Lecture Notes on Quantization and ADC

Overview

• Today's class builds upon the previous discussion of sampling issues, focusing on the operation of quantization.
• The flow of the course includes:
  • Understanding ADC (Analog-to-Digital Converter) at a block level.
  • Exploring the math behind sampling and quantization.
  • Evaluating ADC performance and characterization.
  • Discussing circuit-level implementations.

Key Concepts

Quantization

• Quantization is the process of discretizing the amplitude axis of a signal.
• Incoming signal ranges from -V_reference to +V_reference.
• Instead of storing all possible amplitudes, the range is divided into a finite number of segments (levels).
• Each level is represented by a specific amplitude value.
• Example: With 8 levels, 3 bits are needed to represent them (since 2^3 = 8).

Input-Output Characteristic

• The input signal spans from -V to +V (denoted as Full Scale Range, FSR).
• If quantizing into 4 levels, the step size (Δ) is calculated as:
  • Step size (Δ) = Total Range / Number of Levels = (2V) / 4 = V / 2.
• Normalization is done with respect to the step size for easier analysis.

Quantization Error

• Quantization error occurs as a result of rounding off the input signal to the nearest quantization level.
• The quantization error (Q) can be modeled as the difference between the quantized output (V) and the actual input (Y):
  • Q = V - Y.
• This results in a non-linear operation, as the output does not scale linearly with the input.

Types of Quantizers

• Mid-Rise Quantizer: No quantization level at zero (even number of levels).
• Mid-Tread Quantizer: At least one quantization level at zero (odd number of levels).

Characteristics of Quantization Noise

• The quantization error is often treated as a random signal (noise) under certain assumptions.
• Assumptions made:
  1. Quantization noise is independent of the input signal.
  2. The probability density function (PDF) of the quantization noise is uniform within the range of ±Δ/2.

Signal to Quantization Noise Ratio (SQNR)

• SQNR is a key metric for assessing ADC performance.
• Formula for peak SQNR for sinusoidal signals:
  • SQNR (dB) = 6n + 1.76, where n is the number of bits.
• Example: For a 10-bit ADC, maximum SQNR is about 61 dB.

Effective Number of Bits (ENOB)

• ENOB quantifies the effective resolution of an ADC beyond just the number of bits.
• ENOB is important for determining actual performance despite circuit nonidealities.

Practical Considerations

• The quantization noise power is derived as:
  • Noise Power = Δ² / 12.
• Effective resolution and SQNR calculations are made under sinusoidal signal conditions.
• Frequency domain analysis is emphasized for accurately measuring signal and noise power when using quantizers.

Conclusion

• Quantization is a crucial operation in ADCs, with significant implications for signal processing.
• Understanding quantization errors and noise is essential for designing effective ADC systems.