1. Physical Layer: Signals, Capacity, and Coding
CS 4251: Computer Networking II Nick Feamster Fall 2008

2. This Lecture What’s on the wire? How much will fit?
Frequency, Spectrum, and Bandwidth
How much will fit?
Shannon capacity, Nyquist
How is it represented?
Encoding

3. 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)

4. 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?

5. 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

6. Skipping a Few Steps
Any square wave with amplitude 1 can be represented as:

7. 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…

8. 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.]

9. Analog vs. Digital Signaling
Analog signal: Continuously varying EM wave
Digital signal: Sequence of voltage pulses
Signal
Analog
Digital
Signal occupies same spectrum as analog data
Codec produces bitstream
Digital data encoded using a modem
Signal consists of two voltage levels
Analog
Data
Digital

10. 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

11. 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)

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

19. 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)

20. 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…

21. 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

22. 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

23. 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

24. 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

25. 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