Electric Circuits and Networks

14.5 SYMMETRY PROPERTIES OF FOURIER TRANSFORMS

There is nothing wrong mathematically inv(t) possessing an imaginary part. However, it will not be a physical waveform then. We deal with physical waveforms in Circuit Theory. Hence, we deal with real functions of time exclusively. The Fourier transform of a real v(t) has many interesting symmetry properties.

 

 

Fig. 14.4-10 Waveform and Spectra for Example 14.4-5

14.5.1 Conjugate Symmetry Property

Let v(t) be a real function of t and V(jω) its Fourier transform. Then, V(–jω) = V*(jω). This can be shown as follows:

 

 

This implies that Re(V(–jω)) = Re(V(jω)) and Im(V(–jω)) = –Im(jω). It also ...