Formula Sheets: Z-Transform in Signals & Systems | Signals and Systems - Electrical Engineering (EE) PDF Download

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Formula Sheet for Z-Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Z-Transform: Transforms a discrete-time signal x[n] to the z-domain X(z), used
for analyzing linear time-invariant (LTI) systems.
• Complex Variable: z =re
j? , where r: Magnitude, ? : Angle.
• Region of Convergence (ROC): Set of z values where the transform converges.
2. Z-Transform De?nition
• Unilateral Z-Transform:
X(z) =
8
X
n=0
x[n]z
-n
• Bilateral Z-Transform:
X(z) =
8
X
n=-8
x[n]z
-n
3. Inverse Z-Transform
• Inverse Transform:
x[n] =
1
2pj
I
C
X(z)z
n-1
dz
(Typically evaluated using partial fraction expansion or residue method).
4. Properties of Z-Transform
• Linearity:
ax
1
[n]+bx
2
[n]?aX
1
(z)+bX
2
(z)
• Time Shifting:
x[n-n
0
]?z
-n
0
X(z), x[n+n
0
]?z
n
0
X(z)
• Scaling in Z-Domain:
z
n
0
x[n]?X

z
z
0

• Time Reversal:
x[-n]?X

1
z

• Convolution:
x[n]*h[n]?X(z)H(z)
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Page 2


Formula Sheet for Z-Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Z-Transform: Transforms a discrete-time signal x[n] to the z-domain X(z), used
for analyzing linear time-invariant (LTI) systems.
• Complex Variable: z =re
j? , where r: Magnitude, ? : Angle.
• Region of Convergence (ROC): Set of z values where the transform converges.
2. Z-Transform De?nition
• Unilateral Z-Transform:
X(z) =
8
X
n=0
x[n]z
-n
• Bilateral Z-Transform:
X(z) =
8
X
n=-8
x[n]z
-n
3. Inverse Z-Transform
• Inverse Transform:
x[n] =
1
2pj
I
C
X(z)z
n-1
dz
(Typically evaluated using partial fraction expansion or residue method).
4. Properties of Z-Transform
• Linearity:
ax
1
[n]+bx
2
[n]?aX
1
(z)+bX
2
(z)
• Time Shifting:
x[n-n
0
]?z
-n
0
X(z), x[n+n
0
]?z
n
0
X(z)
• Scaling in Z-Domain:
z
n
0
x[n]?X

z
z
0

• Time Reversal:
x[-n]?X

1
z

• Convolution:
x[n]*h[n]?X(z)H(z)
1
• Multiplication by n:
nx[n]?-z
dX(z)
dz
• Initial Value Theorem:
x[0] = lim
z?8
X(z)
• Final Value Theorem (if poles of (1-z
-1
)X(z) inside unit circle):
lim
n?8
x[n] = lim
z?1
(z-1)X(z)
5. Common Z-Transform Pairs
• Unit Impulse:
d[n]? 1
• Unit Step:
u[n]?
1
1-z
-1
, |z|> 1
• Unit Ramp:
nu[n]?
z
-1
(1-z
-1
)
2
, |z|> 1
• Exponential:
a
n
u[n]?
1
1-az
-1
, |z|>|a|
• Sinusoid:
sin(?
0
n)u[n]?
z
-1
sin? 0
1-2z
-1
cos? 0
+z
-2
, |z|> 1
cos(?
0
n)u[n]?
1-z
-1
cos? 0
1-2z
-1
cos? 0
+z
-2
, |z|> 1
6. System Analysis
• Transfer Function (LTI system):
H(z) =
Y(z)
X(z)
• Impulse Response:
h[n] =Z
-1
{H(z)}
• Output Response:
Y(z) =H(z)X(z)
• Stability: System stable if all poles of H(z) lie inside the unit circle (|z|< 1).
2
Page 3


Formula Sheet for Z-Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Z-Transform: Transforms a discrete-time signal x[n] to the z-domain X(z), used
for analyzing linear time-invariant (LTI) systems.
• Complex Variable: z =re
j? , where r: Magnitude, ? : Angle.
• Region of Convergence (ROC): Set of z values where the transform converges.
2. Z-Transform De?nition
• Unilateral Z-Transform:
X(z) =
8
X
n=0
x[n]z
-n
• Bilateral Z-Transform:
X(z) =
8
X
n=-8
x[n]z
-n
3. Inverse Z-Transform
• Inverse Transform:
x[n] =
1
2pj
I
C
X(z)z
n-1
dz
(Typically evaluated using partial fraction expansion or residue method).
4. Properties of Z-Transform
• Linearity:
ax
1
[n]+bx
2
[n]?aX
1
(z)+bX
2
(z)
• Time Shifting:
x[n-n
0
]?z
-n
0
X(z), x[n+n
0
]?z
n
0
X(z)
• Scaling in Z-Domain:
z
n
0
x[n]?X

z
z
0

• Time Reversal:
x[-n]?X

1
z

• Convolution:
x[n]*h[n]?X(z)H(z)
1
• Multiplication by n:
nx[n]?-z
dX(z)
dz
• Initial Value Theorem:
x[0] = lim
z?8
X(z)
• Final Value Theorem (if poles of (1-z
-1
)X(z) inside unit circle):
lim
n?8
x[n] = lim
z?1
(z-1)X(z)
5. Common Z-Transform Pairs
• Unit Impulse:
d[n]? 1
• Unit Step:
u[n]?
1
1-z
-1
, |z|> 1
• Unit Ramp:
nu[n]?
z
-1
(1-z
-1
)
2
, |z|> 1
• Exponential:
a
n
u[n]?
1
1-az
-1
, |z|>|a|
• Sinusoid:
sin(?
0
n)u[n]?
z
-1
sin? 0
1-2z
-1
cos? 0
+z
-2
, |z|> 1
cos(?
0
n)u[n]?
1-z
-1
cos? 0
1-2z
-1
cos? 0
+z
-2
, |z|> 1
6. System Analysis
• Transfer Function (LTI system):
H(z) =
Y(z)
X(z)
• Impulse Response:
h[n] =Z
-1
{H(z)}
• Output Response:
Y(z) =H(z)X(z)
• Stability: System stable if all poles of H(z) lie inside the unit circle (|z|< 1).
2
7. Region of Convergence (ROC)
• Causal Signal: ROC is outside the outermost pole (|z|>r).
• Anti-Causal Signal: ROC is inside the innermost pole (|z|<r).
• Two-Sided Signal: ROC is an annular ring between poles.
• Stability: ROC must include the unit circle (|z| = 1) for a stable system.
8. Partial Fraction Expansion
• For Distinct Poles X(z) =
N(z)
Q
m
i=1
(1-p
i
z
-1
)
:
X(z) =
m
X
i=1
A
i
1-p
i
z
-1
, A
i
= lim
z?p
i
(1-p
i
z
-1
)X(z)
• Repeated Poles (order m):
X(z) =
m
X
k=1
A
k
(1-pz
-1
)
k
+other terms, A
k
=
1
(m-k)!
lim
z?p
d
m-k
dz
m-k
h
(1-pz
-1
)
m
X(z)
i
9. Frequency Response from Z-Transform
• Frequency Response:
H(e
j? ) =H(z)



z=e
j? (Valid if ROC includes unit circle).
• Magnitude and Phase:
|H(e
j? )|, ?H(e
j? )
10. Design Considerations
• System Analysis: Use transfer function to determine stability, poles, and zeros.
• Transient Response: Analyze using inverse Z-transform.
• Applications: Digital signal processing, discrete-time control systems, ?lter de-
sign.
• ROC: Ensure correct ROC to determine causality and stability.
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