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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