Signals and SystemsEE 112Lt 14 Sil f f Fi T fLecture 14: Special forms of Fourier TransformKhosrow GhadiriElectrical Engineering Department Electrical Engineering Department San Jose State University© Khosrow GhadiriJean Baptiste Charles FourierJean Baptiste Charles Fourier (3/21/17685/16/1830)© Khosrow GhadiriJean Baptiste Charles Fourier (3/21/1768-5/16/1830)2Signal and System EE Dept. SJSUThe Fourier transformFourier series produce discrete line spectra with nonzero values at Fourier series produce discrete line spectra with nonzero values at specific frequencies.  Non-periodic function such as unit step function has continuous frequency spectra.A i di i l b th ht i di i l i hi h th A non-periodic signal can be thought as a periodic signal in which the period extent from to . Then, for a signal that is a function of time with period from to , we form the Fourier integral ftFjtFftedt Which exist for every value of the radian frequency . The Fourier transform is, in general, a complex functionFftedt The notation to express the Fourier transform and its inversejFeF mFFe   ft FF1FftF© Khosrow Ghadiri 3Signal and System EE Dept. SJSUft FFFftFSpecial forms of the Fourier transformThe time function can be expressed as the sum of real and ftThe time function can be expressed as the sum of real and imaginary parts the Fourier integral can be represented as Re Imft f t jf tft By Euler’s identity   Re Imjt jtF f te dt j f te dt  The real part of Fourier transform isRe Im Re Imcos sin sin cosFfttfttdtjfttfttdt  cos sinFfttfttdt The imaginary part of Fourier transform isRe Re Imcos sinFfttfttdtIRIsin cosFfttfttdt© Khosrow Ghadiri 4Signal and System EE Dept. SJSUImReImsin cosFfttfttdtSpecial forms of the inverse Fourier transformThe time function can be expressed in the term of its spectral ftThe time function can be expressed in the term of its spectral components as a real and imaginary parts. In the inverse Fourier transformB th E l ’ id titft  Re Im12jtftFjFedBy the Euler’s identity  Re Im1cos sin2ft F t F tdj  The real part of Fourier transform isRe Imsin cos2jFtF td1cos sinft F tF td  The imaginary part of Fourier transform isReRe Imcos sin2ft F tF td 1sin cosft F tF td © Khosrow Ghadiri 5Signal and System EE Dept. SJSUIm Re Imsin cos2ft F tF td Tabulation of Fourier transformThe correspondence between time to frequency domain for real The correspondence between time to frequency domain for real, imaginary, even, and odd functions in both the time and frequency domains will be shown in tabular form as in table belowfFReal Imaginary Complex Even OddRealftReal and EvenReal and OddImaginaryImaginary and EvanImaginary © Khosrow Ghadiri 6Signal and System EE Dept. SJSUgyand OddReal time functionIf the time function is real time function reduces toftFtIf the time function is real time function reduces to And ft Re RecosFfttdtsinFfttdtReFt In conclusion if is real, is, in general, complex.W idi t thi lt i th tbl i th fll i lid Im ResinFfttdtFftWe indicate this result in the table in the following slide © Khosrow Ghadiri 7Signal and System EE Dept. SJSUReal time functionReal Imaginary Complex Even OddFftReal✸Real and EvenReal and OddImaginaryImaginary and EvanImaginary and Odd© Khosrow GhadiriOdd8Signal and System EE Dept. SJSUReal time and even functionIf the time function is real time function and even:ftIf the time function is real time function and even: For even function , then the product , with respect to t is even, while the product is odd. thenftRe Re Re2cosF f t t d t f t evenRe ReftftRecosfttResinftt AndTherefore if is realRe Re Re02cosF f t t d t f t even  Im Re Resin 0F f t t dt f t even  Ff t evenTherefore, if , is real. To determine whether is even or odd when , we must perform a test for evenness or oddness with respect to , thus, substitution of for in , yieldsFRef t evenFReft evenReF In conclusion if is real and even, is also real and even.We indicate this result in the table in the following slide Re Re Re Re002cos 2cosFfttdtfttdtF   ftF© Khosrow GhadiriWe indicate this result in the table in the following slide 9Signal and System EE Dept. SJSUReal time and even functionReal Imaginary Complex Even OddFftReal✸Real and EvenXXReal and OddImaginaryImaginary and EvanImaginary and Odd© Khosrow GhadiriOdd10Signal