EE521 Analog and Digital Communications

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EE521 Analog and Digital Communications

• James K. Beard, Ph. D.
• jkbeard_at_temple.edu
• Tuesday, March 22, 2005
• http//astro.temple.edu/jkbeard/

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Attendance
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Essentials

• Text Bernard Sklar, Digital Communications,
  Second Edition
• SystemView
• Office
• EA 349
• Tuesday afternoons 330 PM to 430 PM before
  class
• MWF 1030 AM to 1130 AM
• Next quiz March 22
• Final Exam Scheduled
• Tuesday, May 10, 600 PM to 800 PM
• Here in this classroom

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Todays Topics

• Quiz 1
• Gray code MPSK
• Waveform Coding, Part 1
• Waveform coding and structured sequences
• Types of error control
• Structured sequences
• Discussion (as time permits)

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Question 3 Computations
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Gray Codes

• Sometimes called reflected codes
• Defining property only one bit changes between
  sequential codes
• Conversion
• Binary codes to Gray
• Work from LSB up
• XOR of bits j and j1 to get bit j of Gray code
• Bit past MSB of binary code is 0
• Gray to binary
• Work from MSB down
• XOR bits j1 of binary code and bit j of Gray
  code to get bit j of binary code
• Bit past MSB of binary code is 0

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Gray Code MPSK
Defining Characteristic The Hamming distance
between adjacent codes is 1 Result less
opportunity for bit errors gives lower BER See
Sklar 4.9.4 pp. 234-235
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Sklar Chapter 6
From other sources
Essential
Legend
Information source
Message symbols
Optional
Channel symbols
X M I T
Format
Source encode
Encrypt
Channel encode
Multi-plex
Pulse modulate
Bandpass modulate
Freq-uency spread
Multiple access
Bit stream
Synch-ronization
Digital baseband waveform
Digital bandpass waveform
Channel
R C V
Format
Source decode
Decrypt
Channel decode
Demul-tiplex
Detect
Demod-ulate Sample
Freq-uency despread
Multiple access
Channel symbols
To other destinations
Message symbols
Information sink
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Channel Coding Topic Areas

• Overview Waveform Coding and Structured
  Sequences
• Modulation
• M-ary signaling
• Antipodal and orthogonal pulses
• Trellis-coded modulation
• Codes as structured sequences
• Block codes
• Convolutional codes
• Turbo codes

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Waveform Coding and Structured Sequences

• Channel coding
• Structured sequences (EDAC)
• Waveform design
• Structured sequences
• Coding digital sequences for transmission
• Increases the number of bits and provides EDAC
  capability
• Waveform design
• How to code a pulse for RF use
• A design point that selects containment in time
  and frequency regions

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M-ary Signaling

• MPSK or MFSK
• Number of waveforms is M2k
• Advantages of each
• Signals can be orthogonal with MFSK
• MPSK uses one frequency channel
• Additional requirements
• MFSK requires more bandwidth
• MPSK requires more Eb/N0

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The Orthogonality Condition

• Normalized orthogonality
• Orthogonality can be
• Time signals are nonzero at different times
• Functional orthogonal functions
• Codes orthogonal codes
• In frequency see orthogonal functions

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Antipodal and Orthogonal signals

• Antipodal
• Two signals
• One the negative of the other
• Orthogonal
• M signals
• A matched filter for any one produces a near-zero
  result with any other as input
• Orthogonality can be in time, frequency, or code

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Walsh-Hadamard Sequences

• A simple way to formulate orthogonal code
  sequences
• Based on recursive augmentation of Walsh-Hadamard
  matrices

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Properties of Walsh-Hadamard Sequences

• Matrices are symmetrical
• Matrices are self-orthogonal
• Each matrix has rows or columns are a sequence of
  orthogonal sequences of length 2k
• Cross-correlation properties
• Excellent for zero lag
• Poor for other lags

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Bi-Orthogonal Codes

• Made up of rows or columns from half a Hadamard
  matrix
• Codes of order M/22k-1 appended to their
  antipodal opposite
• Slightly improved symbol error performance
• Half the bandwidth of orthogonal codes

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Bi-Orthogonality
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Transformational Codes

• Also called Simplex codes
• Generated from orthogonal sets
• First digit of each code is deleted
• Minimum energy code
• Characterized by

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Summary of Codes

• For large values of M
• All three codes have similar BER performance
• Biorthogonal codes have bandwidth advantage
• Bandwidth requirements
• Grow exponentially with M
• True of all three codes

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Primitive Error Control

• Older schemes were based on terminal connectivity
• Simplex one-way communication
• Half duplex first one direction then the other
• Full duplex both directions simultaneously
• Duplex allows Acknowledgement/negative
  acknowledgement (ACK/NAK) handshake

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Structured Sequences

• Three kinds
• Block codes
• Convolutional codes (later)
• Turbo codes (next semester)
• Increasing M improves symbol error performance
  and bandwidth requrements

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Channel Models

• Discrete memoryless channel (DMC)
• Discrete input and output alphabets
• BER depends only on signal at current epoch
• BER equations are as studied before
• Gaussian channel
• DMC with binary input, continuous output
• Gaussian noise is added to symbols
• Binary symmetric channel
• A DMC with a binary alphabet only 1, 0
• A Gaussian channel with hard decoding on output

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Code Rate and Redundancy

• Begin with k data bits per symbol
• Add EDAC bits to form a symbol of n bits
• Parity bits or check bits
• Generally, redundancy bits
• This is an (n,k) code
• Redundancy is (n-k)/k
• Code rate is k/n

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Parity codes

• Parity check codes
• Single parity bit can detect even number of
  errors
• Useful in triggering NAK with low BER
• Rectangular codes
• Double parity, second on pth bit of k words
• Parity on bit p and word q allows correction of a
  single error

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Parameters in the Trade Space

• Error performance
• Bandwidth vs. data rate
• Power
• Coding gain as defined by decrease in Eb/N0
  required to obtain a specified BER when coding is
  used

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Relationship Between Some Basic Trade Parameters
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Linear Block Codes

• These are (n,k) codes based on polynomials in
  binary arithmetic
• Polynomials are added and subtracted
• Arithmetic is modulo 2
• Polynomial coefficients considered as vectors
• Sets closed on addition are called Vector
  subspaces

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Maximal-Length Sequences

• Bit sequence is essentially random
• Pseudo-random noise (PRN) code
• Codes Construction
• Shift registers with feedback
• Recursive modulo-2 polynomial arithmetic
• PRN codes are then selected for good
  cross-correlation properties

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Desirable PRN Code Properties

• Maximal length 2m codes before repeating
• Balance equal number of (1) and (-1) pulses
• Closed on circular shifts
• Contain shorter subsequences
• Good autocorrelation properties

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Galois Field Vector Extensions of Order 2m

• Polynomials modulo 2 of order m-1
• Arithmetic is done modulo a generating polynomial
  of the form
• Proper selection of generating polynomial
• Sequence of powers produces all 2m elements
• Set is closed on multiplication

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An Important Isomorphism

• Shift registers with feedback
• Bits in shift register are isomorphic with
  polynomial coefficients
• Shift is isomorphic with multiplication by x
• Modulo the generating polynomial is isomorphic to
  multiple-tap feedback
• Shift registers with feedback can produce a
  Galois field in sequence of powers of x
• These codes are also called m-sequences