The Importance of Fourier Theory

What Is Importance of Fourier Theory?

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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