Channel Capacity: Nyquist and Shannon Limits

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Channel CapacityNyquist and Shannon Limits

• Based on Chapter 3 of William Stallings, Data and
  Computer Communication, 8th Ed.

Kevin BoldingElectrical EngineeringSeattle
Pacific University
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Nyquist Limit on Bandwidth

• Find the highest data rate possible for a given
  bandwidth, B
• Binary data (two states)
• Zero noise on channel

Example shown with bandfrom 0 Hz to B Hz
(Bandwidth B)Maximum frequency is B Hz
Period 1/B

• Nyquist Max data rate is 2B (assuming two signal
  levels)
• Two signal events per cycle

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Nyquist Limit on Bandwidth (general)

• 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

Period 1/B
4 signal levels 2 bits/signal

• General Nyquist Max data rate is 2B log2M
• M signal levels, 2 signals per cycle

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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

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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)

• Shannons Formula for maximum capacity in bps
• C B log2(1 SNR)
• Capacity can be increased by
• Increasing Bandwidth
• Increasing SNR (capacity is linear in SNR(dB) )

SNR in linear form
Warning Assumes uniform (white) noise!
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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
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Achieving the Nyquist Limit

• The Nyquist Limit requires two signaling events
  per Hertz
• C2B log2M
• This must be achieved using waveforms with
  frequency components lt 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

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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 orignal data