Bài giảng Digital Signal Processing - Chapter 2: Sampling and Reconstruction - Hà Hoàng Kha

g rate or sampling frequency (samples/second or 
Hz) 
Sampling and Reconstruction 
Digital Signal Processing 
 3. Sampling-example 1 
10 
 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: 
Sampling and Reconstruction 
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Digital Signal Processing 
 3. Sampling-example 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 discrete-time signals ? 
Sampling and Reconstruction 
 at a sampling rate fs=1/T results in a discrete-
time signal x(n). 
Digital Signal Processing 
 3. Sampling-Aliasing of Sinusoids 
12 
 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( ) cos(2 )x t A f t  
 The sinusoids is sampled at fs , resulting in a 
discrete time signal xk(n). 
( ) cos(2 )k kx t A f t  
 If fk=f0+kfs, k=0, ±1, ±2, ., then x(n)=xk(n) . 
Proof: (in class) 
Sampling and Reconstruction 
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Digital Signal Processing 
 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 )x t A t A ft  
 Note that / sec
/ sec
sf samples samples
f cycles cycle
 
 To sample a single sinusoid properly, we must require 2s
f 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 fmax . 
2) The sampling rate fs must be chosen at least twice the maximum 
frequency fmax. max2sf f
Fig: Typical bandlimited spectrum 
 fs=2fmax is called Nyquist rate; fs/2 is called Nyquist frequency; 
 [-fs/2, fs/2] is Nyquist interval. 
Sampling and Reconstruction 
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Digital Signal Processing 
 4. Sampling Theorem 
15 
 The values of fmax 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 Theorem-Spectrum 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 
2
( ) s
j f nt
n
n
s t S e



  2
1 1 1
( ) ( )s
j f nt
n
T T
S t e dt t dt
T T T
    
21
( ) s
j f nt
n
s t e
T



 
21
( ) ( ) ( ) ( ) s
j nf t
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 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 
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 fs/2. 
( ) ( ) ( ) ( )aX f X f H f X f 
 An ideal reconstructor (lowpass filter) 
[ / 2, / 2]
( )
0
s sT f f f
H f
otherwise
 
 

Then 
Sampling and Reconstruction 
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Digital Signal Processing 
 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) ? 
Sampling and Reconstruction 
Digital Signal Processing 
 5. Analog reconstruction-Example 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 
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Digital Signal Processing 
 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) ----------------------> xa(t), fa=f0 
sampling at fs ideal reconstructor 
 xk(t), fk=f0+kfs------------------> x(n) ----------------------> xa(t), fa=f0 
sampling at fs ideal reconstructor 
mod( )a sf f f
 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 
Sampling and Reconstruction 
Digital Signal Processing 
 5. Analog reconstruction-Example 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 
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Digital Signal Processing 
 5. Analog reconstruction-Example 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 
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Digital Signal Processing 
 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. 
s pass stopf f f 
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 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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Digital Signal Processing 
 6. Antialiasing prefilter-Example 
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 prefilter-Example 
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 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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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 fs/2. 
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 
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Digital Signal Processing 
 7. Ideal and practical analog reconstructors 
31 Sampling and Reconstruction 
Fig: Frequency response of staircase reconstructor 
Digital Signal Processing 
 7. Practical reconstructors-antiimage postfilter 
32 Sampling and Reconstruction 
 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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Digital Signal Processing 
 8. Homework 
33 Sampling and Reconstruction 
 Problems: provided in class