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public class MyPolynomial implements Polynomial { private double[] c; // coefficients public MyPolynomial(double[] a) { // a[i] = coeffficient of x^i int n = c.length; c = new double[n]; System.arraycopy(a, 0, c, 0, n); } public int degree() { return c.length - 1; } public Polynomial derivative() { double da[] = new double[c.length-1]; for (int i=0; i<da.length; i++) { da[i] = (i+1)*c[i+1]; } return new MyPolynomial(da); } public boolean equals(Object object) { if (object==this) { return true; } else if (!(object instanceof MyPolynomial)) { return false; } MyPolynomial that = (MyPolynomial)object; return java.util.Arrays.equals(that.c, this.c); } public Polynomial sum(Polynomial p) { if (!(p instanceof MyPolynomial)) { throw new IllegalArgumentException("use a MyPolynomial object"); } MyPolynomial that = (MyPolynomial)p; double[] pc = that.c; int n = Math.max(c.length, pc.length); MyPolynomial q = new MyPolynomial(new double[n]); for (int i=0; i<n; i++) { q.c[i] = c[i] + pc[i]; } return q; } public String toString() { StringBuilder buf = new StringBuilder(); int n = c.length; if (n > 0 && c[0] != 0.0) { buf.append(c[0]); } if (n > 1 && c[1] != 0.0) { buf.append(String.format(" + %.2f", c[1])); }

x.hashCode(): 1671711

To find the N-point DFT of this sequence, it is easier if we write the cosine in terms of complex exponentials:

~ ( k= C x ( n ) e - i % n i = )

CHAP. 1]

At line 4, the Ratio object x invokes the hashCode() method that it inherits from the Object class.

At this point, note that if 9 = 2 r r k o / N ,

e-~nliJi+ul)

for (int i=2; i<n; i++) { if (c[i] != 0.0) { buf.append(String.format(" + %.2f^%d", c[i], i)); } } return buf.toString(); } public double valueAt(double x) { double y = 0.0; for (int i=0; i<c.length; i++) { y += c[i]*Math.pow(x, i); } return y; } }

CHAP. 1]

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Because the first term is a sum of a complex exponential of frequency wo = 2 z ( k - k o ) / N , the sum will be equal to i zero unless k = ko, in which case the sum is equal to N . Similarly, the second sum is equal to zero unless != N - k,,, in which case the sum is equal to N . Therefore. if on= 2 z k o / N . the DFT coefficients are

X(k) =

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.

Another kind of polymorphism occurs with generic methods, where an actual type is substituted in for a type parameter. This is illustrated in Example 1.9. EXAMPLE 1.9 Parametric Polymorphism

In the general case, when wo # 2 n k n / N , we must use the geometric series to evaluate Eq. (6.18):

Note that. unless % is an integer multiple of 2 n / N , X ( k ) is, in general, nonzero for each k . The reason for this difference belween these two cases comes from the fact that X ( k ) corresponds to samples of the DTFT of x ( n ) , which is

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