Lecture 9: Digitization

Overview

• Recap of Lectures 1-7 covering analog modulation.
  • Types: Amplitude Modulation (AM), Angle Modulation.
  • Topics: Modulation/Demodulation, Power, Bandwidth, Applications, Advantages/Disadvantages.
• Introduction to Digitization before moving to Digital Modulation.

Key Concepts in Digitization

Sampling and Quantization

• Sampling: Converts a continuous-time signal into a discrete-time signal.
  • Terminology: Continuous time (analog), Discrete time (sampled but still analog).
  • Requires adherence to the Nyquist-Shannon Criterion for reversible sampling.
• Quantization: Converts sampled signals (discrete-time, continuous-valued) to discrete values (digital).
• Together, sampling and quantization achieve digitization of analog signals.

Nyquist-Shannon Criterion

• If a signal is band-limited with a maximum frequency component of B Hz, the sampling frequency f_s should be at least 2B to avoid aliasing.
• Critical Sampling: f_s = 2B.
• Over Sampling: f_s > 2B.
• Under Sampling: f_s < 2B (leads to aliasing issues).

Examples

• Audio message with frequency components at 1 kHz and 3 kHz.
  • Maximum frequency B = 3 kHz.
  • Nyquist rate 2B = 6 kHz.
  • Suitable sampling rates > 6 kHz (e.g. 8 kHz).
• Signal between 300 Hz and 3.3 kHz.
  • Maximum frequency B = 3.3 kHz.
  • Nyquist rate 2B = 6.6 kHz. Suitable rates could be 10 kHz, 15 kHz, etc.

Aliasing and Anti-Aliasing

• Aliasing: When sampling rate is less than Nyquist rate, leading to spectral folding and distortion.
• Anti-aliasing Filters: Applied to prevent or reduce aliasing.
  • Pre-filtering (before sampling) is generally better than post-filtering (after sampling).

Quantization

• Quantization Levels: Finite set of amplitude values.
  • Example: 2-bit quantizer has 4 levels.
  • Error/Noise: Difference between original and quantized signal (quantization noise).
• Quantization Error: Maximum error is ±(Δ/2) where Δ is the step size.
• Granularity Noise: Noise due to limited number of quantization levels.
• Overload Noise: Noise due to signal exceeding the quantizer’s range.

Mathematical Expressions

• Number of Levels (L): L = 2^n, where n is the number of bits.
  • Example: 8 bits ⇒ 256 levels.
• Step Size (Δ): R / L, where R is the range of the quantizer.

Bit Rate Calculation

• Bit rate (bits per second) = bits per sample × samples per second.

Oversampling and Signal Quality

• Oversampling above Nyquist rate can improve the quality of digitized signals by reducing quantization noise.

Non-Uniform Quantization

• Non-linear Quantization: More levels for low amplitudes, fewer for high amplitudes to match non-uniform probability density functions of real signals.
  • Companding: Applying non-linearity before quantization and its inverse after quantization.

Noise Reduction Techniques

• Historical context: Analog tape recording systems and noise reduction systems like dbx and Dolby.
• Companding in Noise Reduction: Compressing the signal before recording and expanding it during playback to reduce noise and improve dynamic range.

Digital Modulation and Transmission

• Baseband Modulation: Uses a physical medium like fiber optic cables.
  • Covered in Lecture 10.
• Bandpass Modulation: Uses carrier signals for wireless communication.
  • Covered in Lecture 11.

Summary and Next Steps

• Covered: digitization, sampling, quantization, anti-aliasing, and theoretical underpinnings.
• Upcoming: Baseband and bandpass modulation methods in digital communication.

Stay tuned for the next lecture and stay safe!