Explain any two: a) Shannon's theorem for channel capacity b) Time division multiplexing c) Sample and HOLD circuit d) FSK

a) Shannon's Theorem for Channel Capacity

Shannon's theorem, also known as the Shannon-Hartley theorem, is a fundamental principle in information theory that defines the maximum rate at which information can be transmitted over a communication channel without error, given the channel's bandwidth and signal-to-noise ratio (SNR).

Step by Step Explanation:

1. Channel Bandwidth (B): This is the range of frequencies that the channel can carry. It is measured in hertz (Hz).

2. Signal-to-Noise Ratio (SNR): This is the ratio of the power of the signal (useful information) to the power of the noise (unwanted disturbance) affecting the signal. It is usually expressed in decibels (dB).

3. Channel Capacity (C): This is the theoretical maximum data rate (in bits per second) that can be transmitted with an arbitrarily low error rate.

Formula:

The Shannon-Hartley theorem is mathematically expressed as:

[ C = B \log_2(1 + SNR) ]

where:

• ( C ) is the channel capacity in bits per second (bps),
• ( B ) is the bandwidth of the channel in hertz (Hz),
• ( SNR ) is the signal-to-noise ratio (a pure number).

Example:

Suppose we have a channel with a bandwidth of 3 MHz (3,000,000 Hz) and an SNR of 1023 (30 dB, since ( 10 \log_{10}(1023) \approx 30 ) dB). The channel capacity would be:

[ C = 3,000,000 \log_2(1 + 1023) ] [ C \approx 3,000,000 \log_2(1024) ] [ C \approx 3,000,000 \times 10 ] [ C \approx 30,000,000 \text{ bps} ] [ C = 30 \text{ Mbps} ]

b) Time Division Multiplexing (TDM)

Time Division Multiplexing is a method of transmitting and receiving independent signals over a common signal path by means of synchronized switches at each end of the transmission line so that each signal appears on the line only a fraction of time in an alternating pattern.

Step by Step Explanation:

1. Time Slots: The channel is divided into time slots, and each user is allocated a specific time slot during which they can transmit their signal.

2. Multiplexing: During the transmission, signals from multiple users are interleaved in time.

3. Demultiplexing: At the receiving end, the interleaved signals are separated out into their original form, using the timing information.

Example:

Consider a TDM system with 4 users. The channel is divided into 4 time slots, and each user is assigned one time slot. If each time slot is 1 ms, then every 4 ms, each user gets to use the entire channel bandwidth for 1 ms.

c) Sample and HOLD Circuit

A Sample and Hold (S/H) circuit is an electronic device that samples the voltage of a continuously varying analog signal and holds (locks) its value at a constant level for a specified minimum period of time.

Step by Step Explanation:

1. Sampling: The S/H circuit receives an analog input signal and a sampling signal. When the sampling signal is active, the S/H circuit captures the current value of the analog signal.

2. Holding: After sampling, the circuit maintains the captured voltage value constant, even if the input signal changes. This held value can be processed or digitized by an analog-to-digital converter (ADC).

Example:

In an ADC system, the S/H circuit samples an analog signal every 1 microsecond and holds this value steady long enough for the ADC to convert it to a digital value.

d) Frequency Shift Keying (FSK)

Frequency Shift Keying is a frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a carrier wave.

Step by Step Explanation:

1. Binary Data: FSK uses binary data as the input signal.

2. Carrier Frequency: There are typically two frequencies used: one for binary '1' (high frequency) and one for binary '0' (low frequency).

3. Modulation: The carrier frequency is shifted between these two frequencies according to the binary data signal.

Example:

In a binary FSK system, if the carrier frequency is 1 kHz for binary '0' and 2 kHz for binary '1', then the modulated signal will switch between these two frequencies as the input binary data changes.

Differences and Important Points:

Feature	Shannon's Theorem	TDM	S/H Circuit	FSK
Principle	Defines the limit of data transmission rate	Multiplexes signals by dividing time into slots	Samples and holds analog signals for processing	Modulates digital data by changing carrier frequency
Key Parameters	Bandwidth, SNR	Time slots, synchronization	Sampling rate, hold time	Carrier frequencies, binary data
Application	Theoretical limit for communication systems	Telecommunication systems, data networks	ADC systems, signal processing	Digital communication, data transmission
Advantage	Establishes maximum capacity	Efficient use of bandwidth	Allows accurate digitization of analog signals	Robust against noise, simple to implement

Each of these concepts plays a crucial role in the field of communications and signal processing, and understanding them is essential for designing and analyzing modern communication systems.