Transcript Sampling of Analog Signal

Digital Coding of
Analog Signal
Electronics Engineering Department,
Sardar Vallabhbhai National Institute of Technology,
Surat-395007.
Prepared By:
Amit Degada
Teaching Assistant
Outline
• Analog To Digital Converter
• Review of sampling
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Nyquist sampling theory: frequency and time domain
Alliasing
Bandpass sampling theory
Natural Sampling
Aperture Effect
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Quantization.
Quantization Error.
Companding.
Two optimal rules
A law/u law
• Quantization
• Coding
• Differential PCM
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 FM Hz is determined
uniquely by its values at uniform intervals spaced no
greater than (1/2FM) 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
fs(t)
interval of T seconds.
f(t)
A/D
conversion
T
Sampling
Sampling
Sampled waveform
Signal waveform
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Impulse sampler
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Impulse Sampling
with increasing sampling time T
Sampled waveform
Sampled waveform
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Sampled waveform
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Sampled waveform
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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
EE 541/451 Fall 2006
(3.1)
Math
From T able A6.3 we have

g( t )   (t  nTs ) 
n  
1
G( f ) 
Ts


 ( f
m  

 f G( f
m  

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(
)
2W
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
g(
) 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

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
) sin c( 2Wt  n ) , -   t  

2W
n  
(3.9)is an interpolation formulaof g (t )

g(
(3.9)
Interpolation
If the sampling is at exactly the Nyquist rate, then
 t  nTs 

g (t )   g (nTs ) sin c
n  
 Ts 

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)
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.
Natural sampling
(Sampling with rectangular waveform)
Signal waveform
Sampled waveform
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Natural sampler
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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.
Time Division Multiplexing
• Entire spectrum is allocated for a channel (user) for a limited
time.
• The user must not transmit until its
k1
k2
k3
k4
k5
k6
next turn.
• Used in 2nd generation
c
Frequency
f
t
• Advantages:
– Only one carrier in Time
the medium at any given time
– High throughput even for many users
– Common TX component design, only one power amplifier
– Flexible allocation of resources (multiple time slots).
Quantization
• Scalar Quantizer Block Diagram
Quantization Procedure
Quantization Error
Quantization Type
Mid-tread
Mid-rise
Quantization Noise
Quantization Noise
• What happens if no. of representation
level increases?
• >64 distortion is significant
• Quantization
error
is
uniformly
distributed in interval (-∆/2 to ∆/2).
• The Avg. Power of Quantizing error qe
Math
K∆+ ∆/2
K∆
Pq
qe
K∆- ∆/2
Sample of
Amplitude
K∆+ qe
0V
Example
• A sinusoidal Signal of amplitude Am
uses
all
Representation
levels
provided for Quantization in the case
of full load condition. Calculate
Signal to Noise ratio in db assuming
the number of quantization levels to
be 512.
• ANS: 55.8 db.
Example
• SNR for varying number of representation
levels for sinusoidal modulation 1.8+6 X
dB
Number of
representati
on level L
32
64
128
256
Number of
Bits per
Sample, R
5
6
7
8
SNR (dB)
31.8
37.8
43.8
49.8
Companding
• Process of uniform Quantization is not
possible.
• Example: Speech, Video.
• The variation in power from weak signal to
powerful signal is 40 db.
• So Ratio 1000:1
• Excursion in Large amplitude occurs less
frequently.
• This Scenario is cared by Non- Uniform
Quantization.
Non-uniform Quantizer
F: nonlinear compressing function
F-1: nonlinear expanding function
F and F-1: nonlinear compander
X
y
F
Example
F: y=log(x)
Q
^
y
F-1
^
x
F-1: x=exp(x)
We will study nonuniform quantization by PCM example next
A law and  law
Input-Output characteristic
of Compressor
 Law/A Law
• The -law algorithm (μ-law) is a companding algorithm,
primarily used in the digital telecommunication systems of
North America and Japan. Its purpose is to reduce the
dynamic range of an audio signal. In the analog domain,
this can increase the signal to noise ratio achieved during
transmission, and in the digital domain, it can reduce the
quantization error (hence increasing signal to quantization
noise ratio).
• A-law algorithm used in the rest of worlds.
• A-law algorithm provides a slightly larger dynamic range
than the mu-law at the cost of worse proportional
distortion for small signals. By convention, A-law is used for
an international connection if at least one country uses it.
 Law
A Law
EE 541/451 Fall 2006
Implementation of
Compander
• Diode equation
• Piece-wise linear Approach
Coding