multiply two matrices, we will discuss an example from a real life situation. Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city. Let I (resp. S) be the initial population of the inner city (resp. the suburban area). So after one year, the population of the inner part is


0.6 I + 0.3 S


while the population of the suburbs is

0.4 I + 0.7 S


After two years, the population of the inner city is

0.6 (0.6 I + 0.3 S) + 0.3 (0.4 I + 0.7 S)


and the suburban population is given by

0.4 (0.6 I + 0.3 S) + 0.7(0.4 I + 0.7 S)

Blue Matrix 1 Myspace Layout 2.0

Is there a nice way of representing the two populations after a certain number of years? Let us show how matrices may be helpful to answer this question. Let us represent the two populations in one table (meaning a column object with two entries):

\begin{displaymath}\left(\begin{array}{c} I\\ S\\ \end{array}\right)\end{displaymath}


So after one year the table which gives the two populations is

\begin{displaymath}\left(\begin{array}{c} 0.6 I + 0.3 S\\ 0.4 I + 0.7 S\\ \end{array}\right)\end{displaymath}


If we consider the following rule (the product of two matrices)

\begin{displaymath}\left(\begin{array}{cc} a&b\\ c&d\\ \end{array}\right) \lef... ...left(\begin{array}{c} aI + bS\\ cI + dS\\ \end{array}\right),\end{displaymath}


then the populations after one year are given by the formula

\begin{displaymath}\left(\begin{array}{cc} 0.6&0.3\\ 0.4&0.7\\ \end{array}\right) \left(\begin{array}{c} I\\ S\\ \end{array}\right).\end{displaymath}


After two years the populations are

\begin{displaymath}\left(\begin{array}{cc} 0.6&0.3\\ 0.4&0.7\\ \end{array}\rig... ...ght) \left(\begin{array}{c} I\\ S\\ \end{array}\right)\Bigg).\end{displaymath}


Combining this formula with the above result, we get

\begin{displaymath}\left(\begin{array}{cc} 0.6&0.3\\ 0.4&0.7\\ \end{array}\rig... ...imes 0.4&0.4 \times 0.3 + 0.7 \times0.7\\ \end{array}\right). \end{displaymath}


In other words, we have

\begin{displaymath}\left(\begin{array}{cc} a&b\\ c&d\\ \end{array}\right) \lef... ...{array}{cc} ae+ bg&af+bh\\ ce + dg&cf+dh\\ \end{array}\right)\end{displaymath}

In fact, we do not need to have two matrices of the same size to multiply them. Above, we did multiply a (2x2) matrix with a (2x1) matrix (which gave a (2x1) matrix). In fact, the general rule says that in order to perform the multiplication AB, where A is a (mxn) matrix and B a (kxl) matrix, then we must have n=k. The result will be a (mxl) matrix. For example, we have

\begin{displaymath}\left(\begin{array}{ccc} a&b&c\\ d&e&f\\ \end{array}\right)... ...a +b\beta +c\nu\\ d\alpha +e\beta +f\nu\\ \end{array}\right).\end{displaymath}


Remember that though we were able to perform the above multiplication, it is not possible to perform the multiplication

\begin{displaymath}\left(\begin{array}{c} \alpha\\ \beta\\ \nu\\ \end{array}\... ...)\left(\begin{array}{ccc} a&b&c\\ d&e&f\\ \end{array}\right).\end{displaymath}


So we have to be very careful about multiplying matrices. Sentences like "multiply the two matrices A and B" do not make sense. You must know which of the two matrices will be to the right (of your multiplication) and which one will be to the left; in other words, we have to know whether we are asked to perform $A \times B$ or $B \times A$. Even if both multiplications do make sense (as in the case of square matrices with the same size), we still have to be very careful. Indeed, consider the two matrices

\begin{displaymath}\left(\begin{array}{cc} 0&1\\ 0&0\\ \end{array}\right)\;\mb... ...nd}\; \left(\begin{array}{cc} 0&0\\ 1&0\\ \end{array}\right).\end{displaymath}


We have

\begin{displaymath}\left(\begin{array}{cc} 0&1\\ 0&0\\ \end{array}\right)\left... ...ht) = \left(\begin{array}{cc} 1&0\\ 0&0\\ \end{array}\right) \end{displaymath}


and

\begin{displaymath}\left(\begin{array}{cc} 0&0\\ 1&0\\ \end{array}\right)\left... ...ht) = \left(\begin{array}{cc} 0&0\\ 0&1\\ \end{array}\right).\end{displaymath}


So what is the conclusion behind this example? The matrix multiplication is not commutative, the order in which matrices are multiplied is important. In fact, this little setback is a major problem in playing around with matrices. This is something that you must always be careful with. Let us show you another setback. We have

\begin{displaymath}\left(\begin{array}{cc} 0&1\\ 0&0\\ \end{array}\right)\left... ...egin{array}{cc} 0&0\\ 0&0\\ \end{array}\right);\;\mbox{i.e.},\end{displaymath}


the product of two non-zero matrices may be equal to the zero-matrix.




Properties involving Addition. Let A, B, and C be mxn matrices. We have
1.
A+B = B+A
2.
(A+B)+C = A + (B+C)
3.
$A + {\cal O} = A$
where $\cal O$ is the mxn zero-matrix (all its entries are equal to 0);
4.
$A+B = {\cal O}$ if and only if B = -A.

Properties involving Multiplication.

1.
Let A, B, and C be three matrices. If you can perform the products AB, (AB)C, BC, and A(BC), then we have

(AB)C = A (BC)


Note, for example, that if A is 2x3, B is 3x3, and C is 3x1, then the above products are possible (in this case, (AB)C is 2x1 matrix).
2.
If $\alpha$ and $\beta$ are numbers, and A is a matrix, then we have

\begin{displaymath}\alpha (\beta A) = (\alpha \beta) A\end{displaymath}


3.
If $\alpha$ is a number, and A and B are two matrices such that the product $A\cdot B$ is possible, then we have

\begin{displaymath}\alpha (AB) = (\alpha A)B = A (\alpha B)\end{displaymath}


4.
If A is an nxm matrix and $\cal O$ the mxk zero-matrix, then

\begin{displaymath}A {\cal O} = {\cal O}\end{displaymath}


Note that $A {\cal O}$ is the nxk zero-matrix. So if n is different from m, the two zero-matrices are different.

Properties involving Addition and Multiplication.
1.
Let A, B, and C be three matrices. If you can perform the appropriate products, then we have

(A+B)C = AC + BC


and

A(B+C) = AB + AC


2.
If $\alpha$ and $\beta$ are numbers, A and B are matrices, then we have

\begin{displaymath}\alpha (A+B) = \alpha A + \alpha B\end{displaymath}


and

\begin{displaymath}(\alpha +\beta)A = \alpha A + \beta B\end{displaymath}


Example. Consider the matrices



\begin{displaymath}A = \left(\begin{array}{cc} 0&1\\ -1&0\\ \end{array}\right)... ...nd}\; C = \left(\begin{array}{ccc} 0&1&5\\ \end{array}\right).\end{displaymath}



Evaluate (AB)C and A(BC). Check that you get the same matrix.
Answer. We have



\begin{displaymath}AB = \left(\begin{array}{c} -1\\ -2\\ \end{array}\right)\end{displaymath}



so



\begin{displaymath}(AB)C = \left(\begin{array}{c} -1\\ -2\\ \end{array}\right)... ...t(\begin{array}{ccc} 0&-1&-5\\ 0&-2&-10\\ \end{array}\right).\end{displaymath}



On the other hand, we have



\begin{displaymath}BC = \left(\begin{array}{ccc} 0&2&10\\ 0&-1&-5\\ \end{array}\right)\end{displaymath}



so



\begin{displaymath}A(BC) = \left(\begin{array}{cc} 0&1\\ -1&0\\ \end{array}\ri... ...t(\begin{array}{ccc} 0&-1&-5\\ 0&-2&-10\\ \end{array}\right).\end{displaymath}




Example. Consider the matrices



\begin{displaymath}X = \left(\begin{array}{c} a\\ b\\ c\\ \end{array}\right)\... ...ray}{cccc} \alpha & \beta & \nu & \gamma\\ \end{array}\right).\end{displaymath}



It is easy to check that



\begin{displaymath}X = a \left(\begin{array}{c} 1\\ 0\\ 0\\ \end{array}\right... ...) + c \left(\begin{array}{c} 0\\ 0\\ 1\\ \end{array}\right) \end{displaymath}



and



\begin{displaymath}Y = \alpha \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ \end{ar... ...ma \left(\begin{array}{cccc} 0 &0 & 0& 1\\ \end{array}\right).\end{displaymath}



These two formulas are called linear combinations. More on linear combinations will be discussed on a different page.
We have seen that matrix multiplication is different from normal multiplication (between numbers). Are there some similarities? For example, is there a matrix which plays a similar role as the number 1? The answer is yes. Indeed, consider the nxn matrix



\begin{displaymath}I_n = \left(\begin{array}{ccccc} 1&0&0&\cdots&0\\ 0&1&0&\cdo... ...cdot\\ \cdot&&&&\cdot\\ 0&0&0&\cdots&1\\ \end{array}\right).\end{displaymath}



In particular, we have



\begin{displaymath}I_2 = \left(\begin{array}{ccc} 1&0\\ 0&1\\ \end{array}\righ... ...egin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{array}\right).\end{displaymath}



The matrix In has similar behavior as the number 1. Indeed, for any nxn matrix A, we have



A In = In A = A



The matrix In is called the Identity Matrix of order n.
Example. Consider the matrices



\begin{displaymath}A = \left(\begin{array}{cc} 1&2\\ -1&-1\\ \end{array}\right... ...B = \left(\begin{array}{cc} -1&-2\\ 1&1\\ \end{array}\right).\end{displaymath}



Then it is easy to check that



\begin{displaymath}AB = I_2 \;\;\mbox{and}\;\; BA = I_2.\end{displaymath}




The identity matrix behaves like the number 1 not only among the matrices of the form nxn. Indeed, for any nxm matrix A, we have



\begin{displaymath}I_n A = A\;\;\mbox{and}\;\; A I_m = A.\end{displaymath}



In particular, we have



\begin{displaymath}I_4 \left(\begin{array}{c} a\\ b\\ c\\ d\\ \end{array}\ri... ... \left(\begin{array}{c} a\\ b\\ c\\ d\\ \end{array}\right).\end{displaymath}
Posted by 0 comments
Subscribe to: Posts (Atom)