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'system Analysis by Digital Computer. F. F. Kuo and J. F. Kaiser, Eds.. John Wiley and Sons, New York. 1966

31.31 The velocity v of an object moving on the x-axis is cos 3t. The object is at the origin at t = 0. (a) Find a formula for the position x at any time t. (b) Find the average value of the position x over the interval 0 t /3. (c) For what values of t in [0, /3] is the object moving to the right (d) What are the maximum and minimum x-coordinates of the object 31.32 An object moves on a straight line with velocity v = 3t 1, where v is measured in meters per second. How far does the object move in the period of 0 t 2 seconds [Hint: Apply the fundamental theorem.] 31.33 Evaluate: 2 n + sin + + sin n n n + sec2 2 + + sec2 (n 1) +2 (b) lim sec2 n 4n 4n 4n 4n (a)

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Find to the nearest degree the measure of the angle of elevation of the sun when a tree 60 ft high casts a shadow of (a) 10 ft; (b) 60 ft.

A system is said to be allpass filter if the frequency response magnitude is constant: 1H(eJ")l = c An example of an allpass filter is the system that has a frequency response

where ct is any real number with l 1 < I. The unit sample response of this allpass filter is a h(n) = -a8(n)

At a certain time of day, the angle of elevation of the sun measures 34 . Find, to the nearest foot, the length of the shadow cast by (a) a 15-ft vertical pole; (b) a building 70 ft high. (8.8) A light at C is projected vertically to a cloud at B. An observer at A, 1000 ft horizontally from C, notes the angle of elevation of B. Find the height of the cloud, to the nearest foot, if m A 37 . (8.8) A lighthouse built at sea level is 180 ft high (Fig. 8-21). From its top, the angle of depression of a buoy measures 24 . Find, to the nearest foot, the distance from the buoy to the foot of the lighthouse. (8.8)

+ (1 - ct2)an-'u(n - 1)

Fig. 8-21

31.34 (Midpoint Rule) In a Riemann sum (30.1),

Many of the filters that are important in applications have piecewise constant frequency response magnitudes. These include the low-pass, high-pass, bandpass, and bandstop filters that are illustrated in Fig. 2-1. The intervals over which the frequency response magnitude is equal to 1 are called the passbands, and the intervals over which it is equal to 0 are called the stopbands. The frequencies that mark the edges of the passbands and stopbands are the cutofffrequencies.

An observer, on top of a hill 300 ft above the level of a lake, sighted two ships directly in line. Find, to the nearest foot, the distance between the boats if the angles of depression noted by the observer measured (a) 20 and 15 ; (b) 35 and 24 ; (c) 9 and 6 . (8.10) In Fig. 8-22, m A 43 , m BDC 54 , m C the result of (a), find the length of AB. 90 , and DC 170 ft. (a) Find the length of BC. (b) Using (8.10)

w-=a-n -

Fig. 8-22

Fig. 2-1.

Fig. 8-23

1 resulting sum is said to be obtained by the midpoint rule. Use the midpoint rule to approximate 0 x 2 dx, using a division into ve equal subintervals, and compare with the exact result obtained by the fundamental theorem.

Filters are often interconnected to create systems that have desirable properties. Two common types of connections are series (cascade) and parallel. A cascade of two linear shift-invariant systems is shown in the figure below.

H (el") = H l ( e j W ) ~ 2 ( e J W )

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