But now, after transforming our windowed Time Data to Frequency Data and zooming in a little, we see that the fundamental has grown plumper than a Thanksgiving turkey!
Will the real signal line please step forward?! Now there are seven lines where there used to be one. However all of this fits together under Fourier Transform theory. We multiplied two Time Domain signals together: our A/D sample and the Data Window “sample”. That results in a convolution in the Frequency Domain of the A/D and Data Window spectra. Our original Fundamental line has been “smeared” into about seven lines.
So in the original sample, there must have been some other hocus pocus that made the ends match up. Indeed there was. First, we locked together the frequency references of the two signal generators that were used to generate the A/D clock and the analog sine wave. Next, we chose a “Magic Frequency” for our test signal. A magic frequency is just a frequency that falls exactly on the center of one of the DFT bins. These can be calculated by dividing the sample frequency, Fs, by the sample size, N and multiplying by any integer less than N. In this case:
Fs = 5 MHz, N = 8192, choose bin 3200: 5 MHz / 8192 * 3200 = 1.953125 MHz.
Naturally, you need an analog signal generator that lets you enter a lot of decimal places…
If you do no windowing at all, you have actually convolved your sample with a “square window”. And the spectrum of a Square Window is the ” (sine x) / x” shape. Here’s an example. Suppose we have a data sample where the ends don’t match up. In this case, we’ll use ScopeDSP’s Sine Generator function to generate a frequency of 1.952819824 MHz at the same rate of 5 MHz with the same sample size of 8192 points. This frequency is at bin “3199.5”, that is, half way between bin 3200 (as in our A/D sample) and the previous bin number 3199.