Transcript Analog-to-Digital Conversion

Analog-to-Digital Conversion
PAM(Pulse Amplitude Modulation)
PCM(Pulse Code Modulation)
PAM(Pulse Amplitude Modulation)
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Conversion of analog signal to a pulse type
signal where the amplitude of signal denotes
the analog information
Two class of PAM signals
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Natural sampling (gating)
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Easier to generate
Instantaneous sampling
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Flat-top pulse
More useful to conversion to PCM
PAM with natural sampling
W(t)
Ws(t)
t
t
S(t)
Analog bilateral switch

Ts
Ws(t)
=W(t)S(t)
W(t)
t
Duty Cycle D=/Ts=1/3
S(t)
Spectrum of PAM with natural sampling
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Spectrum of input analog signal
Spectrum of PAM
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D=1/3, fs=4B
BT= 3fs = 12B
-B
|Ws(f)|

D
n 
D=1/3
-2fs
-fs
1
f

-3fs
|W(f)|
sin  f
W ( f  nf s )
 f
D
-B
B
fs
B
sin  f
 f
2fs
3fs
PAM with flat-top sampling
W(t)
Ws(t)
t
t

S(t)
Ts
t
Sample and Hold
Spectrum of PAM with flat-top sampling
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|W(f)|
Spectrum of Input
Spectrum of PAM
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1
/Ts=1/3, fs=4B
BT= 3fs = 12B
f
-B
|Ws(f)|
1
H( f )
Ts
-2fs
-fs

 W ( f  nf )
s
n 
 sin  f
Ts  f
D=1/3
-3fs
B
-B
B
fs
2fs
3fs
Summary of PAM
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Require very wide bandwidth
Bad noise performance
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Not good for long distance transmission
Provide means for converting a analog signal to
PCM signal
Provide means for TDM(Time Division Multiplexing)
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Information from different source can be interleaved to
transmit all of the information over a single channel
PCM(Pulse Code Modulation)
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Definition
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PCM is essentially analog to digital conversion of a
signal type where the information contained in the
instantaneous samples of an analog signal is
represented by digital words in a serial bit stream
Analog signal is first sampled at a rate higher than
Nyquist rate, and then samples are quantized
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Uniform PCM : Equal quantization interval
Nonuniform PCM : Unequal quantization interval
Why PCM is so popular ?
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PCM requires much wider bandwidth
But,
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Inexpensive digital circuitry
PCM signal from analog sources(audio, video, etc.) may
be merged with data signals(from digital computer) and
transmitted over a common high-speed digital
communication system (This is TDM)
Regeneration of clean PCM waveform using repeater.
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But, noise at the input may cause bit errors in regenerated PCM
output signal
The noise performance is superior than that of analog
system.
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Further enhanced by using appropriate coding techniques
PCM transmitter/receiver
Analog
signal
LPF
BW=B
Bandlimited
Analog signal
PCM
signal
Encoder
Sampler
& Hold
Quantized
PAM signal
Flat-top
PAM signal
Quantizer
No. of levels=M
Channel, Telephone lines with regenerative repeater
PCM
signal
Decoder
Quantized
PAM signal
Reconstruction
LPF
Analog
Signal
output
Waveforms in PCM
Uniform quantizer
Error signals
Waveform of signals
PCM signal
PCM word
Encoder
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Usually Gray code is used
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Only one bit change for each step change in
quantized level
Single errors in received PCM code word will
cause minimum error if sign bit is not changed
In text, NBC(Natural Binary Coding) is used
Multilevel signal can be used
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Much smaller bandwidth than binary signals
Requires multilevel circuits
Uniform PCM
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Uniform
distribution
Let M=2n is large enough
=2Xmax/M
Xmax
x
xi
Distortion
2
Di 
12M
2
D   Di 
12
i 1
M
-Xmax
/2
-/2
x
xi
x
SQNR of PCM
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Distortion
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2 xmax 2
)
2
2
2
x
x
x

D
 M
 max2  max
 maxn
n 2
12
12
3M
3(2 )
3(4 )
2
(
SQNR
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E[ X 2 ]
Let normalized input : X  xmax
E[ X 2 ] 3M 2 E[ X 2 ] 3(4n ) E[ X 2 ]
SQNR 


 3(4n ) X 2
D
xmax
xmax
SQNR dB  10log10 SQNR  4.77  6.02n  10log10 X 2
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SQNR dB _ pk  4.77  6.02n
Bandwidth of PCM
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Hard to analyze because PCM is nonlinear
Bandwidth of PCM
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If sinc function is used to generate PCM
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1
1
R  nf s
2
2
, where R is bit rate
If rectangular pulse is used
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BPCM 
BPCM  R  nf s
, first null bandwidth
If fs=2B (Nyquist sampling rate)
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Lower bound of BW: BPCM  nB
In practice, BPCM  1.5nB is closer to reality
Performance of PCM
Quantizer
Level, M
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
n bits
M=2n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Bandwidth
>nB
2B
4B
6B
8B
10B
12B
14B
16B
18B
20B
22B
24B
26B
28B
30B
32B
SQNR|dB_PK
4.8+6n
10.8
16.8
22.8
28.9
34.9
40.9
46.9
52.9
59.0
65.0
71.0
77.0
83.0
89.1
95.1
101.1
PCM examples
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Telephone communication
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Voice frequency : 300 ~ 3400Hz
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Encoding with 7 information bits + 1 parity bit
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Bit rate of PCM : R = fs x n = 8K x 8 = 64 Kbits/s
Buad rate = 64Ksymbols/s = 64Kbps
Required Bandwidth of PCM
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Minimum sampling frequency = 2 x 3.4KHz = 6.8KHz
In US, fs = 8KHz is standard
If sinc function is used: B > R/2 = 32KHz
If rectangular is used: B = R = 64KHz
SQNR|dB_PK = 46.9 dB (M = 27)
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Parity does not affect quantizing noise but decrease errors caused by
channels
PCM examples
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CD (Compact Disk)
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For each stereo channel
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16 bit PCM word
Sampling rate of 44.1KHz
Reed-Solomon coding with interleaving to correct burst
errors caused by scratches and fingerprints on CD
High quality than telephone communication
Homework
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Illustrative Problems
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4.9, 4.10, 4.11, 4.12
Problems
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4.14
Nonuniform quantization
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Example: Voice analog signal
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Peak value(1V) is less appears while weak
value(0.1V, 20dB down) around 0 is more
appears (nonuniform amplitude distribution)
Thus nonuniform quantization is used
Implementation of nonuniform quantization
Analog
Input
Compression
(Nonlinear)
filter
PCM with
Uniform
Quantization
PCM
output
Nonuniform Quantization
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Two types according to compression filter
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-law : used in US
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y
ln(1   x )
ln(1   )
sgn( x)
See Figure 4.9, Page 155
A-law : used in Europe
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 Ax
sgn( x),
0 x  1

A
1  ln A
y
1  ln( A x ) sgn( x), 1  x  1
 1  ln A
A
Nonuniform Quantization
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Compandor = Compressor + Expandor
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Compressor: Compression filter in transmitter
Expander: Inverse Compression filter in receiver
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-law : x 
SQNR
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(1   )  1
y

sgn( y )
SQNR dB    6.02n
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Uniform quantizing:   4.77 10log10 X 2
-law:   4.77  20log10 (ln(1   ))
A-law:   4.77  20log10 (1  ln A)
Homework
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Illustrative Problems
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4.13, 4.14
Problems
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4.17