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Bishop, Robert H. - The Mechatronics Handbook [CRC Press 2002] Part 11

Tham khảo tài liệu 'bishop, robert h. - the mechatronics handbook [crc press 2002] part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | TABLE 23.5 Properties of the Fourier Transform Property Signal Description Fourier Transform Linearity ax t by t a b constants aX f bY f Evenness and oddness x -1 x t X f 2 f x t cos 2nft dt x t x t X f -2 f x t sin 2 nft dt J0 Time shift x t t e-2nfTX f Time scale x at ỳ X ff 1 a a J Time reversal x -t X f Duality X t x -f Time convolution x t y t X f Y f Frequency convolution x t y t X f Y f Modulation .Ầ ynM x t e X f - f Time differentiation dn - x t dtn j2 pf nX f Frequency differentiation fx t f j 1 ndLX f I2pj f-X f Integration ftt 1 x T dT J-tt --T--r X f 1 X 0 ổ f j2nf 2 v Correlation Rxy T f y t x t - T dt -tt Y -f X f Parseval s theorem f x t 2dt -tt tt f X f 2df -tt the amplitude spectral density gives a continuous display of the amplitude density spectrum which in this case is in the form of impulses rather than just a number. Similarly the Fourier transform of a train of impulses of the form tt p t y 8 t- nT n -tt 23.27 is given by Tt Y C 1 X 1 C r 7. y X 7-1 1 P f T yS f- kFs Fs T 1 k -tt 1 23.28 Energy and Power Spectral Density6 Suppose x t is an aperiodic signal with a Fourier transform X f then its energy is given by E Rxx 0 f x t 2 dt f X f 2df - tt - tt 23.29 This is Parseval s theorem and it shows that the principle of conservation of energy in the time and frequency domains holds. The amplitude spectrum X f can be expressed as X f X f ZX f 2002 CRC Press LLC TABLE 23.5 Properties of the Fourier Transform Property Signal Description Fourier Transform Linearity ax t by t a b constants aX f bY f Evenness and oddness x -1 x t X f 2 f x t cos 2nft dt x t x t X f -2 f x t sin 2 nft dt J0 Time shift x t t e-2nfTX f Time scale x at ỳ X ff 1 a a J Time reversal x -t X f Duality X t x -f Time convolution x t y t X f Y f Frequency convolution x t y t X f Y f Modulation .Ầ ynM x t e X f - f Time differentiation dn - x t dtn j2 pf nX f Frequency differentiation fx t f j 1 ndLX f I2pj f-X f Integration ftt 1 x T dT J-tt --T--r X f 1 X 0 ổ f j2nf 2 v .