Draw block diagram of PCM system and explain it. Explain the terms sampling, quantization and quantization error.

Block Diagram of PCM System

A Pulse Code Modulation (PCM) system can be represented by the following block diagram:

Input Analog Signal -> [Sampler] -> [Quantizer] -> [Encoder] -> PCM Output

1. Sampler: This block samples the input analog signal at regular intervals.
2. Quantizer: The sampled signal is then quantized, meaning it is approximated by a finite set of levels.
3. Encoder: Finally, the quantized values are encoded into a digital bit stream.

Explanation of PCM System

1. Sampling

Sampling is the process of converting a continuous-time signal into a discrete-time signal. This is done by measuring the amplitude of the signal at uniform intervals, known as the sampling interval (T_s). The rate at which the signal is sampled is called the sampling frequency (f_s), and it is the reciprocal of the sampling interval (f_s = 1/T_s).

According to the Nyquist-Shannon sampling theorem, the sampling frequency must be at least twice the highest frequency component in the input signal to accurately reconstruct the original signal from its samples. This minimum sampling rate is known as the Nyquist rate.

Formula: [ f_s \geq 2 \times f_{max} ]

where ( f_{max} ) is the maximum frequency present in the input signal.

Example: If the input signal has a maximum frequency of 3 kHz, the sampling frequency must be at least 6 kHz.

2. Quantization

Quantization is the process of mapping the infinite set of values of the sampled signal to a finite set of levels. This is necessary to represent the signal with a limited number of bits. Each sampled value is approximated by the nearest level, known as a quantization level.

The difference between the number of possible analog values and the number of quantization levels is determined by the bit depth of the system. For example, an 8-bit system can represent 2^8 = 256 different levels.

Formula: [ Q = \frac{V_{max} - V_{min}}{2^n} ]

where ( Q ) is the quantization step size, ( V_{max} ) and ( V_{min} ) are the maximum and minimum voltages, and ( n ) is the number of bits.

Example: For an 8-bit system with a voltage range from 0 to 5 volts, the quantization step size would be ( Q = \frac{5 - 0}{256} \approx 0.0195 ) volts.

3. Quantization Error

Quantization error, also known as quantization noise, is the difference between the actual sampled value and the quantized value. This error is introduced because the quantization process involves rounding off the sampled values to the nearest quantization level.

The quantization error can be at most half the quantization step size. It is a form of distortion that is introduced during the quantization process and is an irreversible loss of information.

Formula: [ e_q = x(t) - Q[n] ]

where ( e_q ) is the quantization error, ( x(t) ) is the actual sampled value, and ( Q[n] ) is the quantized value.

Example: If the actual sampled value is 2.345 volts and the nearest quantization level is 2.355 volts in an 8-bit system, the quantization error would be ( e_q = 2.345 - 2.355 = -0.01 ) volts.

Differences and Important Points

Term	Description	Formula/Example	Important Points
Sampling	Converting continuous signal to discrete by measuring at uniform intervals.	( f_s \geq 2 \times f_{max} )	Must satisfy Nyquist rate to avoid aliasing.
Quantization	Approximating sampled signal to finite set of levels.	( Q = \frac{V_{max} - V_{min}}{2^n} )	Introduces quantization error/noise.
Quantization Error	Difference between actual sampled value and quantized value.	( e_q = x(t) - Q[n] )	Can be at most half the quantization step size. Irreversible loss of information.

In summary, a PCM system digitizes an analog signal by sampling it at regular intervals, quantizing the sampled values to a finite set of levels, and encoding these levels into a digital bit stream. Sampling determines the temporal resolution, while quantization determines the amplitude resolution of the digital signal. Quantization error is an inherent part of the PCM process and affects the fidelity of the reconstructed signal.