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2017/4/14
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Lecture 8
Discrete‐TimeSignals and Systems
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What is in this chapter?
• Discrete-time signals
• Periodicity, symmetry, energy and power
• Basic discrete-time signals
• Discrete-time LTI systems
• Recursive and non-recursive systems
• Difference representation of systems
• Convolution sum
• Causality and stability
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Discrete-time Signals
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A discrete‐time signal can be thought of as a real‐ orcomplex‐valued function of the integer sample index :
⋅ : → →
Most discrete‐time signals are obtained by samplingcontinuous‐time signals, but there are inherently discrete signals.
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Fibonacci Sequence:
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Discrete‐Time Sinusoidal Sequences
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Stock Price
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Basic Discrete-time Signals
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Discrete-time Sinusoids
Discrete frequency
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cos
If discrete frequency (rad), is periodic. Otherwise, not!
Discrete-time Sinusoids
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Discrete-time Unit-step and Unit-sample Signals
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The unit‐sample (impulse) signal is defined as:
The unit‐step signal is defined as:
1, 00, 0
1, 00, 0
Generic representation of discrete-time signals
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Any discrete‐time signal can be represented as a linearcombination of shifted impulses:
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Discrete-time Systems
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Linearity
Time‐Invariance
Stability
Causality
Discrete-time Systems
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Linear• Scaling: • Additivity:
Time‐Invariant•
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Non‐Linear System:compute square root
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Scaling
Additivity
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Recursive and Non-recursive Discrete-time Systems
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Moving Average: Linear Time‐Invariant
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Running Sum:
1 2 1
3 2 1
Convolution Sum
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Let be the impulse response of an LTI discrete‐time system, i.e. is the output of the system when the input is an impulse .
For any input , we have
For this input , the output of the system is:
≔ ∗
Convolution
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Convolutional Sum
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Commutative
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Moving‐Average Filter
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0 1 2
Impulse Response:
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Moving‐Average Filter
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Convolutional Sum: Graphical Computation
1, 0 40, . .
, 0 60, . .
∗
0, 0
,
,
∈ 0,4∈ 4,6
,0,
∈ 6,1010
Convolutional Sum: Graphical Computation
1, 0 40, . .
, 0 60, . .
∗
0, 0
,
,
∈ 0,4∈ 4,6
,0,
∈ 6,1010
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Linear Filtering
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Non-linear Filtering
Find the Median Value Non‐Linear Operation
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Causality of Discrete-time Systems
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Real‐time processing requires causality!
A discrete‐time system is causal if the output does not dependon the future inputs!
BIBO Stability of LTI Discrete-time Systems
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For a bounded input, the output of an LTI system is bounded as follows:
∞
An LTI discrete‐time system is said to be BIBO stable if its impulse response isabsolutely summable, i.e.
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Periodic and Aperiodic Signals
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Discrete-time Periodic Sinusoids
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Finite Energy and Finite Power Discrete-time Signals
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When Ω :
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When Ω 3.2:
Ω 3.2 Ω
Even and Odd Discrete-time Signals
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Basic Operation:
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What have we accomplished?
• Obtained similar theory for discrete- and continuous-time signals and systems with distinct differences
• Discrete frequency is finite but circular
• Sampling period determines radian frequency of discrete signals
• Discrete sinusoids not necessarily periodic
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Oppenheim Book: Chapter 1 & 2Chaparro Book: Chapter 9 in Edition 2, Chapter 8 in Edition 1
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