Generalized Capacity of Constrained Systems

1. Generalized Capacity of Constrained Systems Syed Ali Jafar and Sriram Vishwanath Wireless Systems Laboratory

2. Outline • Introduction and motivation. • Polynomial (poly.) and no-growth (n-g.) capacity. • Algebraic characterization of poly. and n-g. capacity. • Conclusion.

3. Introduction • Shannon Capacity (Shan.) where is number of sequences of length n in constrained system , and is the PF eigenvalue.

4. Problems with Shannon Capacity. • Finite and positive only for systems with an exponential growth in sequences. • Assigns zero to various practically useful systems with less than exponential growth. • Provides no information about such systems – treats them as “useless”.

5. Polynomial Growth Systems • Battery driven (Energy Constrained) infra red systems : Can produce only “k” pulses over battery lifetime. # sequences for large n.

6. Polynomial and no-growth capacity • Polynomial capacity (poly.) • No-growth capacity (n-g.)

7. Alg. charac. contd. • Lemma 1 : An irreducible graph has PF eigenvalue . • Lemma 2: For an irreducible graph for large n, PF eig. > 1 implies grows exponentially , PF eig. = 1 implies constant.

8. The main theorem • Let be a constrained system with lossless representation , and PF eig. . Then, Case 1 : implies that the Shannon capacity is positive. Poly. and n-g. are infinite.

9. The main theorem contd. • Case 2: . Let be a path that passes through the maximum number of irreducible components ( ) in , then • If , the poly. is positive, Shan. is zero and n-g is infinite. • If , the n-g is finite and poly. and Shan. are zero. • Case 3: . All three capacities are and is zero.

10. Proof of theorem • If, There exists at least one irreducible component with PF eig. . satisfies hence, Shan. is finite, poly. and n-g. are infinite.

11. Proof contd. • If and , then • one can traverse through these irreducible components in ways. • All other paths composed of components can generate at most sequences. • Since , we have poly. equal to , Shan. equals zero and n-g. equals infinity.

12. Proof contd. • If and , we have a special case of case 2 with the number of sequences growing as which is a constant. Clearly, Shan. and poly. are zero and n-g. is finite. • If , by Lemma 1, the graph has no irreducible components, and hence has no sequences for large n.

13. Conclusions • Shannon capacity insufficient to characterize various practical systems. • Polynomial and no-growth capacity defined to study such systems. • Algebraic characterization obtained in terms of the irreducible components of a lossless representation of the system.