1 / 8

Channel Capacity: Nyquist and Shannon Limits

Channel Capacity: Nyquist and Shannon Limits. Based on Chapter 3 of William Stallings, Data and Computer Communication, 7 th Ed. Kevin Bolding Electrical Engineering Seattle Pacific University. Find the highest data rate possible for a given bandwidth, B Binary data (two states)

liliha
Download Presentation

Channel Capacity: Nyquist and Shannon Limits

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Channel Capacity:Nyquist and Shannon Limits Based on Chapter 3 of William Stallings, Data and Computer Communication, 7th Ed. Kevin BoldingElectrical EngineeringSeattle Pacific University

  2. Find the highest data rate possible for a given bandwidth, B Binary data (two states) Zero noise on channel 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 Nyquist Limit on Bandwidth Period = 1/B • Nyquist: Max data rate is 2B (assuming two signal levels) • Two signal events per cycle

  3. If each signal point can be more than two states, we can have a higher data rate M states gives log2M bits per signal point 00 10 00 11 00 00 00 11 01 10 10 01 00 00 11 Nyquist Limit on Bandwidth (general) Period = 1/B 4 signal levels: 2 bits/signal • General Nyquist: Max data rate is 2B log2M • M signal levels, 2 signals per cycle

  4. 4 levels - noise corrupts data 2 levels - better margins Practical Limits • Nyquist: Limit based on the number of signal levels and bandwidth • Clever engineer: Use a huge number of signal levels and transmit at an arbitrarily large data rate • The enemy: Noise • As the number of signal levels grows, the differences between levels becomes very small • Noise has an easier time corrupting bits

  5. Characterizing Noise • Noise is only a problem when it corrupts data • Important characteristic is its size relative to the minimum signal information • Signal-to-Noise Ratio • SNR = signal power / noise power • SNR(dB) = 10 log10(S/N) • Shannon’s Formula for maximum capacity in bps • C = B log2(1 + SNR) • Capacity can be increased by: • Increasing Bandwidth • Increasing SNR (note: capacity linear in SNR(dB) ) Warning: Assumes uniform (white) noise!

  6. From Nyquist: From Shannon: Equating: or SNR is the S/N ratio needed tosupport the M signal levels M is the number of levelsneeded to meet Shannon Limit Shannon meets Nyquist Example: To support 16 levels (4 bits), we need a SNR of 255 (24 dB) Example: To achieve Shannon limit with SNR of 30dB, we need 32 levels

  7. 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 Achieving the Nyquist Limit • The Nyquist Limit requires two signaling events per Hertz • C=2B log2M • This must be achieved using waveforms with frequency components <= B Period = 1/B “Corners” require higher-frequency components • We need a way to represent a ‘1’ with a pulse that has no components greater than B • Must be able to overlap two pulses per Hertz without loss of information

  8. 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -5 -4 -3 -2 -1 0 1 2 3 4 5 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -5 -4 -3 -2 -1 0 1 2 3 4 5 1.5 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0.5 0 -0.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Sinc (Nyquist) Pulses • The Sinc Pulse is defined as sin(x)/x • Sinc pulse at frequency f requires bandwidth f • sin(x 2f)/(x 2f) • Note that the sinc pulse is zero at all multiples of 1/2f except for the singular pulse • Pulses can overlap as long as each one is centered on a multiple of 1/2f • When the pulses are summed, checking the waveform at each multiple of 1/2f gives the original data

More Related