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1、BEIJING JIAOTONG UNIVERSITYSignals and Systems Complex frequency-domain analysis for signalsz-domain representation for D-T signalsUnilateral z-transform of typical signalsProperties of unilateral z-transformInversion of unilateral z-transformimpulse signalstep signalexponential signalimaginary expo
2、nential signalsinusoidal signalUnilateral z-transform of typical signals1.impulse signal kZ Zkkzkk 1 0z00 0)z(mI)z(eRCORThe ROC includes the entire z-plane.Unilateral z-transform of typical signals2.step signal ukz1Z Zu ku k zkk 0zzzzkk,11101COR0 0)z(mI)z(eR 1 Unilateral z-transform of typical signa
3、lsThis is a geometric series of infinite length in the ratio(z1).If the magnitude of the ratio is less than 1,or|z|1,the sum converges.Magnitude of pole3.exponential signal akukZ Za u ka u k zkkkk 0zazakk11()101COR0 0)z(mI)z(eR|a|zaThis is a geometric series of infinite length in the ratio(a/z).If t
4、he magnitude of the ratio is less than 1,or|z|a|,the sum converges.Unilateral z-transform of typical signalsMagnitude of pole4.imaginary exponential signal ej0kukZ Zu kzk1 e1e(j)1(j)00Similarly,we can derive the result.Z Zu ku k zkkkkee0jj00z1zezzkk1 e1(e)1j01j00COR0 0)z(mI)z(eR 1 Unilateral z-trans
5、form of typical signalsMagnitude of pole5.sinusoidal signal cos(0k)uk,sin(0k)ukEuler formulaZZZZZZk u ku ku kkkcos()=e /2e /20jj00zz21e21e111111jj00zzzzzz21 2cos()1 2cos()122cos()1cos()0012120011zzzz2 1cos()jsin()1cos()jsin()11100001111Unilateral z-transform of typical signalsZ Zu kzkzzz)cos()1 2cos
6、(,11 cos()012010Z Zkzzkzuz(s1 2co),1n()sin()si001201COR0 0)z(mI)z(eR 1 zzzz2cos()1,1cos()z0202zzzz2cos()1,1sin()020Unilateral z-transform of typical signals5.sinusoidal signal cos(0k)uk,sin(0k)ukMagnitude of poleAcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.Unilateral z-transform of typical signals