System Identification with a Limited Bandwidth Input Signal and Region of Interest
The reason x[n] must be white is because the solution will effectively spectrally weight the channel response based on the amount of energy present in each spectral frequency location. A white noise source provides equal weight to all frequencies. If energy is not present in any particular frequency bin, a proper solution cannot be found for that frequency.

If you only care about a small band of signals, then I would argue that you can still use the least squared approach. The reason for this is to consider a system that provides band pass filtering: for such a system I could excite the input with a white noise source and use the least squares approach (Wiener-Hopf equations) of comparing the input and output signals which would accurately provide a least squares estimate of the channel. So if you have a band limited signal, as long as that signal is white over your band of interest, it will still provide the accurate solution over that band (and you ignore everything else).