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which corresponds to the case in which cr = N / 2 and B = n / 2 or 3x12. Linear phase filters may be classified into four types, depending upon whether h(n) is symmetric or antisymmetric and whether N is even or odd. Each of these filters has specific constraints on the locations of the zeros in H ( z ) which, in turn, place constraints on the frequency response magnitude. For each of these types, which are described below, it is assumed that h(n) is real-valued, and that h(0) is the first nonzero value of h(n).

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public class TestQuadraticProbing { private static final int MASK = 0x7FFFFFFF; // 2^32-1 private static final int CAPACITY = 11; private static int size = 0; private static boolean[] used = new boolean[CAPACITY]; public static void main(String[] args) { printHash("Rad"); printHash("Uhr"); printHash("Ohr"); printHash("Tor"); printHash("Hut"); printHash("Tag"); printHash("Eis"); printHash("Ast"); printHash("Zug"); printHash("Hof"); printHash("Mal"); } private static void printHash(String word) { System.out.printf("hash(%s) = %d, load = %d%%%n", word, hash(word), 100*size/CAPACITY); } private static int hash(Object object) { ++size;

This program extends the program in Example 8.4. It keeps track of which table components are used and the load factor after each hashing.

A type I linear phase filter has a symmetric unit sample response,

and N is even. The center of symmetry is about the point cr = N/2, which is an integer, as illustrated in Fig. 5-3(a).

65 66 67 68 69 70 71 72 73 74 75 76 77 78

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

h(n)

int h = (object.hashCode() & MASK) % CAPACITY; if (used[h]) { int h0 = h; int jump = 1; while (used[h]) { System.out.printf("%d, ", h); h = (h0 + jump*jump)%CAPACITY; // squared increment ++jump; } } used[h] = true; return h; } }

public class Ex1406 public class TestLinearProbing { private static final int MASK = 0x7FFFFFFF; // 2^32-1 private static final int CAPACITY = 11; private static int size = 0; private static boolean[] used = new boolean[CAPACITY]; public static void main(String[] args) { printHash("Rad"); printHash("Uhr"); printHash("Ohr"); printHash("Tor"); printHash("Hut"); printHash("Tag"); printHash("Eis"); printHash("Ast"); printHash("Zug"); printHash("Hof"); printHash("Mal"); } private static void printHash(String word) { System.out.printf("hash(%s) = %d, load = %d%%%n", word, hash(word), 100*size/CAPACITY); } private static int hash(Object object) { ++size; int h = (object.hashCode() & MASK) % CAPACITY; while (used[h]) { System.out.printf("%d, ", h); h = (h+1)%CAPACITY; } used[h] = true; return h; } }

- -1

h(n)

The output is:

- -1

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A type I1 linear phase filter has a symmetric unit sample response, and N is odd. Therefore, the center of symmetry of h ( n ) occurs at the half-integer value a, = N / 2 , as illustrated in Fig. 5-3(b). The frequency response of a type I1 linear phase filter may be written as

hash(Rad) = 3, load = 9% hash(Uhr) = 4, load = 18% hash(Ohr) = 2, load = 27% hash(Tor) = 8, load = 36% hash(Hut) = 5, load = 45% 3, 4, hash(Tag) = 7, load = 54% 5, hash(Eis) = 6, load = 63% 3, 4, 7, hash(Ast) = 1, load = 72% hash(Zug) = 9, load = 81%

hash(Rad) = 3, load = 9% hash(Uhr) = 4, load = 18% hash(Ohr) = 2, load = 27% hash(Tor) = 8, load = 36% hash(Hut) = 5, load = 45% 3, 4, 5, hash(Tag) = 6, load = 54%

A type I11 linear phase filter has a unit sample response that is antisymmetric,

CHAP. 8]

The essential difference here is in the sequence of index numbers probed within the while loop when a collision occurs. Instead of searching linearly, it uses a squared increment. For example, when the insertion of "Ast" collides at index 3, linear probing continued probing at indexes 4, 5, 6, 7, 8, and 9 (in Example 8.6). But with quadratic probing, only indexes 3, 4, 7, and 1 ( = 12 mod 11) are probed, using successive jumps of 1, 4, and 9 (12, 22, and 32). Linear probing required 50 percent more probes.

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