This assignment is intended to provide an easier analysis of how windows work. Begin by looking at individual spectrum bins as affected by off-bin-centred frequency components with different windows.
A sine wave component of the test signal is bin-centred if it has an integer number of periods in the data segment being analyzed. This almost never happens without planning.
When the signal is periodic and an integer number of periods fill the acquisition time interval, the FFT turns out such that the signal frequency matches one of the FFT frequency bins. When the number of periods in the acquisition is not an integer, the endpoints are discontinuous. The result is the high side lobes seen in the un-windowed spectrum plot. This phenomenon is called Spectral Leakage.
Consider the case of a sampling rate of 1024 Hz being used to look at a 16 Hz. Clearly 16 is a divisor of 1024, but how many samples are needed to uniquely identify the component.
a) using different sample lengths as shown, produce the corresponding FFT of the signal, explaining your results.
b) Repeat this exercise for frequencies 16.5 Hz, 17.0 Hz & 17.5 Hz
It is now necessary to consider the effects of windowing the data set with a view to minimizing leakage effects of a non bin-centered signal on the spectra computed.
c) repeat parts a) & b) by applying windowing to the data sets.
This has been illustrated graphically below using von Hann (Hanning) & Hamming windows, although clearly other could be used.. Graphs are shown for 16 Hz and again for 16.5 Hz.