DSP-Lec 02-Sampling and Reconstruction.pdf

sampling an reconstruction

1. Lecture 02
SAMPLING AND RECONSTRUCTION

2. ❖Sampling
Content
2
❑Sampling theorem
❖Antialiasingprefilter
❑Spectrum of sampling signals
❖Analog reconstruction
❑Ideal prefilter
❑Practical prefilter
❑Ideal reconstructor
❑Practical reconstructor

3. Review of useful equations
3
❖Linear system1
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
FT
f t j f f f f  ⎯→ + − − 0
( ) cos(2 )x t A f t=+ ()xt
Linear system
h(t)
H(f)( ) ( ) ( )y t x t h t= ()Xf ( ) ( ) ( )Y f X f H f=
❖Especially, 0 0 0
1
cos(2 ) [ ( ) ( )]
2
FT
f t f f f f  ⎯→ + + − 0 0 0
( ) | ( )| cos(2 arg( ( )))y t A H f f t H f= + +
❖Fourier transform:
❖Trigonometric formulas: 1
sin( )cos( ) [sin( ) sin( )]
2
a b a b a b= + + −

4. Fourier transform
4
❖A function can be described by a summation of waves with different
amplitudes and phases. () ()exp2Hf ht iftdt 

−
=− () ()exp2ht Hf iftdf

−
=

5. Fourier transform
5

6. Fourier transform
6

7. ❖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.

8. 2. Analog to digital conversion
8
❖Analog to digital (A/D) conversion is a three-step process.
Sampler
Quantizer Coder
t=nT
x
Q(n)
x(t) x(nT)≡x(n) 11010
A/D converter
n
x
Q(n)
000
001
010
011
100
101
110
111
t
x(t)
n
x(n)

9. 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)

10. 3. Sampling-example 1
10
❖The analog signal x(t)=2cos(2πt) with t(s) is sampled at the rate f
s=4
Hz. Find the discrete-time signal x(n) ?
Solution:
❖x(n)≡x(nT)=x(n/fs)=2cos(2πn/fs)=2cos(2πn/4)=2cos(πn/2)
n 0 1 2 3 4
x(n) 2 0 -2 0 2
❖Plot the signal

11. 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 f
s=1 Hz.
Find the discrete-time signals ? 1 1 1
1 7 1 7
( ) ( ) ( ) 2cos(2 ) 2cos( )
8 1 4
s
x n x nT x n n n
f
 = = = 1
2cos((2 ) ) 2cos( )
44
nn

= − = 2 2 2
1 1 1 1
( ) ( ) ( ) 2cos(2 ) 2cos( )
8 1 4
s
x n x nT x n n n
f
 = = =
❖Observation: x
1(n)=x
2(n) →based on the discrete-time signals, we
cannot tell which of two signals are sampled ? These signals are
called “alias”

12. 3. Sampling-example 2
12
f
2=1/8 Hzf
1=7/8 Hz
f
s=1 Hz
Fig: Illustration of aliasing

13. at a sampling rate f
s=1/T results in a discrete-
time signal x(n).
3. Sampling-Aliasing of Sinusoids
13
❖In general, the sampling of a continuous-time sinusoidal signal
❖Remarks: We can see that the frequencies f
k=f
0+kf
s are
indistinguishable from the frequency f
0after sampling and hence they
are aliases of f
00
( ) cos(2 )x t A f t=+
❖The sinusoids is sampled at f
s, resulting in a
discrete time signal x
k(n). ( ) cos(2 )
kk
x t A f t=+
❖If f
k=f
0+kf
s, k=0, ±1, ±2, …., then x(n)=x
k(n) .
Proof: (in class)

14. 4. Sampling Theorem-Sinusoids
14
❖Consider the analog signal where Ωis
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
f
s=8f, f
s=4f, and f
s=2f.( ) cos( ) cos(2 )x t A t A ft =  =
❖Note that / sec
/ sec
s
fsamples samples
f cycles cycle
==
❖To sample a single sinusoid properly, we must require2
s
fsamples
f cycle

Fig: Sinusoid sampled at different rates

15. 4. Sampling Theorem
15
❖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 f
smust be chosen at least twice the maximum
frequency f
max.max
2
s
ff
Fig: Typical bandlimitedspectrum
❖f
s=2f
maxis called Nyquistrate;f
s/2 is called Nyquistfrequency;
[-f
s/2, f
s/2] is Nyquistinterval.

16. 4. Sampling Theorem
16
❖The values of f
maxand f
sdepend on the application
Applicationfmax fs
Biomedical 1 KHz 2 KHz
Speech 4 KHz 8 KHz
Audio 20 KHz 40 KHz
Video 4 MHz 8 MHz

17. 4. Sampling Theorem-Spectrum Replication
17
❖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


=−
= 21 1 1
( ) ( )
s
j f nt
n
TT
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 ()xt
❖Observation: The spectrum of discrete-time signal is a sum of the
original spectrum of analog signal and its periodic replication at the
interval f
s.

18. 4. Sampling Theorem-Spectrum Replication
18
Fig: Typical badlimitedspectrum
❖f
s/2 ≥ f
max
❖f
s/2 < f
max
Fig: Aliasing caused by overlapping spectral replicas
Fig: Spectrum replication caused by sampling

19. 5. Ideal Analog reconstruction
19
Fig: Ideal reconstructoras a lowpassfilter
❖An ideal reconstructoracts as a lowpassfilter with cutoff frequency
equal to the Nyquistfrequency fs/2.( ) ( ) ( ) ( )
a
X f X f H f X f==
❖An ideal reconstructor(lowpassfilter) [ / 2, / 2]
()
0
ss
T f f f
Hf
otherwise
−
=

Then

20. 5. Analog reconstruction-Example 1
20
❖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 x
a(t) ?

21. 5. Analog reconstruction-Example 2
21
❖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 x
a(t) ?

22. 5. Analog reconstruction
22
❖Remarks: x
a(t) contains only the frequency components that lie in the
Nyquistinterval (NI) [-f
s//2, f
s/2].
❖x(t), f
0NI ------------------> x(n) ----------------------> x
a(t), f
a=f
0
sampling at f
s ideal reconstructor
❖x
k(t), f
k=f
0+kf
s------------------> x(n) ----------------------> x
a(t), f
a=f
0
sampling at f
s ideal reconstructormod( )
as
f f f=
❖The frequency f
aof reconstructed signal x
a(t) is obtained by adding
to or substractingfrom f
0(f
k) enough multiples of fsuntil it lies
within the Nyquistinterval [-f
s//2, f
s/2].. That is

23. 5. Analog reconstruction-Example 3
23
❖The analog signal x(t)=10sin(4πt)+6sin(24πt) is sampled at the rate 20
Hz. Find the reconstructed signal x
a(t) ?

24. 5. Analog reconstruction-Example 4
24
❖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
Nyquistrate. Determine the signal x
a(t) that would be aliased with
x(t) ? Plot the spectrum of signal x(n) for this sampling rate?

25. 6. Ideal antialiasingprefilter
25
❖The signals in practice may not bandlimitted, thus they must be
filtered by a lowpassfilter
Fig: Ideal antialiasingprefilter

26. 6. Practical antialiasingprefilter
26
Fig: Practical antialiasinglowpassprefilter
❖The Nyquistfrequency fs/2 is in the middle of transition region.
❖A lowpassfilter: [-f
pass, f
pass] is the frequency range of interest for the
application (f
max=f
pass)
❖The stopbandfrequency f
stopand the minimum stopbandattenuation
A
stopdB must be chosen appropriately to minimize the aliasing
effects.s pass stop
f f f=+

27. 6. Practical antialiasingprefilter
27
❖The attenuation of the filter in decibels is defined as10
0
()
( ) 20log ( )
()
Hf
A f dB
Hf
=−
where f
0is a convenient reference frequency, typically taken to be at
DC for a lowpassfilter.
❖α
10=A(10f)-A(f) (dB/decade): the increase in attenuation when f is
changed by a factor of ten.
❖α
2=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/f
N
for large f, thus α
10=20N
(dB/decade) and α
10=6N (dB/octave)

28. 6. Antialiasingprefilter-Example
28
❖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 prefilterby an anlogprefilterH(f). Then, the output y(t)
of the prefilteris 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:

29. 6. Antialiasingprefilter-Example
29
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 prefilterwith cutoff fs/2=20 KHz.
c)When H(f) is a practical prefilterwith specifications as shown
below:
The filter’s phase response is assumed to be ignored in this example.

30. 7. Ideal and practical analog reconstructors
30
❖An ideal reconstructoris an ideal lowpassfilter with cutoff Nyquist
frequency fs/2.

31. 7. Ideal and practical analog reconstructors
31
❖The ideal recontructorhas the impulse response:
which is not realizable since its impulse response is not casual sin( f t)
()
s
s
ht
ft


=
❖It is practical to use a
staircase reconstructor

32. 7. Ideal and practical analog reconstructors
32
Fig: Frequency response of staircase recontructor

33. 7. Practical reconstructors-antiimagepostfilter
33
❖An analog lowpasspostfilterwhose cutoff is Nyquistfrequency fs/2
is used to remove the surviving spectral replicas.
Fig: Spectrum after postfilter
Fig: Analog anti-image postfilter