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Channel Capacity: Nyquist and Shannon Limits. Based on Chapter 3 of William Stallings, Data and Computer Communication, 8 th Ed. Kevin Bolding Electrical Engineering Seattle Pacific University. Find the highest data rate possible for a given bandwidth, B Binary data (two states)

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Channel capacity nyquist and shannon limits l.jpg

Channel Capacity:Nyquist and Shannon Limits

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

Kevin BoldingElectrical EngineeringSeattle Pacific University


Nyquist limit on bandwidth l.jpg

Find the highest data rate possible for a given bandwidth, B

Binary data (two states)

Zero noise on channel

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Nyquist Limit on Bandwidth

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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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

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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


Practical limits l.jpg

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


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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)

  • Shannon’s 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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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


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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


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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


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