What is meant by quantization?

Quantization is a methodology of carrying out signal modulation by the process of mapping input values from an infinitely long set of continuous values to a smaller set of finite values. Quantization forms the basic algorithm for lossy compression algorithms and represents a given analog signal into digital signals. In other words, these algorithms form the base of an analog-to-digital converter. Devices that process the algorithm of quantization are known as a quantizer. These devices aid in rounding off (approximation) the errors of an input function called the quantized value.

Quantizer

A quantizer is a device which is responsible for altering the sampled input signal into a signal that possesses some predetermined voltage levels. The level of quantization performed by the quantizer depends on the bit value of the encoder.

Advantages of quantizer

• The number of bits needed for the representation of a signal is greatly reduced by the use of a quantizer.
• Quantizer reduces the bitrate which in turn reduces the bandwidth requirement.

However, a quantizer can lead to certain disadvantages like the introduction of certain errors in the original signal during the approximation or rounding off, known as quantization noise.

In the context of digital electronics and electronic instrumentation, the measure of the distance of amplitude between two points in a waveform is known as resolution. A resolution is primarily related to a digital-to-analog converter. It is generally between the maximum measured signal and the part that can be resolved. Resolution refers to the ability of an instrument to detect a change theoretically and is represented as the number of bits.

Types of quantization

Here, in this section, the various types of quantization performed by the quantizer are outlined.

Uniform quantization

Uniform quantization is also known as linear quantization. In this type of quantization, the steps between two quantized levels are maintained at the same level by the quantizer. Based on the position of the presence of origin, the uniform quantization is classified as midtread and midrise quantization.

• Midtread quantization: This type of uniform quantization is characterized by a staircase graph. The center is present at the mid-point of the tread of the graph.
• Midrise quantization: This kind of uniform quantization is characterized by a staircase graph as well, but the mid-point of such graph is present at the rising part of the graph.

Non-uniform quantization

The step size produced by the quantizer is not constant but has a certain degree of variation. In this case, the quantized signal shows non-linearity with the discrete signal.

Quantization of analog signal

The algorithm is primarily implemented in the analog-to-digital converter. The algorithm leads to the achievement of the discrete value of an analog signal. To illustrate the method of quantization. For instance, consider the below-mentioned analog signal in the form of a sinusoidal waveform.

In the process of quantization, the quantizer represents the maximum value of the amplitude of the given analog waveform into a similar waveform which is in the discrete form. This process also depends on the bit level of the encoder. The below image shows the sampling of the signal by the encoder of the quantizer.

The quantizer then transforms the signal into a quantized signal preferably known as signal modulation. A major drawback of the quantized signal developed by the quantizer is the approximation it follows. For instance, if the sampled output has a value of 1.2, then the output of the quantized value is 1. Hence, due to this process, much of the information is lost, which leads to quantization error.

The quantization error is simply evaluated as-

$\epsilon = \varphi -\omega$, where $\epsilon$ represents quantization error,

$\varphi$ denotes quantized signal, and

$\omega$ denotes input signal.

Resolution of an analog-to-digital converter

The resolution of an analog-to-digital converter is normally defined as the minute change in the input analog signal, which changes the output digital signal by a unit count. Resolution is a primary parameter of an analog-to-digital converter and is associated with transfer functions, these are characterized by staircase waveform. The step count of the waveform is equal to the resolution of the waveform. However, with larger bits of signals, the transfer function shows a considerable deviation from the input signal and contributes to large quantization noises.

Quantization noise is primarily associated with a signal when a continuous random variable is converted to a discrete random variable or a discrete random variable is converted to a continuous random variable having fewer magnitudes (levels).

Hence, the quantization noise can be reduced by increasing the level of quantization.

Context and Applications

This topic is widely used in many graduate and postgraduate degree courses of:

• Bachelors in Technology (Electrical Engineering)
• Bachelors in Technology (Electrical and Electronics Engineering)
• Masters in Technology (Electronics Engineering)
• Masters in Technology (Digital Signal Processing)

Practice Problems

Q1. In which of the following quantizations does the output signal show non-linearity with the input signal?

1. Uniform quantization
2. Non-uniform quantization
3. Midrise quantization
4. Midtread quantization

Answer: Option b

Explanation: In the non-uniform quantization the reconstruction and transition levels of the quantized signals show uneven spacing. The quantized output signal has non-linearity with the input signal.

Q2. Which of the following is true for a quantizer?

1. The maximum amplitude of the signal is represented as a simpler discrete signal.
2. It performs the process of quantization.
3. Both a and b
4. None of these

Answer: Option d

Explanation: Quantizers the devices which aid in the signal quantization process and map the input amplitude values into an output value of the required discrete signal.

Q3. By which of the following process quantization noise can be minimized?

1. Increasing the level of quantization
2. Decreasing the level of quantization
3. Increasing rate of sampling
4. Increasing resolution

Answer: Option a

Explanation: The primary cause of quantization noise is the process of approximation or rounding off. Hence, if the discrete amplitudes of the output signals are increased the quantization noise can be reduced.

Q4. Which of the following relation correctly relates quantization error with the input signal?

1. Quantization error = quantized signal - input signal
2. Quantization error = quantized signal + input signal
3. Quantization error = quantized signal - signal amplitude
4. None of these

Answer: Option a

Explanation: The quantization error can be simply evaluated by subtracting the quantized signal from the input signal.

Q5. Resolution is characterized by which of the following parameter?

1. Piece-wise functions
2. Quadratic functions
3. Algebraic functions
4. Transfer functions

Answer: Option d

Explanation: Transfer functions form the basic parameter of resolution associated with an analog-to-digital converter. Transfer function theoretically models the system's output from a set of inputs.