# The Importance of Fourier Theory

## What Is Importance of Fourier Theory?

Fourier analysis allows us to determine not only the sine wave components in any complex signal but also how much bandwidth a particular signal occupies. Although a sine or cosine wave at a single frequency theoretically occupies no bandwidth, complex signals obviously take up more spectrum space. For example, a 1-MHz square wave with harmonics up to the eleventh occupies a bandwidth of 11 MHz. If this signal is to pass unattenuated and undistorted, then all harmonics must be passed.

An example is shown in Fig. 2-63. If a 1-kHz square wave is passed through a lowpass filter with a cutoff frequency just above 1 kHz, all the harmonics beyond the third harmonic are greatly attenuated or, for the most part, filtered out completely. The result is that the output of the low-pass filter is simply the fundamental sine wave at the square wave frequency.

If the low-pass filter were set to cut off at a frequency above the third harmonic, then the output of the filter would consist of a fundamental sine wave and the third harmonic. Such a waveshape was shown in Fig. 2-58(a). As you can see, when the higher harmonics are not all passed, the original signal is greatly distorted. This is why it is important for communication circuits and systems to have a bandwidth wide enough to accommodate all the harmonic components within the signal waveform to be processed. Fig. 2-64 shows an example in which a 1-kHz square wave is passed through a bandpass filter set to the third harmonic, resulting in a 3-kHz sine wave output. In this case, the filter used is sharp enough to select out the desired component.

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