Physical Layer: Signals, Capacity, and Coding

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Physical LayerSignals, Capacity, and Coding

• CS 4251 Computer Networking IINick
  FeamsterSpring 2008

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This Lecture

• Whats on the wire?
• Frequency, Spectrum, and Bandwidth
• How much will fit?
• Shannon capacity, Nyquist
• How is it represented?
• Encoding

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Digital Domain

• Digital signal signal where intensity maintains
  constant level for some period of time, and then
  changes to some other level
• Amplitude Maxumum value (measured in Volts)
• Frequency Rate at which the signal repeats
• Phase Relative position in time within a single
  period of a signal
• Wavelength The distance between two points of
  corresponding phase ( velocity period)

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Any Signal Sum of Sines

• Our building block
• Add enough of them to get any signal f(x) you
  want!
• How many degrees of freedom?
• What does each control?
• Which one encodes the coarse vs. fine structure
  of the signal?

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Fourier Transform

• Continuous Fourier transform
• Discrete Fourier transform
• F is a function of frequency describes how much
  of each frequency f contains
• Fourier transform is invertible

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Skipping a Few Steps

• Any square wave with amplitude 1 can be
  represented as

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Spectrum and Bandwidth

• Any time domain signal can be represented in
  terms of the sum of scaled, shifted sine waves
• The spectrum of a signal is the range of
  frequencies that the signal contains
• Most signals can be effectively represented in
  finite bandwidth
• Bandwidth also has a direct relationship to data
  rate

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Relationship Data Rate and Bandwidth

• Goal Representation of square wave in a form
  that receiver can distinguish 1s from 0s
• Signal can be represented as sum of sine waves
• Increasing the bandwidth means two things
• Frequencies in the sine wave span a wider
  spectrum
• Intervals in the original signal occur more
  often
• Include representation of square wave as sum of
  sine waves here. Derive data rate from
  bandwidth.

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Analog vs. Digital Signaling

• Analog signal Continuously varying EM wave
• Digital signal Sequence of voltage pulses

Signal
Analog
Digital
Analog
Data
Digital
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Transmission Impairments

• Attenuation
• The strength of a signal falls off with distance
  over any transmission medium
• Delay distortion
• Velocity of a signals propagation varies w/
  frequency
• Different components of the signal may arrive at
  different times
• Noise

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Attentuation

• Signal strength attentuation is typically
  expressed as decibel levels per unit distance
• Signal must have sufficient strength to be
• Detected by the receiver
• Stronger than the noise in the channel to be
  received without error
• Note Increasing frequency typically increases
  attentuation (often corrected with equalization)

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Sources of Noise

• Thermal noise due to agitation of electrons,
  function of temperature, present at all
  frequencies
• Intermodulation noise Signals at two different
  frequencies can sometimes produce energy at the
  sum of the two
• Crosstalk Coupling between signals

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Channel Capacity

• The maximum rate at which data can be transmitted
  over a given communication path
• Relationship of
• Data rate bits per second
• Bandwidth constrained by the transmitter, nature
  of transmission medium
• Noise depends on properties of channel
• Error rate the rate at which errors occur
• How do we make the most efficient use possible of
  a given bandwidth?
• Highest data rate, with a limit on error rate for
  a given bandwidth

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Nyquist Bandwidth

• Consider a channel that has no noise
• Nyquist theorem Given a bandwidth B, the highest
  signal rate that can be carried is 2B
• So, C 2B
• But (stay tuned), each signal element can
  represent more than one bit (e.g., suppose more
  than two signal levels are used)
• So C 2B lg M
• Results follow from signal processing
• Shannon/Nyquist theorem states that signal must
  be sampled at twice its highest rate to avoid
  aliasing

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Shannon Capacity

• All other things being equal, doubling the
  bandwidth doubles the data rate
• What about noise?
• Increasing the data rate means shorter bits
• which means that a given amount of noise will
  corrupt more bits
• Thus, the higher the data rate, the more damage
  that unwanted noise will inflict

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Shannon Capacity, Formally

• Define Signal-to-Noise Ratio (SNR)
• SNR 10 log (S/N)
• Then, Shannons result says that, channel
  capacity, C, can be expressed as
• C B lg (1 S/N)
• In practice, the achievable rates are much lower,
  because this formula does not consider impulse
  noise or attenuation

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Example

• Bandwidth 3-4MHz
• S/N 250
• What is the capacity?
• How many signal levels required to achieve the
  capacity?

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Modulation

• Baseband signal the input
• Carrier frequency chosen according to the
  transmission medium
• Modulation is the process by which a data source
  is encoded onto a carrier signal
• Digital or analog data can be modulated onto
  digital and analog signals

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Data Rate vs. Modulation Rate

• Data rate rate, in bits per second, that a
  signal is transmitted
• Modulation rate the rate at which the signal
  level is changed (baud)

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Digital Data, Digital Signals

• Simplest possible scheme one voltage level to
  1 and another voltage level to 0
• Many possible other encodings are possible, with
  various design considerations

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Aspects of a Signal

• Spectrum a lack of high-frequency components
  means that less bandwidth is required to transmit
  the signal
• Lack of a DC component is also desirable, for
  various reasons
• Clocking Must determine the beginning and end of
  each bit position.
• Not easy! Requires either a separate clock lead,
  or time synchronization
• Error detection
• Interference/Noise immunity
• Cost and complexity

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Nonreturn to Zero (NRZ)

• Level A positive constant voltage represents one
  binary value, and a negative contant voltage
  represents the other
• Disadvantages
• In the presence of noise, may be difficult to
  distinguish binary values
• Synchronization may be an issue

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Improvement Differential Encoding

• Example Nonreturn to Zero Inverted
• Zero No transition at the beginning of an
  interval
• One Transition at the beginning of an interval
• Advantage
• Since bits are represented by transitions, may be
  more resistant to noise
• Disadvantage
• Clocking still requires time synchronization

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Biphase Encoding

• Transition in the middle of the bit period
• Transition serves two purposes
• Clocking mechanism
• Data
• Example Manchester encoding
• One represented as low to high transition
• Zero represented as high to low transition

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Aspects of Biphase Encoding

• Advantages
• Synchronization Receiver can synchronize on the
  predictable transition in each bit-time
• No DC component
• Easier error detection
• Disadvantage
• As many as two transitions per bit-time
• Modulation rate is twice that of other schemes
• Requires additional bandwidth