Most signals in life are continuous: pressure waves propogating through air, chemical reactions, body movement. For computers to process these continuous signals, however, they must be converted to digital representations via a Analog-to-Digital Converter ADC.

A digital signal is different from its continous counterpart in two primary ways:. In this lesson, we will use audio data is our primary signal. Sound is a wonderful medium for learning because we can both visualize and hear the signal.

Recall that a microphone responds to air pressure waves. We'll plot these waveforms, manipulate them, and then play them. We suggest plugging in your headphones, so you can really hear the distinctions in the various audio samples. Note : We downsample audio data to 3, Hz and below. We could not get Chrome to play audio at these lower sampling rates but they did work in Firefox. We'll make a note of this again when it's relevant.

To install this package, you have two options. First, from within Notebook, you can execute the following two lines within a cell you'll only need to run this once :. This Notebook was designed and written by Professor Jon E. Froehlich at the University of Washington along with feedback from students.

Quantization refers to the process of transforming an analog signal, which has a continuous set of values, to a digital signal, which has a discrete set. See the figure below from Wikipedia's article on Quantization. This is the tiniest discriminable change you can observe on the Uno's analog input pins.

In contrast, the ESP32 runs on 3. A digitized sample can have a maximum error of one-half the discretization step size i.

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Because when we convert an analog value to a digital one, we round to the nearest integer. Consider a voltage signal of 0. For the examples below, we'll work with pre-digitalized audio waveforms sampled at So, while not a true continuous sample of course not, it's already a digital signal! And we'll "downsample" to investigate the effects of quantization levels and sampling rates.The digitization of analog signals involves the rounding off of the values which are approximately equal to the analog values.

The method of sampling chooses a few points on the analog signal and then these points are joined to round off the value to a near stabilized value. Such a process is called as Quantization. The analog-to-digital converters perform this type of function to create a series of digital values out of the given analog signal.

The following figure represents an analog signal. This signal to get converted into digital, has to undergo sampling and quantizing. The quantizing of an analog signal is done by discretizing the signal with a number of quantization levels. Quantization is representing the sampled values of the amplitude by a finite set of levels, which means converting a continuous-amplitude sample into a discrete-time signal. The following figure shows how an analog signal gets quantized.

The blue line represents analog signal while the brown one represents the quantized signal. Both sampling and quantization result in the loss of information. The quality of a Quantizer output depends upon the number of quantization levels used. The discrete amplitudes of the quantized output are called as representation levels or reconstruction levels.

The spacing between the two adjacent representation levels is called a quantum or step-size. The following figure shows the resultant quantized signal which is the digital form for the given analog signal. The type of quantization in which the quantization levels are uniformly spaced is termed as a Uniform Quantization. The type of quantization in which the quantization levels are unequal and mostly the relation between them is logarithmic, is termed as a Non-uniform Quantization.

There are two types of uniform quantization. They are Mid-Rise type and Mid-Tread type. The following figures represent the two types of uniform quantization. The Mid-Rise type is so called because the origin lies in the middle of a raising part of the stair-case like graph.

The quantization levels in this type are even in number.Forums New posts Search forums. Best Answers. Media New media New comments Search media. Groups Search groups Upcoming events. Log in Register. Search titles only. Search Advanced search….

### Digital Communication - Quantization

JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. How do I solve quantization errors in ADC system? Status Not open for further replies. KrisUK Newbie level 4. How do I work out quantization error in a ADC system?

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I looked around on different sites from a recommendation from another user and came to the conclusion it is the max voltage divided by the number of bits. Is this correct? Thank you. Kral Advanced Member level 4.

The total error includes the quantization error plus scale factor gain error, non-linearity errors. Regards, Jon. Last edited by a moderator: Aug 27, I don't really need to know the theory behind it, just how to work it out for an exam I've got coming up. Quantization Error assuming you're using round-off. Quantization Error OK Thanks everyone. Sorry, missed your post earlier Kral.Learn the methods and applications of modeling the quantization error of an ADC using a noise source.

We also discussed that modeling the error term as a noise signal can significantly simplify the problem of analyzing the effect of the error on system performance.

For example, if the input of a quantizer is a DC value, the quantization error will be constant. As another example, assume that the amplitude of the input is always between two adjacent quantization levels of the quantizer.

In this case, the quantization error is equal to the input minus a DC value. Another interesting case occurs when the input is a sinusoid and the sampling frequency of the quantizer is a multiple of the input frequency. An example is illustrated in Figure 1 below. The left curve of Figure 1 depicts two periods of a bit quantized sine wave.

The right curve shows the quantization error.

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For this example, the ratio of the sampling frequency to the input frequency is Visual inspection of the quantization error reveals periodic behavior one period is indicated by the orange rectangle.

As you can see, there is a correlation between the input and the error signal, whereas a noise source is not correlated with the input. In such cases, we expect the error signal to have considerable frequency components at the harmonics of the input. The error signal does not resemble noise in the above examples. However, in many practical applications, such as speech or music, the input is a complicated signal and exhibits rapid fluctuations that occur in a somewhat unpredictable manner.

In such cases, the error signal is likely to act as a noise source. Experimental measurements and theoretical studies have shown that modeling the quantization error as a noise source is valid if the following four conditions are satisfied:. You can find a more formal way of expressing these conditions in Section 4.

If these necessary conditions are satisfied, we can replace the error signal with an additive noise source as shown in Figure 2. This allows us to use concepts such as signal-to-noise ratio SNR to characterize the effect of the quantization error. However, before that, we need to find a statistical model for the noise source. The first step in characterizing a noise source can be estimating how often a given value is likely to occur.

This amplitude distribution can be obtained by observing the noise signal for a long time and taking samples to create an amplitude histogram. The histogram consists of a number of bins that correspond to the contiguous amplitude intervals spanning the entire possible range of the noise amplitude.

The Audible Effects of Aliasing and Quantization Error

The height of a bin indicates the number of samples that lie in the bin interval. If we apply an eight-bit quantizer to this signal, the quantization error sequence will be as shown in Figure 4. Interestingly, almost the same number of samples lie in the different bin intervals; the height of the bins is close to the total number of samplesdivided by the number of bins Quantizationin mathematics and digital signal processingis the process of mapping input values from a large set often a continuous set to output values in a countable smaller set, often with a finite number of elements.

Rounding and truncation are typical examples of quantization processes.

### Understanding Amplitude Quantization Error for ADCs

Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms. The difference between an input value and its quantized value such as round-off error is referred to as quantization error. A device or algorithmic function that performs quantization is called a quantizer.

An analog-to-digital converter is an example of a quantizer. The essential property of a quantizer is having a countable-set of possible output-values members smaller than the set of possible input values.

## How do I solve quantization errors in ADC system?

The members of the set of output values may have integer, rational, or real values. For the example uniform quantizer described above, the forward quantization stage can be expressed as. This decomposition is useful for the design and analysis of quantization behavior, and it illustrates how the quantized data can be communicated over a communication channel — a source encoder can perform the forward quantization stage and send the index information through a communication channel, and a decoder can perform the reconstruction stage to produce the output approximation of the original input data.

In general, the forward quantization stage may use any function that maps the input data to the integer space of the quantization index data, and the inverse quantization stage can conceptually or literally be a table look-up operation to map each quantization index to a corresponding reconstruction value.

This two-stage decomposition applies equally well to vector as well as scalar quantizers. Because quantization is a many-to-few mapping, it is an inherently non-linear and irreversible process i.

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The set of possible input values may be infinitely large, and may possibly be continuous and therefore uncountable such as the set of all real numbersor all real numbers within some limited range. The set of possible output values may be finite or countably infinite. For example, vector quantization is the application of quantization to multi-dimensional vector-valued input data.

An analog-to-digital converter ADC can be modeled as two processes: sampling and quantization. Sampling converts a time-varying voltage signal into a discrete-time signala sequence of real numbers. Quantization replaces each real number with an approximation from a finite set of discrete values. Most commonly, these discrete values are represented as fixed-point words.

Though any number of quantization levels is possible, common word-lengths are 8-bit levelsbit 65, levels and bit Quantizing a sequence of numbers produces a sequence of quantization errors which is sometimes modeled as an additive random signal called quantization noise because of its stochastic behavior. The more levels a quantizer uses, the lower is its quantization noise power. Rate—distortion optimized quantization is encountered in source coding for lossy data compression algorithms, where the purpose is to manage distortion within the limits of the bit rate supported by a communication channel or storage medium.

The analysis of quantization in this context involves studying the amount of data typically measured in digits or bits or bit rate that is used to represent the output of the quantizer, and studying the loss of precision that is introduced by the quantization process which is referred to as the distortion.

Most uniform quantizers for signed input data can be classified as being of one of two types: mid-riser and mid-tread. The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a stairway.

Mid-tread quantizers have a zero-valued reconstruction level corresponding to a tread of a stairwaywhile mid-riser quantizers have a zero-valued classification threshold corresponding to a riser of a stairway. Mid-tread quantization involves rounding. The formulas for mid-tread uniform quantization are provided in the previous section. Mid-riser quantization involves truncation. The input-output formula for a mid-riser uniform quantizer is given by:. Note that mid-riser uniform quantizers do not have a zero output value — their minimum output magnitude is half the step size.

In contrast, mid-tread quantizers do have a zero output level. For some applications, having a zero output signal representation may be a necessity. In general, a mid-riser or mid-tread quantizer may not actually be a uniform quantizer — i. The distinguishing characteristic of a mid-riser quantizer is that it has a classification threshold value that is exactly zero, and the distinguishing characteristic of a mid-tread quantizer is that is it has a reconstruction value that is exactly zero.

A dead-zone quantizer is a type of mid-tread quantizer with symmetric behavior around 0. The region around the zero output value of such a quantizer is referred to as the dead zone or deadband.An ADC transforms an input value into one of the values from a set of discrete levels and outputs a digital code to specify the quantization level.

The quantization process introduces some error into the system. This article will look at quantization error by applying a ramp input to a quantizer. At the mid-point of these intervals, there is a transition from one digital output value to the next one. The important point here is that a given digital code represents a range of analog input values; the amplitude of the input is quantized.

Therefore, even ideal amplitude quantization introduces some error. The blue line in Figure 4 shows the ramp applied to the input. Moreover, the figure shows the quantized levels that we get in the DAC output in red.

Similarly, we can find the error value for the other quantization levels as shown in Figure 5. Now we can use Figure 5 to calculate the root mean square RMS value of the quantization error for a ramp input. The error introduced by ignoring these two parts decreases as the resolution of the quantizer increases. We obtain:.

The integrals in the above equation correspond to the time-shifted versions of the same signal. Therefore, these integral terms are equal. We can directly calculate the above integral. Therefore, the RMS error is given by the following equation:. To investigate some properties of this error, we applied a ramp input and observed that the RMS of the error is proportional to the LSB value.

Increasing the resolution of the quantizer will reduce the LSB and the error term. Also note that, for a given value of the input, we can calculate the exact value of the error. If we apply an eight-bit quantizer to this signal, the quantization error sequence will be as shown in Figure 8. Comparing the input cosine of this example with the error sequence, we observe the following:. As we observed in the example of the ramp input, we know that the quantization error signal is not really random and can, in fact, be calculated for a given input value.

But what if we could model the quantization error as a random signal under some assumptions? Now, if this low-amplitude signal varies in an unpredictable manner, one may conclude that it resembles the noise sources that we usually encounter in different circuits and systems. Considering the quantization error as a noise source can simplify the problem a lot. We know how to analyze the effect of particular types of noise sources on a linear time-invariant LTI system.

The instantaneous value of a noise source is usually unpredictable, so a time-domain analysis is not possible. However, we can observe the noise for a long time and use measured results to find a statistical model of the noise.

Having the PSD of a noise signal, we can use the Laplace transform for a continuous-time system or the Z transform for a discrete-time system to analyze the effect of the noise on the output spectrum of an LTI system without knowing the instantaneous value of the noise. In this case, we can use the model of Figure 3 to describe the quantization process with an additive noise source, as shown in Figure 9.

As you can see, the quantized value V out is equal to the input V in plus a noise signal that models the quantization error V q.

Even ideal amplitude quantization introduces some error, called quantization error, into the system. It seems that we can model the quantization error as a noise signal under certain assumptions.This can be very confusing, so this type of intro helps to avoid some of the pitfalls and dead ends.

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