A Sine Wave is a Sine Wave, Right?

        In this note we want to look at a sinusoidal signal and what happens when we compute the frequency components in the signal.  We will start with the signal below.

Then, we will take the FFT of this signal.  The result is graphed below.

There is one line in the spectrum of this signal, and it appears at a very low value of the index, i.  We can expand the graph of the FFT, and we get the graph below.

Except for a little "fuzz" the only point in the spectrum is at i = 1, i.e. at the fundamental.  Since we have a data record length of .001 second, that fundamental is at a frequency of 1000 Hz, i.e. 1 KHz.

        Now consider what happens when the sine wave is changed to the one below.  The only difference is that the frequency of the sine wave is now 2000 Hz.

Now, the spectrum looks like the plot below.

At first glance, this plot looks pretty much like the plot for the first signal.  Closer inspection shows that the point on the spectrum is at i = 2 (and the previous one was at i = 1).  Thinking about that you should note the following.         Now, try to predict what will happen when we make the frequency of the sine wave 2.5 KHz.  Here is a plot of the signal.

Notice that there are 2 and a half periods in the data set.  When we take the FFT of this data set we get some interesting results - as displayed in the graph below.

Clearly, even though this is a sine wave, there are several non-zero points in the spectrum.  Or is it really a sine wave.  Remember that we are computing coefficients of a Fourier Series of the signal in the data record assuming that it repeats.  Let's look at the signal just above when it repeats - i.e. plotting it for twice the data record length.

As you can see, when the signal repeats it isn't exactly a sinusoid.  Parts of the signal are sinusoidal, but that double peak in the middle takes it out of the realm of "sinusoidiality".

        And, that's the major point to make here.  When we FFT a data record, we are assuming that the data in the record repeats when we do that FFT.  If you can plot two cycles of the data record and you get something like that almost-sinusoidal signal above, then you have to be careful.  And the time to be careful is right at the beginning before you even take the data.  End of story.