Spectrum Shaping: A New Perspective on Cognitive Radio—Part II: Coexistence with Uncoded Legacy Transmission

Spectrum shaping is examined for cognitive users to enable graceful coexistence with pre-existing legacy services. In Part I of this two-part paper series, the case of a coded legacy system was presented. In Part II, the case of an uncoded, or analog, legacy system is treated. Optimal power spectral densities are characterized for the cognitive user so that the analog legacy user achieves a prescribed distortion constraint as measured by a mean-squared estimation error. As in Part I, it is demonstrated that through spectrum shaping, the cognitive user can transmit at its full available power and thereby achieve logarithmic growth of the cognitive transmission rate. Necessary conditions for the optimality of the spectrum shaping problem are investigated and under special conditions, low-complexity optimization algorithms are developed. A low-complexity sub-optimal solution corresponding to an on-off cognitive power spectral density is analyzed, and shown to be sufficient for attaining the logarithmic growth of cognitive transmission rates. To summarize this two-part paper, a series of general considerations for system implementation of spectrum shaping are discussed.