Lecture 4: Interpretation of the Smoothness of the Function
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Schedule ::
Periodic Signals ::
Fourier Spectrum ::
Symmetry ::
Fundamental Frequency ::
Smoothness of the Function ::
Fourier
Series ::
Exponential Series ::
Spectra ::
Bandwidth ::
Energy
of Signals ::
Geometric Viewpoint
Functions which are smooth (e.g. continuous) have most of their variations at lower frequencies. Functions which are not smooth have variations at higher frequencies. We can look at the rate of decay of the amplitude spectrum to determine something about the smoothness of the function.
For example, the square wave function has abrupt jumps and is not even continuous.
The coefficients of the F.S. decay as 
.
By contrast, the sawtooth function we examined is smoother, since it is continuous.
Its coefficients decay more quickly, decaying down as 
.
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