Outline
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Time Response of a
Discrete-Time System
Response of linear system.
Convolution theorem.
Impulse response.
z-transfer function.
M. S. Fadali
Professor of EE
UNR
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Principle of Superposition
Convolution Summation
• Write input as a weighted sum of deltas
Impulse response sequence
The response of a discrete time system to
a unit impulse.
• Obtain response for an arbitrary input
sequence in terms of the impulse response.
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• Time Response: sum of impulse scaled impulse
responses
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Causal System
• Response to
Response of a LTI System
starts at time
Theorem 2.2: The response of a LTI discrete
time system
to an arbitrary input
is given by the convolution
sequence
summation of the input sequence and the
impulse response sequence of the system
.
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• Change summation variable:
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Response of Causal LTI DT
System to an Impulse at iT
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Input Components &
Corresponding Output Components
Input
(k  i)
Response
{h(ki)}
......
iT
iT
LTI
System
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Example: k = 2
Z-Transfer Function
u(0) h(2)
0
k
u(1) h(1)
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•
k
u(2) h(0)
0
the z-transfer function or transfer
function.
The transfer function & the impulse
response sequence are z-transform
pairs.
Z
k
0
1
2
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Output fo DT System
Convolution Theorem
• The z-transform of the convolution of two
time sequences is equal to the product of
their z -transforms.
(Proved earlier)
• The z-transform of the output of a linear
system is the product of the transfer
function and thei z –transform of the input.
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Use the z-transform to find the system
output:
i. z-transform the input.
ii. Multiply the z-transform of the input by
the z-transfer function.
iii. Inverse z-transform to obtain the output
sequence.
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Example 2.20
(a) From Difference Equation
• Let
Find the impulse response
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(a) From the difference equation.
• By induction
(b) Using z-transformation.
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(b) Using z-transformation
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Example 2.21
Find the system transfer function and its
response to a sampled unit step.
Solution: z-transform
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• Use the delay theorem
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