How about the amplitudes of the fundamental wave and the harmonics of such a sawtooth formed signal?
|Because this is a periodic signal it consists of a fundamental
wave with frequency:|
and harmonics with frequencies 2f, 3f, 4f etc.
So it is sufficient to look at just one cycle of the input signal G(t), as well as to one cycle of the fundamental, and two respectively three cycles (and so on) of the harmonics.
For the Fourier Transform we have to observe the cosines as well. If you do so you will see that, in this case of the sawtooth signal, after multiplication and integration (see later) the results will be always zero. The cosines don't contribute in this case.
Left you see the original signal and the sine of the fundamental having frequency f and the sines of the second and third harmonic.
|Here we see the sawtooth multiplied with the sine of the
fundamental wave. A somewhat strange form. Mirrored it is something like:|
x . sin(x)
of which the surface below the line must be estimated (integrated).
|And this is how it looks for the second harmonic with
Of course the surface below the 0-line must be subtracted, but we see already that the result will be more than zero.
|And this the the third harmonic with frequency 3f.|
Here as well the result will be more than zero.
|Sampling with 12f implies that no higher frequencies may be
present in the signal than 6f. They are surely there, seeing the very sharp
edges, but we leave it for this moment.|
We should be aware that we can't calculate higher harmonics than 6f. If we want to do so, we have to take a higher sampling frequency, e.g. 24f or 48f or even higher. Using the PC this is not at all a problem, but for calculating "by hand" this is quite too much work.
|Of course we have to sample the sines of the fundamental
and the harmonics as well. Here you see the fundamental:|
sin(30°), sin(60°), sin(90°) enz.
Be aware: the vertical scale is stretched in respect to the sawtooth.
|Both sampled wave forms are multiplied sample by
Here you see what you get.
|The wobbling line is the fundamental including harmonics 2 to 5.|
Left we see it projected on top of the sawtooth and it wobbles nicely around it.
Right we see the samples as we took them from the sawtooth. There are quite some abberations visible.
In fact we should have taken the samples from the wobbling line.