Source Coding: Part 1-Formatting

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Source Coding Part 1-Formatting

• Topics covered from
• Chapter 2 (Digital Communications-Bernard
  Sklar)Chapter 3 (Communication Systems-Simon
  Haykin)

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Layering of Source Coding

• Source coding includes
• Formatting (input data)
• Sampling
• Quantization
• Symbols to bits (Encoding)
• Compression
• Decoding includes
• Decompression
• Formatting (output)
• Bits to symbols
• Symbols to sequence of numbers
• Sequence to waveform (Reconstruction)

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Layering of Source Coding
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Formatting

• The first important step in any DCS
• Transforming the information source to a form
  compatible with a digital system

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Formatting of Textual Data (Character Codes)

• A textual information is a sequence of
  alphanumeric characters
• Alphanumeric and symbolic information are encoded
  into digital bits using one of several standard
  formats, e.g, ASCII, EBCDIC

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Character Coding (Textual Information)

• Example 1
• In ASCII alphabets, numbers, and symbols are
  encoded using a 7-bit code

• A total of 27 128 different characters can be
  represented using
• a 7-bit unique ASCII code

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Formatting of Analog Data

• To transform an analog waveform into a form that
  is compatible with a digital communication, the
  following steps are taken
• Sampling
• Quantization and Encoding
• Base-band transmission (PCM)

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Sampling

• Strictly band limited
• Band unlimited

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Sampling in Frequency Domain
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Sampling Theorem

• The sampling theorem for strictly band-limited
  signals of finite energy in two equivalent parts
• Analysis A band-limited signal of finite energy
  that has no frequency components higher than W
  hertz is completely described by specifying the
  values of the signal at instants of time
  separated by 1/2W seconds.
• Synthesis A band-limited signal of finite
  energy that has no frequency components higher
  than W hertz is completely recovered form
  knowledge of its samples taken at the rate of 2W
  samples per second. (using a low pass filter of
  cutoff freq. W)
• Nyquist rate (fs)
• The sampling rate of 2W samples per second for a
  signal bandwidth of W hertz
• Nyquist interval (Ts)
• 1/2W (measured in seconds)

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Type of Sampling

• Ideal
• Natural
• Practical
• Sample and Hold (Flat-top)

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Ideal Sampling ( or Impulse Sampling)
x(t)x?(t)
x(t)
Ts

• Is accomplished by the multiplication of the
  signal x(t) by the uniform train of impulses
• Consider the instantaneous sampling of the analog
  signal x(t)

• Train of impulse functions select sample values
  at regular intervals

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Ideal Sampling
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Practical Sampling

• In practice we cannot perform ideal sampling
• It is not practically possible to create a train
  of impulses
• Thus a non-ideal approach to sampling must be
  used
• We can approximate a train of impulses using a
  train of very thin rectangular pulses

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Natural Sampling
If we multiply x(t) by a train of rectangular
pulses xp(t), we obtain a gated waveform that
approximates the ideal sampled waveform, known as
natural sampling or gating
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Natural Sampling

• Each pulse in xp(t) has width Ts and amplitude
  1/Ts
• The top of each pulse follows the variation of
  the signal being sampled
• Xs (f) is the replication of X(f) periodically
  every fs Hz
• Xs (f) is weighted by Cn ? Fourier Series
  Coeffiecient
• The problem with a natural sampled waveform is
  that the tops of the sample pulses are not flat
• It is not compatible with a digital system since
  the amplitude of each sample has infinite number
  of possible values
• Another technique known as flat top sampling is
  used to alleviate this problem here, the pulse
  is held to a constant height for the whole sample
  period
• This technique is used to realize Sample-and-Hold
  (S/H) operation
• In S/H, input signal is continuously sampled and
  then the value is held for as long as it takes to
  for the A/D to acquire its value

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Flat-Top Sampling
Time Domain
Frequency Domain
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Flat-Top Sampling
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Aliasing

• Aliasing Phenomenon
• The phenomenon of a high-frequency component in
  the spectrum of the signal seemingly taking on
  the identify of a lower frequency in the spectrum
  of its sampled version.
• To combat the effects of aliasing in practices
• Prior to sampling a low-pass anti-alias filter
  is used to attenuate those high-frequency
  components of a message signal that are not
  essential to the information being conveyed by
  the signal
• The filtered signal is sampled at a rate slightly
  higher than the Nyquist rate.
• Physically realizable reconstruction filter
• The reconstruction filter is of a low-pass kind
  with a passband extending from W to W
• The filter has a non-zero transition band
  extending form W to fstop-W
• Thus use Engr. Nyquist formula

Fig. a
Fig. b
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Fig. a Under-sampled Signal
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Fig. b Over-sampled Signal
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Pulse-Amplitude Modulation (PAM)

• Output of Sampling (natural/SH) is known as PAM
• Pulse-Amplitude Modulation (PAM)
• The amplitude of regularly spaced pulses are
  varied in proportion to the corresponding sample
  values of a continuous message signal.
• Two operations involved in the generation of the
  PAM signal
• Instantaneous sampling of the message signal m(t)
  every Ts seconds,
• Lengthening the duration of each sample, so that
  it occupies some finite value T.

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Other forms of Pulse Modulations
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Other forms of Pulse Modulations

• PDM (Pulse-duration modulation)
• Pulse-width or Pulse-length modulation.
• The samples of the message signal are used to
  vary the duration of the individual pulses.
• PDM is wasteful of power
• PPM (Pulse-position modulation)
• The position of a pulse relative to its
  un-modulated time of occurrence is varied in
  accordance with the message signal.

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Other forms of Pulse Modulations
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Quantization
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Quantization

• Amplitude quantizing Mapping samples of a
  continuous amplitude waveform to a finite set of
  amplitudes.

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Qunatization example
amplitude x(t)
111 3.1867
110 2.2762
101 1.3657
100 0.4552
011 -0.4552
010 -1.3657
001 -2.2762
000 -3.1867
Ts sampling time
t
PCM codeword
110 110 111 110 100 010 011 100
100 011
PCM sequence
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Quantization Effect

• Sampling and Quantization Effects
• Quantization (Granularity) Noise Results when
  quantization levels are not finely spaced apart
  enough to accurately approximate input signal
  resulting in truncation or rounding error.
• Quantizer Saturation or Overload Noise Results
  when input signal is larger in magnitude than
  highest quantization level resulting in clipping
  of the signal.
• Timing Jitter Error caused by a shift in the
  sampler position. Can be isolated with stable
  clock reference.

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Non-uniform Quantization

• Nonuniform quantizers have unequally spaced
  levels
• The spacing can be chosen to optimize the
  Signal-to-Noise Ratio for a particular type of
  signal
• It is characterized by
• Variable step size
• Quantizer size depend on signal size

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• M any signals such as speech have a nonuniform
  distribution
• Basic principle is to use more levels at regions
  with large probability density function (pdf)
• Concentrate quantization levels in areas of
  largest pdf
• Or use fine quantization (small step size) for
  weak signals and coarse quantization (large step
  size) for strong signals

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Non-uniform Quantization
Non-uniform quantization is achieved by, first
passing the input signal through a compressor.
The output of the compressor is then passed
through a uniform quantizer. The combined effect
of the compressor and the uniform quantizer is
that of a non-uniform quantizer. At the
receiver the voice signal is restored to its
original form by using an expander. This
complete process of Compressing and Expanding the
signal before and after uniform quantization is
called Companding.
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Non-uniform Quantization (Companding)
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Non-uniform Quantization (Companding)
The 3 stages combine to give the characteristics
of a Non-uniform quantizer.
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• Basically, companding introduces a nonlinearity
  into the signal
• This maps a nonuniform distribution into
  something that more closely resembles a uniform
  distribution
• A standard ADC with uniform spacing between
  levels can be used after the compandor (or
  compander)
• The companding operation is inverted at the
  receiver
• There are in fact two standard logarithm based
  companding techniques
• US standard called µ-law companding
• European standard called A-law companding

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Nonuniform quantization using companding

• Companding is a method of reducing the number of
  bits required in ADC while achieving an
  equivalent dynamic range or SQNR
• In order to improve the resolution of weak
  signals within a converter, and hence enhance the
  SQNR, the weak signals need to be enlarged, or
  the quantization step size decreased, but only
  for the weak signals
• But strong signals can potentially be reduced
  without significantly degrading the SQNR or
  alternatively increasing quantization step size
• The compression process at the transmitter must
  be matched with an equivalent expansion process
  at the receiver

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• The signal below shows the effect of compression,
  where the amplitude of one of the signals is
  compressed
• After compression, input to the quantizer will
  have a more uniform distribution after sampling

• At the receiver, the signal is expanded by an
  inverse operation
• The process of CO M pressing and exPANDING the
  signal is called companding
• Companding is a technique used to reduce the
  number of bits required in ADC or DAC while
  achieving comparable SQNR

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Input/Output Relationship of Compander

• Logarithmic expression Y log X is the most
  commonly used compander
• This reduces the dynamic range of Y

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Types of Companding? -Law Companding Standard
(North South America, and Japan)

• where
• x and y represent the input and output voltages
• ? is a constant number determined by experiment
• In the U.S., telephone lines uses companding with
  ? 255
• Samples 4 kHz speech waveform at 8,000 sample/sec
• Encodes each sample with 8 bits, L 256
  quantizer levels
• Hence data rate R 64 kbit/sec
• ? 0 corresponds to uniform quantization

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A-Law Companding Standard (Europe, China, Russia,
Asia, Africa)

• where
• x and y represent the input and output voltages
• A 87.6
• A is a constant number determined by experiment

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Pulse Code Modulation (PCM)
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Pulse Code Modulation (PCM)

• Pulse Code Modulation refers to a digital
  baseband signal that is generated directly from
  the quantizer and encoder output
• Sometimes the term PCM is used interchangeably
  with quantization

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Figure 3.13(Communication System-Simon
Haykin)The basic elements of a PCM system.
(Topic 3.7)
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Pulse-Code Modulation

• PCM (Pulse-Code Modulation)
• A message signal is represented by a sequence of
  coded pulses, which is accomplished by
  representing the signal in discrete form in both
  time and amplitude
• The basic operation
• Transmitter sampling, quantization, encoding
• Receiver regeneration, decoding, reconstruction
• Operation in the Transmitter
• Sampling
• The incoming message signal is sampled with a
  train of rectangular pulses
• The reduction of the continuously varying message
  signal to a limited number of discrete values per
  second
• Nonuniform Quantization
• The step size increases as the separation from
  the origin of the input-output amplitude
  characteristic is increased, the large end-step
  of the quantizer can take care of possible
  excursions of the voice signal into the large
  amplitude ranges that occur relatively
  infrequently.

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• Encoding
• To translate the discrete set of sample vales to
  a more appropriate form of signal
• A binary code
• The maximum advantage over the effects of noise
  in a transmission medium is obtained by using a
  binary code, because a binary symbol withstands a
  relatively high level of noise.
• The binary code is easy to generate and
  regenerate

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• Regeneration Along the Transmission Path
• The ability to control the effects of distortion
  and noise produced by transmitting a PCM signal
  over a channel
• Equalizer
• Shapes the received pulses so as to compensate
  for the effects of amplitude and phase
  distortions produced by the transmission
• Timing circuitry
• Provides a periodic pulse train, derived from the
  received pulses
• Renewed sampling of the equalized pulses
• Decision-making device
• The sample so extracted is compared o a
  predetermined threshold
• ideally, except for delay, the regenerated signal
  is exactly the same as the information-bearing
  signal
• The unavoidable presence of channel noise and
  interference causes the repeater to make wrong
  decisions occasionally, thereby introducing bit
  errors into the regenerated signal
• If the spacing between received pulses deviates
  from its assigned value, a jitter is introduced
  into the regenerated pulse position, thereby
  causing distortion.

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Fig.5.13
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Receiver

• Operations in the Receivers
• Decoding and expanding
• Decoding regenerating a pulse whose amplitude
  is the linear sum of all the pulses in the code
  word
• Expander a subsystem in the receiver with a
  characteristic complementary to the compressor
• The combination of a compressor and an expander
  is a compander
• Reconstruction
• Recover the message signal passing the expander
  output through a low-pass reconstruction filter

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Line Coder

• The input to the line encoder is the output of
  the A/D converter or a sequence of values an that
  is a function of the data bit
• The output of the line encoder is a waveform

• where f(t) is the pulse shape and Tb is the bit
  period (TbTs/n for n bit quantizer)
• This means that each line code is described by a
  symbol mapping function an and pulse shape f(t)
• Details of this operation are set by the type of
  line code that is being used

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• Goals of Line Coding (qualities to look for)
• A line code is designed to meet one or more of
  the following goals
• Self-synchronization
• The ability to recover timing from the signal
  itself
• That is, self-clocking (self-synchronization) -
  ease of clock lock or signal recovery for symbol
  synchronization
• Long series of ones and zeros could cause a
  problem
• Low probability of bit error
• Receiver needs to be able to distinguish the
  waveform associated with a mark from the waveform
  associated with a space
• BER performance
• relative immunity to noise
• Error detection capability
• enhances low probability of error

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• Spectrum Suitable for the channel
• Spectrum matching of the channel
• e.g. presence or absence of DC level
• In some cases DC components should be avoided
• The transmission bandwidth should be minimized
• Power Spectral Density
• Particularly its value at zero
• PSD of code should be negligible at the frequency
  near zero
• Transmission Bandwidth
• Should be as small as possible
• Transparency
• The property that any arbitrary symbol or bit
  pattern can be transmitted and received, i.e.,
  all possible data sequence should be faithfully
  reproducible

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Summary of Major Line Codes

• Categories of Line Codes
• Polar - Send pulse or negative of pulse
• Uni-polar - Send pulse or a 0
• Bipolar (a.k.a. alternate mark inversion,
  pseudoternary)
• Represent 1 by alternating signed pulses
• Generalized Pulse Shapes
• NRZ -Pulse lasts entire bit period
• Polar NRZ
• Bipolar NRZ
• RZ - Return to Zero - pulse lasts just half of
  bit period
• Polar RZ
• Bipolar RZ
• Manchester Line Code
• Send a 2- ? pulse for either 1 (high? low) or 0
  (low? high)
• Includes rising and falling edge in each pulse
• No DC component

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• When the category and the generalized shapes are
  combined, we have the following
• Polar NRZ
• Wireless, radio, and satellite applications
  primarily use Polar NRZ because bandwidth is
  precious
• Unipolar NRZ
• Turn the pulse ON for a 1, leave the pulse OFF
  for a 0
• Useful for noncoherent communication where
  receiver cant decide the sign of a pulse
• fiber optic communication often use this
  signaling format
• Unipolar RZ
• RZ signaling has both a rising and falling edge
  of the pulse
• This can be useful for timing and synchronization
  purposes

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• Bipolar RZ
• A unipolar line code, except now we alternate
  between positive and negative pulses to send a
  1
• Alternating like this eliminates the DC component
• This is desirable for many channels that cannot
  transmit the DC components
• NoteThere are many other variations of line
  codes (see Fig. 2.22, page 80 for more)

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Commonly Used Line Codes

• Polar line codes use the antipodal mapping
• Polar NRZ uses NRZ pulse shape
• Polar RZ uses RZ pulse shape

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• Unipolar NRZ Line Code (on-off Signaling)
• Unipolar non-return-to-zero (NRZ) line code is
  defined by unipolar mapping
• In addition, the pulse shape for unipolar NRZ is
• where Tb is the bit period

Where Xn is the nth data bit
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• Bipolar Line Codes
• With bipolar line codes a space is mapped to zero
  and a mark is alternately mapped to -A and A

• It is also called pseudoternary signaling or
  alternate mark inversion (AMI)
• Either RZ or NRZ pulse shape can be used

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• Manchester Line Codes
• Manchester line codes use the antipodal mapping
  and the following split-phase pulse shape

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Figure 3.15Line codes for the electrical
representations of binary data. (a) Unipolar NRZ
signaling. (b) Polar NRZ signaling. (c)
Unipolar RZ signaling. (d) Bipolar RZ signaling.
(e) Split-phase or Manchester code.
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Comparison of Line Codes

• Self-synchronization
• Manchester codes have built in timing information
  because they always have a zero crossing in the
  center of the pulse
• Polar RZ codes tend to be good because the signal
  level always goes to zero for the second half of
  the pulse
• NRZ signals are not good for self-synchronization
• Error probability
• Polar codes perform better (are more energy
  efficient) than Uni-polar or Bipolar codes
• Channel characteristics
• We need to find the power spectral density (PSD)
  of the line codes to compare the line codes in
  terms of the channel characteristics

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Comparisons of Line Codes

• Different pulse shapes are used
• to control the spectrum of the transmitted signal
  (no DC value, bandwidth, etc.)
• guarantee transitions every symbol interval to
  assist in symbol timing recovery
• 1. Power Spectral Density of Line Codes (see Fig.
  2.23, Page 90)
• After line coding, the pulses may be filtered or
  shaped to further improve there properties such
  as
• Spectral efficiency
• Immunity to Intersymbol Interference
• Distinction between Line Coding and Pulse Shaping
  is not easy
• 2. DC Component and Bandwidth
• DC Components
• Unipolar NRZ, polar NRZ, and unipolar RZ all have
  DC components
• Bipolar RZ and Manchester NRZ do not have DC
  components

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Differential Encoding
(a) Original binary data. (b) Differentially
encoded data, assuming reference bit 1. (c)
Waveform of differentially encoded data using
unipolar NRZ signaling.
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Differential Coding

• Encoding
• encoded(k) encoded(k 1) XOR original(k)
• where k starts from 0
• Encoded(-1) is called the reference bit which
  can be either 1 or 0
• Decoding
• original(k) encoded (k 1) XOR encoded(k)
• where k starts from 0
• Reference bit remains same for both encoding and
  decoding process

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Sources of Corruption in the sampled, quantized
and transmitted pulses

• Channel Effects
• Channel Noise (AWGN, White Noise, Thermal etc)
• Intersymbol Interference (ISI)
• Sampling and Quantization Effects
• Quantization (Granularity) Noise
• Quantizer Saturation or Overload Noise
• Timing Jitter

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Bits per PCM word and M-ary Modulation

• Section 2.8.4 Bits per PCM Word and Bits per
  Symbol
• L2l
• Section 2.8.5 M-ary Pulse Modulation Waveforms
• M 2k
• Problem 2.14 The information in an analog
  waveform, whose maximum frequency fm4000Hz, is
  to be transmitted using a 16-level PAM system.
  The quantization must not exceed 1 of the
  peak-to-peak analog signal.
• (a) What is the minimum number of bits per
  sample or bits per PCM word that should be used
  in this system?
• (b) What is the minimum required sampling rate,
  and what is the resulting bit rate?
• (c) What is the 16-ary PAM symbol Transmission
  rate?

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Note

• Topics Covered

• Digital Communications-Bernard Sklar
• Chapter 2
• Communication System-Simon Haykin 4th Ed.
• Chapter 3
• 3.1-3.8