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9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Write and test a recursive function that returns the sum of the squares of the first n positive integers. Write and test a recursive function that returns the sum of the first n powers of a base b. Write and test a recursive function that returns the sum of the first n elements of an array. Write and test a recursive function that returns the maximum among the first n elements of an array. Write and test a recursive function that returns the maximum among the first n elements of an array, using at most lg n recursive calls. Write and test a recursive function that returns the power x n. Write and test a recursive function that returns the power xn, using at most 2 lg n recursive calls. Write and test a recursive function that returns the integer binary logarithm of an integer n (i.e., the number of times n can be divided by 2). Write and test a recursive boolean function that determines whether a string is a palindrome. (A palindrome is a string of characters that is the same as the string obtained from it by reversing its letters.)

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Math.sin(x) error 0.0000000000000 0.0000000000000 0.0998334166468 0.0000000000005 0.1986693307951 0.0000000000009 0.2955202066613 0.0000000000069 0.3894183423087 0.0000000000018 0.4794255386042 0.0000000000051 0.5646424733950 0.0000000000120 0.6442176872377 0.0000000000015 0.7173560908995 0.0000000000027 0.7833269096275 0.0000000000043 0.8414709848079 0.0000000000063 Math.cos(x) error 1.0000000000000 0.0000000000000 0.9950041652780 0.0000000000000 0.9800665778412 0.0000000000002 0.9553364891256 0.0000000000021 0.9210609940029 0.0000000000007 0.8775825618904 0.0000000000028 0.8253356149097 0.0000000000082 0.7648421872845 0.0000000000013 0.6967067093472 0.0000000000027 0.6216099682707 0.0000000000054 0.5403023058681 0.0000000000098 This works because on each recursive call x is divided by 2, and eventually it reaches the basis criterion (-0.005 < x && x < 0.005), which stops the recursion. s(x) 0.0000000000000 0.0998334166464 0.1986693307941 0.2955202066544 0.3894183423069 0.4794255385991 0.5646424733831 0.6442176872362 0.7173560908969 0.7833269096232 0.8414709848016 c(x) 1.0000000000000 0.9950041652781 0.9800665778414 0.9553364891277 0.9210609940036 0.8775825618932 0.8253356149179 0.7648421872857 0.6967067093499 0.6216099682760 0.5403023058779

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In previous chapters, we have seen how to represent a sequence in terms of a linear combination of complex exponentials using the discrete-time Fourier transform (DTFT) and how the sequence values may be used as the coefficients in a power series expansion of a complex-valued function of z. For finite-length sequences there is another representation, called the discrete Fourier transform (DFT). Unlike the DTFT, which is a continuous function of a continuous variable, w , the DFT is a sequence that corresponds to samples of the DTFT. Such a representation is very useful for digital computations and for digital hardware implementations. In this chapter, we look at the DFT, explore its properties, and see how it may be used to perform such tasks as digital filtering and evaluating the frequency response of a linear shift-invariant system.

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9.10 Write and test a recursive function that returns a string that contains the binary representation of a positive integer. 9.11 Write and test a recursive function that returns a string that contains the hexadecimal representation of a positive integer. 9.12 Write and test a recursive function that prints all the permutations of the first n characters of a string. For example, the call print("ABC",3) would print

Let K(n) be a periodic sequence with a period N:

CHAP. 9]

Although, strictly speaking,K(n) does not have a Fourier transform because it is not absolutely summable, it can be expressed in terms of a discrete Fourier series (DFS) as follows:

9.13 Implement the Fibonacci function iteratively (without using an array).

which is a decomposition of K(n) into a sum of N harmonically related complex exponentials. The values of the discrete Fourier series coefficients, $(k), may be derived by multiplying both sides of this expansion by e - j Z n n l l Nsumming over one period, and using the fact that the complex exponentials are orthogonal: ,

9.1 9.2 9.3 9.4 9.5

Note that the DFS coefficients are periodic with a period N:

914 Implement the recursive Ackermann function: A(0, n) = n + 1 A(m, 0) = A(m 1, 1), if m > 0 A(m, n) = A(m 1, A(m, n 1)), if m > 0 and n > 0 915 Prove Pascal s recurrence relation (page 170) 916 Trace the recursive implementation of the Euclidean Algorithm (Example 913 on page 171) on the call gcd(385, 231) 917 Implement the Euclidean Algorithm (page 171) iteratively 918 Implement the recursive Euclidean Algorithm using the integer remainder operator % instead of repeated subtraction 919 Implement the Euclidean Algorithm iteratively using the integer remainder operator % instead of repeated subtraction 920 Use mathematical induction to prove that the recursive implementation of the Fibonacci function (Example 96 on page 167) is correct 921 Use mathematical induction to prove that the recursive function in Problem 9.

x(n) =

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