L 1 5 sampling quantizing encoding pcm

sampling

1. Lecture 1.5.
Sampling. Quantizing. Encoding.

2. BASEBAND PULSE AND DIGITAL SIGNALING
•Analog-to-digital signaling (pulse code
modulation ) Binary and multilevel digitals
signals
•Spectra and bandwidths of digital signals
•Prevention of intersymbol interference

3. INTRODUCTION
We will study how to encode analog waveforms into base
band digital signals. Digital signal is popular because of
the low cost and flexibility.
Main goals:
•To study how analog waveforms can be converted to
digital waveforms, Pulse Code Modulation.
•To learn how to compute the spectrum for digital signals.
•Examine how the filtering of pulse signals affects our
ability to recover the digital information. Intersymbol
interference (ISI).

4. PULSE AMPLITUDE MODULATION
Pulse Amplitude Modulation (PAM) is used to describe the
conversion of the analog signal to a pulse-type signal in
which the amplitude of the pulse denotes the analog
information.
The purpose of PAM signaling is to provide another
waveform that looks like pulses, yet contains the information
that was present in the analog waveform.
There are two classes of PAM signals:
•PAM that uses Natural Sampling (gating);
•PAM that uses Instantaneous Sampling to produce a flat-
top pulse.

5. Natural Sampling (Gating)
DEFINTION: If w(t) is an analog waveform bandlimited to B hertz, the PAM
signal that uses natural sampling (gating) is
w
s
(t) =w(t)s(t) Where
S(t) is a rectangular wave switching waveform and f
s
= 1/T
s
≥ 2B.
THEOREM: The spectrum for a naturally sampled PAM signal is:
•Where f
s
= 1/T
s
,

ω
s
= 2π f
s
,
• the Duty Cycle of s(t) is d = τ/T
s
,
•W(f)= F[w(t)] is the spectrum of the original unsampled waveform,
•c
n
represents the Fourier series coefficients of the switching waveform.
sin( )
( ) F[ ( )] ( ) ( )
s s n s s
n n
nd
W f w t cW f nf d W f nf
nd
p
p
¥ ¥
=-¥ =-¥
= = - = -å å

6. Natural Sampling (Gating)
w(t)
w
s
(t) =w(t)s(t)
s(t)

7. Generating Natural Sampling
The PAM wave form with natural sampling can be generated using a
CMOS circuit consisting of a clock and analog switch as shown.

8. Spectrum of Natural Sampling
sin( )
( ) ( )

sin( )

s s
n
nd
W f d W f nf
nd
nd
d
nd
p
p
p
p
¥
=-¥
= -å
• The duty cycle of the switching
waveform is d = τ/T
s
= 1/3.
• The sampling rate is f
s
= 4B.
sin( )
( ) F[ ( )] ( ) ( )
s s n s s
n n
nd
W f w t cW f nf d W f nf
nd
p
p
¥ ¥
=-¥ =-¥
= = - = -å å

9. Recovering Naturally Sampled PAM
At the receiver, the original analog waveform, w(t), can be
recovered from the PAM signal, w
s
(t), by passing the PAM
signal through a low-pass filter where the cutoff frequency is:
B <f
cutoff
< f
s
-B
If the analog signal is under sampled f
s
< 2B, the effect of
spectral overlapping is called Aliasing. This results in a
recovered analog signal that is distorted compared to the
original waveform.
LPF Filter
B <f
cutoff
< f
s
-B

10. Demodulation of PAM Signal
 The analog waveform may be recovered from the PAM
signal by using product detection,

11. Instantaneous Sampling (Flat-Top PAM)
• This type of PAM signal
consists of instantaneous
samples.
• w(t) is sampled at t = kT
s
.
• The sample values w(kT
s
)
determine the amplitude of
the flat-top rectangular
pulses.

12. Instantaneous Sampling (Flat-Top PAM)
DEFINITION: If w(t) is an analog waveform bandlimited
to B Hertz, the instantaneous sampled PAM signal is
given by
–Where h(t) denotes the sampling-pulse shape and, for flat-top
sampling, the pulse shape is,
THEOREM: The spectrum for a flat-top PAM signal is:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
s s s s s s
k k k
w t w kT h t kT h t w kT t kT h t w t t kTd d
¥ ¥ ¥
=-¥ =-¥ =-¥
é ù
= - = * - = * -
ê ú
ë û
å å å
[ ]
1
( ) ( ) ( )
sin
( ) ( )
s s
ks
W f H f W f nf
T
f
H f h t
f
pt
t
pt
¥
=-¥
= -
æ ö
=Á =
ç ¸
è ø
å

13. The spectrum of the flat-top PAM
 Analog signal maybe recovered from the flat-top PAM signal by the use of a LPF.
LPF Response
Note that the recovered signal
has some distortions due to the
curvature of the H(f).
Distortions can be removed by
using a LPF having a response
1/H(f).

14. Some notes on PAM
•The flat-top PAM signal could be generated by using a
sample-and-hold type electronic circuit.
•There is some high frequency loss in the recovered analog
waveform due to filtering effect H(f) caused by the flat top
pulse shape.
•This can be compensated (Equalized) at the receiver by
making the transfer function of the LPF to 1/H(f)
•This is a very common practice called “EQUALIZATION”
•The pulse width τ is called the APERTURE since τ/T
s
determines the gain of the recovered analog signal
Disadvantages of PAM
•PAM requires a very larger bandwidth than that of the original signal;
•The noise performance of the PAM system is not satisfying.

15. Pulse Code Modulation
Pulse Code Modulation
Quantizing
Encoding
Analogue to Digital Conversion
Bandwidth of PCM Signals

16. PULSE CODE MODULATION (PCM)
DEFINITION: Pulse code modulation (PCM) is essentially
analog-to-digital conversion of a special type where the
information contained in the instantaneous samples of an
analog signal is represented by digital words in a serial bit
stream.
The advantages of PCM are:
•Relatively inexpensive digital circuitry may be used extensively.
•PCM signals derived from all types of analog sources may be
merged with data signals and transmitted over a common high-
speed digital communication system.
•In long-distance digital telephone systems requiring repeaters, a
clean PCM waveform can be regenerated at the output of each
repeater, where the input consists of a noisy PCM waveform.
•The noise performance of a digital system can be superior to that of
an analog system.
•The probability of error for the system output can be reduced even
further by the use of appropriate coding techniques.

17. Sampling, Quantizing, and Encoding
The PCM signal is generated by carrying out three basic
operations:
1.Sampling
2.Quantizing
3.Encoding
1.Sampling operation generates a flat-top PAM signal.
2.Quantizing operation approximates the analog values by
using a finite number of levels. This operation is considered in
3 steps
a)Uniform Quantizer
b)Quantization Error
c)Quantized PAM signal output
3.PCM signal is obtained from the quantized PAM signal by
encoding each quantized sample value into a digital word.

18. Analog to Digital Conversion
The Analog-to-digital Converter (ADC)
performs three functions:
–Sampling
•Makes the signal discrete in time.
•If the analog input has a
bandwidth of W Hz, then the
minimum sample frequency
such that the signal can be
reconstructed without distortion.
–Quantization
•Makes the signal discrete in
amplitude.
•Round off to one of q discrete
levels.
–Encode
•Maps the quantized values to
digital words that are n bits long.
If the (Nyquist) Sampling Theorem is
satisfied, then only quantization
introduces distortion to the system.
ADC
Sample
Quantize
Analog
Input
Signal
Encode
111
110
101
100
011
010
001
000
Digital Output
Signal
111 111 001 010 011 111 011

19. Quantization
The output of a sampler is still continuous in amplitude.
–Each sample can take on any value e.g. 3.752, 0.001, etc.
–The number of possible values is infinite.
To transmit as a digital signal we must restrict the number of
possible values.
Quantization is the process of “rounding off” a sample
according to some rule.
–E.g. suppose we must round to the nearest tenth, then:
3.752 --> 3.8 0.001 --> 0

20. Illustration of the Quantization Error

21. PCM TV transmission:
(a)5-bit resolution;
(a)8-bit resolution.

22. Uniform Quantization
•Most ADC’s use uniform
quantizers.
•The quantization levels
of a uniform quantizer
are equally spaced apart.
•Uniform quantizers are
optimal when the input
distribution is uniform.
When all values within
the Dynamic Range of
the quantizer are equally
likely.
Input sample X
Example: Uniform n =3 bit quantizer
q=8 and X
Q
= {±1,±3,±5,±7}
2 4 6 8
1
5
3
Output sample
X
Q
-2-4-6-8
Dynamic Range:
(-8, 8)
7
-7
-3
-5
-1
Quantization Characteristic

23. Quantization Example
Analogue signal
Sampling TIMING
Quantization levels.
Quantized to 5-levels
Quantization levels
Quantized 10-levels

24. PCM encoding example
Chart 1. Quantization and digitalization of a signal.
Signal is quantized in 11 time points & 8 quantization segments.
Chart 2. Process of restoring a signal.
PCM encoded signal in binary form:
101 111 110 001 010 100 111 100 011 010 101
Total of 33 bits were used to encode a signal
Table: Quantization levels with belonging code words
Levels are encoded
using this table
M=8

25. Encoding
•The output of the quantizer is one of M possible signal
levels.
–If we want to use a binary transmission system, then we need to
map each quantized sample into an n bit binary word.
•Encoding is the process of representing each quantized
sample by an n bit code word.
–The mapping is one-to-one so there is no distortion introduced by
encoding.
–Some mappings are better than others.
•A Gray code gives the best end-to-end performance.
•The weakness of Gray codes is poor performance when the
sign bit (MSB) is received in error.
2
2 , log ( )
n
M n M= =

26. Gray Codes
•With gray codes adjacent samples differ only in one bit
position.
•Example (3 bit quantization):
X
Q Natural coding Gray Coding
+7 111 110
+5 110 111
+3 101 101
+1 100 100
-1 011 000
-3 010 001
-5 001 011
-7 000 010
•With this gray code, a single bit error will result in an
amplitude error of only 2.
–Unless the MSB is in error.

27. Waveforms in a PCM system for M=8
M=8
(d) PCM Signal
(c) Error Signal
(b) Analog Signal, PAM Signal, Quantized PAM Signal
(a) Quantizer Input output characteristics
2
2 log ( )
isthenumberof Quantizationlevels
isthenumberof bitspersample
n
M n M
M
n
= =

28. PCM Transmission System

29. Practical PCM Circuits
•Three popular techniques are used to
implement the analog-to-digital converter
(ADC) encoding operation:
1.The counting or ramp, ( Maxim ICL7126 ADC)
2.Serial or successive approximation, (AD 570)
3.Parallel or flash encoders. ( CA3318)
•The objective of these circuits is to generate
the PCM word.
•Parallel digital output obtained (from one of the
above techniques) needs to be serialized
before sending over a 2-wire channel
•This is accomplished by parallel-to-serial
converters [Serial Input-Output (SIO) chip]
•UART,USRT and USART are examples for
SIO’s

30. Bandwidth of PCM Signals
•The spectrum of the PCM signal is not directly related to the spectrum
of the input signal.
•The bandwidth of (serial) binary PCM waveforms depends on the bit
rate R and the waveform pulse shape used to represent the data.
•The Bit Rate R is
R=nf
s
Where n is the number of bits in the PCM word (M=2
n
) and f
s
is the
sampling rate.
•For no aliasing case (f
s
≥ 2B), the MINIMUM Bandwidth of PCM B
pcm(Min)
is:
B
pcm(Min)
= R/2 = nf
s/
/2
The Minimum Bandwidth of nf
s/
/2 is obtained only when sin(x)/x pulse is
used to generate the PCM waveform.
•For PCM waveform generated by rectangular pulses, the First-null
Bandwidth is:

B
pcm
= R = nf
s

31. PCM Noise and Companding
Quantization Noise
Signal to Noise Ratio
PCM Telephone System
Nonuniform Quantization
Companding

32. The process of quantization can be interpreted as an
additive noise process.
•The signal to quantization noise ratio (SNR)
Q=S/N is given
as:
Quantization Noise
Signal
X
Quantized Signal
X
Q
Quantization Noise
n
Q
Average Power{ }
( )
Average Power{ }
Q
Q
X
SNR
n
=

33. Effects of Noise on PCM
Two main effects produce the noise or distortion in the PCM output:
–Quantizing noise that is caused by the M-step quantizer at the PCM transmitter.
–Bit errors in the recovered PCM signal, caused by channel noise and improper
filtering.
•If the input analog signal is band limited and sampled fast enough so that
the aliasing noise on the recovered signal is negligible, the ratio of the
recovered analog peak signal power to the total average noise power is:
•The ratio of the average signal power to the average noise power is
–M is the number of quantized levels used in the PCM system.
–P
e
is the probability of bit error in the recovered binary PCM signal at the receiver
DAC before it is converted back into an analog signal.

34. Effects of Quantizing Noise
•If P
e
is negligible, there are no bit errors resulting from channel noise and no ISI,
the Peak SNR resulting from only quantizing error is:
•The Average SNR due to quantizing errors is:
•Above equations can be expresses in decibels as,
Where, M = 2
n
α = 4.77 for peak SNR
α = 0 for average SNR

35. DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS
•Assume that an analog audio voice-frequency(VF) telephone signal occupies a
band from 300 to 3,400Hz. The signal is to be converted to a PCM signal for
transmission over a digital telephone system. The minimum sampling frequency is
2x3.4 = 6.8 ksample/sec.
•To be able to use of a low-cost low-pass antialiasing filter, the VF signal is
oversampled with a sampling frequency of 8ksamples/sec.
•This is the standard adopted by the Unites States telephone industry.
•Assume that each sample values is represented by 8 bits; then the bit rate of the
binary PCM signal is
• This 64-kbit/s signal is called a DS-0 signal (digital signal, type zero).
• The minimum absolute bandwidth of the binary PCM signal is
8
This B is for a sinx/x type pulse sampling
PCM
2 2
s
nfR
B³ =

36. DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS
•We require a bandwidth of 64kHz to transmit this digital voice PCM signal, whereas
the bandwidth of the original analog voice signal was, at most, 4kHz.
•We observe that the peak signal-to-quantizing noise power ratio is:
Note:
1.Coding with parity bits does NOT affect the quantizing noise,
2.However coding with parity bits will improve errors caused by
channel or ISI, which will be included in P
e
( assumed to be 0).
• If we use a rectangular pulse for sampling the first null bandwidth is given by

37. Nonuniform Quantization
Many signals such as speech have a nonuniform distribution.
–The amplitude is more likely to be close to zero than to be at higher levels.
Nonuniform quantizers have unequally spaced levels
–The spacing can be chosen to optimize the SNR for a particular type of signal.
2 4 6 8
2
4
6
-2
-4
-6
Input sample
X
Output sample
X
Q
-2-4-6-8
Example: Nonuniform 3 bit quantizer

38. Companding
•Nonuniform quantizers are difficult to make and
expensive.
•An alternative is to first pass the speech signal
through a nonlinearity before quantizing with a uniform
quantizer.
•The nonlinearity causes the signal amplitude to be
Compressed.
–The input to the quantizer will have a more uniform
distribution.
•At the receiver, the signal is Expanded by an inverse
to the nonlinearity.
•The process of compressing and expanding is called
Companding.

39. m-Law Companding
•Telephones in the U.S., Canada
and Japan use m-law
companding:
–Where m = 255 and |x(t)| < 1
ln(1 | ()|)
| ()|
ln(1 )
xt
y t
m
m
+
=
+
0 1
1
Input |x(t)|
O
u
t
p
u
t

|
x
(
t
)
|

40. Non Uniform quantizing
•Voice signals are more likely to have amplitudes near zero than at extreme
peaks.
•For such signals with non-uniform amplitude distribution quantizing noise will
be higher for amplitude values near zero.
•A technique to increase amplitudes near zero is called Companding.
Effect of non linear quantizing can be
obtained by first passing the analog
signal through a compressor and then
through a uniform quantizer.
x’
Q(
.
)
y
Uniform Quantizer
C(
.
)
x
Compressor
x’

41. 220023002400250026002700280029003000
-1
-0.5
0
0.5
1
220023002400250026002700280029003000
-1
-0.5
0
0.5
1
010002000300040005000600070008000900010000
-1
-0.5
0
0.5
1
010002000300040005000600070008000900010000
-1
-0.5
0
0.5
1
Example: m-law Companding
x[n]=speech /song/
y[n]=C(x[n])
Companded Signal
Segment of
x[n]
Segment of y[n]
Companded Signal
Close View of the Signal

42. A-law and m-law Companding
•These two are standard companding methods.
•u-Law is used in North America and Japan
•A-Law is used elsewhere to compress digital telephone signals

43. SNR of CompanderSNR of Compander
• The output SNR is a function of input signal level for uniform quantizing.
• But it is relatively insensitive for input level for a compander

44. SNR Performance of Compander
• The output SNR is a function of input signal level for uniform quantizing.
• But it is relatively insensitive for input level for a compander.
• α = 4.77 - 20 Log ( V/x
rms
) for Uniform Quantizer
V is the peak signal level and x
rms
is the rms value
• α = 4.77 - 20 log[Ln(1 + μ)] for μ-law companding
• α = 4.77 - 20 log[1 + Ln A] for A-law companding

45. V.90 56-Kbps PCM Computer modem
•The V.90 PC Modem transmits data at 56kb/s from
a PC via an analog signal on a dial-up telephone
line.
•A μ law compander is used in quantization with a
value for μ of 255.
•The modem clock is synchronized to the 8-
ksample/ sec clock of the telephone company.
•7 bits of the 8 bit PCM are used to get a data rate
of 56kb/s ( Frequencies below 300Hz are omitted
to get rid of the power line noise in harmonics of
60Hz).
•SNR of the line should be at least 52dB to operate
on 56kbps.
•If SNR is below 52dB the modem will fallback to
lower speeds ( 33.3 kbps, 28.8kbps or 24kbps).

46. END