# The effect of time increment $\Delta t$, frequency resolution $\Delta f$ and Gibbs phenomenon on the amplitude of the signal in the frequency domain?

Experiment: Vibration analysis (i.e.: major concern is on transfer function or frequency response).

The input is an impulse (imagine as a knocking effect or knocking force), defined as a half sine curve. The output is the vibration speed of the specimen (measured by laser vibrometer). The vibration is naturally damped, so within $$0.6\ \rm s$$, the vibration speed has greatly decreased - but it is most certainly removed after $$1.2~1.3\ \rm$$s. The results are then computed with FFT to convert into frequency domain (automatic process done by the computer) and then the transfer function would be investigated.

My questions are as followed:

1. Would having long measurement period (i.e.: one sample period) have any effect on the frequency spectrum? This is assuming the number of sampling points (in time domain) or the number of spectral/FFT lines is kept the same. The measurement time can be changed via the selected bandwidth (and thus, the sampling frequency)
2. Is there any proven work or mathematic, saying that for the same bandwidth and same sampling frequency, increasing the frequency resolution (i.e.: make the spectrum coarser) would decrease the height of the peak?
3. Would changing the time increment (time between 2 consecutive sampling points in time domain) change the frequency spectrum?

Though I think that these 3 questions are somewhat the same as each other, it is worth writing all of them down.

Reference graph: different bandwidths (and thus, different frequency resolutions, different measurement time, and different time increment) lead to different frequency response (or transfer function). The most "problematic" are the 2 peaks at 300 and 330 Hz. Normally, by increasing the frequency resolution, the common belief is that the amplitude would be lower.

And then, question 4:

1. Would the Gibbs phenomenon have any (visible) effect on the frequency spectrum?