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int[][] pascal(int n) { int[][] p = new int[n][n]; for (int j = 0; j < n; j++) { p[j][0] = p[j][j] = 1; } for (int i = 2; i < n; i++) { for (int j = 1; j < i; j++) { p[i][j] = p[i-1][j-1] + p[i-1][j]; } } return p; } public class TestSieve { private static final int SIZE=1000; private static boolean[] isPrime = new boolean[SIZE]; public static void main(String[] args) { initializeSieve(); printSieve(); } private static void initializeSieve() { for (int i = 2; i < SIZE; i++) { isPrime[i] = true; } for (int n = 2; 2*n < SIZE; n++) { if (isPrime[n]) { for (int m = n; m*n <SIZE; m++) { isPrime[m*n] = false; } } } } private static void printSieve() int n=0; for (int i = 0; i < SIZE; i++) if (isPrime[i]) { System.out.printf("%5d%s", } } System.out.printf("%n%d primes } } { { i, ++n%16==0 "\n":"");

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and the second is a demodulator,

2.4 2.5 2.6 2.7 2.8 2.9

the cascade is shift-invariant, even though a modulator and a demodulator are shift-varying. Another example is when S l is an up-sampler

less than %d%n", n, SIZE);

public class TestFill { public static void main(String[] args) { Object[] a = new Object[4]; Arrays.fill(a, new Date()); ch02.ex02.DuplicatingArrays.print(a); Arrays.fill(a, 22); ch02.ex02.DuplicatingArrays.print(a); Arrays.fill(a, "Yo!"); ch02.ex02.DuplicatingArrays.print(a); }

y(n) = w(2n)

In this case, the cascade is shift-invariant, and y ( n ) = x ( n ) . However, if the order of the systems is reversed, the cascade will no longer be shift-invariant. Also, if a linear shift-invariant system, such as a unit delay, is inserted between the up-sampler and the down-sampler, the cascade of the three systems will, in general, be shift-varying.

public class TestSieve { private static final int SIZE=1000; private static Vector<Boolean> isPrime = new Vector<Boolean>(SIZE); public static void main(String[] args) { initializeSieve(); printSieve(); } private static void initializeSieve() { isPrime.add(false); // 0 is not prime isPrime.add(false); // 1 is not prime for (int i = 2; i < SIZE; i++) { isPrime.add(true); }

2.2 2.3 2.4

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The first nonzero value of a finite-length sequence x(n) occurs at index n = -6 and has a valuex(-6) = 3, and the last nonzero value occurs at index n = 24 and has a value x(24) = -4. What is the index of the first nonzero value in the convolution y(n) = x(n) * x(n) and what is its value What about the last nonzero value

Because we are convolving two finite-length sequences, the index of the first nonzero value in the convolution is equal to the sum of the indices of the first nonzero values of the two sequences that are being convolved. In this case, the index is n = - 12, and the value is y(-12) = x2(-6) = 9 Similarly, the index of the last nonzero value is at n = 48 and the value is

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