I’m reading data from the microphone and want to perform some analysis on it. I’m attempting to generate a spectrum analyser something like this:

What I have at the moment is this:

My understanding is that I need to perform a Fourier analysis – a Fast Fourier Transform ? – to extract the component frequencies and their amplitudes.
Can someone confirm my understanding is correct and exactly what type of Fourier transform I need to apply?
At the moment, I’m getting frames containing 4k samples from the mic (using NAudio). The buffer I’ve got is 16bits/sample (Signed Short). For reference, the above plot shows approx half a frame
I’m coding in VB so any .Net libraries/examples (preferably on NuGet) would be of most use. I believe implementations vary considerably so the less I have to massage my data, the better.
The top plot is that of a spectrograph, where each vertical time line is colored based on the magnitudes of the result from an FFT (likely windowed) of a slice in time (possibly overlapped) of the input waveform. The number of vertical points to plot (the frequency resolution) is related to the length of the FFT. Almost any FFT will do. If you use the most common complex-to-complex FFT, just set the imaginary portion of each complex input sample to zero, copy a slice in time of samples of your input waveform to the “real” part, FFT, and take the magnitude or log magnitude of each complex result bin, then map these values to colors per your preference.