Transmission Theory

Transmission Theory

In a communication system you need to know the maximum capacity of a channel. The Nyquist theorem and the Shannon theorem are the two theories that describe that capacity, and each applies in a different situation.

1. The Nyquist Theorem

The Nyquist theorem describes how the maximum symbol rate relates to the bandwidth of a noiseless ideal channel.

The Theorem

For a noiseless channel of bandwidth $W$ Hz, the maximum symbol rate is:

$C = 2W \log_2 M$

where:

• $C$: maximum symbol rate (baud)
• $W$: channel bandwidth (Hz)
• $M$: number of symbol states

Key Points

1. Conditions of use: ideal channels that are noiseless or nearly so
2. Symbol rate: the maximum symbol rate is twice the bandwidth
3. Symbol states: the number of symbol states $M$ affects the transmission rate
4. Practical use: applies to high-quality wired transmission

Example

Take a telephone line of bandwidth 3000 Hz:

• With a binary signal ($M = 2$): $C = 2 \times 3000 \times \log_2 2 = 6000 \text{ baud}$

• With a quaternary signal ($M = 4$): $C = 2 \times 3000 \times \log_2 4 = 12000 \text{ baud}$

2. The Shannon Theorem

The Shannon theorem describes how the maximum data rate relates to the bandwidth and the signal-to-noise ratio of a real, noisy channel.

The Theorem

For a channel of bandwidth $W$ Hz and signal-to-noise ratio $S/N$, the maximum data rate is:

$C = W \log_2(1 + \frac{S}{N})$

where:

• $C$: maximum data rate (bps)
• $W$: channel bandwidth (Hz)
• $S$: signal power
• $N$: noise power
• $S/N$: signal-to-noise ratio

Key Points

1. Conditions of use: real channels that carry noise
2. Signal-to-noise ratio: the higher the ratio, the higher the transmission rate
3. Bandwidth: the larger the bandwidth, the higher the transmission rate
4. Practical meaning: it gives the theoretical limit of a real channel

Calculating the Signal-to-Noise Ratio

The ratio is usually expressed in decibels (dB):

$\text{SNR(dB)} = 10 \log_{10}(\frac{S}{N})$

Example

Take a telephone line of bandwidth 3000 Hz and a signal-to-noise ratio of 20 dB:

1. Calculate the signal-to-noise ratio: $\frac{S}{N} = 10^{20/10} = 100$

2. Calculate the maximum data rate: $C = 3000 \times \log_2(1 + 100) \approx 3000 \times 6.66 \approx 19980 \text{ bps}$

3. Comparing the Two Theorems

4. Practical Applications

Applications of the Nyquist Theorem

1. High-quality wired transmission: optical fibre, high-quality cable
2. Short-distance transmission: inside a local area network
3. Low-noise environments: well-shielded transmission environments

Applications of the Shannon Theorem

1. Wireless communication: mobile communications, WiFi
2. Long-distance transmission: telephone lines, cable
3. Noisy environments: industrial settings, urban settings

5. Factors That Affect Channel Capacity

The Effect of Bandwidth

• The larger the bandwidth, the higher the transmission rate
• But bandwidth is expensive to increase
• Cost and performance have to be balanced

The Effect of the Signal-to-Noise Ratio

• The higher the signal-to-noise ratio, the higher the transmission rate
• The ratio can be improved in the following ways:
  • Increase the transmit power
  • Use better transmission media
  • Adopt noise-resistant encoding

The Effect of the Encoding Scheme

• Different encoding schemes have different efficiencies
• Encoding complexity and transmission efficiency have to be balanced
• Modern communication systems use sophisticated encoding techniques

6. Calculating the Real Transmission Rate

The real transmission rate is usually below the theoretical limit, because you have to allow for:

1. Protocol overhead: frame headers, checksums, and so on
2. Encoding efficiency: the redundancy of error-correcting codes
3. Equipment limits: hardware processing capability
4. Network congestion: the load on the network

The formula for the real rate:

Real rate = theoretical rate × encoding efficiency × (1 - protocol overhead ratio)