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for all n. This is equivalent to saying that the sequence repeats itself every N samples. If a signal is periodic with period N , it is also periodic with period 2 N , period 3 N , and all other integer multiples of N. The fundamental period, which we will denote by N , is the smallest positive integer for which Eq. (I . I ) is satisfied. If Eq. (1. I ) is not satisfied for any integer N , x ( n ) is said to be an aperiodic signal.

7.28. In each part of Fig. 7-53, determine the angles that can be used to prove the indicated triangles similar.

xZ(n)= cos(n2) x3(n) = e ~ ~ ' ' l '

1 1 g(x+h) g(x)

(7.19)

If xl (n) is a sequence that is periodic with a period N1,and x2(n)is another sequence that is periodic with a period N2, the sum x(n)=x ~ ( n ) xdn)

Fig. 7-53

where gcd(NI, N2) means the greatest common divisor of N1 and N2. The same is true for the product; that is,

will be periodic with a period N given by Eq. (1.2). However, the fundamental period may be smaller. Given any sequence x ( n ) , a periodic signal may always be formed by replicating x ( n ) as follows:

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7.29. In each part of Fig. 7-54, determine the pair of congruent angles and the proportion needed to prove the indicated triangles similar. (7.20)

A discrete-time signal will often possess some form of symmetry that may be exploited in solving problems. Two symmetries of interest are as follows:

g(x) g(x + h)

Fig. 7-54

Definition: A real-valued signal is said to be even if, for all n ,

7.30. In each part of Fig. 7-55, state the proportion needed to prove the indicated triangles similar.

x(n) = x(-n)

(7.21)

whereas a signal is said to be odd if, for all n ,

h 0 hg(x)g(x + h)

x(n) = -x(-n)

Fig. 7-55

Any signal x ( n ) may be decomposed into a sum of its even part, x,(n), and its odd part, x,(n), as follows:

(7.25)

x(n> = x d n )

Fig. 7-56

[by algebra:

(1.3)

(7.22)

x,(n) = ( x ( n )

+x(-n))

Fig. 7-57

x,(n) = i ( x ( n ) - x ( - n ) )

35. Find a. 84. Find AB. 7. Find d. (7.22)

multiply top and bottom by g(x)g(x + h)] [by algebra] [by Property IV of limits]

Definition: A complex signal is said to be conjugate symmetric3 if, for all n ,

7.33. Find x in each part of Fig. 7-58.

x(n) = x*(-n)

Fig. 7-58

and a signal is said to be conjugate antisymmetric if, for all n ,

1/g(x) h 0 g(x + h)

7.34. A 7-ft upright pole near a vertical tree casts a 6-ft shadow. At that same time, find (a) the height of the tree if its shadow is 36 ft long; (b) the length of the shadow of the tree if its height is 77 ft. (7.23) 7.35. Prove each of the following: (a) In nABC, if AD and CE are altitudes, then AD :CE AB :BC. (7.23)

x(n) = -x*(-n)

Any complex signal may always be decomposed into a sum of a conjugate symmetric signal and a conjugate antisymmeuic signal.

(b) In circle O, diameter AB and tangent BC are sides of nABC. If AC intersects the circle in D, then AD:AB AB : AC. (c) The diagonals of a trapezoid divide each other into proportional segments. (d) In right nABC, CD is the altitude to the hypotenuse AB, then AC:CD 7.36. Prove each of the following: (a) A line parallel to one side of a triangle cuts off a triangle similar to the given triangle. (b) Isosceles right triangles are similar to each other. (c) Equilateral triangles are similar to each other. (d) The bases of a trapezoid form similar triangles with the segments of the diagonals. 7.37. Complete each of the following statements: (7.26) AB :BC. (7.24)

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