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107

Figure 27 properties of Fourier Transformation

This additivity can be understood in terms of how sinusoids behave. Consider adding two

sinusoids with the same frequency but different amplitudes) and phases If the two phases happen

to be same, the amplitudes will add when the sinusoids are added. If the two phases happen to be

exactly opposite, the amplitudes will subtract when the sinusoids are added. When sinusoids (or

spectra) are in polar form, they cannot be added by simply adding the magnitudes and phases.

In spite of being linear, the Fourier transform is not shift invariant. In other words, a shift

in the time domain does not correspond to a shift in the frequency domain. Instead, a shift in the

time domain corresponds to changing the slope of the phase.

2.3.1.5 Examples of Fourier Transforms

Recall the periodic sawtooth function used in section 1, Fourier transforms can be used to

find its frequencies. In Figure 28, the first “peak” in positive frequency domain indicates 0.5 Hz

as the function’s first frequency. Note the offset of the function results in a peak of magnitude at

0 Hz, and the time shift in the function causes the shift in slope of phase. Compare the result

after removing the offset and time shift (shown in Figure 29) with the original result.

-8

0 0.5 1

4*f(x)

0 5 10 15 20 25 30

-2

-1.5

-1

-0.5

0 5 10 15 20 25 30

-4

0 0.5 1

f(x)+g(x)

0 5 10 15 20 25 30

-2

-1.5

-1

-0.5

0 5 10 15 20 25 30

1 2 ... 109 110 111 112 113 114 115 116 117 118 119 ... 142 143

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Vishay 2310 Betriebsanweisung Seite 114