discrete -time signals and systems sampling –ii sampling ...sampling at diffe--rent rates from...
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Yogananda Isukapalli
Discrete - Time Signals and Systems
Sampling – IISampling theorem & Reconstruction
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2
Sampling at diffe--rent ratesFrom these figures,it can be concludedthat it is very important to sample the signaladequately to avoidproblems in reconstruction, which leads us toShannon’s samplingtheorem
Fig:7.1
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Claude Shannon: The man who started the digital revolution
1916-2001
Shannon arrived at the revolutionaryidea of digital representation by sampling the information source at an appropriate rate, and converting the samples to a bit stream
Before Shannon, it was commonly believed that the only way of achievingarbitrarily small probability of errorin a communication channel was to reduce the transmission rate to zero.
All this changed in 1948 with the publication of “A Mathematical Theory of Communication”—Shannon’s landmark work
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Shannon’s Sampling theorem
max
max
A continuous signal ( ) with frequencies no higher than can be reconstructed exactly from its samples [ ] [ ], if the samples are taken at a ra 2 ,te where 1 s
s
s s
x t
x n x nf
f fT
f T³=
=
This simple theorem is one of the theoretical Pillars of digital communications, control and signal processing
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Shannon’s Sampling theorem,• States that reconstruction from the samples is possible, but it doesn’t specify any algorithm for reconstruction• It gives a minimum sampling rate that is dependent only on the frequency content ofthe continuous signal x(t)• The minimum sampling rate of 2fmax is called
the “Nyquist rate”
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Example1: Sampling theorem-Nyquist rate( ) 2cos(20 ), ?x t t find the Nyquist frequencyp=
max
max
( ) 2cos(2 (10) )
10 10
, 2 20snyq
x t tThe only frequency in the continuous timesignal is Hzf Hz
Nyquist sampling rate Sampling ratef f Hz
p=-
\ =
= =
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Continuous-time sinusoid of frequency 10Hz
Sampled at Nyquist rate, so, the theorem states that 2 samples are enough per period. Intuitivelyit doesn’t seem to be enough, there must be a sophisticated algorithm for reconstruction
Fig:7.2
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max
max
( ) cos(10 ) sin(22 ) cos(2 (5) ) sin(2 (11) )
5 11,11 2 22
x t t tt t
The frequencies in the signal are andf Hz
Nyquist frequency f Hz
p pp p
= += +
\ =
= =
Example2: Nyquist rate
Fig:7.3
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Example3: Nyquist rateThe positive side of the spectrum is shown below,find the ‘Nyquist rate’ of sampling for this signal?
0 21 48 f Hz
The range of the spectrum is from 21 to 48fmax=48HzNyquist sampling rate, fs=2 fmax=96Hz
Fig:7.4
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Frequency mapping
0ˆ 2 sf Tw p=
ˆ discrete time frequencyw -
02 f continuous time frequencyw p= -
0
0
0
( ) cos(2 ), [ ] cos(2 )
s
s
Continuous signal with frequency fx t A f t sampled at a rate fx n A f n f
pp
==
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ˆˆ ˆ
s
s
s
TT f
w ww w w\ =
= =
Thus there is a corresponding continuous-timefrequency for every discrete-time frequency
However, the converse is not true, due to aliasingand folding. Principal aliases are the generallyaccepted basis for obtaining a continuous-timefrequency from a discrete-time one
Frequency mapping contd….
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With the “Nyquist sampling rate”:
max
max max
ˆ
' ' ,
2ˆ , 1, 2,3....
2 , 2ˆ2
s s
ii i s
s
s
i ii i s
T fLet there be i frequencies in the continuoustime signal mapping them into discrete timefrequencies
ff i
ff f Nyquist rate
f ff
f f
w w w
pw w
p pw w
= =-
-
= = =
=
= = =
!
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maxmax
max
min
ˆ
ˆ
ˆ
ff
correspondingly for negative frequencies
Thus the discrete - time frequencies are guaranteedto be in the range - <
pw p
w p
p w p
= =
= -
£The above result is the combined effect of Applying‘Nyquist rate’ to the principal alias domain
max max
ˆ
1 1 - -
2 2s i s i
- < also implies that the continuous frequencyobtained from a discrete one is guaranteed to be in the
range of f f f or f f f
p w p£
< £ < £
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Spectrum view of frequency mappingConsider a continuous-time sinusoid x(t) withfrequency f0
0 0( ) cos( ) cos(2 )x t A t A f tw f p f= + = +
12
jAe f
Fig. 7.5
w
12
jAe f-
0w0w-
Notice that the spectrum is plotted against w0
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0
0
0
[ ] ( ) cos(( ) )This discrete-time spectrum has, ˆ 2 , 0, 1, 2,...ˆ 2 , 0, 1, 2,...
s s
s
s
x n x n f A f n
f l l =f l l =
w f
w w pw w p
= = +
= + ± ±
= - + ± ±
The continuous-time signal sampled at a rate fs
Fig. 7.6
*12
jAe f
w
*
0 sfw0 sfw- pp-0 2sfw p- +0
0 2sfw p+0 2sfw p-0 2sfw p- -
*
Mag
nitu
de
fundamental domain obtained with Nyquist rate
Principal aliases
Frequenciesassociated withprincipal +ve frequency, 0 sfw
Frequenciesassociated withprincipal –ve frequency,notice conjugatefor magnitude.folded aliases arealso associated with –ve frequencies
0 sfw-
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Example1: Over Sampling• In most applications sampling rate is chosen to be higher than Nyquist rate to avoid problems in reconstruction
• The sampling rate in CD’s is 44.1kHz. The highest frequency we can hear is 20kHz, so sampling rate is slightly higher than 40kHz
• Consider sampling a 100Hz sinusoid at 500samples/sec
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Fig:7.7
Fig:7.8
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2 100ˆ 0.4500
2(100) 200 , 500
s s
s
s
T f
Nyquist rate Hz but f Hzf Nyquist rate
pw w w p= = = =
= = =
>>
Notice that both principal aliases are well with in the limit of . In the reconstruction only frequencies in this range are used to get continuous frequencies
ˆ- <p w p£
Fig:7.9
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Example2: Under Sampling, aliasing
2(100) 200 , 125
s
s
Nyquist rate Hz but f Hzf Nyquist rate
= = =<<
The effect of under sampling can be seen in the time-domain plot itself Fig:7.10
Fig:7.11
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2 100ˆ 1.6125s sT f pw w w p= = = =
( , ]ˆ
0.4 in 25
Notice that prinipal aliases are beyond the rangeAnd their aliases are in the primary rangeMapping results an analog frequency of Hz
p pp w p
p
-- < £
Fig:7.12
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Example3: Sampling at the rate of signal frequency
Analog spectrum,notice that the signal and the sampling frequencies are same, 100Hz 2(100) 200 , 100
s
s
Nyquist rate Hz but f Hzf Nyquist rate
= = =
<<<
sampling at the signal frequency means picking up the same value from each cycle
Fig:7.13
Fig:7.14
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2 100ˆ 2100s sT f pw w w p= = = =
2 , 2ˆ '0 ' -
Notice that prinipal aliases are at andAnd is their common alias, which is in the range
p pp w p
-< £
The result is obvious as we have the same value for each sample
Fig:7.15
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This movie illustrates the phenomenon of aliasing. A 600 Hz sinusoid is sampled at 500 samples per second.
This movie illustrates the phenomenon of folding. A 600 Hz sinusoid is sampled at 750 samples per second.
A 600 Hz sinusoid is sampled at 2000 samples per second. Since the samples are taken at more than two times the frequency of the cosine wave, there is no aliasing.
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Apparent frequencyIf the Nyquist rate is not followed, the apparentfrequency will not be the actual frequency
max
max
2 ,
0 2 , 2s apparant actual
s s
f f f f
f f folding occurs at f
³ =
£ <
If the sampling rate is not enough, the chirpsignal could sound likethis , remember howan actual chirp is supposed to sound
Fig:7.16
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A Mechanical viewpoint of samplingIn this movie the video camera is sampling at a fixed rate of 30 frames/second. Observe how the rotating phasor aliases to different speeds as it spins faster.
In this movie the video camera is sampling at afixed rate of 30 frames/second. Observe how the rotating phasors alias to a different speed as the disk spins faster. The fact that the four phasors are identical further contributes to the aliasing effect.
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A/DConverter
C/DConverter
A/DConverter
D/CConverter
Signal Sensors
Processing Actuators
Fig:7.17, General System
Samples of continuous time signal are processed as required in this stage, output is also discrete samples
Fig:7.18
Reconstruction
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Ideal Reconstruction
Ideal C-D converter was defined as:( ) [ ] [ ]s sx t x nT x n f= =
Ideal D-C converter is governed by an inverse relation to that of C-D converter
( ) [ ] -
sn f ty t y n n
== ¥ < < ¥
Fig:7.19
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Ideal Reconstruction contd….
The simple substitution of ‘n=fst’, is only valid if the signal consists of one or more sinusoids or if the signal can be expressed as a mathematicalformula
0
0
[ ] cos(2 )( ) cos(2 )
sy A fy An nt tf
Tp fp f
= += +
Most of the real world signals can’t be reducedinto a simple mathematical equation from the discrete samples
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•D-C conversion involves filling in the signal values between sampling instances tn=nTs
•“Interpolation” can be used to approximate the behavior of ideal D-C converter
( ) [ ] ( )sn
y t y n p t nT¥
=-¥
= -å
The above equation describes a broad class of D-C converters. Where p(t) is the characteristicPulse shape of the converter
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Mathematical proof of the sampling theorem gives the ideal pulse shape, which is a sinc function
Fig:7.20
sin( )( )( )
s
s
t Tp tt Tp
p=
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Square PulseThe simplest of all is the ‘square pulse’ defined as,
1 11 ( ) 2 2
0
s sT t Tp t
otherwise
ì - < £ï= íïî
Each term will create flat region of Amplitude y[n] centered at
[ ] ( )sy n p t nT-st nT=
Since the effect of the flat pulse is to hold or replicate each sample for seconds, it is alsoknown as ‘zero-order hold reconstruction’
sT
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Movie Illustrates a similar process
In all the examples sampling rate is greater than Nyquist rate
Fig:7.21
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Obviously the more no. of samples we have the betterwe should be able to reconstruct the signal
Notice that the sampling is much higher than previous case
Movie Illustrates a similar process
Fig:7.22
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Triangular Pulse
1 ( )
0 s s st T T t T
p totherwise
ì - - < £= íî
It is a pulse of 1st order polynomial, defined as,
In this case the output y(t) of the D-C converterat any time ‘t’ is the sum of the scaled pulsesthat overlap at that time instant. The performanceis better than a square pulse
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Movie Movie
More than Nyquist 4 times Nyquist
Fig:7.23 Fig:7.24
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Cubic spline interpolation: A third order polynomialwith exactly a similar process as a triangular pulseMore than Nyquist 4 times Nyquist
Movie MovieThe results with this pulse are extremely close to original !!
Fig:7.25 Fig:7.26
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1.James H. McClellan, Ronald W. Schafer and Mark A. Yoder, “Signal Processing First”, Prentice Hall, 20032. http://www.research.att.com/~njas/doc/ces5.html,
Shannon’s Biography
Reference