Transcript ppt - NOISE
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) • Discrete Fourier transform: n 1 Fk f x e f ( x) e 2ik x dx 2i kn x 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 Analog Data Digital Digital Signal occupies same Codec produces spectrum as analog bitstream data Digital data encoded using a modem Signal consists of two 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