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Thus, the poles of H ( z ) become the zeros of G(z), and the zeros of H ( z ) become the poles of G(z). The region of convergence that is associated with the inverse system is determined by the requirement that H ( z ) and G ( z ) have overlapping regions of convergence.'

which also allows the initial capacity to be set. If you use a constructor that does not take one or the other of these two arguments, then the default values of capacity 101 and load threshold 75 percent will be used.

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There are two possible regions of convergence for ,q(n). The ti rst is IzJ > f ,and the second is lzI i$. Because lzl < f does not overlap the region of convergence for H(z), the only possibility for the inverse system is l z l > {. In this case, the unit sample response is ~ ( n = () u(n) - 0 . 8 ( ~ ) " - ' u ( n- I ) ) 1"

'If this were not the case, H ( r ) G ( z ) would not be the identity system. because the region of convergence would be empty.

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COLLISION RESOLUTION ALGORITHMS The collision resolution algorithm used in the previous examples is called linear probing. When a new item hashes to a table component that is already in use, the algorithm specifies to increment the index until an empty component is found. This may require a wraparound back to the beginning of the hash table. EXAMPLE 8.6 Linear Probing

public class TestHashing { private static final int MASK = 0x7FFFFFFF; // private static final int CAPACITY = 101; public static void main(String[] args) { printHash("Rad"); printHash("Uhr"); printHash("Ohr");

H (z) =

0.5 - Z - ' I - 0.82-'

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.

= 2^32-1

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where the region of convergence may be either (zI > 2 or IzI c 2. Because both regions of convergence overlap the region of convergence of H ( z ) , both are valid inverse systems. The first, which has a region of convergence li( > 2, has a unit sample response ~ ( n = 2(2)"u(n) - 1,6(2)"-'u(n -- I) ) and is causal but unstable. The second, with a region of convergence lzl < 2, has a unit sample response

A linear shift-invariant system with a rational system function may be written in factored form as follows:

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

CHAP. 8]

k Assuming only first-order poles, with a # fraction expansion as follows:

When p I the partial fraction expansion has the form q,

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; } }

printHash("Tor"); printHash("Hut"); printHash("Tag"); }

If p = 0,H ( z ) has only zeros,

These systems are called finite-length impulse response (FIR) filters. If p r 0, H(z) is infinite in length, and these systems are called infinite-length impulse response (IIR) filters. If h(n) is real, H (z) = H *(z*), and the complex poles and zeros of H (z) occur in complex-conjugate pairs. is l For example, if ak = rkeJWAa complex-valued pole, a; = r k e p J W h i l also be a pole. This symmetry implies that the complex terms in Eq. (5.5) may be combined to form terms of the form

The output is:

9 10 11 12 13 14 15 16 17 18 19 20 21

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