By Ray Horak

Geared toward communications pros in components resembling company, strategic making plans, and community administration and research, this paintings bargains an easy clarification of speedy altering, complicated applied sciences, within the context of figuring out the rising structures of the twenty first century. the basics of transmission structures, facts communications protocols and provider recommendations are defined, and the improvement of community convergence, together with interactive functions and the web, is traced. The ebook additionally describes the know-how and regulatory ideas underlying the evolution of up to date voice, information and video networks.

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**Additional resources for Communications Systems & Networks, Voice, Data & Broadband Technologies**

**Example text**

In the next example, we look at nonlinear properties. Example 2 (GCD-LCM). The loop GCD-LCM in Figure 1(b) computes the greatest common divisor (y1 and y2 ) and the least common multiple (y3 + y4 ) of (x1 , x2 ). As in Example 1, our goal is to discover information about the loop. In this case, our strategy is to construct an S-inductive conjunction of quadratic polynomial inequalities. Therefore, in each iteration, we use the parameterized assertion Quadratic(x1 , x2 , y1 , y2 , y3 , y4 ) ≥ 0 .

Manna I<0 I≥0 ¬Π • I<0 I≥0 • χi χi ∃x. χi ∧ I < 0 (a) strengthening ∃x. χi ∧ ¬Π ∧ I < 0 (b) Π-strengthening Fig. 2. Illustration of the two strengthening conditions corresponding to the following sequence of χi ’s: def ⇒ χ0 : true def ⇒ χ1 : k ≥ 0 def ⇒ χ2 : k ≥ 0 ∧ j ≥ 0 def ⇒ χ3 : k ≥ 0 ∧ j ≥ 0 ∧ j ≥ 2k + 2 χ0 = true χ1 = χ0 ∧ ϕ1 χ2 = χ1 ∧ ϕ2 χ3 = χ2 ∧ ϕ3 Conjuncts k ≥ 0 and j ≥ 2k + 2 entail j ≥ 2. Therefore, χ3 as well as j ≥ 2 are S-invariant. Moreover, every χi and j ≥ 2 are S-inductive. The strengthening condition is essential for making progress.

Together, these three conjuncts constrain I ≥ 0 to be inductive relative to χi−1 . The ﬁnal conjunct asserts that there is some χi−1 -state s that is excluded by the new invariant. We call this constraint the strengthening condition. Any solution for (p0 , p1 , p2 ) of this VCP represents an inductive assertion ϕi def that strengthens χi−1 . Set χi = χi−1 ∧ ϕi , and generate the next VCP with χi instead of χi−1 . If a solution does not exist, then halt the incremental construction. Section 4 discusses how to solve the sequence of VCPs.