DiscreteTime Signals and Systems

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Discrete-Time Signals and Systems

• ??????

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Content

• Introduction
• Discrete-Time Signals---Sequences
• Linear Shift-Invariant Systems
• Stability and Causality
• Linear Constant-Coefficient Difference Equations
• Frequency-Domain Representation of Discrete-Time
  Signals and Systems
• Representation of Sequences by Fourier Transform
• Symmetry Properties of Fourier Transform
• Fourier Transform Theorems
• The Existence of Fourier Transform
• Important Transform Pairs

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Discrete-Time Signals and Systems

• Introduction

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The Taxonomy of Signals

• Signal A function that conveys information

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Signal Process Systems
Facilitate the extraction of desired information
e.g.,

• Filters
• Parameter estimation

Signal Processing System
signal
output
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Signal Process Systems
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Signal Process Systems

• A important class of systems

Linear Shift-Invariant Systems.

• In particular, well discuss

Linear Shift-Invariant Discrete-Time Systems.
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Discrete-Time Signals and Systems

• Discrete-Time Signals---Sequences

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Representation by a Sequence

• Discrete-time system theory
• Concerned with processing signals that are
  represented by sequences.

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Important Sequences

• Unit-sample sequence ?(n)

• Sometime call ?(n)
• a discrete-time impulse or
• an impulse

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Important Sequences

• Unit-step sequence u(n)

• Fact

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Important Sequences

• Real exponential sequence

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Important Sequences

• Sinusoidal sequence

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Important Sequences

• Complex exponential sequence

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Important Sequences

• A sequence x(n) is defined to be periodic with
  period N if

• Example consider

must be a rational number
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Energy of a Sequence

• Energy of a sequence is defined by

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Operations on Sequences

• Sum
• Product
• Multiplication
• Shift

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Sequence RepresentationUsing delay unit
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Discrete-Time Signals and Systems

• Linear Shift-Invariant Systems

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Systems
y(n)Tx(n)
x(n)
Mathematically modeled as a unique transformation
or operator.
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Linear Systems
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Examples
Ideal Delay System
Moving Average
Accumulator
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Examples
Are these system linear?
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Examples
A Memoryless System
Is this system linear?
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Linear Systems
??k?impulse ???n?????
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Shift-Invariant Systems
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Shift-Invariant Systems
??/???????????
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Linear Shift-Invariant Systems
??k?impulse ???n?????
???????
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Impulse Response
h(n)T?(n)
x(n)?(n)
T
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Convolution Sum
y(n)
x(n)
convolution
A linear shift-invariant system is completely
characterized by its impulse response.
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Characterize a System
x(n)
x(n)h(n)
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Properties of Convolution Math
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Properties of Convolution Math
These systems are identical.
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Properties of Convolution Math
These two systems are identical.
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Example
y(n)?
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Example
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Example
compute y(0)
compute y(1)
How to computer y(n)?
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Example
Two conditions have to be considered.
nltN and n?N.
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Example
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Example
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Impulse Response ofthe Ideal Delay System
Ideal Delay System
By letting x(n)?(n) and y(n)h(n),
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Impulse Response ofthe Ideal Delay System
?????

• ?(n? nd)??????
• Shift or
• Copy

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Impulse Response ofthe Moving Average
Moving Average
????(n? k)????
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Impulse Response ofthe Accumulator
Accumulator
??????
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Discrete-Time Signals and Systems

• Stability and Causality

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Stability

• Stable systems --- every bounded input produce a
  bounded output (BIBO)
• Necessary and sufficient condition for a BIBO

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ProveNecessary Condition for Stability

• Show that if x is bounded and S lt ?, then y is
  bounded.

where M max x(n)
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ProveSufficient Condition for Stablility

• Show that if S ?, then one can find a bounded
  sequence x such that y is unbounded.

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Example

• Show that the linear shift-invariant system with
  impulse response h(n)anu(n) where alt1 is
  stable.

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Causality

• Causal systems --- output for y(n0) depends only
  on x(n) with n ?n0.
• A causal system whose impulse response h(n)
  satisfies

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Discrete-Time Signals and Systems

• Linear Constant-Coefficient Difference Equations

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N-th Order Difference Equations
Examples
Ideal Delay System
Moving Average
Accumulator
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Compute y(n)
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The Ideal Delay System
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The Moving Average
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The Moving Average
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Discrete-Time Signals and Systems

• Frequency-Domain Representation of
• Discrete-Time Signals and Systems

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Sinusoidal and Complex Exponential Sequences

• Play an important role in DSP

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Frequency Response
eigenvalue
eigenfunction
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Frequency Response
phase
magnitude
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ExampleThe Ideal Delay System
magnitude
phase
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ExampleThe Ideal Delay System
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Periodic Nature ofFrequency Response
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Periodic Nature ofFrequency Response
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Periodic Nature ofFrequency Response
Generally, we choose ?????? To represent one
period in frequency domain.
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Periodic Nature ofFrequency Response
Low Frequency
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Ideal Frequency-Selective Filters
Lowpass Filter
Bandstop Filter
Highpass Filter
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Moving Average
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Moving Average
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Moving Average

• M4
• Lowpass
• Try larger M

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Discrete-Time Signals and Systems

• Representation of Sequences by
• Fourier Transform

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Fourier Transform Pair
Synthesis
Inverse Fourier Transform (IFT)
Analysis
Fourier Transform (FT)
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Prove
n m
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Prove
n ? m
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Prove
x(n)
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Notations
Synthesis
Inverse Fourier Transform (IFT)
Analysis
Fourier Transform (FT)
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Real and Imaginary Parts
Fourier Transform (FT)
is a complex-valued function
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Magnitude and Phase
magnitude
phase
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Discrete-Time Signals and Systems

• Symmetry Properties of Fourier Transform

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Conjugate-Symmetric andConjugate-Antisymmetric
Sequences

• Conjugate-Symmetric Sequence
• Conjugate-Antisymmetric Sequence

an even sequence if it is real.
an odd sequence if it is real.
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Sequence Decomposition

• Any sequence can be expressed as the sum of a
  conjugate-symmetric one and a conjugate-antisymmet
  ric one, i.e.,

Conjugate Symmetric
Conjugate Antisymmetric
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Function Decomposition

• Any function can be expressed as the sum of a
  conjugate-symmetric one and a conjugate-antisymmet
  ric one, i.e.,

Conjugate Antiymmetric
Conjugate Symmetric
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Conjugate-Symmetric andConjugate-Antiymmetric
Functions

• Conjugate-Symmetric Function
• Conjugate-Antisymmetric Function

an even function if it is real.
an odd function if it is real.
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Symmetric Properties
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Symmetric Properties
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Symmetric Properties
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Symmetric Properties
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Symmetric Properties
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Symmetric Properties for Real Sequence x(n)
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Facts 1. real part is even 2. Img. part is
odd 3. Magnitude is even 4. Phase is odd
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Discrete-Time Signals and Systems

• Fourier Transform Theorems

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Linearity
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Time Shifting ? Phase Change
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Frequency Shifting ?Signal Modulation
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Time Reversal
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Differentiation in Frequency
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The Convolution Theorem
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The Modulation or Window Theorem
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Parsevals Theorem
Facts
Letting ?0, then proven.
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Parsevals TheoremEnergy Preserving
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Example Ideal Lowpass Filter
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Example Ideal Lowpass Filter
The ideal lowpass fileter Is noncausal.
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Example Ideal Lowpass Filter
To approximate the ideal lowpass filter using a
window.
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Example Ideal Lowpass Filter
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Discrete-Time Signals and Systems

• The Existence of Fourier Transform

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Key Issue
Does X(ej?) exist for all ??
Synthesis
We need that X(ej?) lt ? for all ?
Analysis
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Sufficient Condition for Convergence
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More On Convergence

• Define

• Uniform Convergence

• Mean-Square Convergence

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Discrete-Time Signals and Systems

• Important Transform Pairs

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Fourier Transform Pairs
Fourier Transform
Sequence
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Fourier Transform Pairs
Fourier Transform
Sequence
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Fourier Transform Pairs
Fourier Transform
Sequence