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Discrete-time Systems
Prof. Siripong Potisuk
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Input-output Description
A DT system transforms DT inputs into DT outputs
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System Interconnection
- Build more complex systems- Modify response of a system
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Response of an LTI System
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(Also referred to as Impulse response)
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Properties of Convolution Sum
A discrete-time LTI system is completely characterized by its impulse response, i.e., completely determines its input-output behavior. There is only one LTI system with a given h[n]
The role of h [n] and x [n] can be interchanged
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Commutative Property
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The Distributive Property
is equivalent to
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The Associative Property
is equivalent to
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- The impulse response of a causal LTI system must be zero before the impulse occurs.- Causality for a linear system is equivalent to the condition of initial rest.
0
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k
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Stability for LTI Systems:
A necessary and sufficient condition for an LTI system tobe BIBO stable is that the impulse response is absolutelysummable.
Causality for LTI Systems:
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Time-domain Description of DT LTI Systems
A general Nth-order linear constant-coefficient differenceequation
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k
N
kkk knyaknxb
any
0 10
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Recursive equation, i.e., expresses the output at time n interms of previous values of the input and output
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Solutions of LCCDE’s
- The complete solution depends on both the causal input x[n] and the initial conditions, y[-1], y[-2],……, y[-N ].
- The solution can be decomposed into a sum of two parts:
response state-zero the
rest) (initialonly input the todue is ][
responseinput -zero the
input) (noonly conditions initial the todue is ][
re whe][][][
1
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Finite Impulse Response (FIR) System
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The equation is nonrecursive, i.e., previously computedvalues of the output are not used to recursively computethe present value of the output.
The impulse response is seen to have finite duration andgiven by
otherwise
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,0
Mnnh a
nb
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Infinite Impulse Response (IIR) System
recursive isequation difference the1,NFor
If the system is initially at rest, the impulse responsewill have infinite duration.
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Example Consider the difference equation
rest. initial assume and ][][ where][]1[2
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