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EXAMPLE 1.17 The number of different permutations of the 11 letters of the word M I S S I S S I P P I, which consists of 1 M, 4 I s, 4 S s, and 2 P s, is

11! 1!4!4!2!

CHAP. 12]

34,650

Fig. 12-9

In a permutation we are interested in the order of arrangement of the objects. For example, abc is a different permutation from bca. In many problems, however, we are interested only in selecting or choosing objects without regard to order. Such selections are called combinations. For example, abc and bca are the same combination. The total number of combinations of r objects selected from n (also called the combinations of n things taken n r at a time) is denoted by nCr or a b. We have (see Problem 1.27) r n r

n! r!(n r)!

Hv1 ! with !x R2 =L1 ; that is,

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Design is the process of applying various techniques and principles for the purpose of de ning a device, a process, or a system in su cient detail to permit its physical realization. 1 Design is also the most artistic or creative part of the software development process. Few rules can be written to guide design. The design process converts the what of the requirements to the how of the design. The results of the design phase should be a document that has su cient detail to allow the system to be implemented without further interaction with either the speci er or the user. The design process also converts the terminology from the problem space of the requirements to the solution space of the implementation. Some authors talk about object-oriented analysis (OOA) objects, which are in the problem/domain space, and object-oriented design (OOD) objects, which are in the solution/implementation space. For example, in the problem space we can talk about a realworld object like a person; in the solution space we can talk about a C class called person. Gunter et al.2 write about the phenomenon in the environment (world) and the phenomenon in the implementation (machine). A phenomenon can be visible or hidden. The user-oriented requirements may be expressed in terms of the phenomenon, hidden or visible, from the environment. However, the speci cation that will be used as the basis for development must sit between the environment and the implementation and must be expressed in terms of a visible phenomenon from each. This speci cation is the starting point for design and will be called the development speci cation in this book.

(29)

The magnitude and angle plots are shown in Fig. 12-10. The voltage transfer function Hv1 approaches zero at high frequencies and unity at ! 0. Hence the name low-pass.

1) c (n r!

(30)

Fig. 12-10

(31)

EXAMPLE 12.2 Obtain the voltage transfer function Hv1 for the open circuit shown in Fig. 12-11. p frequency, in hertz, does jHv j 1= 2 if (a) C2 10 nF, (b) C2 1 nF At what

8 3

Taylor, An Interim Report of Engineering Design, MIT, 1959. Gunter, Gunter, Jackson, and Zave, A Reference Model for Requirements and Speci cations, IEEE Software, May/June 2000, 37 43.

8 7 6 3!

Fig. 12-11

The numbers (29) are often called binomial coefficients because they arise in the binomial expansion n x n 1 y 1

1 jHv j q 1 !=!x 2 p and so jHv j 1= 2 when ! !x or when f 2 104 =2 3:18 kHz. 2 10 4 2 104 rad=s 10 10 9

2 y2

Comparing a and b , it is seen that the greater the value of C2 , the lower is the frequency at which jHv j drops to 0.707 of its peak value, 1; in other words, the more is the graph of jHv j, shown in Fig. 1210, shifted to the left. Consequently, any stray shunting capacitance, in parallel with C2 , serves to reduce the response of the circuit.

4 y4 4

(32)

HALF-POWER FREQUENCIES The frequency !x calculated in Example 12.2, the frequency at which jHv j 0:707jHv jmax

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