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Fast wideband spectrum analysis is expensive in power and hardware resources. We show that the spectrum-analysis architecture used by the biological cochlea is extremely efficient: analysis time, power and hardware usage all scale linearly with N, the number of output frequency bins, versus N log(N) for the Fast Fourier Transform. We also demonstrate two on-chip radio frequency (RF) spectrum analyzers inspired by the cochlea. They use exponentially-tapered transmission lines or filter cascades to model cochlear operation: Inductors map to fluid mass, capacitors to membrane stiffness and active elements (transistors) to active outer hair cell feedback mechanisms. Our RF cochlea chips, implemented in a 0.13 mum CMOS process, are 3 mm times 1.5 mm in size, have 50 exponentially-spaced output channels, have 70 dB of dynamic range, consume <300 mW of power and analyze the radio spectrum from 600 MHz to 8 GHz. Our work, which delivers insight into the efficiency of analog computation in the ear, may be useful in the front ends of ultra-wideband radio systems for fast, power-efficient spectral decomposition and analysis. Our novel rational cochlear transfer functions with zeros also enable improved audio silicon cochlea designs with sharper rolloff slopes and lower group delay than prior all-pole versions.