"Undersampled" Polyphase Filter Bank
To analyze a slice of the spectrum:


Use an IQ mixer to translate the centre of the slice, $F_c$, to 0 Hz. This is done by multiplying the signal with $e^{-j2\pi F_c/F_s}$, where $F_s$ is the sample frequency.
Use a decimation low-pass FIR filter to remove the signal components outside the slice. The pass band of the filter is half the slice width. Output from this process is the down-converted slice sampled at a low sampling rate (just meeting the Nyquist criteria).
Apply your favourite window function and FFT to the time signal.


This process is repeated for each of the wanted slices.