ECE 4371, Fall, 2014
Introduction to Telecommunication
Engineering/Telecommunication Laboratory
Zhu Han
Department of Electrical and Computer Engineering
Class 7
Sep. 17th, 2014
Outline

Analog vs. Digital

ADC/DAC: gateway between analog and digital domains
– Sampling Theorem
– Quantization
– Most important part in communication system
– Most important during interview
– Read books carefully

Examples
Claude Elwood Shannon, Harry Nyquist
Sampling Theory

In many applications it is useful to represent a signal in terms
of sample values taken at appropriately spaced intervals.

The signal can be reconstructed from the sampled waveform
by passing it through an ideal low pass filter.

In order to ensure a faithful reconstruction, the original signal
must be sampled at an appropriate rate as described in the
sampling theorem.
– A real-valued band-limited signal having no spectral
components above a frequency of B Hz is determined
uniquely by its values at uniform intervals spaced no greater
than  2B1  seconds apart.
Sampling Block Diagram

Consider a band-limited signal f(t) having no spectral
component above B Hz.

Let each rectangular sampling pulse have unit amplitudes,
seconds in width and occurring at interval of T seconds.
f(t)
A/D
conversion
T
Sampling
fs(t)
Impulse Sampling
Sampled waveform
Signal waveform
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0
1
201
1
201
Impulse sampler
0
1
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Impulse Sampling
with increasing sampling time T
Sampled waveform
Sampled waveform
0
0
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1
Sampled waveform
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Sampled waveform
0
0
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1
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Introduction
Let g (t ) denotetheideal sampledsignal
g (t ) 

 g (nT )  (t  nT )
n  
s
s
where Ts : samplingperiod
f s  1 Ts : samplingrate
(3.1)
Equation number is
not the same as
in the book
Math
From T able A6.3 we have

g( t )   (t  nTs ) 
n  
1
G( f ) 
Ts



m  

 f G( f
m  
( f 
s
m
)
Ts
 m fs )
g ( t )  f s

 G( f
m  
 m fs )
(3.2)
or we may apply Fourier T ransformon (3.1) t o obt ain
G ( f ) 

 g (nT ) exp( j 2 nf T )
n  
s
or G ( f )  f sG ( f )  f s
s

 G( f
m  
m 0
 m fs )
(3.3)
(3.5)
If G ( f )  0 for f  W and Ts  1
2W

n
j n f
G ( f )   g (
) exp(
)
2
W
W
n  
(3.4)
Math, cont.
With
1.G ( f )  0 for f  W
2. f s  2W
we find from Equation(3.5) that
1
G( f ) 
G ( f ) ,  W  f  W
(3.6)
2W
Substituting (3.4)into(3.6) we may rewriteG ( f ) as
n
jnf
)
exp(

) ,  W  f  W (3.7)

2W
W
n  
n
g (t ) is uniquely determinedby g (
) for    n  
2W
n 

or  g (
)  containsall information of g (t )
 2W 
1
G( f ) 
2W

g(
Interpolation Formula
n 

T o reconstruct g (t ) from  g (
)  , we may have
 2W 

g (t )   G ( f ) exp( j 2ft )df

W

W

1
2W
n
j n f
g(
) exp(
) exp( j 2 f t)df

2W
W
n  

n 

W exp j 2 f (t  2W )df (3.8)

n sin(2 Wt  n )
  g(
)
2W
2 Wt  n
n  
n
1
  g(
)
2W 2W
n  
W

n
  g(
) sin c( 2Wt  n ) , -   t  
2W
n  
(3.9)is an interpolation formulaof g (t )
(3.9)
Interpolation
If the sampling is at exactly the Nyquist rate, then
 t  nTs 

g (t )   g (nTs ) sin c
n  
 Ts 

 t  nTs 

g (t )   g (nTs ) sin c
n  
 Ts 

g (t )
Practical Interpolation
Sinc-function interpolation is theoretically perfect but it can
never be done in practice because it requires samples from
the signal for all time. Therefore real interpolation must
make some compromises. Probably the simplest realizable
interpolation technique is what a DAC does.
g (t )
Sampling Theorem
SamplingT heoremfor strictlyband - limitedsignals
1.a signal which is limited to  W  f  W , can be completely
n 

described by  g (
) .
 2W 
n 

2.T hesignal can be completelyrecoveredfrom  g (
)
 2W 
Nyquist rate  2W
Nyquist interval 1
2W
When thesignal is not band - limited(under sampling)
aliasing occurs.T oavoidaliasing, we may limit the
signal bandwidth or have higher samplingrate.
Under Sampling, Aliasing
Avoid Aliasing

Band-limiting signals (by filtering) before sampling.

Sampling at a rate that is greater than the Nyquist rate.
f(t)
Anti-aliasing
filter
A/D
conversion
T
Sampling
fs(t)
Anti-Aliasing
Aliasing

2D example
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 500 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1500 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 2200 Hz, Sampling frequency = 2000Hz
Natural sampling
(Sampling with rectangular waveform)
Figure 6.10
Signal waveform
Sampled waveform
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401
601
801
1001
1201
1401
1601
1801
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401
2001
Natural sampler
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1
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401
601
801
1001 1201 1401 1601 1801 2001
601
801
1001
1201
1401
1601
1801
20
Bandpass Sampling
(a) variable sample rate
(b) maximum sample rate without aliasing
(c) minimum sampling rate without aliasing
Bandpass Sampling

A signal of bandwidth B, occupying the frequency range
between fL and fL + B, can be uniquely reconstructed from the
samples if sampled at a rate fS :
fS >= 2 * (f2-f1)(1+M/N)
where M=f2/(f2-f1))-N and N = floor(f2/(f2-f1)),
B= f2-f1, f2=NB+MB.
Bandpass Sampling Theorem