Persistent wideband radio frequency (RF) surveillance and spectral analysis is increasingly important, driven by the proliferation of wireless communication and RADAR technology. However, conventional electronic approaches are limited by the ∼1 GHz bandwidth of real-time analog-to-digital converters (ADCs). While faster ADCs exist, high data rates prohibit continuous operation, limiting these approaches to acquiring short snapshots of the RF spectrum. In this work, we introduce an optical RF spectrum analyzer designed for continuous, wideband operation. Our approach encodes the RF spectrum as sidebands on an optical carrier and relies on a speckle spectrometer to measure these sidebands. To achieve the resolution and update rate required for RF analysis, we use Rayleigh backscattering in single-mode fiber to rapidly generate wavelength-dependent speckle patterns with MHz-level spectral correlation. We also introduce a dual-resolution scheme to mitigate the trade-off between resolution, bandwidth, and measurement rate. This optimized spectrometer design enables continuous, wideband (15 GHz) RF spectral analysis with MHz-level resolution and a fast update rate of 385 kHz. The entire system is constructed using fiber-coupled off-the-shelf-components, providing a powerful new approach for wideband RF detection and monitoring.