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If an error happens after 2 days, what is the probability that the system will not fail in 1, 2, 3, and 4 days If an error happens every 2 days, we can use 0.5 as the instantaneous error rate. It can also be used to estimate the failure probability for 1 day. Thus, F(1) = 0.5. Then, R 1 1 F 1 0:5: R 2 0:25: R 3 0:125: R 4 0:0625.

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The average power entering the circuit during one cycle is equal to the average power absorbed by the resistor. It is the same result obtained in Example 10.1. The power exchanged between the source and the circuit during one cycle also agrees with the result obtained in Example 10.2.

eiv(

A 1-V ac voltage feeds (a) a 1-

3.21. Find the characteristic function of the random variable X having density function given by f (x)

E(eivX)

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In the circuit of Fig. 12-65(b), let R 1 ; C1 1:394 F; C2 0:202 F, and C3 3:551 F. Find H s V2 =V1 and show that it approximates the passive third-order Butterworth low-pass lter of Fig. 12-65(a). Ans: H s 1= 0:99992s3 1:99778s2 2s 1 Show that the half-power cuto frequency in the circuit of Fig. 8-42 is !0 1= RC and, therefore, frequency scaling may be done by changing the value of C or R. Ans: V2 2 2 1 ; !0 2 V1 R2 C2 s2 RCs 1 RC s s 1 !0 !0

3.22. Find the characteristic function of the random variable X having density function f (x) ` x ` , where a 0, and c is a suitable constant.

If we can see a trend in the error rates, then we can estimate the error rate better. Instead of using equations to t the data, plots of the failure rate can be used to visualize the behavior. If x is the inter-failure time, 1=x is the instantaneous failure rate. Plot the instantaneous failure rate versus either failure number or the elapsed time of the failure. Try to t a straight line to the points. The value of the line at the current time can be used for the error rate. The intersection of this line with the horizontal axis indicates either the fault number where the failure rate goes to zero, or the amount of time necessary to remove all the faults. When the x-axis is the elapsed time, then the area under the straight line (units are time failure/time) represents the number of faults. Thus, empirical data about how often the software fails during testing or observation is used to estimate the current rate. Theoretical ideas will be used to re ne the predictions for longer periods of time.

Find RLC values in the low-pass lter of Fig. 12-65(a) to move its half-power cuto frequency to 5 kHz. Ans: R 1 ; C 31:83 mF; L 63:66 mH

ce a|x|,

30 e 2

13.1 TERMINALS AND PORTS In a two-terminal network, the terminal voltage is related to the terminal current by the impedance Z V=I. In a four-terminal network, if each terminal pair (or port) is connected separately to another circuit as in Fig. 13-1, the four variables i1 , i2 , v1 , and v2 are related by two equations called the terminal characteristics. These two equations, plus the terminal characteristics of the connected circuits, provide the necessary and su cient number of equations to solve for the four variables.

a > 2. The characteristic function is therefore given by E(eivX )

Fig. 13-1

30 e e (a (a a iv)

a e(a iv)x 2 0 2 a iv ` a 2(a iv)

The terminal characteristics of a two-port network, having linear elements and dependent sources, may be written in the s-domain as V1 Z11 I1 Z12 I2 V2 Z21 I1 Z22 I2 1

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