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Although there has been a lot of research on analysis and synthesis of quantized feedback control systems, most results are developed for the case of a single quantizer (either measurement quantization or control signal quantization). In this technical note, we investigate the case of feedback control systems subject to both input and output quantization. This is motivated by the fact that it is common in remotely controlled systems that measurement and control signals are shared over a single digital network. More specifically, we consider a single-input single-output linear system with memoryless logarithmic quantizers. We firstly show that the output feedback quadratic stabilization problem in this setting can be addressed with no conservatism by means of a sector bound approach. Secondly, we provide a sufficient condition for quadratic stabilization via the solution of a scaled H?? control problem. Finally, we analyze a problem of bandwidth allocation in the communication channel for finite-level input and output quantizers.