Si l d S tSignals and SystemsEE 112Lecture 15: Fourier Transform propertiesLecture 15: Fourier Transform propertiesKhosrow 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. SJSUOutlineDefinitionDefinition Linearity  Symmetry  Time scaling Time shifting Time shifting  Multiplication  Frequency shifting  Signal modulationTi diff ti ti Time differentiation  Frequency differentiation Time integration Conjugation of time and frequency functions Time convolution Frequency convolution Area under  Area under © Khosrow Ghadiri Parseval’s Theorem3Signal and System EE Dept. SJSUFourier transform properties: Linearity If F And a and b are any real or complex scalars, then ftF  Fgt Gyp, Linearity:  Proof: evaluating the Fourier transform ofGbFatgbtfaaftbgtg By linearity of integration fg  jtaftbgtaftbgtedt  Fyyg  jt jta f t b g t a f t e dt b g t e dt   Faft bgt aF bG  F© Khosrow Ghadiri 4Signal and System EE Dept. SJSUaft bgt aF bG  FExample: Application of Linearity  Consider the below signal. Find its Fourier transform.gftft Let’s write as a linear combination of andttft Then 2sinc sincft t t t t f f      FFFfttt f© Khosrow Ghadiri 5Signal and System EE Dept. SJSUsinc sincft t t t t f f FFFFourier transform properties Symmetry:yy If is the Fourier transform of , then, the symmetry property of the Fourier transform states that2FTFt FFft That is, if in , we replace with , we get the Fourier transform pair 2Ft FFt Proof: Since  Then, 12jtftFed2jtft F d Interchanging and , we get 2jtft Fedt2FTjtfFte dft© Khosrow Ghadiri 6Signal and System EE Dept. SJSUffFourier transform properties Time scaling:g If is the Fourier transform of , and is a real constant, then, 1Fffat FFffta That is, the time scaling property of the Fourier transform states that if we replace the variable in the frequency d i b d di id b th b l t l f faaffFfdomain by , and divide by the absolute value of . Proof: Consider both cases and  ForfaFfaa0a  0a 0a 2jftfat fate dtF Let ; then, , and above equation becomes fat fate dtFatadt d2211fjjfaafffedfedF   F© Khosrow Ghadiri 7Signal and System EE Dept. SJSUffedfedFaa a a    FFourier transform properties Time scaling:g For Let ; then, , and above equation becomes0a 2jftfat f at e dt Fatadt d;, , q And making the above substitution, we find that the  2211fjjfaafffedfedFaa a a      Fg,multiplying factor is . Therefore, for we obtain 1 a1 a1Fffat Faa© Khosrow Ghadiri 8Signal and System EE Dept. SJSUExample: Fourier transform properties Time scaling:g If is the Fourier transform of , Find Fourier transform 0f andsincft12sinc22ftF2t2t22sinc2tfF22fF2t121sinc 22f2t222f2sinc 2f Note: compression in one domain results in expansion in the other domain. A narrow pulse in time domain has broad frequency contents (large bandwidth).110f© Khosrow Ghadiri A broad pulse in time domain has low frequency content (small bandwidth)9Signal and System EE Dept. SJSUFourier transform properties Time shifting:g If is the Fourier transform of , then,  The time shifting property of the Fourier transform states that  00FTjtft t F eFftgp p yif we shift the time function by a constant , the Fourier transform magnitude does not change, but the term is added to its phase angle. Pf ft0t0tProof: L t th d thdt dtt00jtftt ftte dt FttLet ; then, , and thusdt d0tt0tt  00 00FTjtjtjtjft t f e d e f e d F e    F© Khosrow Ghadiri 10Signal and System EE Dept. SJSUFourier transform properties Frequency shiftingqy g If is the Fourier transform of , then,ftF  00Fjteft F That is, multiplication of the time function by , where is a constant, results in shifting the Fourier transform by  Proof:ft0jte00 oroojt jtjteft eftedtFoojtjtoeft fte dtFF©