Formula sheet: Z-Transform | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

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Z-T ransform F orm ula S heet for Digital Signal
Pro cessing (GA TE)
Z-T ransform Definition
• Definition : The Z-T ransform of a discrete-time signal x[n] is defined as:
X(z) =
8
X
n=-8
x[n]z
-n
where z is a complex v ariable, and the region of con v ergence (R OC) is the set of z for
whic h the sum con v erges.
• Unilateral Z-T ransform : Used for causal signals (x[n] = 0 for n< 0 ).
X(z) =
8
X
n=0
x[n]z
-n
Prop erties of Z-T ransform
• Linearit y :
Z{ax[n]+by[n]} =aX(z)+bY(z)
• Time Shifting :
Z{x[n-k]} =z
-k
X(z)
• Scaling in Z-Domain :
Z{a
n
x[n]} =X

z
a

• Con v olution :
Z{x[n]*y[n]} =X(z)Y(z)
• Differen tiation in Z-Domain :
Z{nx[n]} = -z
dX(z)
dz
• Initial V alue Theorem :
x[0] = lim
z?8
X(z)
• Final V alue Theorem (if all p oles of (1-z
-1
)X(z) are inside the unit circle):
lim
n?8
x[n] = lim
z?1
(z -1)X(z)
1
Page 2


Z-T ransform F orm ula S heet for Digital Signal
Pro cessing (GA TE)
Z-T ransform Definition
• Definition : The Z-T ransform of a discrete-time signal x[n] is defined as:
X(z) =
8
X
n=-8
x[n]z
-n
where z is a complex v ariable, and the region of con v ergence (R OC) is the set of z for
whic h the sum con v erges.
• Unilateral Z-T ransform : Used for causal signals (x[n] = 0 for n< 0 ).
X(z) =
8
X
n=0
x[n]z
-n
Prop erties of Z-T ransform
• Linearit y :
Z{ax[n]+by[n]} =aX(z)+bY(z)
• Time Shifting :
Z{x[n-k]} =z
-k
X(z)
• Scaling in Z-Domain :
Z{a
n
x[n]} =X

z
a

• Con v olution :
Z{x[n]*y[n]} =X(z)Y(z)
• Differen tiation in Z-Domain :
Z{nx[n]} = -z
dX(z)
dz
• Initial V alue Theorem :
x[0] = lim
z?8
X(z)
• Final V alue Theorem (if all p oles of (1-z
-1
)X(z) are inside the unit circle):
lim
n?8
x[n] = lim
z?1
(z -1)X(z)
1
Common Z-T ransform P airs
• Unit Impulse : x[n] = d[n]
X(z) = 1, R OC: All z
• Unit Step : x[n] = u[n]
X(z) =
1
1-z
-1
, R OC: |z|> 1
• Exp onen tial : x[n] = a
n
u[n]
X(z) =
1
1-az
-1
, R OC: |z|> |a|
• Ramp : x[n] = nu[n]
X(z) =
z
-1
(1-z
-1
)
2
, R OC: |z|> 1
In v erse Z-T ransform
• Definition : The in v erse Z-T ransform reco v ers x[n] from X(z) .
x[n] =
1
2pj
I
X(z)z
n-1
dz
• Practical Metho ds :
– P artial F raction Expansion: Decomp ose X(z) in to simpler terms.
– P o w er Series Expansion: Express X(z) as a series to find x[n] .
– Residue Metho d: Use con tour in tegration for complex p oles.
System Analysis with Z-T ransform
• T ransfer F unction : F or an L TI system with input x[n] and output y[n] :
H(z) =
Y(z)
X(z)
• Difference Equation to Z-T ransform : F or a system describ ed b y:
N
X
k=0
a
k
y[n-k] =
M
X
k=0
b
k
x[n-k]
The transfer function is:
H(z) =
P
M
k=0
b
k
z
-k
P
N
k=0
a
k
z
-k
• Stabilit y : A system is stable if all p oles of H(z) lie inside the unit circle (|z|< 1 ).
2
Page 3


Z-T ransform F orm ula S heet for Digital Signal
Pro cessing (GA TE)
Z-T ransform Definition
• Definition : The Z-T ransform of a discrete-time signal x[n] is defined as:
X(z) =
8
X
n=-8
x[n]z
-n
where z is a complex v ariable, and the region of con v ergence (R OC) is the set of z for
whic h the sum con v erges.
• Unilateral Z-T ransform : Used for causal signals (x[n] = 0 for n< 0 ).
X(z) =
8
X
n=0
x[n]z
-n
Prop erties of Z-T ransform
• Linearit y :
Z{ax[n]+by[n]} =aX(z)+bY(z)
• Time Shifting :
Z{x[n-k]} =z
-k
X(z)
• Scaling in Z-Domain :
Z{a
n
x[n]} =X

z
a

• Con v olution :
Z{x[n]*y[n]} =X(z)Y(z)
• Differen tiation in Z-Domain :
Z{nx[n]} = -z
dX(z)
dz
• Initial V alue Theorem :
x[0] = lim
z?8
X(z)
• Final V alue Theorem (if all p oles of (1-z
-1
)X(z) are inside the unit circle):
lim
n?8
x[n] = lim
z?1
(z -1)X(z)
1
Common Z-T ransform P airs
• Unit Impulse : x[n] = d[n]
X(z) = 1, R OC: All z
• Unit Step : x[n] = u[n]
X(z) =
1
1-z
-1
, R OC: |z|> 1
• Exp onen tial : x[n] = a
n
u[n]
X(z) =
1
1-az
-1
, R OC: |z|> |a|
• Ramp : x[n] = nu[n]
X(z) =
z
-1
(1-z
-1
)
2
, R OC: |z|> 1
In v erse Z-T ransform
• Definition : The in v erse Z-T ransform reco v ers x[n] from X(z) .
x[n] =
1
2pj
I
X(z)z
n-1
dz
• Practical Metho ds :
– P artial F raction Expansion: Decomp ose X(z) in to simpler terms.
– P o w er Series Expansion: Express X(z) as a series to find x[n] .
– Residue Metho d: Use con tour in tegration for complex p oles.
System Analysis with Z-T ransform
• T ransfer F unction : F or an L TI system with input x[n] and output y[n] :
H(z) =
Y(z)
X(z)
• Difference Equation to Z-T ransform : F or a system describ ed b y:
N
X
k=0
a
k
y[n-k] =
M
X
k=0
b
k
x[n-k]
The transfer function is:
H(z) =
P
M
k=0
b
k
z
-k
P
N
k=0
a
k
z
-k
• Stabilit y : A system is stable if all p oles of H(z) lie inside the unit circle (|z|< 1 ).
2
Key Notes
• Alw a ys sp ecify the R OC when computing Z-T ransforms.
• F or GA TE, fo cus on common transform pairs and prop erties lik e linearit y , time shift-
ing, and con v olution.
• Use partial fraction expansion for in v erse Z-T ransforms in most GA TE problems.
• Ensure causalit y and stabilit y conditions are c hec k ed for system analysis.
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