Z-Transform Short notes | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

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Z-T ransform
The Z-T ransform is a p o w erful to ol in signal pro cessing and con trol systems for analyzing
discrete-time signals and systems. It transforms a discrete-time signal in to a complex
frequency domain, facilitating system analysis and design.
• Definition : The Z-T ransform o f a discrete-time signalx[n] is defined as a function
of the complex v ariable z . F orm ula : X(z) =
P
8
n=-8
x[n]z
-n
, where X(z) is the
Z-T ransform, x[n] is the signal, and z is a complex n um b er. The region where the
sum con v erges is c alled the Region of Con v ergence (R OC).
– A pplic ation : Analyzing linear time-in v arian t (L TI) systems.
– Note : The R OC determines the stabilit y and causalit y of the system.
• Prop erties : Key prop erties simplify Z-T ransform op erations.
– Line arity : Z{ax
1
[n]+bx
2
[n]} =aX
1
(z)+bX
2
(z) .
– Time Shifting : Z{x[n-k]} =z
-k
X(z) .
– Sc aling in Z-Domain : Z{a
n
x[n]} =X(a
-1
z) .
– Convolution : Z{x
1
[n]*x
2
[n]} =X
1
(z)X
2
(z) .
– A pplic ation : Simplifies system analysis and filter design.
– Note : Prop erties require the R OC to include the in tersection of individual
R OCs.
• In v erse Z-T ransform : Reco v ers the time-domain signal x[n] from X(z) . F or-
m ula : x[n] =
1
2pj
H
C
X(z)z
n-1
dz , whereC is a closed con tour in the R OC. Common
metho ds include p artial fraction expansion and residue theorem.
– A pplic ation : Reconstructing signals in digital signal pro cessing.
– Note : P artial fraction expansion is practical for rational functions.
• T ransfer F unction : F or an L TI system, the transfer function is the Z-T ransform of
the impulse resp onse. F orm ula : H(z) =
Y(z)
X(z)
, whereY(z) is the output andX(z) is
the input Z-T ransform. F or a difference equation
P
N
k=0
a
k
y[n-k] =
P
M
k=0
b
k
x[n-k] ,
the trans fer function is H(z) =
?
M
k=0
b
k
z
-k
?
N
k=0
a
k
z
-k
.
– A pplic ation : Analyzing system stabilit y and frequency resp onse.
– Note : P oles of H(z) inside the unit circle ensure stabilit y .
• Stabilit y and R OC : A system is stable if the R OC of its transfer function includes
the unit circle (|z| = 1 ). Condition : F or a causal system, all p oles of H(z) m ust
lie inside the unit circle.
– A pplic ation : Designing stable digital filters.
– Note : The R OC dep ends on the signal’s causalit y and gro wth.
• Common Z-T ransform P airs :
– Unit impulse: d[n]? 1 .
1
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Z-T ransform
The Z-T ransform is a p o w erful to ol in signal pro cessing and con trol systems for analyzing
discrete-time signals and systems. It transforms a discrete-time signal in to a complex
frequency domain, facilitating system analysis and design.
• Definition : The Z-T ransform o f a discrete-time signalx[n] is defined as a function
of the complex v ariable z . F orm ula : X(z) =
P
8
n=-8
x[n]z
-n
, where X(z) is the
Z-T ransform, x[n] is the signal, and z is a complex n um b er. The region where the
sum con v erges is c alled the Region of Con v ergence (R OC).
– A pplic ation : Analyzing linear time-in v arian t (L TI) systems.
– Note : The R OC determines the stabilit y and causalit y of the system.
• Prop erties : Key prop erties simplify Z-T ransform op erations.
– Line arity : Z{ax
1
[n]+bx
2
[n]} =aX
1
(z)+bX
2
(z) .
– Time Shifting : Z{x[n-k]} =z
-k
X(z) .
– Sc aling in Z-Domain : Z{a
n
x[n]} =X(a
-1
z) .
– Convolution : Z{x
1
[n]*x
2
[n]} =X
1
(z)X
2
(z) .
– A pplic ation : Simplifies system analysis and filter design.
– Note : Prop erties require the R OC to include the in tersection of individual
R OCs.
• In v erse Z-T ransform : Reco v ers the time-domain signal x[n] from X(z) . F or-
m ula : x[n] =
1
2pj
H
C
X(z)z
n-1
dz , whereC is a closed con tour in the R OC. Common
metho ds include p artial fraction expansion and residue theorem.
– A pplic ation : Reconstructing signals in digital signal pro cessing.
– Note : P artial fraction expansion is practical for rational functions.
• T ransfer F unction : F or an L TI system, the transfer function is the Z-T ransform of
the impulse resp onse. F orm ula : H(z) =
Y(z)
X(z)
, whereY(z) is the output andX(z) is
the input Z-T ransform. F or a difference equation
P
N
k=0
a
k
y[n-k] =
P
M
k=0
b
k
x[n-k] ,
the trans fer function is H(z) =
?
M
k=0
b
k
z
-k
?
N
k=0
a
k
z
-k
.
– A pplic ation : Analyzing system stabilit y and frequency resp onse.
– Note : P oles of H(z) inside the unit circle ensure stabilit y .
• Stabilit y and R OC : A system is stable if the R OC of its transfer function includes
the unit circle (|z| = 1 ). Condition : F or a causal system, all p oles of H(z) m ust
lie inside the unit circle.
– A pplic ation : Designing stable digital filters.
– Note : The R OC dep ends on the signal’s causalit y and gro wth.
• Common Z-T ransform P airs :
– Unit impulse: d[n]? 1 .
1
– Unit step: u[n]?
1
1-z
-1
, |z|> 1 .
– Exp onen tial: a
n
u[n]?
1
1-az
-1
, |z|>|a| .
– A pplic ation : Simplifies transform calculations.
– Note : P airs are v alid within their resp ectiv e R OCs.
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