Electric Circuits, 9th Edition P69. Designed for use in a one or two-semester Introductory Circuit Analysis or Circuit Theory Course taught in Electrical or Computer Engineering Departments. Electric Circuits 9/e is the most widely used introductory circuits textbook of the past 25 years. As this book has evolved over the years to meet the changing learning styles of students, importantly, the underlying teaching approaches and philosophies remain unchanged. | 656 The Fourier Transform The following observations about F w are pertinent The real part of F w that is A w is an even function of w in other words A cd . The imaginary part of F w that is B cd is an odd function of cd in other words 5 w -B -cd . The magnitude of F w that is A2 cd B2 w is an even function of CD. The phase angle of F w that is 0 cd tan-lB to A to is an odd function of cd. Replacing cd by -w generates the conjugate of F cd in other words F cd F cd . Hence if f t is an even function F cd is real and if f t is an odd function F cd is imaginary. If f t is even from Eqs. and A cd 2 f t COS CDt dt and B id 0. If f t is an odd function. A id 0 and B id -2 17-43 We leave the derivations of Eqs. for you as Problems and . If f t is an even function its Fourier transform is an even function and if f t is an odd function its Fourier transform is an odd function. Moreover if f t is an even function from the inverse Fourier integral f t F cD e wldcD A cD e MtdcD 2tt J-oo 2tt 1 f A cd coscd jsincDt dcD 2ir 1 0G A cd cos idI da 0 217 J go 2 2ir A cd COS CDt dll . Operational Transforms 657 Now compare Eq. with Eq. . Note that except for a factor of 1 2tt these two equations have the same form. Thus the waveforms of 4 co and become interchangeable if f t is an even function. For example we have already observed that a rectangular pulse in the time domain produces a frequency spectrum of the form sin a io. Specifically Eq. expresses the Fourier transform of the voltage pulse shown in Fig. . Hence a rectangular pulse in the frequency domain must be generated by a time-domain function of the form sin t t. We can illustrate this requirement by finding the time-domain function corresponding to the frequency spectrum shown in Fig. . From Eq. M W 2 0 W 2 Figure A rectangular frequency spectrum. co 2 . IM smiot f t - Mcosoit du I ---- 2tf Jo 2tt t 0 1 sino i 2 2 M z 2
