Lecture_ch6.pptx

1
Chapter 6
Sampling and Analog-to-
Digital Conversion

2
Content
1. Sampling Theorem
2. Pulse Code Modulation (PCM)
3. Digital Telephony: PCM in T1 Carrier Systems
4. Digital Multiplexing
5. Differential Pulse Code Modulation (DPCM)
6. Adaptive DPCM (ADPCM)
7. Delta Modulation

3
6.1 Sampling Theorem
Significance
Analog signals can be digitized through sampling and
quantization. The sampling rate must be sufficiently
large so that the analog signal can be reconstructed
from the samples with sufficient accuracy. The sampling
theorem, which is the basis for determining the proper
sampling rate for a given signal, has a deep significance
in signal processing and communication theory.

4
The theorem
Any signal whose spectrum is band-limited to B
Hz can be reconstructed exactly (without any
error) from its samples taken uniformly at a rate
fs >= 2B Hz (samples per second). In other
words, the minimum sampling frequency is:
fs = 2B Hz.

To prove the sampling theorem, consider a
signal g(t) (Fig. 6.1 a) whose spectrum is band-
limited to B Hz (Fig. 6.1b). Sampling g(t) at a
rate of fs Hz can be accomplished by
multiplying g(t) by an impulse train δTs(t) (Fig.
6.1c), consisting of unit impulses repeating
periodically every Ts seconds, where Ts = 1/ fs.

6
1/Ts
x
*
Figure 6.1 Sampled signal and its Fourier spectra.
1/Ts
           









n
t
f
jn
s
n
s
s
Ts
s
e
t
g
T
nT
t
nT
g
t
t
g
t
g 

 2
1
  





n
n
t
jn
s
Ts
s
e
T
t 

1
   






n
s
s
nf
f
G
T
f
G
1

7
Since the impulse train Ts(t) is a periodic signal of period Ts , it can be expressed as a
Fourier series. Its exponential Fourier series is given by:
         
 


n
s
s
Ts nT
t
nT
g
t
t
g
t
g 

  





n
n
t
jn
s
Ts
s
e
T
t 

1
s
s
s f
T


 2
2


       






n
t
f
jn
s
T
s
s
e
t
g
T
t
t
g
t
g 
 2
1
Therefore
Taking the Fourier Transform:
   






n
s
s
nf
f
G
T
f
G
1
(Example 2.5 on page 45
of the text book)
(Example 3.5 on page 74 and Example 3.11 on page 86 of the text book )

If we are to reconstruct g(t) from ḡ(t), we should be
able to recover G( f ) from Ḡ ( f ) using a LPF. This is
possible if there is no overlap between successive cycles
of Ḡ ( f ). This requires fs > 2B. Since the sampling
interval is Ts=1/fs this implies
Therefore,
1
2
S
T
B

The minimum sampling rate fs > 2B samples/second
required to recover g(t) from its samples ḡ(t) is called
the Nyquist rate for g(t), and the corresponding sampling
interval Ts=1/(2B) seconds is called the
Nyquist Interval for g(t).

9
Signal Reconstruction: The
Interpolation Formula
  







B
f
T
f
H s
2
The process of reconstructing a continuous-time signal g(t) from
its samples is also known as interpolation. A signal g(t) band-
limited to B Hz can be reconstructed (interpolated) exactly from
its samples. This is done by passing the sampled signal through
an ideal low-pass filter of bandwidth B Hz and gain Ts to recover
the component (1/Ts)g(t) from the sampled signal. Thus, the
reconstruction (or interpolating) filter transfer function is
   
Bt
BT
t
h s 
2
sinc
2


10
Signal Reconstruction: The Interpolation Formula
Assuming the Nyquist sampling rate, that is:
2B = fs = 1/Ts we obtain
h(t) = sinc(2πBt)
  







B
f
T
f
H s
2
   
Bt
BT
t
h s 
2
sinc
2

h(t) = sinc(2πBt)

11
The kth sample of the input ͞g(t) is the impulse g(nTs)δ(t-
nTs); the filter output of this impulse is g(nTs)h(t-nTs).
Hence the filter output ͞g(t), which is g(t), can now be
expressed as a sum,
     
   
 
   



n
Bt
nT
g
nT
t
B
nT
g
nT
t
h
nT
g
t
g
n
s
s
n
s
s
n
s









2
sinc
2
sinc

12
The above equation is the interpolation formula,
which yields values of g(t) between samples as a
weighted sum of all the sample values.
     

 n
Bt
nT
g
t
g
n
s 
  2
sinc

13
Practical Sampling
Practical Sampling
In proving the sampling theorem, we assumed ideal samples
obtained by multiplying a signal g(t) by an impulse train which is
physically nonexistent. In practice, we multiply a signal g(t) by a
train of pulses of finite width, shown in the next figure. We
wonder whether it is possible to recover or reconstruct g(t) from
the sampled signal. Surprisingly, the answer is positive, provided
that the sampling rate is not below the Nyquist rate. The signal
g(t) can be recovered by low-pass filtering the sampled signal as if
it were sampled by impulse train.

14
Practical Sampling
The envelope is FT of a pulse
x
*

Practical Sampling
                 
    






















n
t
f
jn
s
n
s
n
s
n
s
s
e
T
t
p
t
g
nT
t
t
p
t
g
nT
t
t
p
t
g
nT
t
p
t
g
t
g



2
1
*
*
*
~
       
     
   




















n
s
s
s
n
s
s
s
n
s
s
nf
f
G
nf
P
T
nf
f
nf
P
T
f
G
nf
f
T
f
P
f
G
f
G
1
1
~


Fourier Transform (spectrum) of the sampled signal is:
Therefore g(t) can by reconstructed using a low pass filter (LPF) of
B < Bandwidth < fs - B

16
Practical Sampling
               
t
g
t
p
nT
t
nT
g
t
p
nT
t
p
nT
g
t
g
n
s
s
n
s
s *
*
~ 



 
 
Therefore
     






n
s
s
nf
f
G
T
f
P
f
G
1
~
Nevertheless in practical scenarios scaling factor for
amplitude of each pulse can be considered a constant
value over the pulse duration. In this case sampled signal
Figure 6.5 Practical signal reconstruction.

17
Practical Sampling
The equalizer must remove all the shifted replicas
G( f - n fs) for n > 0
The sampling pulse can be approximated to rectangular pulses:
   









B
f
T
B
f
f
f
P
f
E
s
s
0
  






 


p
p
T
T
t
t
p
5
.
0
   
sin p
j fT
p p
P f T c fT e




To recover g(t) a filter (called equalizer) is needed. Let
E(f) is the frequency response of the equalizer such that
           







n
s
s
nf
f
G
T
f
P
f
E
f
G
f
E
f
G
1
~

18
Practical Sampling
As a result E( f ) must be :
 
 
 
 
















B
f
f
B
f
f
B
Flexible
B
f
f
P
T
f
E
s
s
s
0
 
  B
f
e
fT
f
T
f
E p
fT
j
p
s 


|
|
sin



By choosing Tp small:
  B
f
T
T
f
E
p
s

 |
|

19
Maximum Information Rate
A minimum of 2B independent pieces of information
per second can be transmitted, error free, over a
noiseless channel of bandwidth B Hz. This can be
proved by the sampling theorem.

20
Some Applications of the Sampling Theorem
Figure 6.11 Pulse-modulated signals. (a) The unmodulated signal. (b) The PAM signal.
(c) The PWM (PDM) signal. (d) The PPM signal.

21
Some Applications of the Sampling Theorem
Figure 6.12 Time division multiplexing of two signals.

22
6.2 Pulse Code Modulation
PCM is the most used among the pulse modulation
techniques. It is used to convert analog signal to digital
one.
After quantizing a sampled signal we obtain a L-level
pulse train. This signal can be converted to “binary
pulses” by assigning a binary code for each level. Thus
the L-ary signal is converted to a binary signal where
the 0 and 1 can be expressed as -1V and +1V.
Figure 6.13 PCM system diagram.

23
Quantization
Figure 6.14 Quantization of a sampled analog signal.

26
Pulse Code Modulation
Example1: telephony systems
Step1: the bandwidth is limited to 3400Hz (enough for
intelligibility) with low pass filter.
Step2: the resulting signal is sampled with fs = 8000Hz
(> 6.8KHz).
Step3: the samples are quantized to 256 levels = 28.
Step4: the quantized levels are then coded with 8-bit
binary code.  the telephone signal require a rate of
8000x 8 =64000 binary pulses per second (64kbps).

27
Example2: compact disc CD
Step1: the bandwidth is limited to 15kHz, since it is a HIFI
system.
Step2: the resulting signal is sampled with fs = 44.1kHz
(> 30kHz).
Step3: the samples are quantized to 65,536 levels = 216.
Step4: the quantized levels are then coded with 16-bit
binary code.  the binary pulses are then recorded in the
CD

28
2.2 Uniform Quantizing
Figure 6.14 Quantization of a sampled analog signal.

29
2.2 Uniform Quantizing
In digital communication, the detection error is usually
much less than the quantizing error and can be ignored
i.e., the detection error is ignored.
For quantizing, we limit the amplitude of the message
signal m(t) to the range (-mp, mp), as shown in Fig. 6.10.
Note that mp is not necessarily the peak amplitude of m(t).
The amplitudes of m(t) beyond ±mp are chopped off. Thus,
mp is not a parameter of the signal m(t), but is a constant of
the quantizer. The amplitude range (-mp, mp) is divided into
L uniformly spaced intervals, each of width
Δv = 2mp/L.

(a) Quantizer characteristic of a uniform quantizer (b) Error signal
v v v v
v
v
v
v
v/2
v/2
v/2
v/2
v/2
v/2
v/2
v/2
Quantizer
Input x
Quantizer
Output xq
q
x
x
mp
v v v v
v
v
v
v
v/2
Quantizer
Input x
Quantization Error q
v/2
mp
(a)
(b)
Quantizing Error (noise)
Uniform Quantizer
Δv = 2mp/L = x-xq

Example:
A signal m(t) with amplitude variations in the range -mp to
mp is to be sent using PCM with a uniform quantizer. The
quantization error should be less than or equal to 0.1
percent of the peak value mp. If the signal is band limited
to 5 kHz, find the minimum bit rate required by this
scheme.
Ans: 105 bps

32
Quantizing Noise
The mean square of the original signal is:  
t
m
S 2
0 
The mean square of the quantization error is: 2
2
2
0
3
12 L
m
v
N
p



Where L, number of the quantizer levels and mp is the
highest quantizer level. (a constant of the quantizer)
The SNR, signal to noise ratio is given by:
 
2
2
2
0
0
3
p
m
t
m
L
N
S
SNR 

Δv = 2mp/L

33
2.3 Non-uniform Quantizing
Normally we prefer a constant SNR. Unfortunately it is
not the case. The SNR is directly related to the mean
power of the signal varying from one talker to another.
They also vary with the path length. This is due to the
fact that the quantizer step is constant v = 2mp / L. The
quantizing error is directly proportional to the step size.
How to maintain a constant SNR for different signals?

34
Non-uniform Quantizing
Solution
 Taking smaller steps for small signals and larger steps for large
signals.  nonuniform quantizing.
 Same result can be obtained by signal compression followed by
quantization with uniform-step quantizer
e.g. logarithmic compression
Compression function

35
smaller quantization noise for small signal.
SNR independent from the signal power for
large dynamic range
 disadvantage: more noise in strong signals.

36
  










p
m
m
y


1
ln
1
ln
1
1
0 

p
m
m
 
 































1
1
1
ln
ln
1
1
1
0
ln
ln
1
1
p
p
p
p
m
m
A
m
Am
A
A
m
m
m
m
A
y
Compression laws:
law (in North America and Japan)
Alaw (In Europe and the rest of the world)

38
The coefficients  or A define the degree of
compression.
 For 128-level (7 bits) encoding,  = 100 is used
leading to a dynamic range of 40dB.
 For 256-level (8 bits) encoding,  = 255 is used
leading to a dynamic range of 40dB.
 For 256-level (8 bits) encoding, A = 87.6 is used
leading to a comparable results.

• A number of companies manufacture CODEC chips with various
specifications and in various configurations. Some of these
companies are:
Motorola (USA), National semiconductors (USA), OKI (Japan).
• These ICs include the anti-aliasing bandpass filter (300 Hz to 3.4
kHz), receiver lowpass filter, pin selectable A-law / μ -law option.
Listed below are some of the chip numbers manufactured by
these companies.
Motorola: MC145500 to MC145505
National: TP3070, TP3071 and TP3070-X
OKI: MSM7578H / 7578V / 7579
Details about these ICs can be downloaded from their respective websites.

40
At the receiver end an expander must be used to
restore the original signal. The compressor and the
expander are together called the compander.
In  -law compander the SNR is found to be:
 
   
t
m
m
for
L
N
S p
2
2
2
2
2
0
0
~
1
ln
3


 


42
Compander
Based on logarithmic diode characteristics.










s
I
I
q
KT
V 1
ln
Encoder and Decoder
 Sampling, quantization and encoding are performed
using the Analog to Digital Converter
 Digital to Analog Converter can be used for
decoding.

43
Comments on Logarithmic Units
(Decibel)
 Voltage: V = x Volt  VdB = 20log10(x).
 Ratio between voltages:
A = V1/V2  AdB = 20log10(A).
 Power: P = y Watt  PdB = 10log10(y).
P = y mWatt  PdBm = 10log10(y).
PdBm = PdB + 30
 Ratio between Powers:
G = P1/P2 =  GdB = 10log10(G).

45
2.3 Transmission Bandwidth and
the Output SNR
For a binary PCM, we assign a distinct group of n binary digits (bits)
to each of the L quantization levels. Because a sequence of n binary
digits can be arranged in 2n distinct patterns,
L = 2n or n = log2 L (6.37)
Each quantized sample is, thus, encoded into n bits. Because a
signal m(t) band-limited to B Hz requires a minimum of 2B samples
per second, we require a total of 2nB bits per second (bps), that is,
2nB pieces of information per second. Because a unit bandwidth (1
Hz) can transmit a maximum of two pieces of information per
second, we require a minimum channel of bandwidth BT Hz, given
by
BT = nB Hz (6.38)

46
(6.37)
(6.40)
(6.39)
(6.41)
(6.34) (6.36)
(6.34)
(6.36)
(6.38) (6.39)
(6.40)

48
Digital Telephony: PCM in T1 Carrier
Systems
24 analog channels sampled (PAM)  time multiplexed (TDM) 
quantized (256-level)  encoded (8-bit PCM)  sent to transmission
medium with repeaters at each 6000ft.
At the receiver, the signal is decoded (converted to samples) 
the samples are demultiplexed to 24 channels  the signal of each
channel is filtered (LPF) to restore the desired audio signal.
The 1.544MB/s signal of the T1 is called Digital Signal Level 1 (DS1),
used to build higher DS, (DS2, DS3, DS4). It is used in North America
and Japan.
Now the standard is the 30-channel 2.048MB/s PCM system used
around the world, except North America and Japan who stayed using
the T1. Interfaces are developed for international communication.

49
3. Digital Telephony: PCM in T1
Carrier Systems

50
Systems
Synchronization and signaling
Synchronization
Frame: 192 bits = 8 bits x 24 channel (1 sample for each channel).
Sample rate (1 ch.): 8000 Samples/s  each frame takes 125s.
At the receiver end it is difficult to catch the beginning of each
frame. To solve the problem and separate the information bit
correctly, a framing bit is added to the frame  the total number
of bits in a frame is 193.
The framing bits are used in a way that they form a special pattern
unlikely to be used in voice speech.
At the receiver end, the receiver examines the bits until it gets the
pattern  it catches the beginning of the frame. This is called
synchronization. It takes 0.6 to 6 ms.

51
Digital Telephony: PCM in T1 Carrier Systems

52
Systems
Signaling
It consists of sending information relative to the dialing pulses and
the on-hook/off-hook signals. These information are useful for
telephone switching systems.
Instead of allocating time slots for this information we take the LSB
of every sixth sample of a transmitted signal. This is called
encoding and the signaling channel so derived is called robbed-
bit-signaling.  this result in a small acceptable deterioration in
the SNR of the signal.
The rate of the signaling bit for each channel is 8000 / 6 = 1333 b/s
5
7
6

53
Systems
Question:
How can we detect frames that contain the signaling bits at the
receiver end?
Answer:
superframe technique: the pattern used to detect the beginning
of each frame is also used to detect the beginning of each
superframe (12 frames)  2 signaling bits per channel can be
detected in each superframe [equivalent to 4-state signaling
(00, 01, 10, 11)].

Differential Pulse Code Modulation (DPCM)
 DPCM can be treated as a variation of PCM; it also involves
the three basic steps of PCM, namely, sampling, quantization
and coding.
 In DPCM, what is quantized is the difference between the
actual sample and its predicted value.
 DPCM exploits the correlation between adjacent samples of
the signal, to improve A/D conversion efficiency.
 Quite a few real world signals such as speech signals,
biomedical signals (ECG, EEG, etc.), telemetry signals
(temperature inside a space craft, atmospheric pressure, etc.)
do exhibit sample-to-sample correlation .

Correlation  samples m[k] and m[k+1]
(or m[k-1] and m[k]) do not differ significantly.
Given a set of previous N samples m[k-1], m[k-2],…,m[k-N]
m[k] can be predicted with a small error.
Let denote the predicted value of m[k].
Then the error is:
Instead of transmitting m[k], d[k] is transmitted. Peak of d[k]
is significantly smaller than the peak of m[k]. Therefore the
parameter mp of the quantizer is reduced.
 
m̂ k
     
ˆ
d k m k m k
 

For a given L, the quantization interval Δv = 2mp/L is
reduced compared to PCM. Can be used in two ways.
(1) the quantization noise is reduced
 is increased.
(2) On the other hand for a given SNR, L can be reduced
compared to PCM
number of bits/sample, n=log2 L can be reduced.
Decoding
At the receiver the estimate is found using the past
samples and d[k] is added to calculate m[k].
2
2
2
0
3
12 L
m
v
N
p



 
2
2
0
2
0
3
p
m t
S
SNR L
N m
 
 
m̂ k

Deficiency: At the receiver, instead of the past samples
m[k-1], m[k-2],… we only have the quantized version
of the sequence mq[k-1], mq[k-2],… and the quantized
version of the difference dq[k]. Therefore cannot
be predicted. Only the estimate of the quantized sample
can be predicted
Solution: determine at the transmitter from
mq[k-1], mq[k-2],… and find .
Then transmit the quantized version  dq[k]=d[k]+q[k]
where q[k] is the quantization error.
     
ˆq
d k m k m k
 
 
m̂ k
 
ˆq
m k
 
ˆq
m k

     
     
   
ˆ
q q q
q
m k m k d k
m k d k d k
m k q k
 
  
 
Predictor input is indeed mq[k].
Receiver can find the quantized version
of m[k].
 
q
m k
 
ˆq
m k
(a) DPCM transmitter (b) DPCM receiver
m[k]
d[k] dq[k]
dq[k]
 
ˆq
m k
 
q
m k
 
q
m k
Similar to receiver
 
ˆq
m k  
q
m k
     
ˆq
d k m k m k
 
dq[k]=d[k]+q[k]

Delta Modulation (DM)
 Delta modulation is a predictive waveform coding technique and can
be considered as a special case of DPCM.
Uses the simplest possible quantizer, namely a two level (one bit)
quantizer.
The price paid for achieving the simplicity of the quantizer is the
increased sampling rate (much higher than the Nyquist rate).
Over-sampling is used in order to increase the adjacent sample
correlation.
like in DPCM, acts on the difference (prediction error) signals.

Waveforms illustrative of LDM operation
Linear (or non-adaptive) Delta Modulator (LDM)

Discrete-time LDM system (a) Transmitter (b) Receiver
Delta modulation is a special case of DPCM.
m[k]
d[k] dq[k]
 
1
q
m k 
 
q
m k
 
q
m k
 
1
q
m k 

63
Advantages of Digital Communication
(over analog communication)
1. More immunity to noise and distortion.
2. Possibility of regeneration. (repeaters)  avoid
accumulation of signal distortion along the transmission
path. (which is difficult to avoid in analog
Communication systems)
3. Digital hardware implementation is more flexible. 
use of DSP, micro-processors, mini-processor, VLSI etc.
4. Signal can be coded for protection against error and
piracy.
5. More flexibility in exchanging SNR and bandwidth.

Advantages of Digital Communication
(over analog communication)
5. Easier and more efficient in multiplexing several
signals.
6. Easier in storage and search.
7. Reproduction of digital signal is extremely reliable.
Analog signal is more and more deteriorated with
successive reproduction.
8. Cost/performance criteria will cause the digital
communication and data storage to dominate over
the analog systems. (due to the increasing
performance of digital circuits and its decreasing
cost)

Lecture_ch6.pptx

Recommended

Recommended

More Related Content

Similar to Lecture_ch6.pptx

Similar to Lecture_ch6.pptx (20)

Recently uploaded

Recently uploaded (20)

Lecture_ch6.pptx