A spectrum analyzer having a memory function to adopt a digital-data-based frequency sweep scheme while achieving performance comparable to performance of a high-speed FFT spectrum analyzer, and a method of controlling the spectrum analyzer, in which the spectrum analyzer includes: an ADC for converting a BWP signal, which is at least one analog unit frequency band signal, into a digital data sample at a predetermined sample rate according to a span set by a user; a digital sweep part for sweeping the data sample passed through the ADC while digitally decimating the data sample through a decimation processing block having a two-stage cascaded structure, and processing the swept data sample to increase a frequency sweep speed; and a control unit for controlling the digital sweep part according to various items input, set, and selected by the user to perform spectrum analysis and output a spectrum analysis result.