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Posts Tagged ‘quantization’

Today, I taught a guest lecture for a graduate class at the University of Texas at Austin. The topic was Over-Sampling Data Converters with Mismatch-Shaping. The talk covered some basics of delta-sigma modulators within the context of a DAC and multibit quantization, and then explored various mismatch-shaping schemes.

For me, it was a good learning experience on how to present to those unfamiliar with advanced concepts. Historically, my presentation audiences have almost always consisted of peers already very familiar with the material. This was an entirely different approach, as the backgrounds of the students varied widely. One thing to remember is to limit the scope of a discussion to as narrow a range of new concepts as possible, so as to maximize the focus of the audience and enable the presenter to adequately devote attention to each detail. All in all, it was a worthwhile endeavor.

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This is a continuation of a discussion about quantization and analog-to-digital converters. In that discussion, the normalized quantization step through an N-bit ADC was denoted q, where q = 1/2N. The ADC encoder transfer function yielded a quantization error range over the interval [-q/2,+q/2].

Quantization is a highly nonlinear process. Denoting the input and output of a quantizer as u[n] and uq[n], respectively, the error from quantization uq[n]-u[n] can be re-arranged to yield the additive noise model of quantization error: uq[n]=u[n]+e, where e is the quantization error.

The figure below shows the quantization error for a full-scale sine wave over a single period. Also shown is the quantization error for a full-scale sawtooth ramp signal.

Although the quantization error from the sinusoid is signal-dependent and nonlinear, the commonly used additive noise model assumes a stochastic process in order to simplify the analysis. In particular, the error is treated as an independent and identically distributed (i.i.d.) random variable.

If the quantization error is modeled as a random variable with a uniform distribution, the probability density function is given by:

p(e) = \begin{cases}              0, & x \mbox{ \textless } -q/2 \\            1/q, & -q/2 \leq x \leq +q/2 \\              0, & x \mbox{ \textgreater } +q/2          \end{cases}

The root mean-square (RMS) quantization error with such a distribution can thus be derived:

\begin{array}{lcl}  e_{RMS}^2 & = & E(e^2) \\       	  & = & \int\limits_{-q/2}^{+q/2} e^2 \cdot p(e) \,d e \\       	  & = & \frac{1}{q} \int\limits_{-q/2}^{+q/2} e^2 \,d e \\       	  & = & \frac{q^2}{12} \\    e_{RMS} & = & \frac{q}{\sqrt{12}}  \end{array}

The RMS value of a full-scale sinusoid whose peak-to-peak swing has been normalized to unity is given by:

sig_{RMS} = \frac{1}{2\sqrt{2}}

The signal-to-quantization-noise ratio (SQNR) through the ADC can then be computed and expressed in decibels (dB) as:

\begin{array}{lcl}  SQNR_{sig} & = & 10log_{10} \left( \frac{sig_{RMS}}{e_{RMS}} \right)^2 \\             & = & 10log_{10} \left( \frac{1}{2\sqrt{2}} / \frac{q}{\sqrt{12}} \right)^2  \end{array}

Substituting q=1/2N gives:

\begin{array}{lcl}  SQNR_{sig} & = & 10log_{10} \left( 2^N \cdot \sqrt{3/2} \right)^2 \\             & = & 20log_{10} \left( 2^N \right) + 20log_{10} \left( \sqrt{3/2} \right) \\             & = & (6.02N + 1.76) dB  \end{array}

This is the well-known equation for SNR or dynamic range through an N-bit ADC using the additive noise model of quantization error, and in the absence of all other noise sources like thermal noise in the analog circuitry, dither and sampling jitter. Note that no over-sampling is assumed here.

This analysis assumes that quantization errors are uniformly distributed over the quantization interval. In reality, the errors are not uniformly distributed for a sinusoidal input. For example, referring back to the time-domain quantization error from a sinusoid and a sawtooth ramp shown in the figure above, the respective error distributions are shown in the figure below.

The quantization error of the sawtooth wave appears to be uniformly distributed, but that of the sinusoid is clearly not. This is due to the signal-dependence of the sinusoid’s quantization error. Since the sawtooth actually produces uniformly distributed quantization errors, it is instructive to compute the SQNR from quantizing such a signal.

The RMS value of a full-scale sawtooth whose peak-to-peak swing has been normalized to unity is given by:

saw_{RMS} = \frac{1}{2\sqrt{3}}

Using the RMS quantization error derived above for a uniformly distributed quantization error, the SQNR of a sawtooth wave applied to an ADC can be expressed as:

\begin{array}{lcl}  SQNR_{saw} & = & 10log_{10} \left( \frac{saw_{RMS}}{e_{RMS}} \right)^2 \\             & = & 10log_{10} \left( \frac{1}{2\sqrt{3}} / \frac{q}{\sqrt{12}} \right)^2 \\             & = & 10log_{10} \left( 2^N \right)^2 \\             & = & (6.02N) dB  \end{array}

In general, the computed SQNR depends on the signal source and the model used for the quantization error. For sinusoidal inputs, the approximation of uniformly distributed quantization error improves as the ADC precision increases.

The figure below compares the error distribution of the sawtooth with that of four ADC resolutions (3 bits, 6 bits, 9 bits, and 12 bits). Clearly, the distribution approaches the quantization model of a sawtooth as the ADC resolution is increased.

Modeling the SQNR as 6dB per bit of ADC precision is a good approximation, especially as the ADC precision asymptotically increases. For many signal processing applications, the usefulness of approximating the quantization error as an i.i.d. noise source, far exceeds the inaccuracy of the model.

Copyright © 2008 – 2012 Waqas Akram. All Rights Reserved.

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An Analog-to-Digital Converter (ADC) does exactly what the name implies: it converts an analog electrical signal to a digital representation. Specifically, the analog signal is a continuous-time continuous-amplitude signal, and the digital signal representation produced by the typical ADC is a sequence of discrete-time discrete-amplitude samples. The process of conversion from a high-resolution signal to a low-resolution signal is also known as quantization.

The two main types of ADCs are oversampling converters and Nyquist-rate converters, and there are several architectures for these. In most cases, there is some form of uniform quantization being performed on a high-resolution signal, in order to represent it in terms of a finite set of quantization levels. The error that results from quantization is referred to as quantization error. The spectral representation of random quantization error is known as quantization noise.

An ADC requires a clock signal to synchronize the instances when the analog signal is sampled. The clock frequency is referred to as the sampling rate, i.e. the rate at which samples are taken, and can be denoted fs. It is important that the clock have little or no clock jitter, which creates uncertainty in the sampling instant, and hence increases the quantization error.

An ADC also typically requires a reference voltage, denoted by VREF, which determines the valid voltage range over which the analog input signal can be converted. The input range of an ADC that only operates on positive voltages would go from zero volts, or circuit ground, to VREF. If the analog signal takes on values outside this voltage range, a well-designed input circuit will non-catastrophically limit the ADC to either minimum or maximum voltage, depending on the input signal. As expected, this would produce either a minimum or maximum digital value at the output.

The most common representation used for the digital samples produced by an ADC is a string of binary digits (or bits), where 00..0 represents the smallest analog input, and 11..1 represents the largest. These are sometimes referred to as ADC output codes. An N-bit binary number can represent at most 2N unique levels, and therefore, an N-bit ADC can produce 2N unique codes.

The quantization step or width of each ADC code can be denoted q, where q = VREF/2N. The nominal ADC code width is expected to be equal to a single LSB (least-significant bit), which is the right-most bit in a binary word representation. When the code width is normalized to VREF, q = 1/2N.

In the figure above, an example of a 3-bit ADC encoder transfer function is shown on the left, relating the digital output to the analog input. The encoder transfer function is arranged so that any input signal less than q/2 produces the smallest digital code, 000, input signals between q/2 and 3q/2 produce the next digital code 001, and so on. Alternate arrangements are possible, depending on the specific application requirements of the ADC.

The quantization error resulting from using this encoder transfer function is shown in the figure above on the right, and in this case, it takes values over the interval [-q/2,+q/2]. This assumes that the ADC input is appropriately limited and the digital output code is saturated when the input signal goes outside the operating range of the ADC.

The figure below shows the result when a full-scale sine wave is provided at the input to an ADC with this encoder transfer function.

It should be apparent that quantization is a highly non-linear process, and this makes it very difficult to perform an exact analysis of an otherwise linear system. In order to use classical linear analysis, it is necessary to derive a suitable linearized model of the quantizer, and this will be covered in a future post.

Copyright © 2008 – 2012 Waqas Akram. All Rights Reserved.

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