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Special Forms of the Fourier TransformTABLE 8.3 Time Domain and Frequency Domain Correspondence (Refer also to Tables 8.4 − 8.7)ft () F( ω)Real Imaginary Complex Even OddRealReal and Even Real and OddImaginaryImaginary and EvenImaginary and Oddand∞F Re ( ω) =∫f Re () t cos ωt d t = 0 f Re () t = odd–∞∫F Im ( ω) = – 2 f Re () t sinωtdtf Re () t = odd0∞(8.20)(8.21)Therefore, if f Re () t = odd, F( ω)is imaginary.To determine whether F( ω) is even or odd when f Re () t = odd, we perform a test for evennessor oddness with respect to ω. Thus, substitution of – ω for ω in (8.21), yieldsF Im∫(–ω) = – 2 f Re () t sin( – ω)tdt= 2 f Re () t sinωtdt=0∞∫0∞– F Im ( ω)(8.22)Conclusion: If ft () is real and odd, F( ω)is imaginary and odd. We indicate this result inTable 8.4.TABLE 8.4 Time Domain and Frequency Domain Correspondence (Refer also to Tables 8.5 − 8.7)ft () F( ω)Real Imaginary Complex Even OddRealReal and Even Real and Odd ImaginaryImaginary and EvenImaginary and OddSignals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications8−5