Signals & Systems Theorems

Main note

Nyquist Theorem

Refer old lectures

Formula derived from Nyquist Theorem.

Maximum Data or noiseless = $2BLo{g}_{2}Vbits/sec$ What is the max data on noiseless channel frequency bandiwdth = 2F Log_2 V where f is frequency

The formula ( C = 2B \log_2 V ) represents the maximum data rate (channel capacity) of a noiseless channel, where:

• ( C ) is the channel capacity in bits per second (bps).
• ( B ) is the bandwidth in hertz (Hz).
• ( V ) is the number of discrete signal levels.

Key Points:

1. Bandwidth (( B )): Determines the range of frequencies the channel can transmit.
2. Signal Levels (( V )): Number of discrete levels used for encoding data.
3. Logarithmic Relationship: ( \log_2 V ) indicates the bits represented by each symbol.
4. Nyquist Rate: ( 2B ) is the maximum number of symbols per second in a noiseless channel.

Formula Explanation:

• The Nyquist rate is ( 2B ), representing the maximum symbol rate.
• Each symbol can represent ( \log_2 V ) bits.
• The product of the symbol rate and bits per symbol gives the maximum data rate.

Example:

For a bandwidth of 3 kHz and 8 signal levels (( V = 8 )):

• Bits per symbol: ( \log_2 8 = 3 ).
• Nyquist rate: ( 2 \times 3000 = 6000 ) symbols/sec.
• Maximum data rate: ( C = 6000 \times 3 = 18000 ) bps (18 kbps).

This formula shows the dependence of channel capacity on both bandwidth and signal levels.

The Shannon theorem, also known as the Shannon-Hartley theorem, defines the maximum data rate (channel capacity) that can be achieved over a communication channel with a given bandwidth in the presence of noise. It provides a theoretical upper bound on the information rate that can be transmitted with a negligible probability of error.

The formula is: [ C = B \log_2 (1 + \frac{S}{N}) ]

where:

• ( C ) is the channel capacity in bits per second (bps).
• ( B ) is the bandwidth of the channel in hertz (Hz).
• ( S ) is the average signal power.
• ( N ) is the average noise power.
• ( \frac{S}{N} ) is the signal-to-noise ratio (SNR), often denoted as ( SNR ).

Shannon Theorem

Explanation of Shannon Theorem

1. Bandwidth (( B )): This is the range of frequencies that the channel can transmit. The wider the bandwidth, the higher the potential data rate.

2. Signal-to-Noise Ratio (( S/N )): This is a measure of signal strength relative to background noise. A higher SNR indicates a clearer signal with less noise interference, which allows for higher data rates.

3. Logarithmic Relationship: The ( \log_2 (1 + \frac{S}{N}) ) term quantifies the increase in capacity with respect to the SNR. The logarithm base 2 indicates that capacity grows with the ability to distinguish between more discrete levels, directly tied to SNR.

Formula Explanation

• Signal-to-Noise Ratio (SNR): As SNR increases, the channel can support more bits per second because the signal can be more clearly distinguished from the noise.

• Bandwidth: The data rate is directly proportional to the bandwidth. Doubling the bandwidth doubles the capacity, assuming SNR remains constant.

• Logarithmic Effect: The logarithmic function reflects the diminishing returns of increasing SNR. While increasing SNR improves capacity, the rate of improvement decreases as SNR becomes very high.

Example Calculation

Suppose a communication channel has a bandwidth of 3 kHz (3000 Hz) and a signal-to-noise ratio of 30 dB. First, convert the SNR from decibels to a ratio: $30\text{ dB}=10{\log }_{10}\left(\frac{S}{N}\right)$ $\frac{S}{N}=10^{30/10}=10^3=1000$ Then apply the Shannon theorem formula: $C=3000{\log }_2(1+1000)$ $C=3000{\log }_2(1001)$

Using ( \log_2 (1001) \approx 9.97 ): $C\approx 3000\times 9.97\approx 29910\text{ bps}$

So, the maximum data rate for this channel is approximately 29.91 kbps.

Key Insights

• The Shannon theorem sets a fundamental limit on the data rate given bandwidth and noise constraints.
• Increasing bandwidth or improving SNR increases the maximum data rate.
• The logarithmic relationship shows diminishing returns at high SNR values, emphasizing the balance needed between power and bandwidth for efficient communication.

What are types of Links?

There are 6 major Links

1. Ethernet 802.3
2. Token Ring 802.5
3. 802.11 Wireless
4. Point to Point Protocol
5. Frame Relay
6. AIMC Asynchronous Transfer Mode

What are functions of Data Link Layer

1. Link access : Media Access Controls protocols Decides Ruels for frame transmission.

   Link.excalidraw

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   Excalidraw Data

   Text Elements

   Packet

   Packet

   send

   Receive

   Trailer

   Payload

   Frame

   Trailer

   Payload

   Link to original

2. Framing Link: This Laye protocol Creates Datalink FRAMI. Data live frame has three parts like header, payload, Traitor
3. Reliability: Datalink Layer should provide Reliable Communication

   1. Reliability links.excalidraw

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      Excalidraw Data

      Text Elements

      Packet

      ACK

      S

      R

      Link to original

4. i

Problems

Statement 1

If channel is working on 6 khz and noise is 20 decibel

References

Information

• date: 2024.07.30
• time: 12:16