How to do the highest resolution FFT? I wish to do the FFT of streaming audio data coming into a computer. I am just interested in frequencies from 80 Hz up to 1000 Hz but with a frequency resolution of 3 to 4 Hz and no more. 

Basically my question is , as my sampling frequency (Fs) is 44100, will my FFT window size have to be no less than 11025 (N) ?  (because spacing of FFT bins = Fs/N). Does this mean that I am completely limited to taking windows of data any higher than 4 times a second with my desired resolution. 

Unfortunately I need to be able to capture windows of data 8 times a second and also have a resolution of 3 to 4 Hz. 

Is this actually possible? Many thanks in advance.
The FFT size can be longer than the window size.  For instance you can use audio sample windows of 5512 (about 8 per second) and zero-pad them to length 16384 for each FFT.  If noise level is low enough and the separation between frequency peaks is great enough, the interpolation provided by this zero-padding can provide better frequency estimation and plot resolution.

The "highest resolution" FFT reasonable depends on the signal-to-noise ratio as well as the window length, as after a certain point, interpolation and frequency estimation methods will be affected as much by nearby spectral noise as by the spectral peaks that one is trying to estimate at "high resolution".