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# Low-pass Filtered Signals

All periodic signals can be represented through *Fourier analysis* as a sum of sinusoidal oscillations whose frequencies are integral multiples of the fundamental signal frequency. For example, a *square-wave signal **s*(*t*) of a frequency *f*_{0} is represented as follows:

The first sine term with a frequency *f*_{0} is called the *fundamental oscillation*; the subsequent terms involving progressively higher frequencies are called *harmonics*. In the case of a square-wave signal, Fourier analysis yields an infinite series of harmonics each of which are odd multiples of the fundamental signal frequency and the amplitude of which decreases as the frequency factor rises.

The following interactive animation demonstrates a square-wave signal of a frequency *f*_{0} (red curve) and amplitude 1. Using Fourier analysis as described above, this signal is resolved into a series of sinusoidal oscillations of different frequencies. The tool on the right-hand side allows you to activate the fundamental and its harmonics up to a frequency of 35·*f*_{0}. Activate the individual frequency components successively and observe how the resulting series of sine terms approximates to a square-wave oscillation more and more closely.

If such a square-wave signal (or other periodic, non-sinusoidal signal) is passed through a low-pass filter, the individual harmonics are attenuated very strongly by the filter's frequency response. The signal output by the low-pass filter is correspondingly distorted.