Biomedical Signal processing Chapter 4 Sampling of Continuous-Time Signals

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Biomedical Signal processingChapter 4 Sampling
of Continuous-Time Signals

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

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p.html
2020/11/10
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Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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Chapter 4 Sampling of Continuous-Time Signals

• 4.0 Introduction
• 4.1 Periodic Sampling
• 4.2 Frequency-Domain Representation of Sampling
• 4.3 Reconstruction of a Bandlimited Signal from
  its Samples
• 4.4 Discrete-Time Processing of Continuous-Time
  signals

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4.0 Introduction

• Continuous-time signal processing can be
  implemented through a process of sampling,
  discrete-time processing, and the subsequent
  reconstruction of a continuous-time signal.

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4.1 Periodic Sampling
Unit impulse train

• Continuous-time signal

impulse train sampling
T sampling period
Sampling sequence
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Tsample period fs1/Tsample rateOs2p/Tsample
rate
s(t)??????,???T,????????
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4.2 Frequency-Domain Representation of Sampling
Tsample period fs1/Tsample rate Os2p/T
sample rate
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DTFT
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DTFT
Continuous FT
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Nyquist Sampling Theorem

• Let be a bandlimited signal with
  
• . Then
  is
• uniquely determined by its samples
  
• 
  , if
• The frequency is commonly referred as
  the Nyquist frequency.
• The frequency is called the Nyquist
  rate.

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frequency spectrum of ideal sample signal
No aliasing
aliasing
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Example 4.1 Sampling and Reconstruction of a
sinusoidal signal
Compare the continuous-time and discrete-time FTs
for sampled signal
Solution
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Example 4.1 Sampling and Reconstruction of a
sinusoidal signal
continuous-time FT of
discrete-time FT of
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Example 4.2 Aliasing in the Reconstruction of an
Undersampled sinusoidal signal
Compare the continuous-time and discrete-time FTs
for sampled signal
Solution
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4.3 Reconstruction of a Bandlimited Signal from
its Samples
Gain T
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4.4 Discrete-Time Processing of Continuous-Time
signals
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C/D Converter

• Output of C/D Converter

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D/C Converter

• Output of D/C Converter

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4.4.1 Linear Time-Invariant Discrete-Time Systems
Is the system Linear Time-Invariant ?
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Linear and Time-Invariant

• Linear and time-invariant behavior of the system
  of Fig.4.11 depends on two factors
• First, the discrete-time system must be linear
  and time invariant.
• Second, the input signal must be bandlimited, and
  the sampling rate must be high enough to satisfy
  Nyquist Sampling Theorem.(??????)

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effective frequency response of the overall LTI
continuous-time system
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4.4.2 Impulse Invariance
Given
Design
impulse-invariant version of the continuous-time
system
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4.4.2 Impulse Invariance

• Two constraints

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The discrete-time system is called an
impulse-invariant version of the continuous-time
system
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4.5 Continuous-time Processing of Discrete-Time
Signal
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4.5 Continuous-time Processing of Discrete-Time
Signal
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4.5 Continuous-time Processing of Discrete-Time
Signal
Figure 4.18 Illustration of moving-average
filtering. (a) Input signal xn cos(0.25pn).
(b) Corresponding output of six-point
moving-average filter.
Errata
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Review

• What is Nyquist rate?
• What is Nyquist frequency?

• The Nyquist rate is two times the bandwidth of a
  bandlimited signal.
• The Nyquist frequency is half the sampling
  frequency of a discrete signal processing
  system.( The Nyquist frequency is one-half the
  Nyquist rate)

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• What is the physical meaning for the equation
  DTFT of a discrete-time signal is equal to the FT
  of a impulse train sampling .

Review

• DTFT derived from the equation.
• impulse train sampling xs(t) and xn have the
  same frequency component.

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How many factors does the linear and
time-invariant behavior of the system of Fig.4.11
depends on ?
Review

• First, the discrete-time system must be linear
  and time invariant.
• Second, the input signal must be bandlimited, and
  the sampling rate must be high enough to satisfy
  Nyquist Sampling Theorem.(??????)

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Assume that we are given a desired
continuous-time system that we wish to implement
in the form of the following figure, how to
decide hn and H(ejw)?
Review
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Chapter 4 HW

• 4.5

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