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public class MyCircle implements Circle { private Point c; // center private double r; // radius public MyCircle() { } public MyCircle(Point c, double r) { this.c = c; this.r = r; } public double area() { return Math.PI*r*r; } public Point center() { return c; } public double circumference() { return 2*Math.PI*r; } public boolean contains(Point point) { double x = point.xCoordinate(); double y = point.yCoordinate(); return x*x + y*y < r*r; } public boolean equals(Object object) { if (object==this) { return true; } else if (!(object instanceof MyCircle)) { return false; } MyCircle that = (MyCircle)object; return (that.c == this.c && that.r == this.r); } public double radius() { return r; }

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Y(k)=e j 2 k % ~ ( k )

where X ( k ) is the 10-point DFT of x ( n ) .

public static double s(double x) { if (Math.abs(x) < 0.00005) { return x - x*x*x/6; // basis }

public String toString() { return String.format("[Center: %.2fx; Radius: %.2f]", c, r); } }

Y ( k )=X(k)W(k)

w(n)= 1

CHAP. 9]

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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])); }

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( b ) Multiplying X ( k ) by a complex exponential of the form w:' corresponds to a circular shift of x ( n ) by no. In this case, because no = -2, x ( n ) is circularly shifted to the left by 2, and we have

return 2*s(x/2)*c(x/2); } public static double c(double x) { if (Math.abs(x) < 0.00005) { return 1.0 - x*x/2; // basis } return 1 - 2*s(x/2)*s(x/2); // recursion } // recursion

CHAP. 1]

convolution, we may find the linear convolution and alias the result. The linear convolution of x ( n ) with w ( n ) is

These are faster converging implementation of the sine and cosine functions:

Because z(n) and z(n 10) are the only two sequences in the sum that have nonzero values for 0 5 n c 10, using a table to list the values of z(n) and z(n lo), and summing for n = 0, 1,2, . . . , 9 , we have

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; } }

y(n)=I3.3,l,l.1,3.3,2,2,21

public static double s(double x) { if (-0.005 < x && x < 0.005) { return x - x*x*x/6 + x*x*x*x*x/120; } 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 + x*x*x*x*x/24; } return 1 - 2*s(x/2)*s(x/2); // recursion } // basis

Consider the sequence x ( n ) = 4S(n) Let X ( k ) be the six-point DFT of x ( n ) .