Biomedical Signal processing Chapter 2 Discrete-Time Signals and Systems

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Biomedical Signal processingChapter 2
Discrete-Time Signals and Systems

• Zhongguo Liu
• Biomedical Engineering
• School of Control Science and Engineering,
  Shandong University

2013-12-27
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Chapter 2 Discrete-Time Signals and Systems

• 2.0 Introduction
• 2.1 Discrete-Time Signals Sequences
• 2.2 Discrete-Time Systems
• 2.3 Linear Time-Invariant (LTI) Systems
• 2.4 Properties of LTI Systems
• 2.5 Linear Constant-Coefficient Difference
  Equations

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Chapter 2 Discrete-Time Signals and Systems

• 2.6 Frequency-Domain Representation of
  Discrete-Time Signals and systems
• 2.7 Representation of Sequences by Fourier
  Transforms
• 2.8 Symmetry Properties of the Fourier Transform
• 2.9 Fourier Transform Theorems
• 2.10 Discrete-Time Random Signals
• 2.11 Summary

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2.0 Introduction

• Signal something conveys information
• Signals are represented mathematically as
  functions of one or more independent variables.
• Continuous-time (analog) signals, discrete-time
  signals, digital signals
• Signal-processing systems are classified along
  the same lines as signals Continuous-time
  (analog) systems, discrete-time systems, digital
  systems
• Discrete-time signal
• Sampling a continuous-time signal
• Generated directly by some discrete-time process

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2.1 Discrete-Time Signals Sequences

• Discrete-Time signals are represented as
• In sampling,
• 1/T (reciprocal of T) sampling frequency

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Figure 2.1 Graphical representation of a
discrete-time signal
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Sampling the analog waveform
EXAMPLE
Figure 2.2
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Basic Sequence Operations

• Sum of two sequences
• Product of two sequences
• Multiplication of a sequence by a numbera
• Delay (shift) of a sequence

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Basic sequences

• Unit sample sequence (discrete-time impulse,
  impulse)

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Basic sequences
A sum of scaled, delayed impulses

• arbitrary sequence

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Basic sequences

• Unit step sequence

First backward difference
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Basic Sequences

• Exponential sequences

• A and a are real xn is real
• A is positive and 0ltalt1, xn is positive and
  decrease with increasing n
• -1ltalt0, xn alternate in sign, but decrease in
  magnitude with increasing n
• xn grows in magnitude as n increases

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EX. 2.1 Combining Basic sequences

• If we want an exponential sequences that is zero
  for n lt0, then

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Basic sequences

• Sinusoidal sequence

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Exponential Sequences
Exponentially weighted sinusoids
Exponentially growing envelope
Exponentially decreasing envelope
is refered to
Complex Exponential Sequences
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Frequency difference between continuous-time and
discrete-time complex exponentials or sinusoids

• frequency of the complex sinusoid or
  complex exponential
• phase

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Periodic Sequences

• A periodic sequence with integer period N

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EX. 2.2 Examples of Periodic Sequences

• Suppose it is periodic sequence with period N

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EX. 2.2 Examples of Periodic Sequences

• Suppose it is periodic sequence with period N

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EX. 2.2 Non-Periodic Sequences

• Suppose it is periodic sequence with period N

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High and Low Frequencies in Discrete-time signal
(a) w0 0 or 2?
(b) w0 ?/8 or 15?/8
(c) w0 ?/4 or 7?/4
(d) w0 ?
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2.2 Discrete-Time System

• Discrete-Time System is a trasformation or
  operator that maps input sequence xn into a
  unique yn
• ynTxn, xn, yn discrete-time signal

Discrete-Time System
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EX. 2.3 The Ideal Delay System
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EX. 2.4 Moving Average
for n7, M10, M25
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Properties of Discrete-time systems2.2.1
Memoryless (memory) system

• Memoryless systems
• the output yn at every value of n depends
  only on the input xn at the same value of n

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Properties of Discrete-time systems2.2.2
Linear Systems

• If

• and only If

additivity property
homogeneity or scaling ?(?)?? property

• principle of superposition

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Example of Linear System

• Ex. 2.6 Accumulator system

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Example 2.7 Nonlinear Systems

• Method find one counterexample

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Properties of Discrete-time systems2.2.3
Time-Invariant Systems

• Shift-Invariant Systems

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Example of Time-Invariant System

• Ex. 2.8 Accumulator system

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Example of Time-varying System

• Ex. 2.9 The compressor system

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Properties of Discrete-time systems 2.2.4
Causality

• A system is causal if, for every choice of
  , the output sequence value at the index
  depends only on the input sequence value for

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Ex. 2.10 Example for Causal System

• Forward difference system is not Causal
• Backward difference system is Causal

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Properties of Discrete-time systems 2.2.5
Stability

• Bounded-Input Bounded-Output (BIBO) Stability
  every bounded input sequence produces a bounded
  output sequence.

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Ex. 2.11 Test for Stability or Instability
is stable
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Ex. 2.11 Test for Stability or Instability

• Accumulator system

• Accumulator system is not stable

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2.3 Linear Time-Invariant (LTI) Systems

• Impulse response

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LTI Systems Convolution

• Representation of general sequence as a linear
  combination of delayed impulse

• principle of superposition

An Illustration Example(interpretation 1)
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Computation of the Convolution
(interpretation 2)

• reflecting hk about the origion to obtain h-k
• Shifting the origin of the reflected sequence to
  kn

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Ex. 2.12
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• Convolution can be realized by
• Reflecting hk about the origin to obtain h-k.
• Shifting the origin of the reflected sequences to
  kn.
• Computing the weighted moving average of xk by
  using the weights given by hn-k.

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Ex. 2.13 Analytical Evaluation of the Convolution
For system with impulse response
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2.4 Properties of LTI Systems

• Convolution is commutative(????)

• Convolution is distributed over addition

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Cascade connection of systems
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Parallel connection of systems
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Stability of LTI Systems

• LTI system is stable if the impulse response is
  absolutely summable .

Causality of LTI systems
HW proof, Problem 2.62
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Impulse response of LTI systems

• Impulse response of Ideal Delay systems

• Impulse response of Accumulator

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Impulse response of Moving Average systems
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• Impulse response of Forward Difference

• Impulse response of Backward Difference

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Finite-duration impulse response (FIR) systems

• The impulse response of the system has only a
  finite number of nonzero samples.

such as

• The FIR systems always are stable.

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Infinite-duration impulse response (IIR)

• The impulse response of the system is infinite in
  duration.

Stable IIR System
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Equivalent systems
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Inverse system
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2.5 Linear Constant-Coefficient Difference
Equations

• An important subclass of linear time-invariant
  systems consist of those system for which the
  input xn and output yn satisfy an Nth-order
  linear constant-coefficient difference equation.

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Ex. 2.14 Difference Equation Representation of
the Accumulator
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Block diagram of a recursive difference equation
representing an accumulator
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Ex. 2.15 Difference Equation Representation of
the Moving-Average System with
representation 1
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Difference Equation Representation of the System

• An unlimited number of distinct difference
  equations can be used to represent a given linear
  time-invariant input-output relation.

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Solving the difference equation

• Without additional constraints or information, a
  linear constant-coefficient difference equation
  for discrete-time systems does not provide a
  unique specification of the output for a given
  input.

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Solving the difference equation

• Output

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Solving the difference equation recursively

• If the input and a set of auxiliary value

are specified.
y(n) can be written in a recurrence formula
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Example 2.16 Recursive Computation of Difference
Equation
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Example 2.16 Recursive Computation of Difference
Equation
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Example for Recursive Computation of Difference
Equation

• The system is noncausal.
• The system is not linear.
• The system is not time invariant.

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Difference Equation Representation of the System

• If a system is characterized by a linear
  constant-coefficient difference equation and is
  further specified to be linear, time invariant,
  and causal, the solution is unique.
• In this case, the auxiliary conditions are stated
  as initial-rest conditions(??????).

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Summary

• The system for which the input and output satisfy
  a linear constant-coefficient difference
  equation
• The output for a given input is not uniquely
  specified. Auxiliary conditions are required.

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Summary

• If the auxiliary conditions are in the form of N
  sequential values of the output,

• later value can be obtained by rearranging the
  difference equation as a recursive relation
  running forward in n,

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Summary

• and prior values can be obtained by rearranging
  the difference equation as a recursive relation
  running backward in n.

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Summary

• Linearity, time invariance, and causality of the
  system will depend on the auxiliary conditions.
• If an additional condition is that the system is
  initially at rest, then the system will be
  linear, time invariant, and causal.

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Example 2.16 with initial-rest conditions
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Discussion

• If the input is , with
  initial-rest conditions,

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2.6 Frequency-Domain Representation of
Discrete-Time Signals and systems

• 2.6.1 Eigenfunction and Eigenvalue for LTI

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Eigenfunction and Eigenvalue

• Complex exponentials is the eigenfunction for
  discrete-time systems. For LTI systems

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Frequency response

• is called as frequency response of
  the system.

• Real part, imagine part

• Magnitude, phase

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Example 2.17 Frequency response of the ideal Delay
From defination(2.109)
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Example 2.17 Frequency response of the ideal
Delay
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Linear combination of complex exponential
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Example 2.18 Sinusoidal response of LTI systems
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Sinusoidal response of the ideal Delay
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Periodic Frequency Response

• The frequency response of discrete-time LTI
  systems is always a periodic function of the
  frequency variable with period

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Periodic Frequency Response

• The low frequencies are frequencies close to
  zero
• The high frequencies are frequencies close to

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Example 2.19 Ideal Frequency-Selective Filters
Frequency Response of Ideal Low-pass Filter
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Frequency Response of Ideal High-pass Filter
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Frequency Response of Ideal Band-stop Filter
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Frequency Response of Ideal Band-pass Filter
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Example 2.20 Frequency Response of the
Moving-Average System
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Frequency Response of the Moving-Average System
M1 0 and M2 4
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2.6.2 Suddenly applied Complex Exponential Inputs

• In practice, we may not apply the complex
  exponential inputs ejwn to a system, but the more
  practical-appearing inputs of the form
• xn ejwn ?
  un

• i.e., xn suddenly applied at an arbitrary time,
  which for convenience we choose n0.
• For causal LTI system

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2.6.2 Suddenly applied Complex Exponential Inputs
For causal LTI system

• For n0

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2.6.2 Suddenly applied Complex Exponential Inputs

• Steady-state Response

• Transient response

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2.6.2 Suddenly Applied Complex Exponential Inputs
(continue)

• For infinite-duration impulse response (IIR)

• For stable system, transient response must become
  increasingly smaller as n ? ?,

Illustration of a real part of suddenly applied
complex exponential Input with IIR
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2.6.2 Suddenly Applied Complex Exponential Inputs
(continue)

• If hn 0 except for 0? n ? M (FIR), then the
  transient response ytn 0 for n1 gt M.
• For n ? M, only the steady-state response exists

Illustration of a real part of suddenly applied
complex exponential Input with FIR
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2.7 Representation of Sequences by Fourier
Transforms

• (Discrete-Time) Fourier Transform, DTFT,
• analyzing

• Inverse Fourier Transform, synthesis

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Fourier Transform
rectangular form
polar form
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Principal Value(??)
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Impulse response and Frequency response

• The frequency response of a LTI system is the
  Fourier transform of the impulse response.

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Example 2.21 Absolute Summability

• The Fourier transform

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Discussion of convergence

• Absolute summability is a sufficient condition
  for the existence of a Fourier transform
  representation, and it also guarantees uniform
  convergence.
• Some sequences are not absolutely summable, but
  are square summable, i.e.,

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Discussion of convergence

• Is called Mean-square Cconvergence

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Discussion of convergence

• Mean-square convergence

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Example 2.22 Square-summability for the ideal
Lowpass Filter
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Example 2.22 Square-summability for the ideal
Lowpass Filter
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Gibbs Phenomenon
M3
M1
M19
M7
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Example 2.22 continued

• As M increases, oscillatory behavior at
  
• is more rapid, but the size of the
  ripple does not decrease. (Gibbs Phenomenon)

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Example 2.22 continued
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Example 2.23 Fourier Transform of a constant

• The sequence is neither absolutely summable nor
  square summable.

• The impulses are functions of a continuous
  variable and therefore are of infinite height,
  zero width, and unit area.

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Example 2.23 Fourier Transform of a constant
proof
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Example 2.24 Fourier Transform of Complex
Exponential Sequences

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Example Fourier Transform of Complex Exponential
Sequences
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Example Fourier Transform of unit step sequence
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2.8 Symmetry Properties of the Fourier Transform

• Conjugate-symmetric sequence
• Conjugate-antisymmetric sequence

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Symmetry Properties of real sequence

• even sequence a real sequence that is
  Conjugate-symmetric
• odd sequence real, Conjugate-antisymmetric

real sequence
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Decomposition of a Fourier transform
Conjugate-symmetric
Conjugate-antisymmetric
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xn is complex
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xn is real
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Ex. 2.25 illustration of Symmetry Properties
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Ex. 2.25 illustration of Symmetry Properties

• Real part

• Imaginary part

• a0.75(solid curve) and a0.5(dashed curve)

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Ex. 2.25 illustration of Symmetry Properties

• Its magnitude is an even function, and phase is
  odd.

• a0.75(solid curve) and a0.5(dashed curve)

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2.9 Fourier Transform Theorems

• 2.9.1 Linearity

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Fourier Transform Theorems

• 2.9.2 Time shifting and frequency shifting

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Fourier Transform Theorems

• 2.9.3 Time reversal

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Fourier Transform Theorems

• 2.9.4 Differentiation in Frequency

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Fourier Transform Theorems

• 2.9.5 Parsevals Theorem

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Fourier Transform Theorems

• 2.9.6 Convolution Theorem

if
HW proof
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Fourier Transform Theorems

• 2.9.7 Modulation or Windowing Theorem

HW proof
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Fourier transform pairs
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Fourier transform pairs
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Fourier transform pairs
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Ex. 2.26 Determine the Fourier Transform of
sequence
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Ex. 2.27 Determine an inverse Fourier Transform of
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Ex. 2.28 Determine the impulse response from the
frequency respone
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Ex. 2.29 Determine the impulse response for a
difference equation

• Impulse response

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Ex. 2.29 Determine the impulse response for a
difference equation
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2.10 Discrete-Time Random Signals

• Deterministic each value of a sequence is
  uniquely determined by a mathematically
  expression, a table of data, or a rule of some
  type.
• Stochastic signal a member of an ensemble of
  discrete-time signals that is characterized by a
  set of probability density function.

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2.10 Discrete-Time Random Signals

• For a particular signal at a particular time, the
  amplitude of the signal sample at that time is
  assumed to have been determined by an underlying
  scheme of probability.

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2.10 Discrete-Time Random Signals

• The collection of random variables is called a
  random process.
• The stochastic signals do not directly have
  Fourier transform, but the Fourier transform of
  the autocorrelation and autocovariance sequece
  often exist.

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Fourier transform in stochastic signals

• The Fourier transform of autocovariance sequence
  has a useful interpretation in terms of the
  frequency distribution of the power in the
  signal.
• The effect of processing stochastic signals with
  a discrete-time LTI system can be described in
  terms of the effect of the system on the
  autocovariance sequence.

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Stochastic signal as input
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Stochastic signal as input

• In our discussion, no necessary to distinguish
  between the random variables Xn andYn and their
  specific values xn and yn.

• mXn Exn , mYn E(Yn, can be written as

mxn Exn, myn E(yn.

• The mean of output process

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Stochastic signal as input

• The autocorrelation function of output

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Stochastic signal as input
the power spectrum
DTFT of the autocorrelation function of output
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Total average power in output

• provides the motivation for the term power
  density spectrum.

• ????

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For Ideal bandpass system
Since is a real, even, its FT
is also real and even, i.e.,
so is

• ????

• the power density function of a real signal is
  real, even, and nonnegative.

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Ex. 2.30 White Noise

• A white-noise signal is a signal for which

• Assume the signal has zero mean. The power
  spectrum of a white noise is

• The average power of a white noise is

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Color Noise

• A noise signal whose power spectrum is not
  constant with frequency.

• A noise signal with power spectrum
  can be assumed to be the output of a LTI system
  with white-noise input.

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Color Noise

• Suppose ,

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Cross-correlation between the input and output
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Cross-correlation between the input and output

• If ,

• That is, for a zero mean white-noise input, the
  cross-correlation between input and output of a
  LTI system is proportional to the impulse
  response of the system.

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Cross power spectrum between the input and output

• The cross power spectrum is proportional to the
  frequency response of the system.

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2.11 Summary

• Define a set of basic sequence.
• Define and represent the LTI systems in terms of
  the convolution, stability and causality.
• Introduce the linear constant-coefficient
  difference equation with initial rest conditions
  for LTI , causal system.
• Recursive solution of linear constant-coefficient
  difference equations.

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2.11 Summary

• Define FIR and IIR systems
• Define frequency response of the LTI system.
• Define Fourier transform.
• Introduce the properties and theorems of Fourier
  transform. (Symmetry)
• Introduce the discrete-time random signals.

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Chapter 2 HW

• 2.1, 2.2, 2.4, 2.5, 2.7, 2.11, 2.12,2.15, 2.20,
  2.62,

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