Window Choice of FFT

Question

I am working with data from a TeraFlash THz Spectrometer from Toptica, which gives me the pulse in the time domain and the spectrum in the frequency domain

I have been experimenting with the FFT to get to the spectrum myself and I see some discrepancies compared to the spectrum from the software. The manufacturer told me that they use an adapted blackman window that only changes the first pulse sample and the last two pulse samples, so I assume that everything in-between will be weighted with 1 and only the beginning and end will have a smooth transition to 0. Apparently, this is done such that the pulse will not be altered by the window if it is not centered.

In the attached plot I have included the window (adapted blackman window) as well as the pulse and the two spectra (the software's and mine). I am particularly interested in the absorption lines in the spectrum that are not identical between the two calculation methods...

As I am new to DSP, I am not sure if this is a significant difference or if both spectra should suffice. In this setup, only the absorption lines air (mainly water) are visible, but for further measurements I am interested in the absorption lines of different materials.

EDIT: the other window functions shown in the legend of the plot result in even more discrepancies than the adapted blackman window, thus I chose not to show them.

I used numpy FFT calculation:

def get_fft(t, p, padding=True):
    if padding:
        t, p = zero_padding(t, p)  # apply zero padding to get 100001 datapoints on the spectrum
    p = p * blackman(len(t))  # apply blackman window
    SAMPLE_RATE = len(t) / (t[-1] - t[0]) * 1e12
    N = len(p)
    a = np.abs(rfft(p))
    f = rfftfreq(N, 1 / SAMPLE_RATE) / 1e12
    return f, a

Answer 1

1. The modified Blackman would have higher side-lobes that traditional Hamming or Hann or regular Blackman windows, so more spectral leakage, which tells me your software isn't too worried about lower-valued fft coefficients (loss of weak components shadowed by neighboring stronger ones). If you are worried about that, I'd suggest using a window with more taper.
2. Your results are so close that I'm guessing your implementation of the modified Blackman window is slightly different from theirs. Maybe try playing around with how many samples you taper at the start and end of the window.
3. I wouldn't worry too much about trying to match exactly the software's black-box. Clearly your implementation is correct, just not exactly the same as theirs.