Copyright (c) 2016 - 2019 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Laurent Series and z-Transform - Geometric Series Applications A 20191123 Sat
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Copyright (c) 2016 - 2019 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of theGNU Free Documentation License, Version 1.2 or any later version published by the Free SoftwareFoundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy ofthe license is included in the section entitled "GNU Free Documentation License".
Laurent Series and z-Transform - Geometric Series Applications A
20191123 Sat
causal
causalanti-causal
causalanti-
causal causalanti-
causal causalanti-
Simple Pole Form
Geometric Series Form
Geometric Series Form Simple Pole Form
Geometric Series Form Combinationswith a unit start term
Geometric Series Form Combinationswith non-unit start term
Geometric Series with a unit start term Laurent Series
Geometric Series with a unit start term z-Transform
Geometric Series with a unit start term Laurent Series vs. z-Transform
Laurent Series
z-Transform
Laurent Series
z-Transform
Laurent Series
z-Transform
Laurent Series
z-Transform
Geometric Series with a non-unit start term
Laurent Series
Geometric Series with a non-unit start term
z-Transform
Geometric Series with a non-unit start term
Laurent Series vs. z-Transform
Laurent Series
z-Transform
Laurent Series
z-Transform
Laurent Series
z-Transform
Laurent Series
z-Transform
Complemnt ROC Pairs -Original Geometric Series Form Combinations
unit
non-unit
unit
non-unit
unit
non-unit
unit
non-unit
start term
Complemnt ROC Pairs - Shifted Geometric Series Form Combinations
Complemnt ROC Pairs - ReducedShifted Geometric Series Form Combinations
scale(a) scale(a)
scale(1/z) scale(z)
scale(1/a) scale(1/a)
scale(1/z) scale(z)
Comp.ROC
Comp.ROC
Comp.ROC
Comp.ROC
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
-Comp.Rng
-Comp.Rng
-Comp.Rng
-Comp.Rng
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
-Comp.Rng
-Comp.Rng
-Comp.Rng
-Comp.Rng
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
-Comp.Rng
-Comp.Rng
-Comp.Rng
-Comp.Rng
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
-Comp.Rng
-Comp.Rng
-Comp.Rng
-Comp.Rng
SHL.Seq SHR.Seq
SHL.Seq SHR.Seq
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
-Comp.Rng
-Comp.Rng
-Comp.Rng
-Comp.Rng
SHL.Seq, SHL.ROC SHR.Seq, SHR.ROC
SHL.Seq Shift Right(Sequence Function)SHR.Seq Shift Right(Sequence Function)SHL.ROC Shift Right(Region of Convergence)SHR.ROC Shift Right(Region of Convergence)
u(n) u(n-1)
u(-n-1) u(-n)
u(n-1) u(n)
u(-n) u(-n-1)
SHL.ROC
SHL.ROC
SHR.ROC
SHR.ROC
u(n)
u(-n-1)
u(-n)
u(n-1)
Complement
SHL.Seq SHR.Seq
SHL.Seq SHR.Seq
SHL.Seq SHR.Seq
SHL.Seq SHR.Seq
Simple Pole Form
Geometric Series :
Geometric Series :
the same algebraic formulabut the complement ROC's
the same algebraic formulabut the complement ROC's
two variable function
two variable function
inverse a
inverse a
associated simple pole forms
sign signcomp(z) comp(z)sign comp(n) sign comp(n)
sign signcomp(z) comp(z)sign comp(n) sign comp(n)
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
neg(n) Sym(n)
Sym(n)
Sym(n)
Sym(n)
(1) (2)(3) (4)(5) (6)(7) (8)
(1) (2)(3) (4)(5) (6)(7) (8)
neg(n)
neg(n)
neg(n)
inv(a) inv(a)inv(a) inv(a)inv(a) inv(a)
inv(a) inv(a)inv(a) inv(a)inv(a) inv(a)
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
neg(n)sym(n)
sym(n)
sym(n)
sym(n)
(1) (2)(3) (4)(5) (6)(7) (8)
(1) (2)(3) (4)(5) (6)(7) (8)
neg(n)
neg(n)
neg(n)
sign, inv(a,z) sign, inv(a,z)comp(z) comp(z)
sign, inv(a,z) sign, inv(a,z)comp(z) comp(z)
sign comp(n) sign comp(n)
sign comp(n) sign comp(n)
(1) (2)(3) (4)(5) (6)(7) (8)
(1) (2)(3) (4)(5) (6)(7) (8)
a unit nominator
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
neg(n)sym(n)
sym(n)
sym(n)
sym(n)
neg(n)
neg(n)
neg(n)
inv(a) inv(a)
inv(a) inv(a)
inv(a)
inv(a) inv(a)
inv(a)inv(a) inv(a)
inv(a) inv(a)
(1) (2)(3) (4)(5) (6)(7) (8)
(1) (2)(3) (4)(5) (6)(7) (8)
a unit nominator
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
inv(z) inv(z)
neg(n)sym(n)
sym(n)
sym(n)
sym(n)
neg(n)
neg(n)
neg(n)
id
id
id
id
id
id
id
id
id
id
S(c(n))
S(c(n))
id
id
s(c(n))
s(c(n))
Simple Pole Forms Geometric Series Forms
id
id
id
id
id
id
S(c(n))
S(c(n))
Simple Pole Forms Geometric Series Forms
id
id
id
id
id
id
s(c(n))
s(c(n))
Simple Pole Forms Geometric Series Forms
inv(z) inv(z)
neg(n)sym(n)
inv(z) inv(z)
neg(n)sym(n)
ids(c(n))
ids(c(n))
id
s(c(n))
id
s(c(n))
id
s(c(n))
id
s(c(n))
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