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A class member may be a field, a constructor, a method, a nested class, a nested interface, or an enum type. The implicit argument of a method is the object to which the method invocation is bound. The toString() method returns a String object that represents the state of its implicit argument. The equals() method returns true if and only if states (contents) of its implicit and explicit arguments are the same. A class member that is declared public is accessible from any other class. A class member that is declared protected is accessible only from its own class and subclasses (extensions). A class member that is declared private is accessible only from its own class. A package is a namespace, that is, a name for a group of classes, interfaces, and enum types that can be used to distinguish those from other classes, interfaces, and enum types with the same name. An abstract class is a Java class with at least one abstract method a method that has no body. An abstract data type is a specification of a type s operations that could be implemented in any object-oriented programming language. A constructor is a member function of a class that is used to create objects of that class. It has the same name as the class itself, has no return type, and is invoked using the new operator. A method is an ordinary member function of a class. It has its own name, a return type (which may be void), and is invoked using the dot operator. A class method is declared static and is invoked using the class name. For example,

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In Chap. 6 we saw that the discrete Fourier transform (DFT) could be used to perform convolutions. In this chapter we look at the computational requirements of the DFT and derive some fast algorithms for computing the DFT. These algorithms are known, generically, asfast Fourier fransforms (FFTs). We begin with the radix-2 decimation-in-time FFT, an algorithm published in 1965 by Cooley and Tukey. We then look at mixed-radix FFT algorithms and the prime factor FFT.

Translate this ADT into a Java interface:

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Because x(n) may be either real or complex, evaluating X(k) requires on the order of N complex multiplications and N complex additions for each value of k. Therefore, because there are N values of X(k), computing an N-point DFT requires N* complex multiplications and additions. The basic strategy that is used in the FFT algorithm is one of "divide and conquer." which involves decomposing an N-point DFT into successively smaller DFTs. To see how this works, suppose that the length of x(n) is even (i.e., N is divisible by 2). If x(n) is decin~aredinto two sequences of length N/2, computing the N/2-point DFT of each of these sequences requires approximately ( N 12)' multiplications and the same number of additions. Thus, the two DFTs require 2 ( ~ / 2 ) '= { N' multiplies and adds. Therefore, if it is possible to find the N-point DFT of s ( n ) from these two N/2-point DFTS in fewer than N 2 / 2 operations, a savings has been realized.

double y = Math.abs(x);

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The decimation-in-time FFT algorithm is based on splitting (decimating) x(n) into smaller sequences and finding X ( k ) from the DFTs of these decimated sequences. This section describes how this decimation leads to an efficient algorithm when the sequence length is a power of 2. Let x(n) be a sequence of length N = 2", and suppose that x(n) is split (decimated) into two subsequences, each of length N/2. As illustrated in Fig. 7-1, the first sequence, ~ ( I z is, formed from the even-index terms, )

ADT: Circle area(): Real center(): Point circumference(): Real contains(Point): Boolean equals(Circle): Boolean radius(): Real toString(): String

+ 1)

1.11 1.12

wiik= wik, Eq. ( 7 . 2 )may be written as ,

CHAP. 1]

Note that the first term is the N/2-point DFT of g ( n ) , and the second is the N 12-point DFT of h(n):

invokes the class method abs() that is defined in the Math class. An instance method is declared without the static modifier and is invoked using the name of the object to which it is bound. For example,

Translate this ADT into a Java interface:

Although the N/2-point DFTs of g ( n ) and h ( n ) are sequences of length NL2, the periodicity of the complex exponentials allows us to write

and it is only necessary to form the products W ; H ( k ) for k = 0. 1, . . . , N/2 - 1. The complex exponentials multiplying H ( k ) in Eq. ( 7 . 3 ) are called twiddle factors. A block diagram showing the computations that are necessary for the first stage of an eight-point decimation-in-time FFT is shown in Fig. 7-2. If N / 2 is even, g ( n ) and h ( n ) may again be decimated. For example, G ( k ) may be evaluated as follows:

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