digital communications 3units
TRANSCRIPT
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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY KAKINADA
III Year B. Tech. Electronics and Communication Engineering – I Sem.
DIGITAL COMMUNICATIONS
UNIT I-PULSE DIGITAL MODULATION:
Elements of digital communication systems, advantages of digital communication systems,Elements of PCM: Sampling, Quantization & Coding, Quantization error, Compading in PCMsystems. Differential PCM systems (DPCM).
UNIT II-DELTA MODULATION :
Delta modulation, its draw backs, adaptive delta modulation, comparison of PCM and DM
systems, noise in PCM and DM systems.
UNIT III-DIGITAL MODULATION TECHNIQUES :
Introduction, ASK, FSK, PSK, DPSK, DEPSK, QPSK, M-ary PSK, ASK, FSK, similarity of BFSK and
BPSK.
UNIT IV-DATA TRANSMISSION :
Base band signal receiver, probability of error, the optimum filter, matched filter, probabilityof error using matched filter, coherent reception, non-coherent detection of FSK, calculation
of error probability of ASK, BPSK, BFSK,QPSK.
UNIT V-INFORMATION THEORY :
Discrete messages, concept of amount of information and its properties. Average information,
Entropy and its properties. Information rate, Mutual information and its properties,
UNIT VI-SOURCE CODING :
Introductions, Advantages, Shannon’s theorem, Shanon-Fano coding, Huffman coding,
efficiency calculations, channel capacity of discrete and analog Channels, capacity of a
Gaussian channel, bandwidth –S/N trade off.
UNIT VII-LINEAR BLOCK CODES :
Introduction, Matrix description of Linear Block codes, Error detection and error correction
capabilities of Linear block codes, Hamming codes, Binary cyclic codes, Algebraic structure,encoding, syndrome calculation, BCH Codes.
UNIT VIII-CONVOLUTION CODES :
Introduction, encoding of convolution codes, time domain approach, transform domainapproach. Graphical approach: state, tree and trellis diagram decoding using Viterbi
algorithm.
TEXT BOOKS :
1. Digital communications - Simon Haykin, John Wiley, 2005
2. Principles of Communication Systems – H. Taub and D. Schilling, TMH, 2003
REFERENCES :
1. Digital and Analog Communication Systems - Sam Shanmugam, John Wiley, 2005.
2. Digital Communications – John Proakis, TMH, 1983. Communication Systems Analog &
Digital – Singh & Sapre, TMH, 2004.
3. Modern Analog and Digital Communication – B.P.Lathi, Oxford reprint, 3rd edition,2004.
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UNIT I-PULSE DIGITAL MODULATION
ELEMENTS OF A DIGITAL COMMUNICATION SYSTEM
The analysis and design of digital communication systems Involves the
transmission of information in digital form from a source that generates the information
to one or more destinations.
The source output may be either an analog signal, such as an audio or video signal, or
a discrete signal, such as the output of a teletype machine, that is discrete in time and has
a finite number of output characters.
In a digital communication system, the messages produced by the source are
converted into a sequence of binary digits. The process of efficiently converting the
output of either an analog or discrete source into a sequence of binary digits is called
source encoding or data compression.
The sequence of binary digits from the source encoder, which we call the information
sequence, is passed to the channel encoder
FIGURE 1. Basic elements of a digital communication system.
. The purpose of the channel encoder is to introduce, in a controlled manner, some
redundancy in the binary information sequence that can be used at the receiver to
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overcome the effects of noise and interference encountered in the transmission of the
signal through the channel. This increases the reliability of the received data and
improves the fidelity of the received signal.
The binary sequence at the output of the channel encoder is passed to the digital
modulator, which serves as the interface to the communication channel. Since nearly all
the communication channels encountered in practice are capable of transmitting electrical
signals (waveforms), the primary purpose of the digital modulator is to map the binary
information sequence into signal waveforms.
To elaborate on this point, let us suppose that the coded information sequence is to be
transmitted one bit at a time at some uniform rate R bits per second (bits/s). The digital
modulator may simply map the binary digit 0 into a waveform So(t) and the binary digit 1
into a waveform S1(t). In this manner, each bit from the channel encoder is transmitted
separately. We call this binary modulation.
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 may be the
atmosphere (free space). On the other hand, telephone channels usually employ a variety
of physical media, including wire lines, optical fiber cables, and wireless (microwave
radio).
Whatever the physical medium used for transmission of' the information, the essential
feature is that 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,
e.g., electrical lightning discharges during thunderstorms.
At the receiving end of a digital communication system, the digital demodulator
processes the channel-corrupted transmitted waveform and reduces the waveforms to a
sequence of numbers that represent estimates of the transmitted data symbols. This
sequence of numbers is passed to the channel decoder, which 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. More precisely, the average probability of a
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bit-error at the output of the decoder is a measure of the performance of the demodulator
decoder combination.
In general, the probability of error is a function of the code characteristics, the types of
waveforms used to transmit the information over the channel, the transmitter power, the
characteristics of the channel, and the method of' demodulation and decoding.
The source decoder accepts the output sequence from the channel decoder and, from
knowledge of the source encoding method used attempts to reconstruct the original
signal.
Because of channel decoding errors and possible distortion introduced by the source
encoder, and perhaps, the source decoder, the signal at the output of the source decoder is
an approximation to the original source output. The difference or some function of the
difference between the original signal and the reconstructed signal is a measure of the
distortion introduced by the digital communication system.
The points worth noting are:
The source coding algorithm plays important role in higher code rate
The channel encoder introduced redundancy in data
The modulation scheme plays important role in deciding the data rate and immunity ofsignal towards the errors introduced by the channel
Channel introduced many types of errors like multi path, errors due to thermal noise etc.
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ADVANTAGES OF DIGITAL COMMUNICATION OVER ANALOG MODULATION:
There are many advantages of using Digital Communication over analog
communication. Some of them are listed as below:
1. The Digital communication has mostly common structure of encoding a signal so
devices used are mostly similar.
2. The Digital Communication's main advantage is that it provides us added security to
our information signal.
3. The Digital Communication system has more immunity to noise and external
interference.
4. Digital information can be saved and retrieved when necessary while it is not possible
in analog.
5. Digital Communication system is cheaper than Analog Communication.
6. The configuring process of digital communication system is simple as compared to
analog communication system.
7. In Digital Communication System, the error correction and detection techniques can be
implemented easily.
8. Digital hardware implementation is flexible & permits the use of microprocessors,
digital switching elements & layer scale.
9. Digital systems are relatively less expensive than analog systems
10. Transmission rate can be changed easily.
11. Easy for processing and applying multiplexing techniques.
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ANALOG-TO-DIGITAL CONVERSION
A digital signal is superior to an analog signal because it is more robust to noise
and can easily be recovered, corrected and amplified. For this reason, the tendency today
is to change an analog signal to digital data. In this section we describe two techniques,
pulse code modulation and delta modulation.
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.
PCM consists of three steps to digitize an analog signal:
1.
Sampling
2. Quantization
3. Binary encoding
Before we sample, we have to filter the signal to limit the maximum frequency of the
signal as it affects the sampling rate.
Filtering should ensure that we do not distort the signal, ie remove high frequency
components that affect the signal shape.
PCM Transmitter
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SAMPLING
Analog signal is sampled every TS secs. Ts is referred to as the sampling interval.
f s = 1/Ts is called the sampling rate or sampling frequency. According to the Nyquist
theorem, the sampling rate must be at least 2 times the highest frequency contained in the
signal. There are 3 sampling methods:
Ideal - an impulse at each sampling instant
Natural - a pulse of short width with varying amplitude
Flat top - a pulse of short width with constant amplitude
Usually Flat top sampled signal is generated by sampler.
Three different sampling methods for PCM
Nyquist sampling rate for low-pass and bandpass signals
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QUANTIZATION
Sampling results in a series of pulses of varying amplitude values ranging between
two limits: a min and a max. The amplitude values are infinite (or many) between the two
limits. We need to map the infinite amplitude values onto a finite set of known values.
This is achieved by dividing the distance between min and max into q zones, each of
height
rangeofinputsignal
noofQuantiztionlevels
max minv v
q
The midpoint of each zone is assigned a value from 0 to q-1 (resulting in q values). Each
sample falling in a zone is then approximated to the value of the midpoint. That is
quantization is a process of rounding-off each sampled value to the nearest value.
The reason for approximating to the mid point is that minimizes the maximum
quantization error.
Example:
Assume we have a voltage signal with amplitudes Vmin= -20V and Vmax=+20V.
We want to use q=8 quantization levels. Then zone width = (20 - -20)/8 = 5
The 8 zones are: -20 to -15, -15 to -10, -10 to -5, -5 to 0, 0 to +5, +5 to +10, +10 to +15,
+15 to +20
The midpoints are: -17.5, -12.5, -7.5, -2.5, 2.5, 7.5, 12.5, 17.5
Each zone is then assigned a binary code.
The number of bits required to encode the zones, or the number of bits per sample as it is
commonly referred to, is obtained as follows: v = log2 q
Hence no of bits required to represent each sample are v = 3
The 8 zone (or level) codes are therefore: 000, 001, 010, 011, 100, 101, 110, and 111
Assigning codes to zones: 000 will refer to zone -20 to -15; 001 to zone -15 to -10, etc
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Note: If suppose quantization levels are 16 (2v), no of bits required to represent each
sample are 4 (v bits).
If no of quantization levels are not in the power of 2, for example to distinguish 10 > 23
(
> 2
v
) quantization levels, 4 bits (v+1) are required. Possible no of 4 bit code words are16, use any 10 code words out of 16 for representing samples.
Quantization error is defined as the difference between actual sample and quantized
sample. i.e ( ) ( ) s q s x nT x nT
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TYPES OF QUANTIZERS
1. Uniform Quantizers
Types: a) Symmetrical type of mid rise quantizer
b) Symmetrical type of mid tread quantizer
2.
Non uniform Quantizers
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 ie when all
values within the Dynamic Range of the quantizer are equally likely.
Symmetrical type of mid rise quantizer
a) Symmetrical type of mid rise Quantizer b) Quantization error
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Origin lies in the middle of a rising part of the staircase graph like. Note that in mid rise
type, any input value in between 0 to Δ is mapped to an output value of Δ/2, any input
value between Δ to 2Δ is mapped to an output value of 3Δ/2 and so on.
Mid rise characteristic is desirable because of symmetry and because it uses the 2v levels
of a v bit coder efficiently. A disadvantage of this mid rise characteristic is that it cannot
represent a zero output level.
Symmetrical type of mid tread quantizer
a) Symmetrical type of mid tread Quantizer b) Quantization error
Origin lies in the middle of a tread of a staircase like graph. Note that in mid tread type,
any input value in between -Δ/2 to + Δ/2 is mapped to an output value of zero, any inputvalue between + Δ/2 to 3 Δ/2 is mapped to an output value of Δ and so on.
Unfortunately, this characteristic has an odd number of levels (if it is symmetric) or it
must be non symmetric about zero. Therefore it does not use the 2 v possible levels of a v
bit coder efficiently.
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Illustration of Quantization process for an analog signal & discrete time signal and
error signal in the approximations
Fig.: (a) An analog signal and its quantized version (b) The error signal
Fig: (c) Equispaced samples of m(t ) (d) Quantized sample sequence
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NOISE IN PCM SYSTEMS
The performance of a PCM system is influenced by two major sources of noise.
1. TRANSMISSION NOISE: It is introduced anywhere between the transmitter output
and the receiver input. The effect of transmission noise is to introduce bit errors into
the received PCM wave, with the result that, in case of a binary system, a symbol 1
occasionally is mistaken for a symbol 0, or vice versa. Clearly, the more frequently
such errors occur, the more dissimilar the receiver output becomes compared with the
original message signal.
2. DISTORTION DUE TO QUANTIZING
There are two types of distortions associated with a quantizer:
1. Overload or clipping distortion: Overload distortion occurs when the input signal
exceeds the quantizer's input range, then output will remain at its maximum (or
minimum) value until the input falls within the quantizer's input range. Overload
distortion results in a clipped output signal. To avoid clipping, a quantizer is
matched to the input signal.
2. Quantization distortion: Figure below shows the error signal introduced by the
quantizer. From this figure, it can be seen that quantization error occurs when the
input signal is within the input range of the quantizer. It arises because of the
difference between the input amplitude and the quantized sampled amplitude and
because of the limited sampling rate. The quantization error signal produces
quantization noise or distortion in the reconstructed message signal. Its frequency
spectrum covers a large bandwidth. Low-pass filtering which is used to smooth the
waveform will remove most of the quantization error above its cutoff frequency.
However, some of the quantization error is in the signal band, and that cannot be
removed by the low-pass filter. This will produce a gritty sound at the output of a
PCM system called quantization noise.
Figure: Characteristic of quantization and overload errors.
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Quantization noise is the result of the quantization process. Since the quantization
process adjusts the height of each sample, the original waveform cannot be exactly
reconstructed using a low-pass filter as is the case with PAM signals and the classical
sampling theorem. The sampling rate will also affect the quantization noise since the
quantization error will become larger as the sampling rate decreases.
Figure below shows an analog input signal and its quantized waveform. Shown
below this is the resulting quantization error signal. The maximum amplitude of this error
signal is half a quantization interval. The overall amplitude variation is from half a
quantization interval to minus half a quantization interval. During a period of small
intervals, the error signal appears to be a sawtooth wave.
Figure: Analog input signal, quantized waveform, and quantization error waveform.
Quantization error is another reason for using compressed encoding for digitizing
a voice signal. Compressed encoding allows a higher signal-to-quantization-noise ratio
(SNQR) than linear encoding. This ratio defined as where S is the voice signal level and
NQ is noise due to the quantization error. Clearly, keeping the quantization error small is
key to keeping a high SNQR. As signal amplitude gets smaller, NQ must get smaller to
keep SNQR from dropping. Compression accomplishes this by forcing quantization error
magnitude to decrease with lower amplitudes.
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Illustration of how quantization error is reduced by increasing quantization levels
When a signal is quantized, we introduce an error since the coded signal is an
approximation of the actual amplitude value. The difference between actual and coded
value (midpoint) is referred to as the quantization error.
The more zones, the smaller which results in smaller errors. But, the more zones, the
more bits required to encode the samples which leads higher bit rate.
Example:
In the above example, increasing the no of quantization levels from 5 to 10, decreases the
step size by 2, there by decreases the quantization error.
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NON UNIFORM QUANTIZING
In this step size of the quantizer is not fixed over entire input range and it varies
according to the input signal. i.e step size of the quantizer is reduced at low levels and
increased at high levels.
Non uniform Quantizer of 8 levels
Importance of Non uniform Quantization: Voice signals are more likely to have
amplitudes near zero than at extreme peaks.. Signals with lower amplitude values will
suffer more from quantization error as the error range: /2, is fixed for all signal levels.
Non linear quantization is used to alleviate this problem. The Goal is to keep SNQR fixed
for all sample values.
Two approaches for obtaining Non uniform Quantization:
Direct approach:
The quantization levels follow a logarithmic curve. Smaller ‟s at lower amplitudes and
larger ‟s at higher amplitudes. But this process of varying directly is very difficult.
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Indirect approach:
An Effect of non linear quantizing can be can be obtained by first passing the
sample values through a compressor at the sender, then through a uniform quantizer. This
technique increase amplitudes near zero. To compensate the effects happened at the
sender, pass the sample values through an expander at the receiver. The process of
compression, uniform quantization and expansion is called Companding.
A-law and µ -law Companding• These two are standard companding methods.
• µ -Law is used in North America and Japan
• A-Law is used elsewhere to compress digital telephone signals
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Two types according to compression filter
– -law : used in US
– A-law : used in Europe
A-law & µ-law compression curve
Similarities between A−law and µ −law
Both are linear approximations of logarithmic input/output relationship.
Both are implemented using eight−bit code words (256 levels, one for each
quantization interval).
Eight−bit code words allow for a bit rate of 64 kilobits per second (kbps). This is
calculated by multiplying the sampling rate (twice the input frequency) by the size
of the code word (2 x 4 kHz x 8bits = 64 kbps).
Both break a dynamic range into a total of 16 segments
ln(1 )sgn( )
ln(1 )
x y x
1sgn( ), 01 ln
1 ln( )1sgn( ), 1
1 ln
A x x x
A A y
A x x x
A A
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Differences Between A−law and µ −law
Different linear approximations lead to different lengths and slopes.·
A−law provides a greater dynamic range than u−law.
u−law provides better signal/distortion performance for low level signals than
A−law.
A−law requires 13−bits for a uniform PCM equivalent. u−law requires 14−bits for
a uniform PCM equivalent
SNR of Compander
Example: µ-law Companding
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ENCODING
The output of the quantizer is one of “q” possible signal levels. If we want to use
a binary transmission system, then we need to map each quantized sample into a “v” bit
binary word.
Encoding is the process of representing each quantized sample by an 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.
With gray codes adjacent samples differ only in one bit position.
The weakness of Gray codes is poor performance when the sign bit (MSB) is
received in error.
Example (3 bit quantization):
With this gray code, a single bit error will result in an amplitude error of only 2.
Unless the MSB is in error.
There are several ways by which binary symbols 1 and 0 can be represented by electrical
signals:
Unipolar NRZ (on-off signaling): Symbol 1 is represented by transmitting a pulse of
constant amplitude for the duration of symbol, and symbol 0 is represented by switching
off the pulse. This type of signal is referred to as an on-off signaling or Unipolar non
return to zero.
Polar NRZ: Symbols 1 and 0 are represented by pulses of equal positive and negative
amplitudes. This type of signal is referred to as a polar Non Return to Zero signal.
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Unipolar RZ: A rectangular pulse (half symbol wide) is used for a 1 and no pulse for a
0. This type of signal is called Unipolar Return to zero.
Bipolar RZ: Positive and negative pulses are used alternatively for symbol 1, and no
pulse for symbol 0. This type of signal is called a bipolar signal.
Manchester or Split-phase code: Symbol 1 is represented by a positive pulse followed
by a negative pulse, with both pulses being of equal amplitude and half-symbol wide; for
symbol 0, the polarities of these pulses are reversed. This type of signal is called a split
phase or Manchester code.
Electrical representations of binary data
Reasons and advantages of different encodings will be discussed in UNIT 4
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Bit rate of PCM
The bit rate of a PCM signal can be calculated form the number of bits per sample x the
sampling rate. i.e. Bit rate = v x f s
Bandwidth requirements of PCM
The bandwidth of (serial) binary PCM waveforms depends on the bit rate R and the
waveform pulse shape used to represent the data.
For no aliasing case (f s≥ 2B), the MINIMUM Bandwidth of PCM is:
B pcm(Min) = R/2 = vf s//2.
The Minimum Bandwidth of vf 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
A digitized signal will always need more bandwidth than the original analog signal. Price
we pay for robustness and other features of digital transmission.
EXAMPLE: 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 anti aliasing 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 Bit rate = v x f s = 8 x 8k = 64k bit/sec
This 64-kbit/s signal is called a DS-0 signal (digital signal, type zero).
The minimum absolute bandwidth of the binary PCM signal when sin(x)/x pulse is
used to generate is B pcm(Min) = R/2 = vf s//2 = 32k bit/sec
If we use a rectangular pulse for sampling the first null bandwidth is given by
B pcm(Min) = R = vf s = 64k bit/sec
We require a bandwidth of 64 kHz to transmit this digital voice PCM signal, whereas the
bandwidth of the original analog voice signal was, at most, 4 kHz.
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APPLICATIONS OF PCM
With the advent of fibre optic cables, PCM is used in telephony.
In space communication, space craft transmits signal to earth. Here the ransmitted
power is quite small and the distances are very large.Hence due to high noise
immunity, only pcm systems can be used in such applications.
ADVANTAGES OF PCM
• 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.
DRAW BACKS OF PCM
Encoding, Decoding and quantizing circuitry of PCM is complex
PCM requires a large bandwidth as compared to other systems
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QUANTIZATION ERROR/NOISE IN PCM
Quantization error is defined as the difference between actual sample and quantized
sample. i.e ( ) ( ) s q s x nT x nT
If the step size of the quantizer (mid rise or mid tread) is , then maximum quantization
errormax is
2
and the range of quantization error is ,
2 2
.
As the error is equally likely in the range ,2 2
, it is better to assume error as uniform
random variable. The probability density function of this error is given by
1 1( )
2 2
f
Mean Square value of this quantization error (Noise power) is given by
22 22 2 2
2 2
1[ ] ( )
12 E f d d
SIGNAL TO QUANTIZATION NOISE RATIO IN PCM
Case 1: input signal is sinusoidal signal ( ) sinm m x t A t
SNR = Signal Power (rms) / Quantization noise power =
2
22
12
m A
Where2 2
2
m m
v
A Arangeofinputsignal
noofQuantiztionlevels q
SNR = Signal Power / Quantization noise power
=
2
2
2
12
m A
=
2
2
2
32
222
2
12
m
v
mv
A
A
or23
22
m
BWpcm f
SNR in decibels = 2103
10 log ( 2 ) 1.76 62
v v
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Case 2: Input signal is a DC signal ranging between – Am to +Am
SNR= Signal Power / Quantization noise power =2
2
12
m A
Where2 2
2
m m
v
A Arangeofinputsignal
noofQuantiztionlevels q
SNR =2
2
12
m A
=2
2
2 3* 2
2
2
12
vm
m
v
A
A
SNR in decibels = 21010 log (3* 2 ) 4.76 6v v .
Note: Signal to noise ratio of PCM system improved by 6db for every one bit increase.
PCM TRANSMISSION PATH & REGENERATION
The path between the PCM transmitter and PCM receiver over which the PCM signal
travel, is called as PCM transmission path. The most important feature of PCM system
lies in its ability to control the effects of distortion and noise when the PCM signal travels
on the channel. This capability is accomplished by reconstructing the PCM wave by
means of a chain of regenerative repeaters located at sufficiently close spacing along the
transmission route.
There are three basic functions are performed by a regenerative repeater, namely
1.
Equalization
2. Timing
3. Decision Making
The equalizer shapes the received pulses so as to compensate for the effects of amplitude
and phase distortions produced by the transmission characteristics of the channel.
The timing circuitry provides a periodic pulse train, derived from the received pulses, for
sampling the equalized pulses at the instants of time where the signal to noise ratio is a
maximum.
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The decision making device makes a decision in the favor of 1, if the equalized pulse plus
noise is above the threshold level and it makes a decision in the favor of 0 if the equalized
pulse plus noise is below the threshold level.
PCM RECEIVER
The first operation in the receiver is to regenerate the received pulses. These clean
pulses are then regrouped into code words and decoded into a quantized PAM signal. The
decoding process involves generating a pulse the amplitude of which is the linear sum of
all the pulses in the code word, with each pulse weighted by its place value in the code.
The final operation in the receiver is to recover the signal wave by passing the
decoder output through a low-pass reconstruction filter whose cutoff frequency is equal
to the message bandwidth W. Assuming that the transmission path is error free, the
recovered signal includes no noise with the exception of the initial distortion introduced
by the quantization process
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TIME DIVISION MULTIPLEXING PAM SYSTEM
Normally, in PAM system, the duration of the pulse is much less than the time
period of pulses Ts. Thus no information is being transmitted through the system for most
of the time. The time space Ts – can be utilized to transmit information from other
signals. The signal numbers 2, 3 and 4 are transmitting information with the help of
samples numbered 2, 3 and 4 respectively. This is along with the samples numbered 1 of
the signal number 1.
The time period Ts is equally divided between the four signals, thus allocating a
time slot of
4to each signal. Thus the duration of time slot is such that
4 > . Thus,
there is a guard time
4- between all successive sampling pulses, ensuring that there is
less cross talk between signals. The arrangement by which the information from morethan one signal is transmitted in this manner is known as time division multiplexing.
A TDM PAM system is shown in figure below, which transmits information from
n signals. The switch 1 and switch 2 respectively known as commutator and
decommutator are synchronized electronic switches which rotate at the same speed of 2f M
rotations per second. The commutator samples and combines the samples, while the
decommutator seperates the samples belonging to individual signals.
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Synchronization is the most crucial in TDM system. Thus, for example, if the
commutator is at position 2, the decommutator must also be in position 2. To provide
synchronization, a synchronizing puse is transmitted in every frame (time interval
between two successive samples of the same signal, i.e Ts).
Thus to multiplex n channels, n+1 time slots are provided in a frame; n for
channels and 1 for the synchronizing pulse. The synchronizing pulse is chosen in such a
way that it is easily distinguishable. For this purpose, one of its properties is adjusted in
such a way that it is never attained by the other pulses. For example, in case of PAM, its
amplitude is made larger than the amplitudes of all the other pulses.
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DIFFERENTIAL PULSE CODE MODULATION
When a voice or video signal is sampled at a rate slightly higher than the nyquist
rate, the resulting sampled signal is found to exhibit a high correlation between adjacent
samples. The meaning of this high correlation is that, in an average sense, the signal does
not change rapidly from one sample to next with the result that the difference between
adjacent samples has a variance that is smaller than the variance of the signal itself.
When these highly correlated samples are encoded, as in standard PCM system,
the resulting encoded signal contains redundant information. By removing this
redundancy before encoding, we obtain a more efficient coded signal.
For example, we can observe that the samples taken at 4Ts, 5Ts and 6Ts are
encoded to same value of 110. This information can be carried only by one sample. But
three samples are carrying the same information means that it is redundant. Cosider
another example of samples taken at 9Ts and 10Ts. The difference between these samples
only due to last bit and first two bits are redundant, since they do nit change.
If this redundancy is reduced, then overall bit rate will decrease and number of
bits will decrease and number of bits required to transmit one sample will also be
reduced.
Redundant information in PCM
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DPCM TRANSMITTER
DPCM can be treated as a variation of PCM; it also involves the three basic steps
of PCM, namely, sampling, quantization and coding. But, in the case of DPCM, what is
quantized is the difference between the actual sample and its predicted value, as
explained below.
Let x(t) represent the analog signal that is to be DPCM coded, and let it be
sampled with a period Ts. The sampling frequency fs = 1/Ts is such that there is no
aliasing in the sampling process. Let x(nTs) = m(t) at t= nTs. Quite a few real world
signals such as speech signals, biomedical signals (ECG, EEG, etc.), telemetry signals
(temperature inside a space craft, atmospheric pressure, etc.) do exhibit sample-to-sample
correlation. This implies that x(n) and x(n + 1) (or x( (n) and x (n − 1)) do not differ
significantly. In fact, given a set of previous M samples, say x (n − 1), x (n − 2),…… x (n
− M ) , it may be possible for us to predict (or estimate) x (n) to within a small percentage
error.
DPCM transmitter
Let ( ) s x nT denote the predicted value of x(nTs) and let( ) ( ) ( )
s s se nT x nT x nT
Which is the difference between the unquantized input sample m(nTs) and a prediction
of it, denoted by ( ) s x nT . This predicted value is produced bu using a prediction filter
whose input, as we will see, consists of a quantized version of the input signal x (nTs).
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The difference signal ( ) se nT is called a prediction error, since it is the amount by which
the prediction filter fails to predict the input exactly.
In DPCM, error sequence is quantized, coded and obtained a variation of PCM,
which is known as differential pulse code modulation.
The quantizer output may be expressed as ( ) ( ) ( )q s s e se nT e nT q nT , where
( )e sq nT is the quantization error.
According to fig, the quantizer output ( )q se nT is added to the predicted value
( ) s x nT to produce the prediction filter input ( ) ( ) ( )q s s q s x nT x nT e nT .
( ) ( ) ( ) ( )q s s s e s x nT x nT e nT q nT
( ) ( ) ( )q s s e s x nT x nT q nT That is irrespective of the properties of the prediction filter, the quantized signal
( )q s x nT at the prediction filter input differs from the original input signal ( ) s x nT by the
quantizing error ( )e sq nT . Accordingly if prediction is good, the variance of the prediction
error ( )q se nT will be smaller than the variance of ( ) s x nT .
DPCM RECEIVER
The receiver for reconstructing the quantized version of the input is shown in figure. 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 as used in the transmitter.
DPCM receiver
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Output Signal to Noise ration of the DPCM system:
By definition,
(SNR)o =Variance of the input signal/Variance of the quantized noise =2
2
X
Q
=2 2
2 2* X E
E Q
Where 2 E
is the variance of the prediction error.
(SNR)o =2 2
2 2* X E
E Q
= G p * prediction error to quantization noise ratio.
Where Gp is the predictive gain. This prediction gain must be high as possible. This Gp is
maximized by minimizing the variance
2
E of the prediction error.
THE PREDICTION FILTER
The predicted value ( ) s x nT is modeled as a linear combination of past values of the
quantized input as shown below
1
( ) ( ) p
s k q s s
k
x nT w x nT kT
Where the tapped delay line weights w1, w2, w3………w p define the desired predictionfilter coefficients and p is order of the prediction filter.
Tapped delay line filter used as prediction filter
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The prediction error1
( ) ( ) ( ) p
s s k q s s
k
e nT x nT w x nT kT
The variance of the prediction error is therefore 2 E
= 2[ ( )] s E e nT
2 2
1
[ ( )] [{ ( ) ( )} ]
p
s s k q s s
k
E e nT E x nT w x nT kT
In order to choose a set of weights that minimize the variance 2 E
, we must differentiate
2
E with respect to each weight and then put the resulting derivatives equal to zero.
ADVANTAGES OF DPCM
1. As the difference is being encoded and transmitted by the DPCM technique, a
small difference voltage is to be quantized and encoded.
2. This will require less number of quantization levels and hence less number of bits
to represent them
3. Thus signaling rate and bandwidth of a DPCM system will be less than that of
DPCM.
COMPARISON BETWEEN PCM AND DPCM
Parameter of
comparison
Pulse code modulation Differential Pulse Code
Modulation
Number of bits It can use 4, 8 or 16 bits per sample Bits can be more than one but
are less than PCM
Quantization error Quantization error depends on
number of levels used.
Quantization error is present
Transmission
Bandwidth
Highest bandwidth is required since
number of bits are high
Bandwidth requires is lower
than PCM
Feed back There is no feedback in transmitter
and receiver
Here, feedback exists
Complexity of
implementation
System Complex Simple
Signal to noise ratio Good Fair
Applications Audio and video telephony Speech and video
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Communication Systems- Simon haykin 4th edition Exercise Problems
Problem1: A speech signal has a total duration of 10 s. It is sampled at the rate of 8 kHz
and the encoded. The signal to Quantization noise ratio is required to be 40db. Calculate
the minimum storage capacity needed to accommodate this digitized speech signal.
Solution: The minimum number of bits per sample is 7 for a signal to quantization noise
ratio of 40 dB. Hence
The number of samples in a duration of 10 s = 8000*10 = 8*104 samples.
The minimum storage is therefore = 7*8*104= 560 Kbits
Problem 2 : A PCM system uses a uniform quantizer followed by a 7 bit binary encoder.
The bit rate of the system is equal to 650 10 /b s .
(a) What is the maximum message bandwidth for which the system operates
satisfactorily?
(b) Determine the output signal to quantization noise ratio when a full- load
sinusoidal modulating wave of frequency 1 MHz is applied to the input.
Solution: (a)
Bit rate of the PCM system is given by s R vf
For the system to operate satisfactorily, sampling rate must be atleast equal to the
nyquist rate. Hence max2 s R vf v f
650 10 /b s = max7 2 f
max f = 3.57*106 Hz
(c) The output signal to Quantizing noise ratio is given by SNR in dB= 1.8 + 6v=
43.8 dB
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Problem 3: (a) A sinusoidal signal, with an amplitude of 3.25 volts, is applied to a
uniform quantizer of the mid rise type whose output takes on the values 0, ±1, ±2,±3
volts. Sketch the waveform of the resulting quantizer output for one complete cycle of the
input.
(b) Repeat this evaluation for the case when the quantizer is of the midrise type
whose output takes on the values 0.5, ±1.5, ±2.5, ±3.5 volts.
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Problem 4: The signal m (t) = 6sin (2πt) volts is transmitted using a 4 bit binary PCM
system. The quantizer is of the midrise type, with a step size of 1 volt. Sketch the
resulting PCM wave for one complete cycle of the input. Assume a sampling per second,
with samples taken at t = ±18, ±3/8, ±5/8,……, seconds.
Solution:
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UNIT2- DELTA MODULATION
Delta modulation, like DPCM is a predictive waveform coding technique and can
be considered as a special case of DPCM. It uses the simplest possible quantizer, namely
a two level (one bit) quantizer. The price paid for achieving the simplicity of the
quantizer is the increased sampling rate (much higher than the Nyquist rate) and the
possibility of slope-overload distortion in the waveform reconstruction, as explained in
greater detail later on in this section. In DM, the analog signal is highly over-sampled in
order to increase the adjacent sample correlation. The implication of this is that there is
very little change in two adjacent samples, thereby enabling us to use a simple one bit
quantizer, which like in DPCM, acts on the difference (prediction error) signals. In its
original form, the DM coder approximates an input time function by a series of linear
segments of constant slope. Such a coder is therefore referred to as a Linear (or non-
adaptive) Delta Modulator (LDM). Subsequent developments have resulted in delta
modulators where the slope of the approximating function is a variable. Such coders are
generally classified under Adaptive Delta Modulation (ADM) schemes. We use DM to
indicate either of the linear or adaptive variety.
LINEAR DELTA MODULATION
PRINCIPAL OF WORKING
The principle of operation of an LDM system can be explained with the help of
Fig 2.1 below. The signal x (t), band limited to W Hz is sampled at the rate 2 s f W .
If x(nTs) denote the sample of x(t) at t= nTs. The staircase approximation to x(t), denote
by ( ) s x nT is arrived as follows. One notes, at t=nT s, the polarity of the difference
between x(nTs) and the latest approximation to it; that is ( ) s x nT at t= nTs.
The difference between the input and the previous approximation is quantized
into only two levels, namely, , corresponding to positive and negative differences,
respectively. Thus, if the approximation falls below the signal at any sampling epoch, it is
increased by . If on the other hand, the approximation lies above the signal, it is
diminished by . Provided that the signal does not change too rapidly from sample to
sample, we find that the staircase approximation remains within of the input signal.
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Figure 2.1 Illustration of Delta Modulation
If ( ) 0 se nT then ( )q se nT & ( ) 1 sb nT
If ( ) 0 se nT then ( )q se nT & ( ) 0 sb nT
DELTA MODULATION TRANSMITTER
The principal virtue of delta modulation is its simplicity. It may be generated by applying
the sampled version of the incoming baseband signal to a modulator that involves a
summer, quantizer and accumulator interconnected as shown in figure 2.2.
Fig.2.2 Delta Modulation Transmitter
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Denoting the input signal as x(t) and the staircase approximation as xq(t), the basic
principal of delta modulation may be formalized in the following set of discrete-time
relations.
( ) ( ) ( ) 1
( ) sgn( ( )) 2
s s q s s
q s s
e nT x nT x nT T eq
e nT e nT eq
& ( ) ( ) ( ) 3q s q s s q s x nT x nT T e nT eq
where Ts is the sampling period; e(nTs) is an error signal representing the difference
between the present sample value x(nTs) of the input signal and the latest approximation
to it. Namely, ( ) ( ) s q s s x nT x nT T ; and ( )q se nT is the quantized version of e(nTS).
The quantized output ( )q se nT is finally coded to produce the desired DM wave.
Figure 2.1 illustrates the way in which the staircase approximation ( )q x t follows
variations in the input signal x (t) in accordance with above equations and it also displays
the corresponding binary sequence at the delta modulator output
Working of Accumulator (Stair case wave form generator)
1. In particular, quantizer consists of a hard limiter with input and output relation
defined by eq2 which is depicted in fig 2.2.1. The quantizer output is applied to an
accumulator, producing the result1 1
( ) sgn( ( )) ( )n n
q s s q s
i i
x nT e iT e iT
.
Fig 2.2.1: Input – output characteristic of quantizer for DM system
2. Thus at the sampling instant nTs, the accumulator increments the approximation
by a step in a positive or negative direction, depending upon the algebraic sign
of error signal e(nTs).
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3. If the input signal x(nTs) is greater than the most recent approximation ( ) s x nT , a
positive increment + is applied to the approximation.
4. If on other hand, the input signal is smaller, a negative increment - is applied to
the approximation.5. In this way the accumulator does the best it can to track the input samples by one
step at a time.
DELTA MODULATION RECEIVER
In the receiver, the staircase approximation xq(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. Then pass this staircase waveform
through a low pass filter (with a bandwidth equal to Original signal bandwidth) to recover
the original signal.
Fig.2.3 Delta Modulation Receiver
In comparing the DPCM and DM networks, we note that they are basically similar,
except for two important differences, namely, the use of a one-bit quantizer in delta
modulator and the replacement of the prediction filter by a single delay element.
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QUANTIZING NOISE
Delta modulation systems are subject to two types of quantizing error.
(1) Slope overload distortion
(2) Granular Noise
SLOPE OVERLOAD DISTORTION:
This distortion arises because of large dynamic range of the input signal. The rate of
rise of input signal x(t) is so high that the staircase signal cannot approximate it. The
slope overload is said to occur when the step size „Δ‟ is too small to follow steep
segment of the input waveform x(t). To reduce this error, the step size must be
increased when slope of the signal x(t) is high. Since the step size of delta modulator
remains fixed, its maximum or minimum slopes occur along straight lines. Therefore
this modulator is also known as Linear Delta Modulator.
Quantiztion errors in delta modulation for an arbitrary input
To reduce this slope overload distortion, the slope of the quantizer must be greater
than the maximum slope of the input signal.
Ie.max
( )
imum s
dm t
T dt
GRANULAR NOISE (IDLE NOISE):
Granularity, on other hand refers to a situation where the stair case function ( ) s x nT
hunts around a relatively flat segment of the input function, with a step size that is too
large relative to local slope characteristic of the input. This means that for very small
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DRAWBACKS OF DELTA MODULATION
The Delta Modulation has two major drawbacks as under;
(i) Slope overload distortion
(ii)
Granular Noise
SIMON HAYKIN Problem1: Given a sine wave of frequency f m and amplitude Am
applied to a delta modulator having step size Δ. Show that the slope overload will
occur if2
m
m s
A f T
here Ts is the sampling period.
Solution: Let us consider that the sine wave is represented as ( ) sin(2 )m m x t A f t
Maximum slope of delta modulator is given as sT
.
We know that, the slope overload distortion will take place if slope of the sine wave
is greater than slope of delta modulator i.e.,( )
max dx t
dt >
sT
sin(2 )max m m
s
dA f t
dt T
max 2 cos(2 )m m m s
f A f t T
2 m m f A > sT
2m
m s
A f T
Note:
To avoid slope overload distortion, the condition that must be satisfied is2
m
m s
A f T
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QUANTIZATION ERROR/NOISE IN DELTA MODULATION
Quantization error is defined as the difference between actual sample and quantized
sample. i.e ( ) ( ) s q s x nT x nT
If the step size of the quantizer is , then maximum quantization error max is and the
range of quantization error is , .
As the error is equally likely in the range , , it is better to assume error as uniform
random variable. The probability density function of this error is given by
1 1
( )2
f
Mean Square value of this quantization error (Noise power) is given by2
2 2 2 1[ ] ( )3
E f d d
SIGNAL TO QUANTIZATION NOISE RATIO IN DELTA MODULATION
Case 1: input signal is sinusoidal signal ( ) sinm m x t A t
SNR = Signal Power (rms) / Quantization noise power =
2
2
2
3
m A
No slope overload distortion occurs, for2
m
m s
A f T
, then substituting into the above
equation gives SNR =
2
2
2
3
m A
=
2
2
1
2 2
3
m s f T
This noise power2
3
is uniformly distributed over the frequency band upto s f (which is
more than m f ). Then the output quantization power within the bandwidth BWLPF f is given
by2
'
3
BWLPF q
s
f N
f
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In the receiver, at the output of Low pass filter of Bandwidth BWLPF f
SNR =
2
3
2 2 2
1
2 2 3
8
3
m s s
BWLPF BWLPF m
s
f T f
f f f
f
ADAPTIVE DELTA MODULATION
To reduce slope overload distortion, a large step size is required to accommodate
wide dynamic range of the input signal and small steps are required to reduce granular
noise. In fact, adaptive delta modulation is the modification to overcome these errors.
Finally, we should mention that a delta modulator may also be made adaptive,
wherein the variable step size increases during a steep segment of the input signal and
decreases when the modulator is quantizing an input signal with a slowly varying
segment. In this way the step size is adapted to the level of the input signal. The resulting
system is called an adaptive delta modulator.
The problem in adaptive delta modulation, of course, is to specify suitable rules
for step size variation. Figure below illustrates the operation of ADM.
Waveforms illustrative of ADM operation
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ADAPTIVE DELTA MODULATION TRANSMITTER:
The logic for step size control is added in the diagram. The step size increases or
decreases according to a specified rule depending on one bit quantizer output. As an
example, if one bit quantizer output is high (i.e. 1), then the step size may be doubled for
next sample. If one bit quantizer output is low, then step size may be reduced by one step.
ADAPTIVE DELTA MODULATION RECEIVER:
In the receiver of Adaptive delta modulator shown in figure, there are two portions. The
first portion produces the step size from each incoming bit. Exactly the same process is
followed as that in transmitter. The previous input and present input decides the step size.
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It is then applied to an accumulator which builds up staircase waveform. The low pass
filter then smoothens out the staircase waveform to reconstruct the original signal.
ADVANTAGES OF ADAPTIVE DELTA MODULATION
1.
Signal to Noise ratio becomes better than ordinary delta modulation because of
the reduction in slope overload distortion and idle noise.
2. Because of the variable step size, the dynamic range of ADm is wider than simple
DM.
3. Bandwidth required for the transmission through channel is also less.
Results have been reported in the literature which compares the (SNR)o performance of
μ -law PCM and the ADM scheme discussed above. One such result is shown in Fig.
below for the case of band pass filtered (200-3200 Hz) speech. For PCM telephony, thesampling frequency used is 8 kHz. As can be seen from the figure, the SNR comparison
between ADM and PCM is dependent on the bit rate. An interesting consequence of this
is, below 50 kbps, ADM which was originally conceived for its simplicity, out-performs
the logarithmic PCM, which is now well established commercially all over the world. A
60 channel ADM (continuous adaptation) requiring a bandwidth of 2.048 MHz (the same
as used by the 30 channel PCM system) was in commercial use in France for sometime.
French authorities have also used DM equipment for airborne radio communication and
air traffic control over Atlantic via satellite. However, DM has not found wide-spread
commercial usage simply because PCM was already there first!
Performance of PCM and ADM versus bit rate
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sT
> Aβ sech2(βt)
> AβTs since the maximum value of sech(βt) is 1 at t=0
Problem4: Consider a DM system designed to accommodate analog message signals
limited to bandwidth W= 5kHz. A sinusoidal test signal of amplitude A=1 volt and
frequency f m = 1 kHz is applied to the system. The sampling rate of the system is 50 kHz
(a) Calculate the step size required to minimize slope overload distortion.
(b) Calculate signal to Quantization noise ratio of the system for the specified
sinusoidal test signal.
Solution: (a) To avoid slope overload distortion s
T > 2 m m f A
Therefore = 2 m m s f A T = 2 m m
s
f A
f
=
2 1 1
2 50
k v
k
=0.126 v
(c) Signal to Quantization noise ratio
SNR
3
2 2
3
8
s
BWLPF m
f
f f =
3
2 2
3 (50 )
8 5 (1 )
k
k k
= 475
SNR in db = 10 47510log = 26.8 db
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Problem 5: Consider a low pass signal with a bandwidth of 3 kHz. A linear delta
modulation system with step size Δ=0.1 v is used to process this signal at a sampling
rate ten times the nyquist rate.
(a) For linear delta modulation, the maximum amplitude of a sinusoidal test signal of
frequency 1 kHz which can be processed by the system without slope-overload
distortion.
(b)For the specifications given in part a, evaluate the output signal to noise ratio
under (i) prefilterd and (ii) postfiltered conditions
Solution: (a) For linear delta modulation, the maximum amplitude of a sinusoidal test
signal that can be used without slope overload distortion is
2m
m s
A f T
=
2
s
m
f
f
=
0.1 10 2 3
2 1
k
k
=0.95 v
(b) (i) Under the pre-filtered condition, it is reasonable to assume that the granular
quantization noise is uniformly distributed between – Δ and +Δ. Hence the
variance of the quantization noise is
22 2 2 1[ ] ( )
3 E f d d
When input signal is sinusoidal signal ( ) sinm m x t A t
SNR =
2
22
3
m A
=
2
2
0.95
20.1
3
= 135= 21.3 db
(iii) The signal to noise ratio under the post filtered condition is
SNR
3
2 2
3
8
s
BWLPF m
f
f f
3
2 2
3 (60 )
8 3 (1 )
k
k k
=1367== 31.3 db
The filtering gain in signal to noise ratio due to the use of a reconstruction
filter at the demodulator output is therefore 31.3 db- 21.3 db= 10db.
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Problem 6: A linear delta modulator has a step size of 100 mV and the minimum output
amplitude is + 50 mV. A signal s(t ) = 0.5 u(t ) is applied to the input of the delta
modulator. Show how the modulator tracks the input indicating the distortions in the
waveform. Sketch the waveform for 12 clock cycles, beginning at least 2 clock cycles
before t = 0. Also, sketch the output waveform in NRZ format.
Solution:
Figure (a) below shows the sketch of the delta modulator input and the tracking
distortions.
The input is a step signal of amplitude 0.5 volts beginning at t = 0 as shown by the heavy
line. The input for t < 0 is 0 volts. Initial amplitude of the DM predictor, at clock instant
1, is assumed to be + 50 mV. The clock instants are shown in (b).
At the clock instant 2 the predictor output is higher than the input (0 V) and hence, a
negative step (-100mV) is added to the predictor output. At clock instant 3 the predictor
output is lower (- 50 mV) than the input (0.5 V) and hence, a positive step is added to the
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predictor. At clock instant 4 the predictor output (+50 mV) is still lower than the input.
Hence, a 100 mV step is added. At clock instants 4, 5, 6, 7, and 8 the predictor output is
lower than input and at each instant a 100 mV step is added to the previous predictor out.
At clock instant 9 the predictor output (550 mV) is found higher than the input. Hence, a
100 mV step is subtracted from the predictor output. At clock instant 10 a 100 mV step is
added. The DM output waveform is shown in figure (c)
Problem 7: A segment of a delta modulated data stream is a sequence given below.
0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1
This sequence is applied to a linear delta demodulator having a step size of 100 mV.
Assuming initial output of the demodulator is 0 V, show the output sample voltages at
each input bit and sketch the waveform.
Solution:
The output of demodulator is obtained by adding the step voltage for input 1 and
subtracting the step voltage for input 0. The output is shown in the table below for the
input bits given above.
Input 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1
Output -0.1 0 -0.1 0 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.3 0.4 0.3 0.2 0.1 0.2
The waveform of the output is shown in figure below.
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Problem 8:
An instantaneously companded delta modulator employs the following step
size adaptation algorithm.
where Sk and Sk-1 are the current and previous step sizes, Bk and Bk-1 are the
current and previous output bits, Bk and 1k B have opposite polarity. The
minimum step size is 100 mV, so the amplitude of the steps when the input
is zero is ±50 mV. If a step input x(t ) = 1.2 V is applied to the modulator at
t =0 show how the predictor output tracks the input by sketching the
waveform. Sketch the binary output waveform of the delta modulator.
Solution:
We can show the step size Sk , predictor output Pk and modulator outputs for some clock
cycles in a tabular form as below. The input of 1.2 V is applied to the modulator at t = 0
and we start at t = -2.
The waveforms of the predictor output and the modulator output are plotted in the figure
below.
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Problem 9: A linear delta modulator is used to digitize speech signal band limited to 3.4
kHz. An output filter with 4 kHz cutoff frequency is used. Find the sampling frequency
required to get a performance equivalent to that of a 6-bit linear PCM coder. Compare the
information rates for PC and DM outputs.
Solution:The S/N obtained from a linear PCM coder is
max
6 1.8 6 6 1.8 37.8S
v db N
The S/N obtained from a linear DM coder is SNR
3
2 2
3
8
s
BWLPF m
f
f f
In db we can write the above equation as SNR dB = 10 log
3
2 2
3
8
s
BWLPF m
f
f f
To get DM performance equivalent to PCM technique SNR of DM must be equal to SNR
of PCM.
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37.8 =2
310 log 30 log 10 log 20 log
8 s BWLPF m f f f
37.8 = 3 323
10 log 30 log 10 log 4 10 20 log 3.4 108
s f
f s = 191 kb/s
Assuming 8kHz sampling the information rate for PCM data is R PCM=8*6=48kb/s
For DM the information rate is same as the sampling rate, hence R DM = 191 kb/s
Thus, DM requires approximately four times the data rate compared to 6-bit PCM for
similar performance.
Problem 10: A stereo music signal is sampled at 44.1 kHz and digitized with 16 bits for
recording on CD. If the CD stores 80 minutes of music find the total capacity of the CD
in bytes. What is the quality of the music if it has an RMS value 15 dB below the peak
value of the quantizer?
Solution:
We have fs = 44.1 kHz, n = 16 and number of channels is 2.
Hence, the bit rate
R = 2.n. fs = 2 x 16 x 44.1 x 103 = 1411.2 kb/s
The capacity of the CD is
C = 80 x 60 x 1411.2 kbits
= 6773760 kbits
= 846.72 Mbytes
The signal to noise ratio is
S / Nq = 6.n +1.8 -15 = 6 x16 +1.8 -15 = 82.8 dB
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Important Questions
1. a) Consider a test signal m(t)= A tanh(ßt) defined by a hyperbolic tangent function.
Where A and ß are constants. Determine the minimum step size ∆ for delta modulation of
this signal, which is required to avoid slope overload.
b) Comparison between PCM and DM.
2. a) A signal to transmitted is of the form S(t)=10COS 1000πt+5COS1500πt.
i) Choose an appropriate f and step size for delta modulator.
ii) Find the SNR for your design.
b) Draw the block diagram for Adaptive delta modulation system and explain each
block?
3. (a) Explain the noise effects in delta modulation
(b) A DM system is designed to operate at 3 times the Nyquist rate for a signal with a
3 KHz BW. The quantization step size is 250mv
(i) Determine the maximum amplitude of a 1 KHz sinusoid for which delta modulator
does not show slope overload.
(ii) Determine post filtered output SNR for the signal at part (i)
4. (a) Derive the condition for step size of the quantizer in a DM system to avoid slope
over load distortion for the message signal x(t) = A cos(wt)
(b) Explain the major drawback of the DM system with relevant waveforms.
5. A sinusoidal modulating signal is represented by m(t)= A cos (wmt) where wm=2πf m.
Derive the expression for the maximum output signal to quantization noise ratio in the
DM system system with no slope overload distortion and also determine maximum
output signal to quantization noise ratio at the post receiver?
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UNIT 3- DIGITAL MODULATION TECHNIQUES
Modulation is defined as the process by which some characteristics of a carrier is
varied in accordance with a modulating signal. In digital communications, the modulating
signal consists of binary data or an M-ary encoded version of it. The data is used to
modulate a carrier wave (usually sinusoidal) with fixed frequency.
The modulation process involves switching or keying the amplitude, frequency or
phase of the carrier in accordance with the input data.
Thus there are three basic modulation techniques for the transmission of digital
data. They are known as amplitude-shift keying (ASK), frequency shift keying (FSK) and
phase shift keying (PSK).
If the amplitude of the carrier is switched depending on the input digital signal,
then it is called amplitude shift keying (ASK). This process is quite similar to analog
amplitude modulation.
If the frequency of the sinusoidal carrier is switched depending upon the input
digital signal, then it is known as the frequency shift keying. This is very much similar to
the analog frequency modulation.
If the phase of the carrier is switched depending upon the input digital signal, then
it is called phase shift keying. This is similar to phase modulation.
ASK, PSK & FSK waveforms (with sine as carrier signal)
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Since the phase and frequency modulation has constant amplitude envelope,
therefore FSK and PSK, the effect of non-linearities, noise interference is minimum on
signal detection. However, these effects are more pronounced on ASK. Therefore FSK
and PSK are preferred over ASK.
Figure shows the waveforms for amplitude-shift keying, phase shift keying and
frequency shift keying. In these waveforms, a single feature of the carrier (i.e. amplitude,
phase or frequency) undergoes modulation.
In digital modulations, instead of transmitting one bit at a time, we transmit two
or more bits simultaneously. This is known as M-ary transmission. This type of
transmission results in reduced channel bandwidth.
However, sometimes, we use two quadrature carriers for modulation. This process is
known as Quadrature modulation.
Thus we see that there are a number of modulation schemes available to the designer of a
digital communication system required for data transmission over a bandpass channel.
Every scheme offers system trade-offs of its own. In particular choice is made in favour
of a scheme which possesses as many of the following design characteristics as possible:
(i) Maximum data rate
(ii) Minimum probability of symbol error
(iii) Minimum transmitted power
(iv) Minimum channel bandwidth
(v) Maximum resistance to interfering signals.
(vi) Minimum circuit complexity.
DEFINITIONS AND TERMINOLOGY
There are basically two types of transmission of digital signals
i) BASEBAND DATA TRANSMISSION:
The digital data is transmitted over the channel directly. There is no carrier or any
modulation. This is suitable for transmission over short distances.
A signal whose frequency content (i.e. its spectrum) is in the vicinity of zero (i.e.,
f = 0 or dc) is said to be a baseband signal.
Original source signal are sometimes said to be baseband. Baseband systems
transmit baseband signals.
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This is usually not an effective means of communication.
ii) PASSBAND (BAND PASS OR NARROW BAND) DATA TRANSMISSION: The digital data
modulates high frequency sinusoidal carrier. Hence it is also called digital CW
modulation. It is suitable for transmission over long distances.
Types of passband Modulation are ASK, PSK, FSK and etc.
Bandpass signal spectrum is nonzero in some band of frequency with BW = 2B
centered about f = ±fc, where fc >> 0.
Effective transmission of signal usually requires bandpass signal.
Bandpass transmission involves some translation of the baseband signal to some
band of frequency centered around fc.
Types of Reception for Passband transmission
There are two types of methods for detection of passband signals.
i) COHERENT (SYNCHRONOUS) DETECTION: In this method, the local carrier generated at
the receiver is phase locked with the carrier at the transmitter. Hence it is also called
synchronous detection.
ii) NON COHERENT (ENVELOPE) DETECTION: In this method, the receiver carrier no need
to be phase locked with transmitter carrier. Hence it is also called envelope detection.
Noncoherent detection is simple but it has higher probability of error.
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Types of Digital Modulation techniques (Classification based on envelope)
Types of Digital Modulation techniques (Classification based on coherence)
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DIGITAL MODULATION TECHNIQUES
As mentioned earlier, the binary (i.e. Digital) modulation has three basic forms
amplitude-shift keying(ASK), phase-shift keying(PSK) and frequency shift keying
(FSK).
BINARY AMPLITUDE SHIFT KEYING (ON-OFF KEYING)
Definition:
Amplitude shift keying (ASK) or ON-OFF keying (OOK) is the simplest digital
modulation technique. In this method, there is only one unit energy carrier and it is
switched on or off depending upon the input binary sequence.
a) Binary Modulating signal and b) BASK signal
a) Modulating signal b) spectrum of „a‟ c) Spectrum of BASK signal
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EXPRESSION:
( ) 2 cos(2 ) s c s t P f t To transmit symbol „1‟
( ) 0 s t To transmit symbol „0‟ i.e. no signal is transmitted.
Signal s(t) contains some complete cycles of carrier frequency „f c‟. Hence ASK waveform looks like an ON-OFF of the signal. Therefore it is also known as
the ON-OFF keying (OOK).
SIGNAL SPACE DIAGRAM (CONSTELLATION DIAGRAM) OF BASK
Study of signal spaces provides us with a geometrical method of conceptualizing the
modulation process.
The ASK waveform of equation for symbol 1 can be represented as,
1
2( ) cos(2 ) ( ) s b c s b
b
s t PT f t PT t T
This means that there is only one carrier function 1( )t which is a unit energy signal over
(0, T b). The signal space diagram will have two points on 1( )t . One will be at zero and
other will be at s b P T . The collection of all possible signal points is called the signal
constellation.
Thus, the distance between the two signal points is d= s b P T = b E
The decision boundary is determined by the threshold value λ. If x lies in the region Z 1,
then a decision of a “1” is made. If x lies in the region Z 2, then a decision of a “0” is
made.
One advantage in using the signal space representation is that it is much easier to identify
the “distance” between signal points. The distance between two signal points will be
increased which makes the received signal point less probable be located in the wrong
region.
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GENERATION OF BASK SIGNAL
ASK signal may be generated by simply applying the incoming binary data and the
sinusoidal carrier to the two inputs of a product modulator. The resulting output will be
the ASK waveform.
Generation of BASK signal
BASK RECEPTION:
COHERENT DETECTION OR DEMODULATION OF BINARY ASK SIGNAL
The demodulation of BASK waveform can be achieved with the help of coherent detector
as shown in figure.
It consists of a product modulator which is followed by an integrator and Decision
making device. The incoming ASK signal is applied to one input of the product
modulator. The other input of the product modulator is supplied with a sinusoidal carrier
which is generated with the help of a local oscillator.
The output of the product modulator goes to input of the integrator. The integrator
operates on the output of the multiplier for successive bit intervals and essentially
performs a low-pass filtering action. The output of the integrator goes to the input of a
decision making device.
.
Coherent detection of Binary ASK signal
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Now, the decision making de