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Java is a strongly typed language, which means that every variable must be declared to have a type which determines how that variable can be used. A char cannot be used where a boolean is expected, and a Ratio object cannot be used where a Person object is expected. But there are some situations where a variable of one type can be used where another type is expected. This is called polymorphism (literally, many forms ) because the object appears to have more than one type. There are several kinds of polymorphism. The simplest kind is called inclusion polymorphism. Illustrated in Example 1.8, it refers to the ability of an object to invoke a method that it inherits. EXAMPLE 1.8 Inclusion Polymorphism

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

3. Let x 2 ( n ) be the N-point sequence that is extracted from x ( n ) with the first L - 1 values overlapping with those of x, ( n ) . 4. Perform an N -point circular convolution of x 2 ( n ) with h ( n ) by forming the product H ( k ) X 2 ( k ) and taking the inverse DFT. The first L - I values of y 2 ( n ) are discarded and the final N - L + 1 values are saved and concatenated with the saved values of yl ( n ) :

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The reason for the name overlap-save is that x ( n ) is partitioned into overlapping sequences of length N and, after performing the N-point circular convolution, only the last N - L 1 values are saved.

1 2 3 4 5 6

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 }

Because i ( n ) is periodic with a period N = 4, the DFS coefficients may be found by evaluating the sum

Alternatively, ~fwe express i ( n ) in terns of complex exponentials,

public class TestRatio { public static void main(String[] args) { Ratio x = new Ratio(22, 7); System.out.println("x.hashCode(): " + x.hashCode()); } }

This is a recursive implementation of the tangent function:

we may pick efthe DFS coefficients directly as follows. Using the periodicity of the complex exponentials,

The output is:

If i ( n ) is a periodic sequence with a period N ,

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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 }

2 ( n ) is also periodic with period 2N. Let g ( k ) denote the DFS coefficients when f ( n ) is considered to be periodic with a period N , and let 2 z ( k ) be the DFS coefficients when the period is assumed to be 2N. Express the DFS coefficientsg 2 ( k ) in terms of g ( k ) .

x.hashCode(): 1671711

Note that the term in brackets is equal to 2 when k is even, and it is equal to zero when k is odd. When k is even,

This is a recursive evaluation of a polynomial function:

If i ! ( n )and i z ( n ) are periodic with period N with DFS coefficients2 ( ( k )and x 2 ( k ) ,respectively, show that the sequence with DFS coefficientsf ( k ) = f I ( k ) X z ( k )is equal to the periodic convolution of ,( n ) i I and i 2 ( n ) :

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

public static double p(double[] a, double x) { // returns a[0] + a[1]*x + a[2]*x*x + ... return p(a, x, 0); } private static double p(double[] a, double x, int k) { // returns a[k] + a[k+1]*x + a[k+2]*x*x + ... if (k == a.length) { return 0.0; // basis } return a[k] + x*p(a, x, k+1); // recursion }

i(n)= xil(k),iz(n -k )

Rearranging the sums and combining the exponentials yields I f(n) =

CHAP. 1]

The implementation symbol is used to show that a class implements an interface. EXAMPLE 1.7 Implementing the Comparable Interface

that is sampled with a sampling frequency f, = 1 kHz.Find the DFS coefficients of the sampled signal

Z ( n ) = xa(nTT).

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

Change line 1 of Example 1.5 on page 6 to:

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