UNIT-1.pdf

UNIT-I
PULSE DIGITAL MODULATION TECHNIQUES
 Elements Of Digital Communication System
 Comparison of Digital and Analog Communication Systems
 Waveform Coding: Analog to Digital Conversion
 Quantization and Encoding techniques
 PCM
 Companding in PCM systems – μ-law and A-law
 Applications of PCM
 Introduction to Linear Prediction Theory
 Modulation and demodulation of DPCM, DM and ADM
 Comparison of PCM, DPCM, DM and ADM
 SNRQ of PCM and DM

UNIT-1
PULSE DIGITAL MODULATION TECHNIQUES
INTRODUCTION
In digital communication system, the information bearing digital signal is processed such
that it can be represented by a sequence of binary digits (discrete messages). Then it is used for
ON/OFF keying of some characteristic of a high frequency sinusoidal carrier wave, such as
amplitude, phase or frequency. If the input message signal is in analog form, then it is converted
to digital form by the processes of sampling, quantizing and encoding. Computer data and
telegraph signals are some examples of digital signal. The key feature of a digital communication
system is that it deals with a finite set of discrete messages.
Digital communication systems are becoming increasingly attractive due to the ever-
growing demand for data communication. Because digital transmission offers data processing
options and flexibilities not available with analog transmission. Further, developments in digital
techniques have led to more and more powerful microprocessors, larger and larger memory
devices and a number of programmable logic devices. Availability of these devices has made the
design of digital communication systems highly convenient. The word digital comes from the
Latin word digit and digitus (the Latin word for finger), as fingers.
What is Digital Communication?
Digital communication is a mode of communication where the information or the thought
is encoded digitally as discrete signals and electronically transferred to the recipients.
Examples of Digital Communication
Manager wanted to meet all his team members at the Conference room to discuss their key
responsibility areas and areas of expertise. He didn’t have the time to go to their workstations and
invite them individually. Instead he opted an easier and cheaper mode to communicate his idea.
He sent an email marking a cc to all the participants, inviting them for the meeting. This is an
example of Digital communication where the information was sent electronically.
Digital communication covers a broad area of communications techniques including:
• Digital transmission is the transmission of digital pulses between two or more points in a
Communication system.
• Digital radio is the transmitted of digital modulated analog carriers between two or more points
in a communication system.
Basic Digital Communication Nomenclature
 Textual Message: information comprised of a sequence of characters.
 Binary Digit (Bit): the fundamental information unit for all digital systems.
 Symbol (mi where i =1, 2,…M): for transmission of the bit stream; groups of k bits are
combined to form new symbol from a finite set of M such symbols; M=2k
.
 Digital Waveform: voltage or current waveform representing a digital symbol.
 Data Rate: Symbol transmission is associated with a symbol duration T. Data rate R=k/T
[bps].
 Baud Rate: number of symbols transmitted per second [baud]

ELEMENTS OF DIGITAL COMMUNICATION SYSTEM
The block diagram of a typical digital communication system with only the
essential blocks is shown in the Figure 1.1.
The functions of encryption, multiplexing, spreading, multiple access and equalization are
optional. The upper blocks- Formatter, Source encoder, channel encoder, Baseband
processor/ Band pass modulator- denote signal transformations from the source to
the transmitter. The lower blocks-Baseband decoder/Band pass demodulator,
channel decoder, source decoder, Deformatter – denote signal transformations from
the receiver to the sink. The lower blocks essentially reverse the signal processing
steps performed by the upper blocks.
We shall discuss the basic functions of each of these blocks.
Fig 1.1 Block Diagram of Typical Digital Communication System
TRANSMITTER SECTION
1) Information source
The Source is where the information to be transmitted, originates. The information /
message may be available in digital form (eg: computer data, tele-type data). If the information /
message available is a non-electrical signal, (eg: video signal, voice signal) then it is first converted
into a suitable electrical signal using an input transducer. Then the analog electrical signal is
sampled and digitized using an analog to digital converter to make the final source output to be in
digital form.
2) Formatter
Formatting transforms the source information into binary digits (bits). The bits are then
grouped to form digital messages or message symbols. Each such symbol (mi, where i =
1,2,3……M) can be regarded as a member of a finite alphabet set containing M members. Thus
for M=2, the message symbol mi is binary (it constitutes just a single bit). For M>2, such symbols
are each made up of a sequence of two or more bits (M-ary)
3) Source encoder

The process of efficiently converting the output of either an analog or digital source into a
sequence of binary digits is called source encoding or data compression. Source coding produces
analog-to-digital (A/D) conversion for analog sources. It also removes redundant (unneeded)
information. By reducing data redundancy, source codes can reduce a system’s data rate (i.e.,
reduced bandwidth).Formatting and source coding are similar processes, in that they both involve
data digitization. However, source coding involves data compression in addition to digitization.
Hence, a typical digital communication system would either use formatter, (for digitizing alone)
or source encoder (for both digitizing and compressing).
4) Channel encoder
The channel encoder introduces some redundancy in the binary information sequence, in a
controlled manner. Such introduction of controlled redundancy can be used at the receiver to
provide error correction capability to the data being transmitted. This minimizes the effects of
noise and interference encountered in the transmission of the signal through the channel. Hence
channel coding increases the reliability of the received data and improves the fidelity of the
received signal. Channel coding is used for reliable transmission of digital data
5) Base band processor
For low speed wired transmission, each symbol to be transmitted is transformed from a
binary representation (voltage levels representing binary ones and zeros) to a baseband waveform.
The baseband refers to a signal whose frequency range extends from DC up to a few MHz. The
baseband processor is a pulse modulation circuit. When pulse modulation is applied to binary
symbols, the resulting binary waveform is called Pulse Code-Modulation (PCM) waveform. In
telephone applications, the PCM waveforms are often called as Line codes. After pulse
modulation, each message symbol takes the form of a baseband waveform, gi(t), where i=1,2…M.
6) Band pass Modulator
For transmission of high speed digital data (e.g. Computer communication systems), the
digital signal needs to be modulated. The primary purpose of the digital modulator is to map the
binary information sequence into high frequency analog signal waveforms (carrier signals).The
term band pass is used to indicate that the baseband waveform gi(t) is frequency translated by a
carrier wave to a frequency that is much larger than the spectral content of gi(t). The digitally
modulated signal is a band pass waveform Si(t), where i=1,2,…..M. The digital modulator may
simply map the binary digit 0 into a waveform S1(t) and the binary digit 1 into a waveform S2(t).
We call this as binary modulation (M=2).
Alternatively, the modulator may transmit K coded information bits at a time by using
M=2K distinct waveforms Si(t), i=1,2,……..M, one waveform for each of the 2Kpossible bit
sequences. We call this as M-ary modulation (M>2). The band pass modulator is used for efficient
transmission of digital data. The baseband processor block is not required, if the band pass
modulator block is present. Therefore, these two blocks are shown as mutually exclusive blocks.
CHANNEL
The communication channel is the physical medium that is used to send the signal from
the transmitter to the receiver. In wireless transmission, the channel maybe the atmosphere (free
space). On the other hand, telephone channels usually employ a variety of physical media,
including wirelines, optical fiber cables, and wireless (microwave radio).The transmitted signal is

corrupted in a random manner by a variety of possible mechanisms, such as additive thermal noise
generated by electronic devices, man-made noise, e.g., automobile ignition noise and atmospheric
noise, eg., electrical lightning discharges during thunderstorms.
As the transmitted signal Si(t) propagates over the channel, it is impacted by the channel
characteristics, which can be described in terms of the channel’ impulse response hc(t). Also, at
various points along the signal route, additive random noise n(t) distorts the signal. Hence the
received signal x(t) must be termed as the corrupted version of the transmitted signal Si(t). The
received signal x(t) can be expressed as
x(t)= Si(t) hc(t) + n(t) i=1,2…..M,
Where * represents a convolution operation and n(t) represents a noise process.
RECEIVER SECTION
1) Baseband decoder
The baseband decoder block converts back the line coded pulse waveform to transmitted
data sequence.
2) Band pass demodulator
The receiver front end and/or the demodulator provides frequency down conversion for
each of the received band pass waveform x(t). Digital demodulation is defined as recovery of a
waveform (base band pulse). The demodulator restores x(t) to an optimally shaped baseband pulse
z(t) in preparation for detection. Detection is defined as decision-making regarding the digital
meaning of that waveform.
Typically there are several filter are associated with receiver and demodulator
(i) Filtering to remove unwanted high frequency terms(in frequency down conversion
of band pass waveforms)
(ii) Filtering of pulse shaping.
(iii) Filtering option by equalization to reverse any degrading effects on the signal
caused by the poor impulse response of the channel.
Finally the detector transforms the shaped pulse to an estimate of the transmitted data
symbols (binary or M-ary).Demodulator is typically accomplished with the aid of reference
waveforms. When the reference used is a measure of the entire signal attributes (particularly
phase), the process is termed coherent. When phase information is not used, the process is termed
non-coherent.
3) Channel decoder
The estimates of the transmitted data symbols are passed to the channel decoder. The
channel decoder attempts to reconstruct the original information sequence from knowledge of the
code used by the channel encoder and the redundancy contained in the received data. A measure
of how well the demodulator and decoder perform is the frequency with which errors occur in the
decoded sequence. This is the important measure of system performance called as Probability of
bit error (Pe).
4) Source decoder
The source decoder accepts the output sequence from the channel decoder. From the
knowledge of the source encoding method used, it attempts to reconstruct the original signal from
the source. Because of channel decoding errors and possible distortion introduced by the source

decoder, the signal at the output of the source decoder is an approximation to the original source
output. The difference of this estimate and the original digital signal is the distortion introduced
by the digital communication system.
5) Deformatter
If the original information source was not in digital data form and the output of the receiver
needs to be in the original form of information, a deformatter block is needed. It converts back the
digital data to either discrete form (like keyboard characters) or analog form (speech signal).
6) Information sink
If an analog output is needed in non-electrical form, the output transducer converts the
estimate of digital signal to the required analog output. The information sink may be computer,
data terminal equipment or a user.
7) Synchronization
Synchronization and its key element, a clock signal, is involved in the control of all signal
processing within the digital communication system. It actually plays a role in regulating the
operation of almost every block. Synchronization involves the estimation of both time and
frequency. Coherent systems need to synchronize their frequency reference with the carrier in both
frequency and Phase. For non-coherent systems, phase synchronization is not needed.
BASIC DIGITAL COMMUNICATION TRANSFORMATIONS
The basic signal processing functions which may be viewed as transformations can be
classified into the following nine groups as shown in Figure 1.2.
1. Formatting and Source Coding:
Formatting and source coding are similar processes, in that they both involve data
digitization. Source coding also involves data compression in addition to digitization.
2. Baseband Signaling
Baseband signaling process involves generation of PCM waveforms or line codes.
3. Bandpass signaling
During demodulation, when the references used are a measure of all the signal attributes
(particularly phase), the process is termed coherent. When phase information is not used, the
process is termed non coherent.
4. Equalization
An equalization filter is needed for those systems where channel induced ISI (Inter symbol
interference) can distort the signals.
5. Channel Coding
Waveform coding and structured sequences are the two methods of channel coding.
Waveform coding involves the use of new waveforms. Structured sequences involve the use of
redundant bits.
6. Multiplexing and multiple access
Multiplexing and multiple access both involve the idea of resource sharing. Multiplexing
takes place locally and multiple access takes place remotely.

7. Spreading
Spreading is used in military applications for achieving interference protection and privacy.
Signals can be spread in frequency, in time, or in both frequency and time.
8. Encryption
Encryption and decryption are the basic goals, which are communication privacy and
authentication. Maintaining privacy means preventing unauthorized persons from extracting
information (eavesdropping) from the channel. Establishing authentication means preventing
unauthorized persons from injecting spurious signals (spoofing) into the channel.
9. Synchronization
Synchronization involves the estimation of both time and frequency. Coherent systems
need to synchronize their frequency reference with the carrier in both frequency and phase. For
non-coherent systems, phase synchronization is not needed.
Fig 1.2 Basic digital Communication Transformations
Performance criteria
A digital communication system transmits signals that represent digits. These digits form
a finite set or alphabet, and the set is known a priori to the receiver. A figure of merit for digital
communication systems is the probability of incorrectly detecting a digit or the probability of error
(Pe).

CHANNELS FOR DIGITAL COMMUNICATION
The transmission of information across a communication network is accomplished in the
physical layer by means of a communication channel. One common problem in signal transmission
through any channel is additive noise. Other types of signal degradation that may occur over the
channel are signal attenuation, amplitude and phase distortion, and multipath distortion.
The two channel characteristics, Power and Bandwidth, constitute the primary
communication resources available to the designer. The effects of noise may be minimized by
increasing the power in the transmitted signal. However, equipment and other practical constraints
limit the power level in the transmitted signal. There is limitation for available channel bandwidth
also. This is due to the physical limitations of the medium and the electronic components used to
implement the transmitter and receiver. These two limitations constrain the amount of data that
can be transmitted reliably over any communication channel.
Depending on the mode of transmission used, we may distinguish two basic groups of
communication channels.
I. Channels based on guided propagation - Telephone channels, coaxial cables, and
optical fibers.
II. Channels based on free space (unguided) propagation – wireless broadcast
channels, mobile radio channels, and satellite channels.
Here, we describe some of the important characteristics of three channels used for digital
communication – Telephone, optical fiber and satellite channels.
Telephone channel
The telephone network makes extensive use of wirelines for voice signal transmission, as
well as voice and data transmission. Earlier the telephone channel is built using twisted wire pairs
for signal transmission. Twisted pair wirelines are basically guided electromagnetic channels that
provide relatively modest bandwidths. Now, Telephone channel has evolved into a worldwide
network that encompasses a variety of transmission media (open-wire lines, coaxial cables, optical
fibers, microwave radio, and satellites) and a complex of switching systems. This makes the
telephone channel an excellent choice for data communication over long distances.
The telephone channel has a bandpass characteristic occupying the frequency range 300Hz
to 3400Hz. It has a high signal-to-noise ratio of about 30dB, and approximately linear response.
The channel has a flat amplitude response over the pass-band. But data and image transmissions
are strongly influenced by phase delay variations. Hence an equalizer is designed to maintain a flat
amplitude response and a linear phase response. Transmission rates up to 16.8 kilobits per seconds
(kbps) have been achieved over telephone lines. Telephone channels are naturally susceptible to
electromagnetic interference (EMI), the effects of which are mitigated through twisting the wires.
Optical Fiber channel
An optical fiber is a dielectric waveguide that transports light signals from one place to
another. It consists of a central core within which the light signal is confined. It is surrounded by
a cladding layer having a refractive index slightly lower than the core. The core and cladding are
both made of pure silica glass. Optical fiber communication system uses a light source (LED or

Laser) as the transmitter or modulator. At the receiver, the light intensity is detected by a
photodiode, whose output is an electrical signal. Sources of noise in fibre optical channels are
photodiodes and electronic amplifiers. Optical fibres have unique characteristics that make them
highly attractive as a transmission medium. They are
• Enormous potential bandwidth, resulting from the use of optical carrier frequencies
around 2 x 1014
Hz.
• Low transmission losses, as low as 0.1dB/km.
• Immunity to electromagnetic interference and hence no cross talk.
• Small size and weight.
• Highly reliable photonic devices available for signal generation and signal detection.
• Ruggedness and flexibility.
These unique characteristics have resulted in a rapid deployment of optical fibre channels for
telecommunication services including voice, data, facsimile and video.
Satellite channel
A satellite channel consists of a satellite in geostationary orbit, an uplink from a ground
station, and a downlink to another ground station. Typically, the uplink and the downlink operate
at microwave frequencies with the uplink frequency higher than the downlink frequency. The most
popular frequency band for satellite communications is 6GHz for the uplink and 4GHz for the
downlink. Another popular band is 14/12GHz. On board the satellite there is a low-power
amplifier, which is usually operated in its non-linear mode for high efficiency.
Thus, the satellite channel may be viewed as a powerful repeater in the sky. It permits
communication (from one ground station to another) over long distances at high bandwidths and
relatively low cost. The non-linear nature of the channel restricts its use to constant envelope
modulation techniques (i.e, Phase modulation, frequency modulation).
Communications satellites in geostationary orbit offer the following unique system capabilities:
• Broad area coverage.
• Reliable transmission links
• Wide transmission bandwidths.
In the 6/4GHz band, a typical satellite is assigned a 500MHz bandwidth. This bandwidth is divided
among 12 transponders on board the satellite. Each transponder, using approximately 36MHz of
the satellite bandwidth, can carry at least one color TV signal, 1200 voice circuits, or digital data
at a rate of 50Mbps.
We may classify communication channels in different ways.
1. A channel may be Linear (e.g., wireless radio channel) or non-linear (e.g., satellite
channel).
2. A channel may be time invariant (e.g. Optical Fiber Channel) or time varying (e.g.,
mobile radio channel).
3. A channel may be bandwidth limited (e.g., telephone channel) or power limited
(e.g., optical fiber channel and satellite channel).

Comparison of Digital and Analog Communication Systems:
Digital communications have several advantages over analog communication.
(a) Complexity: Digital systems are difficult to conceptualize but are simpler and easy to
build, whereas Analog systems are easy in conceptual terms but are complex and difficult
to build.
(b) Cost: improvement in VLSI technology and availability of ICs made the cost of digital
systems low when compared to analog systems.
(c) Robustness: components and subsystems used in digital systems are more robust because
they are insensitive to variations in atmospheric conditions and are not effected by
mechanical vibrations. The problem of ageing of components and subsystems does not
arise as it does in analog systems.
(d) Storage and Retrieval: storage and retrieval of voice, data or video at intermediate points
is easy and is inexpensive in terms of storage space.
(e) Flexibility: Digital communication offers considerable flexibility.
(i) Data, voice and video can be multiplexed using TDM and transmitted over the same
channel.
(ii) Signal processing and image processing operations like compression of voice and
image signals etc. can easily be carried out.
(iii) Digital modulating waveforms, for channels with known characteristics can be
chosen so as to make the system more tolerant to channel impairments.
(iv) Adaptive equalization can be implemented.
(f) Effect of Noise and Interference: error-correction codes used in channel coding ensure
fairly good protection against noise and interference in case of digital systems.
(g) Long-haul Communication using a number of Repeaters:
In analog systems, the repeaters at each stage will amplify the signal and noise an
add some more noise. Because of this SNR becomes lower and lower at each stage. When
the signal reaches destination after going through a large number of repeaters it is
dominated by noise.
In digital systems, we use ‘regenerative repeaters’ which consists of a receiver and
a transmitter connected back to back. So the received signal is decoded to a baseband level
at each repeater with practically no errors and this baseband signal is then used by the
transmitter part of the repeater to transmit a strong error-free signal at good power level.
(h) Secrecy of Communication: very powerful encryption and decryption algorithms are used
to maintain high level of secrecy of communication. Spread-spectrum techniques of
transmission if used will further enhance the secrecy and protect against eavesdropping or
jamming by the enemy.
Digital communication systems suffer from the following disadvantages:
(a) Digital Communication Systems generally need more bandwidth than analog systems.
(b) Digital components generally consume more power as compared to analog components.

Waveform Coding: Analog to Digital Conversion
Digital transmission refers to transmission of digital signals between two or more points in
a communication system. If the original signal is in analog form then it needs to be converted
to digital pulses prior to transmission and converted back to analog signals in the receiver. The
conversion of analog signal to digital pulses is known as waveform coding. The digitized
signals may be in the form of binary or any other form digital transmission of discrete level
digital pulses.
Analog information sources can be transformed into digital sources through
the use of sampling, quantization and encoding as shown in Figure 1.3. Sampling converts a
continues time signal to discrete time signal, quantization mechanism converts discrete time
continuous amplitude signal to discrete time discrete amplitude signal and then a suitable
binary symbols are encoded and transferred for communicating , called A/D conversion.
Fig 1.3 Analog to Digital (A/D) conversion Transformation
Sampling
Sampling of the signals is the fundamental operation in signal-processing. A continuous-
time signal is first converted to discrete-time signal by sampling process. Sampling theorem gives
the complete idea about the sampling of signals. The output of the sampling process is called pulse
amplitude modulation (PAM).Because the successive output intervals can be described as a
sequence of pulses with amplitudes derived from the input waveform samples. The analog
waveform can be approximately retrieved from a PAM waveform by simple low-pass filtering.
What is sampling?
The process of converting continuous time –continuous amplitude signals into equivalent
discrete time-continuous amplitude signals, can be termed as sampling.

Sampling Theorem: The statement of sampling theorem can be given in two parts as below
(i) A band limited signal of finite energy, which has no frequency components higher than
fm Hertz, is completely described by its sample values at uniform intervals less than or
equal to 1
2 m
f
seconds apart.
(ii) A band limited signal of finite energy, which has no frequency components higher than
fm Hertz, may be completely recovered from the knowledge of its samples taken at the
rate of 2fm samples per second.
Combining the two parts, the uniform sampling theorem may be stated as follows:
“A continuous-time signal may be completely represented in its samples and recovered back if the
sampling frequency is 2
s m
f f
 .
When the sampling rate becomes exactly equal to 2fm samples per second, then it is called
as Nyquist rate. Nyquist rate is also defined as the minimum sampling rate required for sampling.
It is given by 2
s m
f f
 .
The maximum sampling interval called Nyquist Interval,
1 1
sec
2
s
s m
T
f f
  .
The restriction of fs ≥ 2fm, stated in terms of the sampling rate, is known as
the Nyquist criterion. The Nyquist criterion is a theoretically sufficient
condition to allow an analog signal to be reconstructed completely from a
set of uniformly spaced discrete time samples.
The effect of various Nyquist criteria is shown in Figure 1.4
Fig 1.4 Spectrum of x(t) with different Nyquist Criteria’s
Aliasing effect: If sampling is done less than the Nyquist rate, i.e., 2
s m
f f
 then high frequency
components of x(t) re-appear as low frequency components (spectral overlapping). This
phenomenon is referred as ‘aliasing’ also called ‘frequency folding effect’.

Low pass sampling Theorem: let x(t) be a bandlimited low pas signal, bandlimited to W Hz; i.e.,
X(f)=0 for f W
 . Then it is possible to recover x(t) completely, without any distortion
whatsoever from its samples , if the sampling interval
1
2
s
T
W
 . Specifically, x(t) can be expressed
in terms of its samples ( )
s
x kT as follows:
, ( )
s
where B is any frequency such that W B f W
   .
The sampling of a continuous-time signal is done in several ways. Basically, there are three types
of sampling techniques. They are:
Impulse sampling/Ideal Sampling /Instantaneous sampling: In this the sampling function ( )
x t

is a train of unit impulses and is given by ( ) (t )
s
n
x t nT
 


 
 The spectrum of the sampled version
of the signal is (f) X(f )
s s
n
X f nf



 
 ,shown in Figure 1.5.
Fig 1.5 Ideal sampling phenomenon in time and frequency domains
Natural sampling: In this, the sampling function is a sequence of rectangular narrow pulses, i.e
t
( ) s
n
kT
s t




 
  
 
 and the sampling process is modelled as the product of w(t) and s(t).
( ) 2 ( )sin 2 ( )
s s s
k
x t BT x kT c B t kT


 


Signal w(t) can be recovered without distortion from its samples, given as by below equation and
is shown in Figure 1.6.
Fig 1.6 PAM Natural sampling and its spectrum
Flat-top Sampling: in this sampling of analog signal x(t) is done at s
nT and holding the result
until the tome of next sample ( 1) s
n T
 . The Sample-and-Hold circuit which produces flat-top
samples is shown in Figure 1.7.
Fig 1.7 (a) S/H circuit for Flat-Top sampling (b) signal x(t) and S/H output
,where
t / 2
( )
p t



 
 
 
Mathematically,
sin
( ) ( ) ,
s s
n s
nd
W f d W f nf where d
nd T
 



  

( ) ( ) ( )
s s s
n
x t x kT p t kT


 


Quantization
The next step in the digitization of an analog signal is the discretization of the amplitudes of the
samples obtained through the sampling process.
Def: It is the process of assigning to each one of the sample values of the message signal, a discrete
value from a prescribed set of finite number of such discrete values called ‘Quantum Value’.
The total dynamic range of the analog signal is
divided into equal number of finite number of
levels or segments. We round off a sample
value falling within a particular segment to the
value represented by the prescribed level
passing through the middle of the level. This
process is called Quantization as shown in
Figure 1.8(a). The characteristic function of
quantization process is ‘staircase approach’
shown in Figure 1.8(b). It is a memory –less
process and is irreversible, such that the
rounding off the sample solely depends upon
the actual values only.
Fig 1.8 Quantization Process
Step size,
max min
( ) 2
p p p
V V V
V V
Q Q Q
 

    ,
where , Q is called quantum levels, 2n
Q  and
n-is the number of bits.
Fig 1.8 (b) Input-output characteristics
For any given dynamic range of the analog signal, if the number of the ‘prescribed levels’,
or the ‘quantized levels’, is increased, the interval between two successive levels, called the ‘step
size’, becomes smaller and so the error due to quantization, which can be at the most ±0.5 (step
size), also becomes smaller.
2
2
p
n
V
 

Fig 1.9 Illustration of Quantization Process with example
Quantizers: Quantization can be performed by feeding the samples of the analog signal to a ‘
quantizer’, which transforms each of the samples fed to it into a ‘quantized sample’ having an
amplitude corresponding to the ‘prescribed level’ used for representing any sample value falling
within the pertinent interval (step) into which that analog signal sample falls. This is shown
diagrammatically in Figures. 1.10 (a) and (b).
Fig 1.10 (a) & (b) Action of Quantizers
A quantization process in which the quantization levels are uniformly spaced is called
‘Uniform Quantization’, and a quantizer which performs uniform quantization is called a
‘Uniform Quantizer’. A quantization process in which the quantization levels are not uniformly
spaced, is called a ‘Non Uniform Quantization’, and the quantizer performing the non-uniform
quantization, is called a ‘Non-Uniform Quantizer’.

Uniform Quantization: These are of two types depending on the shape of the input-output
characteristics: (1) Mid-Tread (2) Mid-Rise
Fig1.11 (a) Transfer characteristics of Mid-Tread (b) Transfer characteristics of Mid-Tread
The only difference is that in the mid-tread characteristic, the origin is in the ‘tread’ portion
and that is why it is called mid-tread; whereas for the other one, the origin is in the ‘rise’ portion
of the staircase type of characteristic and so it is known as the mid-rise type of quantizer.
In Mid-tread, if any sampled value falls between
2

 , then the corresponding output is
Zero, for 3
2 2
to 

  output is  and so on.
In the Mid-rise, if any sampled value is Zero then corresponding output is either
2 2
or
 
  , for 0to , output is
2

 and 0
to
 , output is
2

 and so on.
Quantization Noise
The difference between the actual sampled values to the approximated quantized value is called
quantization error.
 
value
actual

 
value
quantized

Let us assume that the message signal is x(t) is zero mean Random Process has a sampled value
( )
s
x nT fed to the quantizer and if ( )
q s
x nT represents the quantized value and above expression
gives the quantization error.
( ) ( )
e s q s
Q x nT x nT
 

Since the maximum quantization
error should not exceed
2


i.e
2 2
e
Q
 
    . Since e
Q is a
random variable which is
uniformly distributed over the
interval
2

 , then its probability
density function is given by
Fig.1.12(b)
These random errors caused by
quantization in the successive
samples, appear as noise, called
the quantization noise.
Fig 1.12(a) Illustration of Quantization Error
Mean square value of the error = Avg.Power in quantization noise.
2
2 2
2
( )d
e e e e
Q Q f Q Q




 
2
2
2
1
d
e e
Q Q




 


Fig 1.12 (b) pdf of quantization noise
3 2
2
1
3
e
Q




 
  
  
Let the sampled value of the signal x(t) is assumed to lie in range (-Vp ,Vp ), then the step size of
uniform quantizer is
2
2
p
n
V
 
Hence,
2 2
1
2
3
n
q p
N V 

2
12
q
N

 

Let P denote the average power of the message signal x(t), then the output SNR of uniform
quantizer is
The above expression says, the SNR of quantizer increases exponentially with increasing the
number of bits per sample.
Now , if we assume that input x(t) is normalized , i.e VP=1
Then ,
2
3 2 n
S P
N
  
Also if the destination signal power ‘P’ is normalized i.e P≤1watt.
Then ,
2
3 2 n
S
N
 
Thus , the signal to quantization noise ratio for normalized values of power ‘P’ and amplitudes of
input x(t) is,
 
2
10
10log 3 2 n
q dB
S
N
 
 
 
 
2
10 10
10log (3) 10log (2 )
n
 
For Normalized input and power,
Non-Uniform Quantization:
For a typical voice signal, smaller amplitudes, such as silent periods, low voices, and weak
talk, predominate in speech (i.e., there is a higher probability they will occur), whereas larger
amplitudes, such as screams and loud voices, are relatively rare.
Uniform quantization is thus wasteful for speech signals, as many
of the quantizing levels are hardly ever used. An efficient scheme
for such signals is to employ a non-uniform quantizer using a
variable step size, where small step sizes are used for small
amplitudes having transfer characteristic is shown in Figure 1.13.
A non-uniform quantization practically gives a SNR ratio that
remains essentially constant for wide r range of input voltage
levels. A non-uniform quantizer is called Robust Quantizer
Fig1.13(a)Transfer characteristics
2
2
3
[ ] 2 n
o
p
P
SNR
V
 
  
 
 
 
4.77 6.02
q dB
S n dB
N
 
  
 
 

Companding: In this process a low amplitude signals are expanded and high amplitude signals
are compressed at the transmitter & at receiver reverse operation is done to recover the original
signal.
“The process of compressing message signals like speech at the transmitter and expanding them
at receiver is called Companding”
Fig 1.13 (b) Companding Process
Non-uniform quantization is equivalent to first subjecting the samples of the message speech signal
to amplitude compression by passing them through a compressor and then applying uniform
quantization to these compressed samples. Compression of the input samples is accomplished
according to a specific law governing the relationship between amplitudes of the input and output
samples. There are two different compression laws in vogue and they are:
(i) μ-law Companding: In North America and Japan , μ-law Companding is used. The
compression characteristics is
where
1 for x 0
sgn
1 for x<0
x
 

 


where μ is positive constant x and y represent input and
output voltages, and Xmax and Ymax are the maximum
positive excursions of the input and output voltages,
respectively.
Fig 1.14 μ-law characteristics
It may be noted that when μ = 0, it corresponds to uniform quantization and that compression of
the larger sample values is higher for larger values of μ. The most recent digital transmission
systems used 8-bit PCM code and μ=255. When μ-law compander is used, the output SNR is
 
 
2
2
3
ln(1 )
o
Q
SNR



,Q-number of quantum levels
 
max
max
ln 1
sgn
ln 1
x
x
y y x


 
 

 
 
 
 



(ii) A-law Companding: European countries and India prefer A-law companding to
approximate true logarithmic Companding. The compression characteristic for A-law
comapnding is
Fig 1.15 A-law characteristics
Clearly, A = 1 corresponds to uniform quantization. Larger the value of the constant A, more is
the compression. The standard value of A used in digital telephony is A=87.6.
Fig 1.16 Compressor and expander characteristics for
both positive and negative values of input
Line Coding: It is the process by which digital symbols are transformed into wave forms
(shaped pulses in case of PCM, DM or ADM) that are compatible with the characteristics of
baseband channel. Classification of various Line coding techniques is shown in Fig 1.17
max
max
max
max
max
max
1
sgn 0
1 lnA
1 ln
1
sgn 1
1 lnA
x
A
x x
y x
x A
y
x
A
x x
y x
x
A
  
 
  
 
 
 
  
 
 
 
 
   
 
  
 
  
 

Fig 1.17 Line Encoding Techniques
Unipolar Non-Return-to-Zero (UP-NRZ): In this binary data 1 is represented by a positive or
negative voltage level and binary data 0 is represented by ground voltage level or vice versa.
Mathematically,
'0' v(t) 0 0 t T
For binary '1' v(t) Vor V 0 t T
b
b
For binary   
    
Unipolar Return-to-Zero (UP-RZ): In this
binary data 1 is represented by a half-bit
wide pulse and binary data 0 is represented
by absence of pulse for whole bit interval.
Mathematically,
'0' v(t) 0 0 t T
T
V 0 t
2
For binary '1' v(t)
T
0 t
2
b
b
b
b
For binary
T
  
  
    
  
 
  


Polar NRZ-L (Level): In this binary data 1 is represented by one voltage level and binary data 0
is represented by another voltage level.
Mathematically,
'0' v(t) 0 t T
'1' v(t) 0 t T
b
b
For binary V
For binary V
   
   
Polar NRZ-M (Mark): In this binary data
1(mark) is represented by change in
voltage level from its previously held
voltage level and binary data 0 is
represented by no change in voltage level from its previous one.
Polar NRZ-S (Space): In this binary data 1 is represented by no change in voltage level from its
previously held voltage level and binary data 0(space) is represented by change in voltage level
from its previous one.
Bi-Phase line encoding is a encoding that uses one cycle of a pulse at 0o
phase to present a binary
data 1 and another cycle of pulse 180o
phase to represent binary data 0. These are called
Manchester Encoding techniques. In this transition occurs in middle of bit.
Bi-phase level (Bi-ϕ-L): Mathematical Expressions,
T
V/ 2 0 t
2
For binary '1' v(t)
T
V/ 2 t
2
b
b
b
T
  

 
  

T
V/ 2 0 t
2
For binary '0' v(t)
T
V/ 2 t
2
b
b
b
T
   

 
  


Bi-phase Mark (Bi-ϕ-M): In this a binary data 1 is represented by second transition which takes
place at one-half-bit-wide pulse and binary data 0 is represented by no second transition.
Bi-phase Space (Bi-ϕ-S): In this a binary data 1 is represented by no second transition and binary
data 0 is represented by second transition which takes place at one-half-bit-wide pulse.
Differential Manchester Encoding: In this a bit (1or 0) representation is defined by transition for
binary 0 and no transition for binary 1 at the beginning of the bit, in addition to transition at the
middle of the bit interval for synchronization.

Multilevel Binary Encoding: In this three voltage levels (+V, 0,-V) are used to encode the binary
data.
Bipolar NRZ: In this format alternating +V and –V voltage levels are used to represent binary
data 1 and no voltage level for binary data 0.
Bipolar RZ: In this format the binary data 1 as well as binary data 0 are represented by
opposite-level pulses that are one-half bit wide.
BP- RZ-AMI (Alternate Mark Inversion): In this format the binary data 1 is represented by
opposite-level pulses that are one-half bit wide and binary data 0 with no pulse.

PULSE CODE MODULATION (PCM)
PCM is known as Digital Pulse Modulation Technique. Pulse Code Modulation
(PCM) refers to the class of baseband signals obtained from the quantized PAM signals by
encoding each quantized sample into a digital word. The essential operations in the transmitter of
a PCM system are sampling, quantizing and encoding as shown in the Figure 1.18.
Fig 1.18 PCM System
The basic elements of PCM are transmitter, Transmission path and Receiver. The essential
operations of transmitter are sampling, quantizing and encoding. The essential operations of
receiver are regeneration of impaired signals, decoding and demodulation of the train of quantized
samples.
(a) PCM Transmitter: The analog input waveform x(t) is lowpass filtered and sampled to
obtain x(KTs ) at a rate 𝑓
𝑠 ≥ 2𝑓
𝑚. A quantizer rounds off the sample values to the nearest
discrete value in a set of q quantum levels. The resulting 𝑥𝑞(𝑘𝑇𝑠) quantized samples are
discrete in and discrete in amplitude as shown in the Figure 1.19.
Fig 1.19 PCM Transmitter

Encoder translates the quantized samples into digital code words. The encoder works
with M-ary digits and produces for each sample a code word consisting of n digits in parallel.
Finally, successive codewords are read out serially to constitute the PCM waveform, an M-ary
digital signal. The PCM generator thereby acts as an ADC, performing analog-to-digital
conversions at the sampling rate 𝑓
𝑠 = 1
𝑇𝑠
⁄ . A timing circuit coordinates the sampling and parallel-
to-serial readout. Figure 1.20 shows how quantization process is done and converted to PCM
waveform.
Fig 1.20 Mechanism of A/D conversion and to PCM
Let us illustrate with an example, how an analog signal is converted to digital signal (PCM).
Assume that an analog signal x(t) is limited in its excursions to the range -4 to +4 V. The step size
between quantization levels has been set at 1 V. Thus, eight quantization levels are employed;
these are located at -3.5, -2.5, ... , +3.5 V. We assign the code number 0 to the level at -3.5 V, the
code number l to the level at -2.5 V, and so on, until the level at 3.5 V, which is assigned the code
number 7. Each code number has its representation in binary arithmetic, ranging from 000 for code
number 0 to 111 for code number 7.

Fig 1.21 Illustration of A/D conversion
The ordinate in Figure 1.21 is labeled with quantization levels and their code numbers.
Each sample of the analog signal is assigned to the quantization level closest to the value of the
sample. Beneath the analog waveform x(t) are seen four representations of x(t), as follows: the
natural sample values, the quantized sample values, the code numbers, and the PCM sequence.
(b)PCM Transmission Path: The most important feature of PCM system lies in its ability
to control the effects of distortion and noise. During transmission through the channel, the PCM
signal will be
1. Attenuated
2. Distorted due to the finite bandwidth of the channel
3. Corrupted by the additive noise introduced in the channel
In order to get a clean PCM wave, a chain of regenerative repeaters are placed sufficiently
close to each other along the transmission route. The basic operations of regenerative repeaters are
Equalization, Timing and Decision Making as shown in Figure 1.22
Fig 1.22 Block diagram of Regenerative Repeater

Equalization: Shapes the received pulses to compensate the effects of amplitude and phase
distortion produced by the transmission characteristic of channel.
Timing: The timing circuitry provides a periodic pulse train derived from the received pulse for
sampling the equalized pulses at instants time where SNR is maximum.
Decision Making: Each sample extracted from the timing circuitry is then compared to a pre-
determined threshold in the decision making device. Usually threshold is set 50% of the maximum
amplitude attained by the received pulse. At each bit interval a decision is made whether a received
symbol is 1 or 0 on the basis if the threshold is exceeded or not. If the threshold is exceeded a new
pulse representing bit 1 is transmitted to next repeater else no pulse, bit 0 is transmitted.
Consider the signal received at the output of the channel, will appear somewhat as shown
in Figure 1.23 assuming that the code word as shown is transmitted. On receiving the signal, is to
make a considered decision, during each time slot, as to whether what has been received during
that time slot is a1’ or a ‘0’, i.e., a ‘pulse’, or ‘no pulse’. If there were to be no noise, it would be
quite easy to make this decision without committing an error.
Fig 1.23 Received signal
A simple though not an elegant way of taking this decision may be to take a sample of the
received signal during each time slot, preferably at the center of the time slot, compare the sample
amplitude with a fixed pre-set threshold and declare it as a ‘1’ if it exceeds the threshold and
declare it as a ‘0’ during that time slot, if it is less than the threshold. The threshold may be set at
50% of the maximum amplitude attained by the received pulse once it is decided that it is a 1 in a
time slot, a local pulse generator in the receiver is triggered and it gives a clean, noise-free
rectangular pulse. Thus, although the received pulses were distorted and corrupted by additive
noise, in their place, through this process of detection of the presence of a pulse and generating a
clean pulse locally in the receiver, a process which is known as ‘regeneration’. The possibility of
errors being committed in the decision making process will be negligibly small, provided the
received signal (pulse) amplitude is large enough compared to noise.
(C)PCM Receiver: The sequence of clean ‘pulses’ and no-pulses’, obtained through the process
of regeneration, are then fed to a decoder which converts each code word into the corresponding
quantized sample value as shown in Figure 1.24(a). The DAC operations of serial-to-parallel
conversion, M-ary decoding, and sample-and-hold generate the analog waveform 𝑥𝑞(𝑡) drawn in
Figure 1.24(b). This waveform is a “staircase” approximation of x(t), similar to flat-top sampling
except that the sample values have been quantized. Low pass filtering then produces the smoothed
output signal, which differs from the message x (t) to the extent that the quantized samples differ
from the exact sample values x(KTs ) .

Fig 1.24 (a) PCM Receiver (b) Reconstructed Signal (Filtered)
Perfect message reconstruction is therefore impossible in PCM, even when random noise has no
effect. A PCM system using a binary code for representing the quantization level numbers is
referred to as binary PCM.
Transmission Bandwidth in PCM:
If in a PCM system, an n-bit binary code word is used to represent each quantized sample
value, then it is referred to as an n-bit binary PCM system. Since each sample value must be
transmitted before the next sample is taken and since the sampling interval is 𝑇𝑠 =
1
𝑓𝑠
, an n-bit
code word must be accommodated in 𝑇𝑠 sec or the maximum time available for each bit is
𝑇𝑠
𝑛
𝑠𝑒𝑐.
The time interval available for each bit (1 or 0), is called one ‘time-slot’, and is of T sec. duration.
Time-slot or bit duration (𝑇𝑏) =
𝑇𝑠
𝑛
𝑠𝑒𝑐.
The bandwidth of the binary PCM depends upon the bit-rate and the waveform pulse shape used
to represent the data.
Bit-Rate (𝑅𝑏) = No.of samples /sec * No.of bits /samples
= 𝑓𝑠*n bits/sec

When a sequence of pulses is transmitted through the channel, adequate time-domain resolution
of the pulses at the output of the channel would require channel bandwidth𝐵𝑇.
1
, is pulse width
2
T
B where 


For an n-bit binary PCM, if fs is the sampling frequency employed, n time slots are to be provided
in one sampling interval Ts.
1
s
s
T
n nf
  
Hence, the minimum bandwidth of an n-bit PCM is employed only when sin(x)/x pulse is used to
generate PCM waveform.
min
2
b
T
R
B nW
  Where W is message signal bandwidth
For PCM generated by Rectangular Pulses, the First-null bandwidth is given by
max
T b s
B R nf
 
In Practice, ( )
2
b
PCM b
R
BW R
 
SNR Calculation of PCM System:
The quantization noise power which is uniformly distributed over interval ,
2 2
 
 

 
 
is given by
2
12
q
N

 (for derivation refer Quantization)
Case (i): Consider n-bit binary PCM, a message signal which is uniformly distributed between
max max
x and x
 , is transmitted.
The mean-squared value of a random variable X which is uniformly
distributed between max max
x and x
 is
max
max
2 2
( )
x
X
x
X x f x dx

 
S =
max
max
2
2 2 max
max
1
2 3
x
x
x
X x dx
x

 
 
 
 

1
2 2
b
T s
R
B nf
 
  
 
 

Noise Power is given as
2
12
q
N

 ,
max max
2 2
,
2n
x x
where
Q
  
2 2
max max
2 2
4
2 12 2 3
q n n
x x
N
   
  
   
 
   
2
10
10log (2 )
n
q dB
S
N
 

 
 
Case (ii): Let us assume that a message signal is of Sinusoidal nature, then ( ) cos2
m m
x t A f t


max min m
2
V V A
Q Q

  
m
2
2n
A

2
m
2
3 2
q n
A
N
 

The signal power is given as,
2
m
2
A
.
Then
2
m
2
2
m
2
3
2 2
2
3 2
n
q only quantization
n
A
S
A
N
 
 
 
  
 
 
   
   

 
2
10
3
10log 2
2
n
q dB
S
N
   
 
 
 
 
 
or
 
1.76 6.02
q dB
S n dB
N
 
 
 
 
 
1.8 6
q dB
S n dB
N
 
 
 
 
 
2
max
2
2
max
2
3 2
2 3
n
q
n
x
S
N x
 
 
 
 
 
 
   
 

 
6
q dB
S n dB
N
 
 
 
 

Case (iii): Suppose that the baseband signal x(t) is modelled as the sample function of a Gaussian
random Process of zero mean and that the amplitude range of x(t) at the quantizer input extends
from -4Arms to 4Arms.
The average resulting signal Power is 2
rms
A .
The resulting quantizer step size is,
8
2
rms
n
A
 
2
2
2
2
3
2
64 16
12 2
n
rms
rms
q
n
A
S
A
N
 
 
 
  
 
 
   
   
  
2
10
3
10log 2
16
n
q dB
S
N
   
 
 
 
 
 
Table 1.1 Comparison of SNR of three cases
Number of
quantizing levels
Q
Number of
binary digits
n
Signal-to-noise ratio(dB)
Uniformly
modulating
wave
Sinusoidal
modulating
wave
Gaussian
modulating
wave
32 5 30 31.8 22.8
64 6 36 37.8 28.8
128 7 42 43.8 34.8
256 8 48 49.8 4.08
From the table it is observed that a 1-bit increase per sample increases SNR by 6dB. In general as
number of bits per sample increases from n to n+k, the SNR increases by “6ndB/6kdB”. Hence it
is called “6ndB rule”.
Noise in PCM systems:
The quantization noise introduced deliberately at the transmitter by rounding off the actual
values of the samples of the message signal to the nearest quantization levels so as to discretize
the message signal in amplitude.
The channel noise, contributed by the electronics of the transmitter and receiver and partly
by noises from external agencies entering into the channel.
When a sequence of binary digits are transmitted owing to various noises, there is a
possibility of interpreting a 1 as a 0 and a 0 as a 1.An ‘error’ is said to have occurred in that time-
slot and the bit transmitted during that time slot, is received erroneously, called bit-error. If we
consider a very long sequence of transmitted bits, the average rate at which bit errors occur, is
 
6 7.2
q dB
S n dB
N
 
  
 
 

referred to as the ‘bit-error rate’. A typical bit-error rate may be 1 in 108
bits, called the bit-error
probability, Pe, is 10-8
.
Consider an n-bit code word. The probability of error for each one of the bits is Pe and so
the probability that any one of the n bits of the code word is erroneous, is nPe. Let us consider as
one of the code words transmitted that this code word is affected by the channel noise and one of
its bits is changed from 0 to 1 or 1 to 0 in the decision-making process, if the bit 1
n
b  the one which
is erroneously interpreted, the error caused in the value of the codeword is 2
2n
. Since all the n-
bits of the affected codeword have equal probability of being affected, this probability is (1/n).
Mean-squared value of the code word is,        
2 2 2 2
2 1 2 1 0
1
2 2 2 2
n n
n
 
 
     
 
 
Each code word represents one of the Q quantization levels, the step size Δ = 2/Q. Since the
codeword transmitted represents a quantization level number, when the codeword 
1 2 1
th
n n
b b b b

is transmitted, this quantization level corresponds to a sample value of  
1 2 1
2
n n
b b b b
Q

 
 
 
for the
normalized message signal.
Hence, when this code word is affected by noise has a mean-squared error,
2 2 2 2
2 1 2 1 0
1 2 2 2 2
2 2 2 2
n n
n Q Q Q Q
 
 
       
         
 
       
 
       
 
1
2
2
0
4 4
2
3
n
k
k
nQ n


 

For n-bit codeword mean-square error caused by channel noise,
2 4 4
3 3
e e
nP P
n
   
The mean-squared error due to quantization noise is,
2
2
1
3
q
N
Q

The two noise process are independent, the mean-squared value of the total noise is given by the
sum of their individual mean-squared values.
2
2
2
1 4
3 3
1 4
3
D e
e
D
N P
Q
Q P
N
Q
 


2 2
2
3
1 4
D e
BinaryPCM
S Q x
destinationSNR
N Q P
 
 
 

 

Trading of SNR with Bandwidth: clubbing the SNR expressions for uncompanded and
companded PCM,
 
2
2
l
SNR c

 
2
2
2
( )
3 ( )
,
3
( )
ln 1
p
m t
uncompanded
m
where c
companded




 

 
 
 

Where l-number of bits, 2
( )
m t -power contained in signal m(t) and 2
p
m -is peak quantization level
of uniform quantizer.
The SNR of PCM signal interms of transmission bandwidth is,
   
2
2
2 2
T
B
b
B
SNR c c
 
, exp
T
B
b called bandwidth ansion factor
B

It is clear that as n increases, SNR increases rapidly (exponentially). But at the same time, the
required transmission bandwidth BT also increases, but only linearly with n. Thus, without
increasing the transmitter power we can just increase n and get an improved destination SNR – but
at a price. Thus, we can save power at the cost of bandwidth and vice versa or in other words, there
is a power- bandwidth trade-off possible in PCM.
For voice telephony applications, SNR requirement is 30dB, PCM outperforms both FM as well
as PM for identical values of b. But for radio broadcasting applications, SNR required is 60dB for
a PCM systems but with b=6, the same SNR is achieved by FM but for PCM b needs to be
increased beyond 8 to achieve 60dB SNR. Hence, in broadcasting applications FM is preferred
over PCM.

Advantages of PCM:
1. It offers a very efficient power-bandwidth exchange.
2. It is very robust since it is almost immune to channel noise and interference.
3. Because of the possibility of the use of regenerative repeaters, it is extremely useful in
long-Haul communication.
4. It makes possible to integrate baseband signals of different types, like audio, video, etc.,
into a common format for easy multiplexing using TDM.
5. Because of its digital nature, coding techniques are available for PCM signals for efficient
Compression, encryption and error-correction.
6. In a TDM system, it is relatively easy to either add or drop any message signal and PCM
signals can easily be TDM-ed.
Disadvantages:
1. PCM signal generation and reception involve complex processes and require the use of
somewhat complex systems. But nowadays because of the availability of VLSI chips for
performing these various operations is not a real problem.
2. For the same message bandwidth, PCM requires a much larger transmission bandwidth
than some of the analog modulation schemes.
PCM-TDM SYSTEM: T1 CARRIER SYSTEM (APPLICATION)
When a large number of PCM signals are to be transmitted over a common channel,
multiplexing of these PCM signals are required. Figure 1.25(a) shows the TDM scheme called T1-
digital system or T1 carrier system.
Fig 1.25(a) A PCM-TDM: T1 Carrier System

Working Operation: This system is designed to accommodate 24 voice channels. Each signal is
bandlimited to 3.3 KHz and sampling is done at 8KHz.these voice signals are selected one by one
and connected to a PCM transmitter by the commutator switch SW1. Each sampled signal is
applied to A/D conversion and Companding and resulting digital waveform is transmitted over co-
axial cable. At the destination, the signal is companded, decoded and demultiplexed using PCM
receiver. The PCM receiver output is connected to different LPF via the decommutator switch
SW2. Synchronization between the commutators is essential for proper communication.
The frame structure of T-1 carrier system is shown in figure 1.25(b). Instead of using a separate
channel for signaling purposes, the LSB slots normally used for voice information, are themselves
used once in six frames, for the purpose of signaling. This arrangement is therefore referred to as
‘channel associated signaling’.
Fig 1.25(b) T1 Carrier System Frame structure
Frame synchronization: Only when there is proper synchronization, the correct receptions of
signals can be done. For this purpose, as shown in Fig. 1.25(c), one frame bit is included at the
beginning of every frame. The pattern formed by 12 such frame bits occurring in 12 successive
frames gives a 12-bit code called the frame sync word, which is known a priori to the receiver,
and used by it for synchronization. This 12-bit code(1000 1101 1100) is repeatedly transmitted
once every 12 frames.
Fig 1.25(c) T1 Carrier Frame details

As shown in figure 1.25(c) in addition to information and framing bits, we need to transmit
signaling bits corresponding to dialing pulses, as well as telephone on-hook/off-hook signals. In
order to create extra time slots for this information, the LSB bit of every sixth sample of signal to
transmit this information. i.e every sixth frame has 7*24= 168 information bits, 24 signaling bits
and 1 framing bit. In all other frames there are 192 information bits and 1 framing bit. This
encoding is called 7
5
6
bit encoding and signaling channel is called robbed-bit signaling.
In T1 carrier system,
Each signal sampling frequency = 8 KHz
1 frame duration =
1
8000
𝑠𝑒𝑐 = 125μsec.
Number of bits per frame= (24*8) + 1(synchronization bit) =193bits
Transmission rate (Bit rate) in 1-sec=
193
125μsec
=1.544Mbits/sec
The bandwidth of PCM system depends on the bit duration (bit time slot), calculated as
Sampling frequency = 2 m
f
Sampling period=1/ 2 m
f
As there are n-channels and N bits per sample and one synchronizing bit
Total number of bits/sampling period (frame) = nN+1
Bit duration,
 
1
l n 1 2
b
m
sampling period
T
tota umber of bits nN f
 

For evaluating the bandwidth, it is assumed that 1’s and 0’s occur alternatively and bit stream is
equivalent to square of the pulse width Tb.
Hence, practical BW=
1
b
T
Therefore the bandwidth of PCM =  
1 2 m
nN f
 Hz
If 1 1
N and n
  , 2 m
BW nNf
 Hz

Linear Prediction Theory
When adjacent samples of a message have good correlation, as in the case of audio and
video message samples encoded using PCM, it is possible to predict the value of a future sample
by making use of the present and some previous samples.
Suppose if we want to predict ( )
s
x nT the th
n sample, we use ‘p’ previous samples,
( 1 ), ( 2 ), ( )
s s s
x n T x n T x n pT
   . Linear combination ( )
s
x nT gives estimate of ( )
s
x nT .
i.e 1 2
( ) ( 1 ) ( 2 ) ( )
s s s p s
x nT h x n T h x n T h x n pT
      
Where 1 2
, , p
h h h real numbers are called Weights. Since a linear combination of the previous
sample values is used for obtaining the predicted value, the prediction process is called ‘Linear
Prediction’. A simple linear combination of the ‘p’ previous samples can be implemented using a
simple FIR digital filter, generally called a “Transversal Filter” as shown in Fig. 1.26(a).
Fig 1.26(a) pth order prediction filter
In the receiver, the predictor is used for the
reverse operation, i.e., obtaining x(nTs) from the e(nTs) which
is given as the output of the decoder. The predictor feedback
loop is configured as shown in Fig. 1.26(b).
The predictor weights or coefficients must be
chosen that the ‘prediction error’ is minimized in some sense.
Usually the error is minimized in the ‘mean-square’ sense, i.e.,
the mean-squared value of the error is minimized by an
appropriate choice of the ‘p’ weights, or coefficients of the
linear combination.
Fig 1.26(b) prediction feedback loop of filter

DIFFERENTAIL PULSE CODE MODUCLATION (DPCM)
When a voice or video signal is sampled at a rate slightly higher than the Nyquist rate (over
sampling), the resulting sampled signal is found to exhibit a high degree of correlation between
adjacent samples, i.e in an average sense, the signal does not change rapidly from one sample to
the next, with the result that the difference between adjacent samples has an average power that is
smaller than the average power of the signal itself. When these highly correlated samples are
encoded as in a standard PCM system, the resulting encoded signal contains redundant
information. Redundancy means that symbols that are not absolutely essential to the transmission
of information are generated as a result of the encoding process. By removing this redundancy
before encoding, we obtain a more efficient encoded signal, compared to PCM.
The DPCM system, in fact, employs a predictor, which predicts the present sample value
making use of a few immediate past sample values by taking the linear combination of those past
samples. The predicted value of the present sample is compared to the actual value of the present
sample and the difference between the two is pulse-code modulated. The advantage in this lies in
the fact that if the prediction is reasonably good, the difference between the actual value and the
predicted value, called the error, will have a much smaller dynamic range than the original message
itself and therefore needs far fewer bits per each error sample than what would have been needed
for the original samples themselves.
Figure 1.27(a) shows the DPCM System. In the transmitter, the current sample, x (nTs) is
compared to the predicted value 𝑥
̂(𝑛𝑇𝑠) and the difference is quantized using an appropriate
number of quantization levels, and then encoded and transmitted in the form of a stream of binary
pulses.
Mathematical Analysis,
The prediction error, given by, ( ) ( ) ( )
s s s
e nT x nT x nT
  (1)
Which is the amount by which prediction filter fails to predict actual input value exactly at an
instant of time.
Fig 1.27(a) DPCM Transmitter

The quantizer output is given as,
( ) ( ) ( )
q s s e s
e nT e nT q nT
  (2)
Where ( )
e s
q nT is the quantization error. According to Fig. 1.27( a), the quantizer output ( )
q s
e nT
is added to predicted value ( )
s
x nT to produce the prediction filter input.
( ) ( ) ( )
q s s q s
x nT x nT e nT
  (3)
Substituting eq (2) in to eq (3) we get,
 
( ) ( ) ( )
s s e s
e nT x nT q nT
  
( ) ( ) ( )
q s s e s
x nT x nT q nT
   (4)
Therefore eq(4) reveals that irrespective of the properties of prediction filter, the quantized signal
at the prediction filter input differs from the original input signal by quantizing error.
Fig 1.27 (b) DPCM Receiver
The receiver for reconstructing the quantized version of the message signal is shown in
Fig.1.27 (b). It consists of a decoder to reconstruct the quantized error signal. The quantized
version of the original input is reconstructed from the decoder output using the same prediction
filter in the transmitter of Fig 1.24 (a). In the absence of channel noise, the encoded signal at the
receiver input is identical to the encoded signal at the transmitter output. Receiver output is equal
to ( )
q s
x nT which differs from the original input ( )
s
x nT only by the quantization error ( )
e s
q nT
incurred as a result of quantizing the prediction error ( )
s
e nT . Finally, estimate of the original
message signal x (t) is obtained by passing the sequence ( )
q s
x nT through a low-pass reconstruction
filter.
Processing Gain of DPCM and Comparison with PCM
For a DPCM system, the output signal-to-quantization noise ratio is defined as
2
2
,
variance of message
variance of quantization error
X
D Q Q
S
N



 

 

 
Let 2
P
 denote the variance of the prediction error in DPCM.
Then,
2 2 2
2 2 2
,
X X P
D Q Q P Q
S
N
  
  
 
 
 
   
 
   
    

The ratio
2
2
P
Q


 
 
 
 
is the usual SNR of PCM. The processing gain is defined by the ratio
2
2
X
P


 
 
 
.
i.e Processing gain ,
2
2
X
P
P
G


 
  
 
is obtained because of differential quantization
.
The quantity Gp, the processing gain, may be greater than one, or less than one, depending
upon how good the prediction is, which in turn depends upon our selection of the weights of the
predictor. With a prediction filter order of 5, it is found that DPCM gives about 11 dB improvement
in the (SNR) D, Q as compared to PCM. For a sampling rate of 8 kHz, DPCM may give a saving in
bit rate to the extent of 1 to 2 bits/sample, i.e., about 8 to 16 kbps, as compared to PCM.
Slope-overload Noise in DPCM: If the signal x(nTs) changes so fast that the predicted
signal ( )
q s
x nT cannot follow , the system noise increases. This phenomenon is called slope
overloading. Let ( )
q s
x nT changes by  
1
N
   from sample to sample and sampling is done
every
1
s
f
seconds. The maximum slope of ( )
q s
x nT is,
 
 
max
( ) 1
1
1
q s
s
s
dx nT N
f N
dt
f
  
 
    
 
 
To prevent slope overload, it is required that the predicted
signal slope should not be less than the signal slope. i.e
Delta Modulation (DM)
In delta modulation (DM), an incoming message signal is oversampled (i.e., at a rate much
higher than the Nyquist rate) to purposely increase the correlation between adjacent samples of the
signal. The increased correlation is done so as to permit the use of a simple
quantizing strategy for constructing the encoded signal. It is a 1-bit DPCM scheme.
DM provides a staircase approximation to the oversampled version of the message signal.
The difference between the input signal and its approximation is quantized into only two levels
namely,  corresponding to positive and negative differences. If the approximation falls below
the input signal at any sampling epoch, it is increased by . If the approximation lies above the
signal, it is diminished  , provided the input signal does not change too rapidly from sample to
sample, and we find that the staircase approximation remains within  of the input signal.
2
2
,
P
P
D Q Q
S
G
N


 
 
   
   
   
 
max
1 (t)
s
f N x

  

Fig 1.28 (a) DM Transmitter
A delta modulator simply consists of a comparator, a single-bit quantizer (hard limiter) and an
accumulator, connected together as shown in Figure. 1.28(a). Let the input signal is m(t) and the
staircase approximation as ( )
q
m t , the basic principle of DM is given by set of discrete-time
relations as:
e(nT ) ( ) ( )
s s q s s
m nT m nT T
   (1)
 
e (nT ) sgn e(nT )
q s s
  (2)
( ) ( ) e (nT )
q s q s s q s
m nT m nT T
  
 
sgn e(nT ) ( )
s q s s
m nT T
    (3)
Where s
T is sampling period, e(nT )
s is error signal representing the difference between the present
sample value ( )
s
m nT of the input signal and the latest approximation, ( )
q s s
m nT T
 and e (nT )
q s is
the quantized version of e(nT )
s . The quantizer output e (nT )
q s is encoded as, if e (nT )
q s   a
binary ‘1’ encoded and if e (nT )
q s   a binary ‘0’ is encoded. This sequence of e (nT )
q s is
encoded by the encoder whose output waveform is shown in Figure. 1.28(b) and transmitted over
the channel.

Fig 1.28 (b) Illustration of DM operation
The quantizer output is applied to an accumulator, then
 
1
( ) sgn e(iT )
n
q s s
i
m nT

 
1
e (iT )
n
q s
i
  (4)
Thus, at the sampling instant s
nT the accumulator increments the approximation by the increment
 in a positive or negative direction, depending on the algebraic sign of the error signale(nT )
s . If
the input signal ( )
s
m nT is greater than the most recent approximation ( )
q s
m nT a positive
increment  is applied to the approximation. If, on the other hand, the input signal is smaller, a
negative increment  is applied to the approximation.
Fig 1.28 (c) DM Receiver
Figure 1.28(c) shows the block diagram of DM receiver. The staircase approximation ( )
q
m t is
reconstructed by passing the sequence of positive and negative pulses, produced at the decoder
output, through an accumulator in a manner similar to that used in the transmitter. An LPF is used
to remove step variations to get a smoot reconstructed message signal m(t).

Drawbacks in Delta Modulation: DM is subject to two types of quantization error: slope overload
distortion and granular noise
(i) Slope overload Distortion:
Since the delta modulator using a fixed step size, is sometimes referred to as a linear
delta modulator (LDM). As shown in Figure 1.28(d) , the average rate of change of
the staircase approximation ( )
q
m t , is given by (Δ/Ts). Because of the fixed step size,
the rate of change of message signal will be larger the rate of change of approximated
signal and cannot track the message signal m(t). This inability of the LDM to correctly
track the message signal, x(t) when x(t) has steep changes, is referred to as ‘slope
overload’ condition.
Slope overload distortion arises if the slope of the message signal is greater than
the slope of the stair case approximated signal,
Condition for slopeover load
max
( )
s
d
m t
dt T


Fig 1.28(d) Illustration of quantization errors –slope overload and granular noise
(ii) Granular noise: As shown in Fig 1.28(d) granular noise or Hunting Noise occurs
when the step size  is too large relative to the local slope characteristics of the message
signal m(t), thereby causing the staircase approximation ( )
q
m t to hunt around a
relatively flat segment of m(t). In other words, Granular noise arises if the slope of the
message signal is much less than the slope of stair case approximated signal.

SNR calculation of DM System
Consider DM with uniform step size Δ with sampling period of Ts, then the maximum slope of
step size is
s
T

. Assume that no slope overload distortion.
Condition for no slope over load distortion,
max
( )
s
d
m t
T dt


Consider a sinusoidal message signal ( ) sin 2 m
m t A f t

 , then for no slope overload distortion to
occur,
max
sin 2 m
s
d
A f t
T dt



max
2 cos2
m m
s
A f f t
T
 


2 m
s
A f
T



2 m s
A
f T



2
s
m
f
A
f



The average signal power for sinusoidal signal is given by,
2
2
A
S 
2
2 2
2 2
2
2 8
s
m s
m
f
f f
S
f


 

 

 
  
The maximum quantization error lies within  . Assume quantization error is uniformly
distributed then its pdf is given as
The quantization noise power is given by,
2
( )
q e e e
N Q f Q dQ


 
2
1
2
e e
Q dQ



 
2
3
q
N

 

The delta modulation receiver consists of a LPF having a cut-off frequency of W Hz. Assume
that the avg. noise power is uniformly distributed over
1 1
s s
to
T T
  . Therefore, the amount of
noise present in the filter bandwidth is,
2 2
3 3
q s
s
W
N W T
f
 
     .
2 2
2 2
2
8
3
s
m
q DM
s
f
f
S
N
W T

 

 
   

 
   

   
 
 
2 2 3
3
8 m s
f WT


If filter bandwidth W = message bandwidth (fm), then,
Bit rate (Rb) = 1/n*(Rb)PCM
1
s s
nf f
n
  
Transmission bandwidth,
2
s
T
f
B 
3
2 2
3
8
s
q m
DM
f
S
N f W

 
 
 
 
 
3 3
2
3 3
8 80
s s
q m m
DM
f f
S
N f f

     
 
     
 
   
 

ADAPTIVE DELTA MODULATION (ADM/CVSDM)
Adaptive delta modulation (ADM) is a delta modulation where the step size Δ of the
staircase waveform is varied depending upon the slope or amplitude characteristics of the analog
input signal. When the DM output is string of consecutive 1s or 0s, indicates the staircase
waveform is not tracking the analog waveform and possibility of slope overload distortion. When
the alternating 1s and 0s is occurring indicates possibility of granular noise. A typical schematic
AM transmitter is shown in Figure 1.29.
Fig 1.29 Adaptive Delta Modulator Transmitter
In ADM, the step size may be made to vary with steepness of variation of the message
signal either continuously, or in a discrete manner. In both the types, the sensing of the steepness
of the message signal is done in the same way. As shown in Fig.1.26 , except during the stat-up
time, whenever the signal is changing steeply, the binary output from the modulator continues to
be the same – a series of 1s if signal is steeply rising and a series of 0s if the signal is steeply
decreasing. On the other hand, when the signal is changing very slowly, or is constant the binary
output from the Delta modulator is alternate 1s and 0s.
The pulse generator produces narrow pulses of fixed amplitude at a rate equal to the desired
sampling rate. The modulator consists of a hard limiter followed by a product device, or a
multiplier. Whatever may be the actual value of e(t), the hard limiter output will be +1 if e(t) is
positive and –1 if e(t) is negative. So the polarity of the pulse 𝑝0(𝑡) depends on the sign of e(t).
The subsystems within the dotted-line box are for ‘adaptation’.
Assume this part is not there and point marked (A) is directly connected to the input of the
integrator. Let us approximate the narrow pulses in the pulse train 𝑝0(𝑡) by impulses. Since
integration of an impulse gives a step, integration of the train of impulses occurring at regular
intervals of Ts = 1/fs will result in a staircase signal approximation of the message signal x(t). The
step size Δ in this staircase approximation depends only on the amplitude of the pulses in 𝑝0(𝑡)
and the gain of the integrator. So we get a staircase approximation with a fixed step size.

Assume that the ‘adaptation’ circuit shown in the dotted-line box is connected. The pulses
in the pulse train 𝑝0(𝑡) try to charge the capacitor C through the resistance R. For a short segment
of time, the pulses are alternatively positive and negative – and this happens when the message
signal is either not changing at all, or is changing very slowly, there will not be any charge
accumulation on the capacitor and the voltage across it will be zero, or negligible. So the gain
control voltage is almost zero or is zero and there will not be any change in the amplitude of the
pulses at the output of the variable-gain amplifier. As the gain of this amplifier is adjusted initially
to be low when the gain-control voltage level is zero, the amplitude of the pulses fed as input to
the integrator and so, the step size of 𝑥(𝑡)
̃, the staircase approximation, will be small.
If x(t) is steeply rising or falling for some time, the consecutive pulses in the pulse train
𝑝0(𝑡) will be either all positive or all negative over that segment of time. So the capacitor will be
charged. Irrespective of whether it is charged positively or negatively, the square law device output
which is the gain-control voltage, will be positive and its value will depend upon the length of time
for which the polarity of the gain of the amplifier and consequently the step size, will go on
increasing till the rate of change of x (t) becomes less and the gain of the amplifier and the step
size will reduce automatically.
The receiver just consists of a decoder (a decision device followed by a pulse generator)
and an integrator followed by a low pass filter with cut-off frequency, W Hz, the band limiting
frequency of x(t). In the absence of any decoding errors due to channel noise, the output pulse train
from the pulse generator part of the decoder will be an exact replica of the transmitted pulse
train 𝑝0(𝑡). These impulse-like narrow pulses, when fed to the integrator, produce a staircase
approximation of x(t) and the LPF, the last stage, removes the out-of-band frequency components
from this staircase approximation to give an estimate of x(t) as shown in Figure 1.30.
Fig 1.30 Adaptive Delta Modulator Receiver
In practice, the staircase step size is bounded by the lower and upper limits as ∆𝑚𝑖𝑛≤
∆(𝑛𝑇𝑠) ≤ ∆𝑚𝑎𝑥. The lower limit ∆𝑚𝑖𝑛 controls the amount of granular noise and the upper limit
∆𝑚𝑎𝑥 controls the amount of slope overload distortion.
The step size will be initially∆𝑚𝑖𝑛. If ∆(𝑛𝑇𝑠) is the step size at the nth
sampling instant, it
is so arranged that

 
( ) if b( ) ( )
( )
1 ( ) if b( ) ( )
s s s s s
s
s s s s s
K nT T nT b nT T
nT
nT T nT b nT T
K
   


  
   


Where b(nTs) is the binary pulse at t = nTs. Hence, if two consecutive binary pulses in the output
binary pulse sequence 𝑝0(𝑡) are alike indicates that x(t) is steeply changing, the step size is
increased by a factor K compared to its previous value. K is generally taken as 1.5 for speech and
image signals. If two consecutive binary pulses of 𝑝0(𝑡) are not alike, which indicates that x(t) is
varying slowly, the step size is decreased by the factor K. For a wide range of bit-rate values like
20 kbps to 60 kbps, a value of K = 1.5 is quite satisfactory and that this type of ADM system with
K = 1.5 gives about 10 dB better (SNR)D as compared to an LDM system for which the step size
is fixed. Figure 1.31(a) shows the waveform signal obtained in the case of ADM and Figure 1.31(b)
shows the comparative waveform of ADM and DM.
Fig 1.31(a) Waveform of ADM Fig 1.31(b) Comparative waveforms of ADM & DM
Comparison of Waveform Coding Techniques
S.No Parameter PCM DPCM DM ADM
1 Number of bits Uses 4, 8 or 16
bits/sample
Less than PCM 1-bit/sample 1-bit/sample
2 Levels and
step size
Levels depends upon
bits and fixed step
size
Fixed number of
levels are used
Step size is
fixed and two
levels
Variable step
size.
3 Error and
distortion
Quantization error Quantization error,
slope overload
distortion
Slope overload
, granular noise
Quantization
error
4 Transmission
BW
High Lessthan PCM Lowest Lowest
5 Feedback No YES YES YES
6 Complexity System is complex Simple Simple Simple

Drill Problems
1. A television signal having a bandwidth of 10.2MHz is transmitted using binary PCM
system. Given that the number of quantization levels is 512. Determine:
(i) Code word length
(ii) Transmission bandwidth
(iii) Final bit rate
(iv) Output SNR
Solution: Given, 4.2
m
f MHz

9
512
2 512 2
n
Q
Q

  
Code word length, 9
n 
Transmission Bandwidth, 9 4.2 37.8
m
BW nf MHz MHz
   
Bit rate,
6 6
2 2 9 4.2 10 75.6 10 / sec
b s m
R nf nf bits
       
Output SNR, (1.8 6 ) (1.8 6 9) 55.8
dB
S
n dB dB dB
N
 
     
 
 
2. A compact disc recording system samples each of the two stereo signals with 16 bit
A/D converter at 44.1Kb/sec.
(i) Determine output S/N ratio for a full scale sinusoid.
(ii) The bit stream of digitized data is augmented by addition of error correcting
bits, clock extraction bits etc. and these additional bits represent 100%
overhead. Determine output bit rate of CD system.
(iii) The CD can record an hours’ worth of music. Determine number of bits
recorded on CD.
Solution: Given 2-stereo channels, n=16, fs=44.1Kbits/sec
(i) Full scale sinusoid, (1.8 6 ) (1.8 6 16) 97.8
dB
S
n dB dB dB
N
 
     
 
 
(ii) Bit rate, b s
R nf
 , then bit rate for 2-stereo channel
3
2 2 16 44.1 10 1.4112 / sec
b s
R nf Mbits
     
Now with including additional 100% overhead,
Output bit rate = 2*1.4112Mbits/sec=2.822Mbits/sec
(iii) Since the CD is recorded on hours’ worth,
Therefore, number of bits recorded = bit rate * Number of sec/hr
=2.822Mbits*3600
=10.16gigabits
3. A DM transmitter with a fixed step of 0.5 V, is given a sinusoidal message signal. If
the sampling frequency is twenty times the Nyquist rate, determine (a) the maximum
permissible amplitude of the message signal, if slope overload is to be avoided, and
(b) the maximum destination SNR under the above condition.
Solution: Given  
20 2 40
s m m
f f f
   , 0.5
 

40
s
m
f
f
 
(a) Maximum permissible value of A to avoid slope overload is
max
2
s
m
f
A
f

 

  
 
0.5 40
3.18
2
V


 
(b) Maximum destination SNR is given by,
3
2
3
8
s
m
f
S
N f

 
   
  
   
    
 
3
2
3
40 2432
8
 
 
 
 
10log2432 33.85
dB
S
dB
N
 
 
 
 
4. Determine the output signal to noise ratio of a linear delta modulation system for 2
KHz sinusoidal input signal sampled at 64 KHz. Slope overload distortion is not
present and the post reconstruction filter has a bandwidth of 4 KHz.
Solution: Given, 2
m
f KHz
 , 64
s
f KHz
 and 4
W KHz

3
2 2
3
8
s
q m
DM
f
S
N f W

 
 
 
 
 
 
 
3
3
2
2 3 3
64 10
3
622.52
8 2 10 4 10

 

  
 
  
    
 
10log(622.52) 27.94
dB
S
dB
N
 
 
 
 
5. 24 telephone channels, each bandlimited to 3.4 KHz are to be TDM-ed using PCM.
Calculate the bandwidth of PCM system for 128 quantization levels and an 8 KHz
sampling frequency.
Solution: Given, channels, n=24
fm= 3.4KHz
Quantum levels, Q=128
2 128
N

Number of bits, 2
log 128 7
N  
 
1 2 m
BW nN f
  
(24 7 1) 8000
   
1.352MHz


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