Quantized Probability Density Functions

Understanding the concept, mathematical functions, and applications of quantized probability density functions (PDFs).

What Are Quantized Probability Density Functions?

Quantized probability density functions (PDFs) describe the distribution of a random variable after it has been discretized or quantized. Instead of representing a continuous range of values, the random variable is mapped to discrete levels, which simplifies calculations and reduces data representation complexity.

• Key Difference: Unlike continuous PDFs, quantized PDFs have discrete values.
• Example: Quantizing a continuous signal into discrete amplitude levels for digital storage.

Key Insight: Quantized PDFs are widely used in digital signal processing, image compression, and machine learning.

Mathematical Representation of Quantized PDFs

Quantized PDFs map continuous values into discrete bins, each represented by a probability. Consider a continuous random variable \( X \) with a PDF \( f_X(x) \). After quantization, the random variable \( Q \) has a quantized PDF, \( f_Q(q) \), defined as:

\( f_Q(q_k) = \int_{q_k - \Delta/2}^{q_k + \Delta/2} f_X(x) \, dx \)

Here:

• \( q_k \): The \( k \)-th quantized level.
• \( \Delta \): The quantization step size.
• \( f_X(x) \): The original continuous PDF.

This equation computes the probability mass for each quantized level by integrating the original PDF over the range corresponding to that level.

Example: A uniform distribution quantized into 4 levels will have equal probabilities for each level, assuming uniform bin width.

Visualizing Quantized PDFs

Quantized PDFs can be visualized as histograms where each bin corresponds to a quantized level. For example, consider a normal distribution quantized into 5 levels:

The continuous PDF:

\( f_X(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

After quantization, the discrete probabilities are distributed as:

• \( q_1: P(q_1) = \int_{-\infty}^{q_1 + \Delta/2} f_X(x) \, dx \)
• \( q_2: P(q_2) = \int_{q_1 + \Delta/2}^{q_2 + \Delta/2} f_X(x) \, dx \)
• \( \dots \)
• \( q_5: P(q_5) = \int_{q_4 + \Delta/2}^{\infty} f_X(x) \, dx \)

Each probability corresponds to the area under the curve for a given bin.

Applications of Quantized PDFs

Quantized PDFs are widely used in various fields:

• Digital Signal Processing: Quantizing audio or video signals for efficient storage and transmission.
• Image Compression: Representing pixel intensities as discrete values in formats like JPEG.
• Machine Learning: Quantizing weights in neural networks to reduce model size and inference time.
• Statistics: Approximating continuous distributions with discrete counterparts for computational efficiency.

Example: In lossy compression, quantized PDFs are used to minimize the storage size while retaining the essential characteristics of the original signal.

Final Thoughts: Quantized probability density functions simplify the representation of continuous variables, making them invaluable in fields like digital signal processing, compression, and machine learning. By understanding the mathematical foundations of quantization, you can leverage this technique to optimize performance and efficiency in various applications.