Using this tool you can select a variety of periodic signals. The signal is shown in the lower graph, and its Fourier series shown in the upper graph as a magnitude and phase representation.
The Fourier series is given by
where
is the period of the signal.
The animation buttons take the basic signal shape, and increase the period of the signal by spacing out the repetitions of the basic shape in time. For each click of the single step button, the period of the waveform is doubled. You can then observe that in the frequency representation, increasing the period introduces more spectral lines. Each doubling of the period introduces a spectral line between two currently existing lines. Repeatedly increasing the period results in a very dense frequency domain representation.
Note also that the scale of the frequency domain representation decreases as the period is increased. This is due to a reduction in power as the period is increased. (The basic signal shape has a fixed energy associated with it. As the period is increased, the energy in one period remains the same, but as the length has increased, the power decreases).
Taking this to its extreme, if the period is increased to infinity, making the signal aperiodic, then the spacing of the specral lines becomes infinitely small, and a continuous frequency response results. The magnitude of the response will be zero, but from this derivation, the Fourier transform can be obtained by removing the normalization factor at the front of the Fourier series equation, and
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