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An array is an object that consists of a sequence of elements that are numbered 0, 1, 2, The element numbers are called index numbers Array elements can be accessed by their index numbers using the subscript operator [], as a[0], a[1], a[2], and so on Arrays are widely used because they are so efficient PROPERTIES OF ARRAYS Here are the main properties of arrays in Java: 1 Arrays are objects 2 Arrays are created dynamically (at run time) 3 Arrays may be assigned to variables of type Object 4 Any method of the Object class may be invoked on an array 5 An array object contains a sequence of variables 6 The variables are called the components or elements of the array 7 If the component type is T, then the array itself has type T[] 8 An array type variable holds a reference to the array object 9.

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Y( k )= w i k x ( k ) (6) Find the finite-length sequence w ( n ) that has a six-point DFT that is equal to the real part of X ( k ) ,

These are mutually recursive implementations of the hyperbolic sine and cosine functions:

W ( k )= R e ( X ( k ) )

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The component type may itself be an array type 10 An array element is a component whose type is not an array type 11 An element s type may be either primitive or reference 12 The length of an array is its number of components 13 An array s length is set when the array is created, and it cannot be changed 14 An array s length can be accessed as a public final instance variable 15 Array index values must be integers in the range 0.. length 1 16 An ArrayIndexOutOfBoundsException is thrown if Property 15 is violated 17 Variables of type short, byte, or char can be used as indexes 18 Arrays can be duplicated with the Objectclone() method 19 Arrays can be tested for equality with the Arraysequals() method 20 Array objects implement Cloneable and javaioSerializable Property 3 follows from Property 1.

Q ( k )= X(2k)

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public static double s(double x) { if (-0.005 < x && x < 0.005) { return x + x*x*x/6; // basis } return 2*s(x/2)*c(x/2); // recursion } public static double c(double x) { if (-0.005 < x && x < 0.005) { return 1.0 + x*x/2; // basis } return 1 + 2*s(x/2)*s(x/2); // recursion }

k = 0, 1,2

Although array types are not classes, they behave this way as extensions of the Object class Property 7 shows that array types are not the same as class types They are, in fact, derived types: For every class type T there is a corresponding array type T[] Also, for each of the eight primitive types, the corresponding array type exists..

( a ) The sequence y(n) is formed by multiplying the DFT of x(n) by the complex exponential W:k. Because this corresponds to a circular shift of x ( n ) by 4,

This is a recursive implementation of the tangent function:

R e { X ( k ) J= f [ X ( k )

CHAP. 2]

To find the inverse DFT of R e { X ( k ) ) ,we need to evaluate the inverse DFT of X * ( k ) . Because

public static double t(double x) { if (Math.abs(x) < 0.5e-10) { return x + x*x/3 + x*x*x*x/5; // basis } double tx2 = t(x/2); return 2*tx2/(1 - tx2*tx2); // recursion }

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