Xử lý tín hiệu số 2 Sampling and Reconstruction

8202014 1 Click to edit Master subtitle style Sampling and Reconstruction Chapter 2 Ha Hoang Kha, Ph.D. Ho Chi Minh City University of Technology Email: hhkhahcmut.edu.vn Digital Signal Processing  Sampling Content 2  Sampling theorem Sampling and Reconstruction  Antialiasing prefilter  Spectrum of sampling signals  Analog reconstruction  Ideal prefilter  Practical prefilter  Ideal reconstructor  Practical reconstructor CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 2 Digital Signal Processing Review of Analog Signals 3  Fourier transform X(Ω) of x(t) is frequency spectrum of the signal Sampling and Reconstruction  inverse Fourier transform  Response of a linear system to an input signal x(t):     X ()  x(t)e j tdt   2f       x(t) X ( )ej t d2   y(t)  h(t t)x(t)dt Y()  H()X() Digital Signal Processing Review of Analog Signals 4  Frequency response of the system, defined as the Fourier transform of the impulse response h(t) Sampling and Reconstruction  The steadystate sinusoidal response of the filter, defined as its response to sinusoidal inputs     H()  h(t)e j tdt j t x(t)  e  y(t)  H()ejt  H()ejt jarg H () CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 3 Digital Signal Processing Review of Analog Signals 5  If the input consists of the sum of two sinusoids of frequencies Sampling and Reconstruction  After filtering, the steadystate output will be j t j t x t Ae 1 A e 2 ( )  1   2  j t j t y(t)  A1H(1)e 1  A2H(2)e 2 Digital Signal Processing Review of useful equations 6  Linear system Sampling and Reconstruction 1 sin( )sin( ) cos( ) cos( ) 2 a b a b a b      1 cos( )cos( ) cos( ) cos( ) 2 a b a b a b     0 0 0 1 sin(2 ) ( ) ( ) 2    f t j f f f f     FT x t A f t ( ) cos(2 )     0 x t ( ) Linear system h(t) H(f) y t x t h t ( ) ( ) ( )   X f ( ) Y f X f H f ( ) ( ) ( )   Especially, 0 0 0 1 cos(2 ) ( ) ( ) 2    f t f f f f     FT y t A H f f t H f ( ) | ( ) | cos(2 arg( ( )))    0 0 0    Fourier transform:  Trigonometric formulas: 1 sin( )cos( ) sin( ) sin( ) 2 a b a b a b     CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 4 Digital Signal Processing  A typical signal processing system includes 3 stages: 1. Introduction 7  The digital processor can be programmed to perform signal processing operations such as filtering, spectrum estimation. Digital signal processor can be a general purpose computer, DSP chip or other digital hardware. Sampling and Reconstruction  The analog signal is digitalized by an AD converter  The digitalized samples are processed by a digital signal processor.  The resulting output samples are converted back into analog by a DA converter. Digital Signal Processing 2. Analog to digital conversion 8  Analog to digital (AD) conversion is a threestep process. Sampler Quantizer Coder t=nT x x(t) x(nT)≡x(n) Q(n) 11010 AD converter n x Q(n) 000 001 010 011 100 101 110 111 t x(t) n x(n) Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 5 Digital Signal Processing 3. Sampling 9  Sampling is to convert a continuous time signal into a discrete time signal. The analog signal is periodically measured at every T seconds  x(n)≡x(nT)=x(t=nT), n=….2, 1, 0, 1, 2, 3……..  T: sampling interval or sampling period (second);  fs=1T: sampling rate or sampling frequency (samplessecond or Hz) Sampling and Reconstruction Digital Signal Processing 3. Samplingexample 1 10  The analog signal x(t)=2cos(2πt) with t(s) is sampled at the rate fs=4 Hz. Find the discretetime signal x(n) ? Solution: Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 6 Digital Signal Processing 3. Samplingexample 2 11  Consider the two analog sinusoidal signals Solution: 1 7 ( ) 2cos(2 ), 8 x t t   2 1 ( ) 2cos(2 ); ( ) 8 x t t t s   These signals are sampled at the sampling frequency fs=1 Hz. Find the discretetime signals ? Sampling and Reconstruction at a sampling rate fs=1T results in a discretetime signal x(n). Digital Signal Processing 3. SamplingAliasing of Sinusoids 12  In general, the sampling of a continuoustime sinusoidal signal  Remarks: We can that the frequencies fk=f0+kfs are indistinguishable from the frequency f0 after sampling and hence they are aliases of f0 x t A f t ( ) cos(2 )     0  The sinusoids is sampled at fs , resulting in a discrete time signal xk(n). x t A f t k k ( ) cos(2 )      If f k=f0+kfs, k=0, ±1, ±2, …., then x(n)=xk(n) . Proof: (in class) Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 7 Digital Signal Processing 4. Sampling TheoremSinusoids 13  Consider the analog signal where Ω is the frequency (rads) of the analog signal, and f=Ω2π is the frequency in cycless or Hz. The signal is sampled at the three rate fs =8f, fs=4f, and fs=2f. x t A t A ft ( ) cos( ) cos(2 )      Note that sec sec fs samples samples f cycles cycle    To sample a single sinusoid properly, we must require fs 2 samples f cycle  Fig: Sinusoid sampled at different rates Sampling and Reconstruction Digital Signal Processing 4. Sampling Theorem 14  For accurate representation of a signal x(t) by its time samples x(nT), two conditions must be met: 1) The signal x(t) must be bandlimitted, i.e., its frequency spectrum must be limited to f max . 2) The sampling rate fs must be chosen at least twice the maximum frequency fmax. f f s  2 max Fig: Typical bandlimited spectrum  f s=2fmax is called Nyquist rate; fs2 is called Nyquist frequency; fs2, fs2 is Nyquist interval. Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 8 Digital Signal Processing 4. Sampling Theorem 15  The values of f max and fs depend on the application Sampling and Reconstruction Application fmax fs Biomedical 1 KHz 2 KHz Speech 4 KHz 8 KHz Audio 20 KHz 40 KHz Video 4 MHz 8 MHz Digital Signal Processing 4. Sampling TheoremSpectrum Replication 16  Let where ( ) ( ) ( ) ( ) ( ) ( ) n x nT x t x t t nT x t s t         ( ) ( ) n s t t nT        s(t) is periodic, thus, its Fourier series are given by ( ) n j f nt 2 s n s t S e      n 1 1 1 ( ) ( ) j f nt 2 s T T S t e dt t dt T T T          1 2 ( ) j f nt s n s t e T      1 2 ( ) ( ) ( ) ( ) j nf t s n x t x t s t x t e T       1 ( ) ( ) s n X f X f nf T      where Thus, which results in  Taking the Fourier transform of yields x t ( )  Observation: The spectrum of discretetime signal is a sum of the original spectrum of analog signal and its periodic replication at the interval f s. Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 9 4. Sampling TheoremSpectrum Replication Digital Signal Processing 17 Fig: Typical badlimited spectrum  f s2 ≥ fmax  f s2 < fmax Fig: Aliasing caused by overlapping spectral replicas Fig: Spectrum replication caused by sampling Sampling and Reconstruction Digital Signal Processing 5. Ideal Analog reconstruction 18 Fig: Ideal reconstructor as a lowpass filter  An ideal reconstructor acts as a lowpass filter with cutoff frequency equal to the Nyquist frequency fs2. X f X f H f X f a( ) ( ) ( ) ( )    An ideal reconstructor (lowpass filter) 2, 2 ( ) 0 T f f f s s H f otherwise       Then Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 10 Digital Signal Processing 5. Analog reconstructionExample 1 19  The analog signal x(t)=cos(20πt) is sampled at the sampling frequency fs=40 Hz. a) Plot the spectrum of signal x(t) ? b) Find the discrete time signal x(n) ? c) Plot the spectrum of signal x(n) ? d) The signal x(n) is an input of the ideal reconstructor, find the reconstructed signal xa(t) ? Sampling and Reconstruction Digital Signal Processing 5. Analog reconstructionExample 2 20  The analog signal x(t)=cos(100πt) is sampled at the sampling frequency fs=40 Hz. a) Plot the spectrum of signal x(t) ? b) Find the discrete time signal x(n) ? c) Plot the spectrum of signal x(n) ? d) The signal x(n) is an input of the ideal reconstructor, find the reconstructed signal xa(t) ? Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 11 Digital Signal Processing 5. Analog reconstruction 21  Remarks: x a(t) contains only the frequency components that lie in the Nyquist interval (NI) fs2, fs2.  x(t), f0  NI > x(n) > xa(t), fa=f0 sampling at fs ideal reconstructor  x k(t), fk=f0+kfs> x(n) > xa(t), fa=f0 sampling at fs ideal reconstructor f f f a s  mod( )  The frequency fa of reconstructed signal xa(t) is obtained by adding to or substracting from f0 (fk) enough multiples of fs until it lies within the Nyquist interval fs2, fs2.. That is Sampling and Reconstruction Digital Signal Processing 5. Analog reconstructionExample 3 22  The analog signal x(t)=10sin(4πt)+6sin(16πt) is sampled at the rate 20 Hz. Find the reconstructed signal xa(t) ? Sampling and Reconstruction CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 12 Digital Signal Processing 5. Analog reconstructionExample 4 23  Let x(t) be the sum of sinusoidal signals x(t)=4+3cos(πt)+2cos(2πt)+cos(3πt) where t is in milliseconds. Sampling and Reconstruction a) Determine the minimum sampling rate that will not cause any aliasing effects ? b) To observe aliasing effects, suppose this signal is sampled at half its Nyquist rate. Determine the signal xa(t) that would be aliased with x(t) ? Plot the spectrum of signal x(n) for this sampling rate? Digital Signal Processing 6. Ideal antialiasing prefilter 24  The signals in practice may not bandlimitted, thus they must be filtered by a lowpass filter Sampling and Reconstruction Fig: Ideal antialiasing prefilter CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 13 Digital Signal Processing 6. Practical antialiasing prefilter 25 Sampling and Reconstruction Fig: Practical antialiasing lowpass prefilter  The Nyquist frequency fs2 is in the middle of transition region.  A lowpass filter: fpass, fpass is the frequency range of interest for the application (fmax=fpass)  The stopband frequency fstop and the minimum stopband attenuation A stop dB must be chosen appropriately to minimize the aliasing effects. f f f s pass stop   Digital Signal Processing 6. Practical antialiasing prefilter 26 Sampling and Reconstruction  The attenuation of the filter in decibels is defined as 10 0 ( ) ( ) 20log ( ) ( ) H f A f dB H f   where f 0 is a convenient reference frequency, typically taken to be at DC for a lowpass filter.  α 10 =A(10f)A(f) (dBdecade): the increase in attenuation when f is changed by a factor of ten.  α 2 =A(2f)A(f) (dBoctave): the increase in attenuation when f is changed by a factor of two.  Analog filter with order N, |H(f)|~1fN for large f, thus α10 =20N (dBdecade) and α10 =6N (dBoctave) CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 14 Digital Signal Processing 6. Antialiasing prefilterExample 27 Sampling and Reconstruction  A sound wave has the form where t is in milliseconds. What is the frequency content of this signal ? Which parts of it are audible and why ? ( ) 2 cos(10 ) 2 cos(30 ) 2 cos(50 ) 2 cos(60 ) 2 cos(90 ) 2 cos(125 ) x t A t B t C t D t E t F t             This signal is prefilter by an analog prefilter H(f). Then, the output y(t) of the prefilter is sampled at a rate of 40KHz and immediately reconstructed by an ideal analog reconstructor, resulting into the final analog output ya(t), as shown below: Digital Signal Processing 6. Antialiasing prefilterExample 28 Sampling and Reconstruction Determine the output signal y(t) and ya(t) in the following cases: a)When there is no prefilter, that is, H(f)=1 for all f. b)When H(f) is the ideal prefilter with cutoff fs2=20 KHz. c)When H(f) is a practical prefilter with specifications as shown below: The filter’s phase response is assumed to be ignored in this example. CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 15 Digital Signal Processing 7. Ideal and practical analog reconstructors 29 Sampling and Reconstruction  An ideal reconstructor is an ideal lowpass filter with cutoff Nyquist frequency fs2. Digital Signal Processing 7. Ideal and practical analog reconstructors 30 Sampling and Reconstruction  The ideal reconstructor has the impulse response: which is not realizable since its impulse response is not casual sin( f t) ( ) s s h t f t     It is practical to use a staircase reconstructor CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 16 Digital Signal Processing 7. Ideal and practical analog reconstructors 31 Sampling and Reconstruction Fig: Frequency response of staircase reconstructor Digital Signal Processing 7. Practical reconstructorsantiimage postfilter 32 Sampling and Reconstruction  An analog lowpass postfilter whose cutoff is Nyquist frequency fs2 is used to remove the surviving spectral replicas. Fig: Spectrum after postfilter Fig: Analog antiimage postfilter CuuDuongThanCong.com https:fb.comtailieudientucntt8202014 17 Digital Signal Processing 8. Homework 33 Sampling and Reconstruction  Problems: provided in class CuuDuongThanCong.com https:fb.comtailieudientucntt

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Click to edit Master subtitle style

Sampling and Reconstruction

Chapter 2

Ha Hoang Kha, Ph.D

Ho Chi Minh City University of Technology Email: hhkha@hcmut.edu.vn

 Sampling

Content

 Sampling theorem

 Antialiasing prefilter

 Spectrum of sampling signals

 Analog reconstruction

 Ideal prefilter

 Practical prefilter

 Ideal reconstructor

 Practical reconstructor

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Digital Signal Processing

Review of Analog Signals

3

 Fourier transform X(Ω) of x(t) is frequency spectrum of the signal

Sampling and Reconstruction

 inverse Fourier transform

 Response of a linear system to an input signal x(t):













 x t e dt













2 ) ( )







 ( ') ( ') ' )

 Frequency response of the system, defined as the Fourier transform

of the impulse response h(t)

 The steady-state sinusoidal response of the filter, defined as its response to sinusoidal inputs













 h t e dt

t j

e t

x( )  () () jt  () jtjargH()

e H e

H t y

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5

 If the input consists of the sum of two sinusoids of frequencies

 After filtering, the steady-state output will be

t j t j

e A e A t

2 1

)

t j t

j

e H A e H A t

) ( )

( )

Review of useful equations

 Linear system

1 sin( ) sin( ) [cos( ) cos( )]

2

1 cos( ) cos( ) [cos( ) cos( )]

2

1

2

FT

0

( )

h(t) H(f)

( ) ( ) ( )

y t x t h t

( )

 Especially,

1

2

FT

( ) | ( ) | cos(2 arg( ( )))

 Fourier transform:

 Trigonometric formulas:

1 sin( ) cos( ) [sin( ) sin( )]

2

Trang 4

 A typical signal processing system includes 3 stages:

1 Introduction

7

 The digital processor can be programmed to perform signal processing operations such as filtering, spectrum estimation Digital signal processor can be

a general purpose computer, DSP chip or other digital hardware

 The analog signal is digitalized by an A/D converter

 The digitalized samples are processed by a digital signal processor

 The resulting output samples are converted back into analog by a D/A converter

2 Analog to digital conversion

 Analog to digital (A/D) conversion is a three-step process

Sampler Quantizer Coder t=nT

xQ(n)

A/D converter

n

xQ(n)

000

001

010 011

100

101

110

111

t

x(t)

n x(n)

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3 Sampling

9

 Sampling is to convert a continuous time signal into a discrete time signal The analog signal is periodically measured at every T seconds

 x(n)≡x(nT)=x(t=nT), n=….-2, -1, 0, 1, 2, 3……

 T: sampling interval or sampling period (second);

 fs=1/T: sampling rate or sampling frequency (samples/second or Hz)

3 Sampling-example 1

 The analog signal x(t)=2cos(2πt) with t(s) is sampled at the rate fs=4

Hz Find the discrete-time signal x(n) ?

Solution:

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3 Sampling-example 2

11

 Consider the two analog sinusoidal signals

Solution:

1

7 ( ) 2 cos(2 ),

8

These signals are sampled at the sampling frequency fs=1 Hz

Find the discrete-time signals ?

at a sampling rate fs=1/T results in a discrete-time signal x(n)

3 Sampling-Aliasing of Sinusoids

 In general, the sampling of a continuous-time sinusoidal signal

 Remarks: We can that the frequencies fk=f0+kfs are indistinguishable from the frequency f0 after sampling and hence they are aliases of f0

0

 The sinusoids is sampled at fs , resulting in a discrete time signal xk(n)

 If fk=f0+kfs, k=0, ±1, ±2, …., then x(n)=xk(n) Proof: (in class)

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4 Sampling Theorem-Sinusoids

13

 Consider the analog signal where Ω is the frequency (rad/s) of the analog signal, and f=Ω/2π is the frequency in cycles/s or Hz The signal is sampled at the three rate

fs=8f, fs=4f, and fs=2f

( ) cos( ) cos(2 )

 Note that / sec

/ sec

s

 To sample a single sinusoid properly, we must require f s 2samples

Fig: Sinusoid sampled at different rates

4 Sampling Theorem

 For accurate representation of a signal x(t) by its time samples x(nT), two conditions must be met:

1) The signal x(t) must be bandlimitted, i.e., its frequency spectrum must

be limited to fmax

2) The sampling rate fs must be chosen at least twice the maximum frequency fmax f s 2fmax

Fig: Typical bandlimited spectrum

 fs=2fmax is called Nyquist rate;fs/2 is called Nyquist frequency;

[-fs/2, fs/2] is Nyquist interval

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4 Sampling Theorem

15

 The values of fmax and fs depend on the application

Biomedical 1 KHz 2 KHz

4 Sampling Theorem-Spectrum Replication

 Let where ( ) ( ) ( ) ( ) ( ) ( )

n



n



 s(t) is periodic, thus, its Fourier series are given by

2

( ) j f nt s

n n





( ) j f nt s ( )

n



2

1 ( ) j f nt s

n

T







2

1 ( ) ( ) ( ) ( ) j nf t s

n

T







1

n

T





where Thus,

which results in

 Taking the Fourier transform of yields x t( )

 Observation: The spectrum of discrete-time signal is a sum of the original spectrum of analog signal and its periodic replication at the interval fs

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4 Sampling Theorem-Spectrum Replication

Digital Signal Processing 17

Fig: Typical badlimited spectrum

 fs/2 ≥ fmax

 fs/2 < fmax

Fig: Aliasing caused by overlapping spectral replicas Fig: Spectrum replication caused by sampling

5 Ideal Analog reconstruction

Fig: Ideal reconstructor as a lowpass filter

 An ideal reconstructor acts as a lowpass filter with cutoff frequency equal to the Nyquist frequency fs/2

( ) ( ) ( ) ( )

a

 An ideal reconstructor (lowpass filter) ( ) [ / 2, / 2]

0

H f

otherwise

 



 



Then

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5 Analog reconstruction-Example 1

19

 The analog signal x(t)=cos(20πt) is sampled at the sampling frequency fs=40 Hz

a) Plot the spectrum of signal x(t) ? b) Find the discrete time signal x(n) ? c) Plot the spectrum of signal x(n) ? d) The signal x(n) is an input of the ideal reconstructor, find the reconstructed signal xa(t) ?

 The analog signal x(t)=cos(100πt) is sampled at the sampling frequency fs=40 Hz

a) Plot the spectrum of signal x(t) ? b) Find the discrete time signal x(n) ? c) Plot the spectrum of signal x(n) ? d) The signal x(n) is an input of the ideal reconstructor, find the reconstructed signal xa(t) ?

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5 Analog reconstruction

21

 Remarks: xa(t) contains only the frequency components that lie in the Nyquist interval (NI) [-fs//2, fs/2]

 x(t), f0 NI -> x(n) -> xsampling at fs ideal reconstructor a(t), fa=f0

 xk(t), fk=f0+kfssampling at f -> x(n) -> xs ideal reconstructor a(t), fa=f0

mod( )

 The frequency fa of reconstructed signal xa(t) is obtained by adding

to or substracting from f0 (fk) enough multiples of fs until it lies within the Nyquist interval [-fs//2, fs/2] That is

 The analog signal x(t)=10sin(4πt)+6sin(16πt) is sampled at the rate 20

Hz Find the reconstructed signal xa(t) ?

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23

 Let x(t) be the sum of sinusoidal signals x(t)=4+3cos(πt)+2cos(2πt)+cos(3πt) where t is in milliseconds

a) Determine the minimum sampling rate that will not cause any aliasing effects ?

b) To observe aliasing effects, suppose this signal is sampled at half its Nyquist rate Determine the signal xa(t) that would be aliased with x(t) ? Plot the spectrum of signal x(n) for this sampling rate?

6 Ideal antialiasing prefilter

 The signals in practice may not bandlimitted, thus they must be filtered by a lowpass filter

Fig: Ideal antialiasing prefilter

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6 Practical antialiasing prefilter

25 Sampling and Reconstruction

Fig: Practical antialiasing lowpass prefilter

 The Nyquist frequency fs/2 is in the middle of transition region

 A lowpass filter: [-fpass, fpass] is the frequency range of interest for the application (fmax=fpass)

 The stopband frequency fstop and the minimum stopband attenuation

Astop dB must be chosen appropriately to minimize the aliasing effects

6 Practical antialiasing prefilter

 The attenuation of the filter in decibels is defined as

10 0

( )

H f

 

where f0 is a convenient reference frequency, typically taken to be at

DC for a lowpass filter

 α10 =A(10f)-A(f) (dB/decade): the increase in attenuation when f is changed by a factor of ten

 α2 =A(2f)-A(f) (dB/octave): the increase in attenuation when f is changed by a factor of two

 Analog filter with order N, |H(f)|~1/fN for large f, thus α10 =20N (dB/decade) and α10 =6N (dB/octave)

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6 Antialiasing prefilter-Example

 A sound wave has the form

where t is in milliseconds What is the frequency content of this signal ? Which parts of it are audible and why ?

( ) 2 cos(10 ) 2 cos(30 ) 2 cos(50 )

2 cos(60 ) 2 cos(90 ) 2 cos(125 )

This signal is prefilter by an analog prefilter H(f) Then, the output y(t) of the prefilter is sampled at a rate of 40KHz and immediately reconstructed by an ideal analog reconstructor, resulting into the final analog output ya(t), as shown below:

6 Antialiasing prefilter-Example

Determine the output signal y(t) and ya(t) in the following cases:

a)When there is no prefilter, that is, H(f)=1 for all f

b)When H(f) is the ideal prefilter with cutoff fs/2=20 KHz

c)When H(f) is a practical prefilter with specifications as shown below:

The filter’s phase response is assumed to be ignored in this example

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7 Ideal and practical analog reconstructors

 An ideal reconstructor is an ideal lowpass filter with cutoff Nyquist frequency fs/2

 The ideal reconstructor has the impulse response:

which is not realizable since its impulse response is not casual

sin( f t)

s

h t

f t





 It is practical to use a staircase reconstructor

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Fig: Frequency response of staircase reconstructor

7 Practical reconstructors-antiimage postfilter

 An analog lowpass postfilter whose cutoff is Nyquist frequency fs/2

is used to remove the surviving spectral replicas

Fig: Spectrum after postfilter Fig: Analog anti-image postfilter

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8 Homework

 Problems: provided in class

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