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Find the exponential Fourier series for the waveform shown in Fig. 17-53 and plot the spectrum. Convert the trigonometric series coe cients found in Problem 17.20 into exponential series coe cients and compare them with the coe cients of the series obtained here. Ans: 1 j2!t 1 1 1 j2!t e e f t V j j e j!t j e j!t j 2 2

12.25 12.25 12.25 2.25 6.25 1.00 6(72.50) 12(122 1)

Find the exponential Fourier series for the waveform shown in Fig. 17-54 and plot the spectrum. Convert the coe cients to trigonometric series coe cients, write the trigonometric series, and compare it with that obtained in Problem 17.21. 1 j3!t 2 1 j!t 2 1 j!t 2 j 2 j e e e 3 1 e j3!t 3

6 a d2 n(n2 1)

0.7027 obtained in Problem 8.26(b).

Ans:

METRIC 1: WEIGHTED METHODS PER CLASS (WMC)

Probability interpretation of regression and correlation 8.39. Derive (39) from (37).

Fig. 17-54

For the least-squares regression line we must consider E5[Y (a bX)]26 E5[(Y E[(Y s2 Y mY) mY)2] b2s2 X b(X 2bsXY mX) (mY (mY mX)2] bmX bmX 2bE[(X a)2 a)]26 mX)(Y mY)] (mY bmX a)2

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where we have used E(X mX) 0, E(Y mY) 0. Denoting the last expression by F(a, b), we have 'F 'a 2(mY bmX a), 'F 'b 2bs2 X 2sXY 2mX(mY bmX a)

Fig. 17-55

Setting these equal to zero, which is a necessary condition for F(a, b) to be a minimum, we find mY Therefore, if y a bx, then y mY y or y a b(x mY mY sY bmX mX) or sXY (x s2 X ra x mX) mX sX b bs2 X sXY

Find the exponential Fourier series for the square wave shown in Fig. 17-55 and plot the line spectrum. Convert the trigonometric series coe cients of Problem 17.22 into exponential series coe cients and compare with the coe cients in the result obtained here. Ans: f t 2V 1 e j5!t 1 e j3!t e j!t e j!t 1 e j3!t 1 e j5!t 5 3 3 5

The similarity of the above proof for populations, using expectations, to the corresponding proof for samples, using summations, should be noted. In general, results for samples have analogous results for populations and conversely.

The weighted methods per class metric is based on the intuition that the number of methods per class is a signi cant indication of the complexity of the software. Further consideration may be necessary to avoid giving full weight to trivial methods, for example, get and set methods. The WMC includes the provision for weighting the methods. Let C be a set of classes each with the number of methods M1 ; . . ., Mn . Let c1 ; . . ., cn be the complexity (weights) of the classes (the value that is to be used for ci is not de ned in the paper).

8.40. The joint density function of the random variables X and Y is f (x, y)

Find the exponential Fourier series for the waveform shown in Fig. 17-56 and plot the line spectrum. V 2 j2!t V V V sin e j!t sin e j!t sin e Ans: f t 2 6 6 6 6 V 2 j2!t sin e 2 6

Find the least-squares regression curve of (a) Y on X, (b) X on Y. (a) The marginal density function of X is f1(x)

Fig. 17-56

2y 1

and the least-squares regression curve of Y on X is given by y E(Y Z X x) 3 yf2(y Z x) dy 2y b dy 1 3x 6x 4 6 0 or x 1.

Fig. 17-57

Neither f2( y Z x) nor the least-squares regression curve is defined when x (b) For 0 y 1, the marginal density function of Y is f2( y) Hence, for 0 y

Find the exponential Fourier series for the half-wave-recti ed sine wave shown in Fig. 17-57. Convert these coe cients into the trigonometric series coe cients, write the trigonometric series, and compare it with the result of Problem 17.24. Ans: f t V j4!t V j2!t V j!t V V j!t V j2!t V j4!t e e e e e e 15 3 4 4 3 15

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