Physical Layer: Signals, Capacity, and Coding
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Transcript of Physical Layer: Signals, Capacity, and Coding
Physical Layer:Signals, Capacity, and Coding
CS 4251: Computer Networking IINick Feamster
Fall 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:
• 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?
xAsin(
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
dxexfk xikxf
2)()(F )(F
1
0
2kF
n
x
xix
nk
ef
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 occupies same spectrum as analog data
Codec produces bitstream
Digital data encoded using a modem
Signal consists of two voltage levels
Analog Digital
Analog
Digital
Data
Signal
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
Attenuation
• 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