PROPERTIES OF Z-TRANSFORM

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PROPERTIES OF Z-TRANSFORM:

PROPERTIES OF Z-TRANSFORM Pamarthy.Chennarao HOD ECE Dept. PALADUGU PARVATHI DEVI COLLEGE OF ENGINEERING

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PROPERTIES OF Z- TRANSFORM 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 2

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LINEAR PROPERTY: Let x 1 (n), x 2 (n) are two discrete sequences and ZT[ x 1 (n) ] = X 1 (z), ZT[ x 2 (n) ] = X 2 (z), then according to linear property of z transform ZT[ a x 1 (n) + b x 2 (n) ] = a X 1 (z) + b X 2 (z) PROOF: From basic definition of z transform of a sequence x(n) Replace x(n) by a x 1 (n) + b x 2 (n) 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 3

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TIME SHIFTING PROPERTY: Let x(n) be a discrete time sequences and ZT[ x(n) ] = X(z), then according to time shifting property of z transform ZT[ x(n – n 0 ) ] = z -no X(z) PROOF: From basic definition of z transform of a sequence x(n) Replace x(n) by x(n – n o ) 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 4

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TIME REVERSAL PROPERTY : Let x(n) be a discrete time sequence and ZT[ x(n) ] = X(z), then according to time reversal property of z transform ZT[ x(– n) ] = X(1/z) PROOF: From basic definition of z transform of a sequence x(n) Replace x(n) by x(– n) 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 5

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TIME CONVOLUTION THEOREM: Let x 1 (n), x 2 (n) are two discrete time sequences and ZT[ x 1 (n) ] = X 1 (z), ZT[ x 2 (n) ] = X 2 (z), then according to time convolution theorem of z transform ZT[ x 1 (n)  x 2 (n) ] = X 1 (z) X 2 (z) PROOF: From basic definition of z transform of a sequence x(n) Replace x(n) by x 1 (n)  x 2 (n) 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 6

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CONJUGATE PROPERTY : Let x(n) be a discrete time sequence and ZT[ x(n) ] = X(z), then according to conjugate property of z transform ZT[ x*(n) ] = X*(z*) PROOF: From basic definition of z transform of a sequence x(n) Replace x(n) by x*(n) 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 7

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DERIVATIVE PROPERTY : Let x(n) be a discrete time sequence and ZT[ x(n) ] = X(z), then according to derivative property of z transform ZT[ n x(n) ] = - d/dz [ X(z) ] PROOF: From basic definition of z transform of a sequence x(n) Differentiate w.r.t z 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 8

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INITIAL VALUE THEOREM : Let x(n) be a discrete time causal sequence and ZT[ x(n) ] = X(z), then according to initial value theorem of z transform PROOF: From basic definition of z transform of a sequence x(n) Apply as z   2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 9

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FINAL VALUE THEOREM : Let x(n) be a discrete time causal sequence and ZT[ x(n) ] = X(z), then according to final value theorem of z transform PROOF: From basic definition of z transform of a causal sequence x(n) Replace x(n) by x(n) – x(n – 1) Apply as z  1 2/3/2011 P.CHENNARAO,HOD ECE DEPT,PPDCET 10