Quantization is a key part of the digital signal processing chain where continuous analog signals are converted into discrete digital signals. Such conversion is essential for many applications in digital communications, audio processing, image compression and more. Engineers and practitioners who design digital systems should understand these nuances since quantization affects the accuracy and efficiency with which signals can be represented.

There are two kinds of quantization namely uniform and non-uniform. They have different properties, advantages and uses of each type. Uniform quantization is simple to implement, it has equal step sizes and is really best suited for signals with high signal-to-noise ratio (SNR). Variable step sizes, known as nonuniform quantization, on the other hand delivers improved performance with signals having a non-uniform amplitude distribution and in low SNR. This article will go into depth with the ideas of quantization, differences between uniform and non-uniform quantization as well as their use cases and benefits.

What is Quantization?

Quantization is the process of mapping a large set of input values to a (countable) smaller subset, such as limited component resolution or digital signal sampling. It is a very important step in analog-to-digital conversion (ADC) when the continuous range of an analogue signal is divided into discrete levels and each level has its own digital code which corresponds to it. While Quantization brings quantization error, the difference between true analog value and it’s quantized digital form. The objective is to reduce this error while representing the signal in an efficient way.

What is Uniform Quantization?

Uniform quantization is a technique in which the range of an input signal is divided into number of zones equally spaced. Because the quantization levels are linearly spaced, you get a uniform step size between each of them. This method is very simple and easy to install. This uniform property makes it ideal for signals with a high SNR (compared to all the sounds that have ever existed, these tones are very loud!) and a uniform noise level distribution within amplitude.

Types of Uniform Quantization

• Mid-rise type Uniform Quantization: This discrete quantized signal comes from the midpoint of the “rise” which is located at an ascending portion of a stair-like graph.
• Uniform Quantization of Mid-tread type: The stair-like graph has it “treads” in the center and signifies that for any quantized signal, their flat portion; mid-be part ought to lie above every stage.

Graph of Mid-Rise and Mid-Tread type uniform quantization

What is Non-uniform Quantization?

Non-Uniform quantization on the other hand, uses variable size steps. Instead of equally spaced intervals it uses smaller steps where the signal amplitude is most likely to be and larger step with lower probability. This method is especially one way to overcome the quantization error in case of signals with nonuniform amplitude distributions, such as speech (such amplitudes depend a lot on each word). Non-uniform quantization can be more difficult to implement but provides better performance for specific applications.

Formula and Derivation of Uniform and Non-Uniform Quantization

Uniform Quantization

1. Formula

In uniform quantization, the range of the input signal is divided into [Tex]{L}[/Tex] equally spaced levels. The step size [Tex](\Delta)[/Tex] is calculated as:

[Tex] \Delta = \frac{V_{max} – V_{min}}{L} [/Tex]

where:

• [Tex]{V_{max}}[/Tex] is the maximum value of the input signal.
• [Tex]{V_{min}}[/Tex] is the minimum value of the input signal.
• [Tex]{?}[/Tex] is the number of quantization levels.

2. Derivation

Let’s derive the formula for uniform quantization:

• The range of the input signal is [Tex]{V_{max} – V_{min}}[/Tex]
• Now, this range is divided into [Tex]{?}[/Tex] equal intervals.
• Each interval (or step size) is given by the total range divided by the number of levels, resulting in [Tex]\Delta[/Tex].

Once the step size is determined, each input value [Tex]{x}[/Tex] can be quantized to the nearest level using:

[Tex] Q(x) = \Delta \cdot \left\lfloor \frac{x – V_{min}}{\Delta} + 0.5 \right\rfloor + V_{min} [/Tex]

This equation ensures that each input value is mapped to the nearest quantization level.

Non-Uniform Quantization

1. Formula

Non-uniform quantization often uses a companding approach, where the signal is first compressed, uniformly quantized, and then expanded. The compression and expansion functions can vary, but a common method is to use a logarithmic companding function. For example, the [Tex]\mu[/Tex]-law companding used in telephony is given by:

[Tex]y = \text{sgn}(x) \cdot \frac{\ln(1 + \mu |x|)}{\ln(1 + \mu)}[/Tex]

where:

• [Tex]{x}[/Tex] is the input signal.
• [Tex]{y}[/Tex] is the compressed signal.
• [Tex]\mu[/Tex] is the companding parameter.
• [Tex]\text{sgn}(x)[/Tex] is the sign function, preserving the sign of [Tex]{x}[/Tex].

2. Derivation

The steps for non-uniform quantization using [Tex]\mu[/Tex] law companding are as follows:

1. Compression: Apply the compression function to the input signal [Tex]{x}[/Tex]:

[Tex]y = \text{sgn}(x) \cdot \frac{\ln(1 + \mu |x|)}{\ln(1 + \mu)}[/Tex]

2. Uniform Quantization: Quantize the compressed signal [Tex]{y}[/Tex] using uniform quantization.

3. Expansion: Apply the inverse of the compression function to the quantized value [Tex]{y_q}[/Tex] :

[Tex] x_q = \text{sgn}(y_q) \cdot \left( \frac{1}{\mu} \left( (1 + \mu)^{|y_q|} – 1 \right) \right)[/Tex]

This process ensures that the quantization error is minimized in regions where the signal amplitude is more likely to occur, providing better performance for non-uniformly distributed signals.

Waveform of Uniform and Non-Uniform Quantization

Uniform Quantization Waveform

In uniform quantization, the quantized output signal consists of equally spaced steps. Each step corresponds to a specific range of the input signal. Here’s a conceptual representation:

• The x-axis represents the continuous input signal.
• The y-axis represents the quantized output signal.
• The horizontal segments indicate the quantization levels.
• The vertical jumps occur at the boundaries between quantization intervals.

Non-Uniform Quantization Waveform

In non-uniform quantization, the quantized output signal consists of variably spaced steps. The steps are smaller in regions where the signal amplitude is more likely, resulting in finer quantization and reduced error in those regions. Here’s a conceptual representation:

• The x-axis represents the continuous input signal.
• The y-axis represents the quantized output signal.
• The horizontal segments indicate the quantization levels.
• The vertical jumps occur at the boundaries between quantization intervals, which are not uniformly spaced.

The non-uniform quantization waveform shows smaller steps in regions with higher probability of occurrence and larger steps in less likely regions. This allows for more precise representation of critical parts of the signal, reducing the overall quantization error.

Difference Between Uniform and Non-Uniform Quantization

In the following table some of the differences of Uniform Quantization and Non-Uniform Quantization is mentioned.