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					       Physical Layer:
Signals, Capacity, and Coding
   CS 4251: Computer Networking II
           Nick Feamster
            Spring 2008
This Lecture
 • What’s on the wire?
   – Frequency, Spectrum, and Bandwidth
 • How much will fit?
   – Shannon capacity, Nyquist
 • How is it represented?
   – Encoding
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)
Any Signal: Sum of Sines
 • Our building block:
    Asin( x   
 • 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?
Fourier Transform
 • Continuous Fourier transform:

           F(k )  F  f ( x)  
                                     
                                                    2ik x
                                         f ( x) e            dx
                                   
 • Discrete Fourier transform:
                           n 1
                     Fk   f x e
                                     2i k x
                                         n


                           x 0
 • F is a function of frequency – describes how much of
   each frequency f contains
 • Fourier transform is invertible
Skipping a Few Steps
 • Any square wave with amplitude 1 can be
   represented as:
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…
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.]
Analog vs. Digital Signaling
 • Analog signal: Continuously varying EM wave
 • Digital signal: Sequence of voltage pulses
                                    Signal
                    Analog                    Digital

                 Signal occupies same Codec produces
       Analog    spectrum as analog   bitstream
Data             data

                 Digital data encoded   Signal consists of two
       Digital   using a modem          voltage levels
Transmission Impairments
 • Attenuation
   – The strength of a signal falls off with distance over
     any transmission medium


 • Delay distortion
   – Velocity of a signal’s propagation varies w/ frequency
   – Different components of the signal may arrive at
     different times


 • Noise
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)
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
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
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
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
Shannon Capacity, Formally
 • Define Signal-to-Noise Ratio (SNR):
   – SNR = 10 log (S/N)

 • Then, Shannon’s 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
Example
 • Bandwidth: 3-4MHz
 • S/N: 250

 • What is the capacity?
 • How many signal levels required to achieve the
   capacity?
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
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)
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…
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
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
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
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
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

				
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