Biblio5

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probably mean “the power expressed in dBW.” We might discuss the velocities of two cars. The velocity of car 1 may be expressed as V1, and the velocity of car 2 as V2. The use of the subscript allows us to distinguish between the two cars. Subscripts are used widely in the text.

Independent and Dependent Variables. In the selection of symbols to solve a particular problem, we must distinguish between constants and variables specific to the problem at hand. Consider the volume of a circular cone where we keep its height constant (h). The formula for its volume is V pr2h/ 3, where r is the radius of the base circle. The height, h, of course, is constant cfor this problem, and just for this problem alone. It is referred to as a parameter. The radius, r, is the independent variable and V is the dependent variable.

B.2.2 Function Concept

P c 75 × 2 × 2 c 300 W.

That is to say, that the variable P depends on the variable I.

In general, we can say that if a variable y depends on another variable x so that for each value of x (in a suitable set), a corresponding value of y is determined. The variable y then is called a function of x. In symbols this is written as:

y c f (x).

Read this as “y is the function of x” or just “y equals f of x.” Remember here that f is not a qualitative symbol but an operative symbol.

This is Eq. (9.12) from Chapter 9 of the text. Note the use of an absolute constant, in this case Boltzmann’s constant. Pn is the independent variable; T, the noise temperature in kelvins, and B, the bandwidth in Hz, are dependent variables. We could set the bandwidth at 1 MHz. Then Pn c f(T).

B.2.3 Using the Sigma Notation

The Greek letter S (capital sigma) indicates summation. The index of summation limits the number of items to be summed or added up. The letter i is often used for this purpose. Values for i are placed above and below sigma. The initial value is placed below, and the final value is placed above the letter sigma. For example, we could have

Swi c w1 + w2 + w3 + w4 . . . + w9 + w10.

Here the initial value is 1 and the final value is 10.

B.3 INTRODUCTORY ALGEBRA

B.3.1 Review of the Laws of Signs

•If we multiply a +factor by another +factor, the product will have a + sign.

•If we divide a +factor by another +factor, the quotient will have a + sign.

B.3.2 Conventions with Factors and Parentheses

Two symbols placed together imply multiplication. Each symbol or symbol grouping is called a factor. Examples:

xy means x multiplied by y;

x( y + 1) means x multiplied by the quantity ( y + 1);

abc means a multiplied by b that is then multiplied by c.

The use of parentheses is vital in algebra. If a parentheses pair has a sign in front of it, that sign is operative on each term inside the parentheses. If the sign is just a +, then consider that each term inside the parentheses is multiplied by +1. Here we just have a flow through.

An algebraic expression may have parentheses inside brackets. The rule is to clear

the “outside” first. In other words, clear the brackets. Then clear the parentheses in that order.

1A factor is a value, a symbol that we multiply by. To factor an expression is to break the expression up into components and when these components are multiplied together, we derive the original expression. For example 12 and 5 are factors of 60 because 12 × 5 c 60.

this case it is 24. The reason: 24 is divisible by 4, 8, and 12. The three terms are now converted to 24ths or:

Subtraction of fractions follows the same procedure, just follow the rules of signs.

Example: Calculate 1/ 4 − 1/ 5 + 2/ 3.

Adding and Subtracting Exponents. An exponent is a number at the right of and above a symbol. The value assigned to the symbol with this number is called a power of the symbol, although power is sometimes used in the same sense as exponent. If the

A negative exponent indicates, in addition to the operations indicated by the numerical value of the exponent, that the quantity is to be made a reciprocal.

Further, when addition is involved, and the numbers have a common exponent, we

particularly easy to do with a scientific calculator using the Xy function, where y can even be a decimal such as −3.7.

B.3.3 Simple Linear Algebraic Equations

An equation is a statement of equality between two expressions. Equations are of two types: identities and conditional equations (or usually simply equations). A conditional

Equation (9.20) in Chapter 9 of the text is an identity. It states:

G/ T c G − 10 log T,

where G is gain and T is noise temperature. Actually the right-hand side of the equation is just a restatement of the left-hand side; it does not tell us anything new. In many cases, identities such as this one are very useful in analysis.

There are various rules for equations. An equation has a right-hand side and a lefthand side. Maintaining equality is paramount. For example, if we add some value to the left side, we must add the same value to the right side. Likewise, if we divide the (entire) left side by a value, we must divide the (entire) right by the same value. As we might imagine, we must carry out similar procedures for subtraction and multiplication.

A linear equation is of the following form: AX + B c 0 (A 0). This is an equation

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with one unknown, X. A will be a fixed quantity, a number; so will B. However, in the parentheses it states that A may not be 0.

Let us practice with some examples. In each case calculate the value of X.

X + 5 c 7

Clue: We want to have X alone on the left side. To do this we subtract 5 from the left side, but following the rules, we must also subtract 5 from the right side. Thus:

Another example is

3X + 7 c 31.

Again, we want X alone. But first we must settle for 3X. Subtract 7 from both sides of the equation.

Again, we want X alone. To do this we can divide by 3 (each side).

Still another example:

z 2 + 1 c 65 (solve for z). Subtract 1 from each side to get z 2 alone.

Take the square root of each side. Thus

z c 8.

More Complex Equations

Solve for R.

Another example: Solve for x.

B.3.4. Quadratic Equations

Quadratic equations will have one term with a square (e.g., X 2) and they take the form:

where A, B, and C are constants (e.g., numbers). A quadratic equation should always be set to 0 before a solution is attempted. For instance, if we have an equation that is

the equation statement to be true. Keep in mind that anything multiplied by 0 will be 0. So there are two solutions to the equation:

x − 1 c 0, thus x c 1 or x + 1 c 0 and x c −1

Proof that these are correct answers is by substituting them in the equation. Solve for x in this example:

x 2 − 5x − 6 c 0, factors into (x − 6, )(x + 1). c 0 Then x − 6 c 0 x c 6 or x + 1 c 0 x c −1

Quadratic Formula. This formula may be used on the conventional quadratic equation in the generic form of

The value of x is solved by simply manipulating the constants A, B, and C. The quadratic formula is stated as follows:

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x c [−B ± (B2 − 4AC)1/ 2]/ 2A or, rewritten with the radical sign:

Just as we did with the factoring method, the quadratic formula will produce two roots (two answers): one with the plus before the radical sign and one with the minus before the radical sign.

B.3.5 Solving Two Simultaneous Linear Equations with Two Unknowns

There are two methods of solving two simultaneous equations:

1. The graphical method, where both equations are plotted and the intersection of the line derived is the common solution; and

2. The algebraic solution.

We will concentrate on the algebraic solution. There are two approaches to solving two simultaneous equations by the algebraic solution:

1. Elimination; and

2. Substitution.

Elimination Method. With this method we manipulate one of the equations such that when the two equations are either added or subtracted, one of the unknowns is eliminated. We then solve for the other unknown. The solution is then substituted in one of the original equations and we solve for the other unknown.

Example:

2x + 3y − 8 c 0. 4x − 5y + 6 c 0

Multiply each term by 2 in the upper equation, and we derive the following new equation:

4x + 6y − 16 c 0.

Place the second equation directly below this new equation, and subtract.

If we subtract the lower equation from the upper, we eliminate the 4x term. Now solve for y.

+11y −1122y c 022

y cc 2.

Example:

3x − 2y − 5 c 0. 6x + y + 12 c 0

There are several possibilities to eliminate one of the unknowns. This time let us multiply each term in the lower equation by two and we get:

12x + 2y + 24 c 0.

Place this new equation below the original upper equation:

Add the two equations and we get 15x + 19 c 0. Solve for x.

Substitution Method. Select one of the two simultaneous equations and solve for one of the unknowns in terms of the other. Example: (repeating the first example from above)

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2x + 3y − 8 c 0. 4x − 5y + 6 c 0

We can select either equation. Select the first equation. Then:

Substitute this value for x in the second equation.

4(8 − 3y)/ 2 − 5y + 6 c 0

16 − 6y − 5y 11+ 6y c 0 22

y−c 2. c −

B.4 LOGARITHMS TO THE BASE 10

B.4.1 Definition of Logarithm

If b is a positive number different from 1, the logarithm of the number y, written loga y, is defined as follows: if ax c y, then x is a logarithm of y to the base a, and we write:

This shows, therefore, that a logarithm is an exponent—the exponent to which the base is raised to yield the number. The expression loga y is read: “logarithm of y to the base

B.4.2 Laws of Logarithms

In Section B.3 we discussed the laws of adding and subtracting exponents. From these laws we can derive the laws of logarithms. Let us say that the generalized base of a logarithm is a, which is positive, and that x and y are real numbers. Here we mean they are not imaginary numbers (i.e., based on the square root of 1).

logarithm by using the “log” button. There will also probably be an “ln” button. This button is used to obtain logarithms to the natural base, where a c 2.71828183+.

Law 1. The logarithm of the product of two numbers equals the sum of the logarithms of the factors. That is:

loga xy c loga x + loga y.

Law 2. The logarithm of the quotient of two numbers equals the logarithm of the dividend minus the logarithm of the divisor. That is:

loga x/ y c loga x − loga y.

Law 3. The logarithm of the nth power of a number equals n times the logarithm of the number. That is:

loga xn c n loga x.

Law 4. The logarithm of the pth root of a number is equal to the logarithm of the number divided by n. That is:

The characteristic is an integer and the mantissa is a decimal. If the number in question is 10 or its multiple, the logarithm has a characteristic only, and its mantissa is .000000++. Consider Table B.1 containing selected the powers of 10.

A scientific calculator gives both the characteristic and the mantissa when a number is entered and we press the “log” button. On most calculators, just enter the number then press the “log” button. Its logarithm (to the base 10) will then appear in the display.

Table B.2 gives ten numbers and their equivalent logarithms. One of the logarithms is blatantly in error. Identify it.

Deriving one type of logarithm may be tricky. This group consists of decimals, in other words, numbers less than one (< 1). First check column 4 of Table B.1.

Find the logarithm of 0.00783. In scientific notation we can express the number as

wish to find the number that generated that logarithm. This is shown on the calculator, usually on the second level, log−1. Sometimes in the literature it is indicated as the antilog.

Example. Given the logarithm, find its corresponding number. Use a scientific calculator.