## Lecture 8

## Channel capacity

- Nyquist channel capacity
- Binary transmission: 1 bit per pulse => transmission rate = 1 / duration of the pulse = 2Wbps
- Number of bits represented by one pulse = log
_{2}(M) = m - Nyquist signalling rate in bps = m * (1 / duration of the pulse) = 2mW
- Data rate is linear in BW under no noise
- C = 2wlog
_{2}(M)bps- Where '2W' is the bandwidth
- And M is the discrete signal (voltage levels)

- Where '2W' is the bandwidth
- Increasing m, increases the transmission rate. Is there an upper limit?

- Binary transmission: 1 bit per pulse => transmission rate = 1 / duration of the pulse = 2Wbps
- Noise limits accuracy
- Receiver makes decision based on transmitted pulse level + noise
- Error rate depends on relative value of noise amplitude an spacing between signal levels
- Large (positive or negative) noise values can cause wrong decision

- Receiver makes decision based on transmitted pulse level + noise
- Noise distribution
- Noise is characterized by probability density of amplitude samples
- Likelihood that certain amplitude occurs
- Thermal electronic noise is inevitable ( due to vibrations of electrons)
- Noise distribution is Gaussian (bell shaped)

- Noise is characterized by probability density of amplitude samples
- Channel capacity
- Is the maximum bit rate supported by a channel
- Can the channel capacity C be made infinite by increasing m?
- NO, there are other constraints introduced by noise and channel interference

- NO, there are other constraints introduced by noise and channel interference

- Is the maximum bit rate supported by a channel
- Shannon capacity
- Data rate cs noise and error rate
- Data rate DR (goes up) then bit time (goes down) and bit rate (goes up) and error rate (goes up)
- Noise affects more bits
- High data more errors for the same given noise
- Signal power or strength vs noise power
- Signal power / noise power

- Signal power / noise power
- SNR
_{db}= 10log_{10}{signal power / noise power} - By increasing m the difference between adjacent levels is reduced affecting SNR
- Reduction in SNR affects the channel capacity (C).
- Shannon channel capacity theorem provides an upper bound on the channel capacity in terms of bandwidth for a noisy channel
- C = Wlog
_{2}(1+SNR)bps

- C = Wlog

- Data rate cs noise and error rate
- Line coding
- Converts a binary sequence into a digital signal
- Unipolar NRZ(non return to zero): bit 1 is represented by a +A volts
- Bit 0 is represented by 0 volts

- Unipolar NRZ(non return to zero): bit 1 is represented by a +A volts
- Average transmitted power per pulse = ½ * A
^{2}+ ½ * (0) = A^{2}/2 - Average value of a signal = A
^{2}/2 Volts

- Polar NRZ : bit 1 is represented by +A/2 volts
- Bit 0 is represented by –A/2 volts
- Average transmitted power per pulse = ½ * (a/2)
^{2}+ ½ * (-A/2)^{2}= A^{2}/ 4 - Half the power used as compared to unipolar NRZ with same distance between levels
- Average value of signal = 0 volts

- NRZ – inverted
- First bit 1 is represented by +A/2 volts
- No change for bit 0; flip to the opposite voltage for the next bit 1
- Average transmitted power per pulse = A
^{2}/4 - Average value of signal = 0 volts

- First bit 1 is represented by +A/2 volts
- Bipolar encoding
- Bit 0 is represented by 0 volts
- Bit 1 is represented by consecutive values of A/2 and –A/2
- Average transmitted power per pulse = A
^{2}/ 8 - Average value of signal = 0 volts

- Bit 0 is represented by 0 volts
- Manchester encoding
- Bit 0 is the change from –A/2 to A/2
- Bit 1 is the change from A/2 to –A/2
- Got popular among Ethernet / token ring networks that were close together and cost was more important than bandwidth

- Bit 0 is the change from –A/2 to A/2

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