93 lines
3.4 KiB
Markdown
93 lines
3.4 KiB
Markdown
+++
|
||
title = "Analog to Digital Converters"
|
||
author = ["Thomas Dehaeze"]
|
||
keywords = ["electronics"]
|
||
draft = false
|
||
+++
|
||
|
||
Tags
|
||
: [Electronics]({{< relref "electronics" >}})
|
||
|
||
|
||
## Types of Analog to Digital Converters {#types-of-analog-to-digital-converters}
|
||
|
||
<https://dewesoft.com/daq/types-of-adc-converters>
|
||
|
||
- Delta Sigma ([Baker 2011](#org60f0e22))
|
||
- Successive Approximation
|
||
|
||
|
||
## Power Spectral Density of the Quantization Noise {#power-spectral-density-of-the-quantization-noise}
|
||
|
||
This analysis is taken from [here](https://www.allaboutcircuits.com/technical-articles/quantization-nois-amplitude-quantization-error-analog-to-digital-converters/).
|
||
|
||
Let's note:
|
||
|
||
- \\(q = \frac{\Delta V}{2^n}\\) the quantization in [V], which is the corresponding value in [V] of the least significant bit
|
||
- \\(\Delta V\\) is the full range of the ADC in [V]
|
||
- \\(n\\) is the number of ADC's bits
|
||
- \\(f\_s\\) is the sample frequency in [Hz]
|
||
|
||
Let's suppose that the ADC is ideal and the only noise comes from the quantization error.
|
||
Interestingly, the noise amplitude is uniformly distributed.
|
||
|
||
The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
|
||
Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#orgee08810)).
|
||
|
||
<a id="orgee08810"></a>
|
||
|
||
{{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}}
|
||
|
||
Now, we can calculate the time average power of the quantization noise as
|
||
|
||
\begin{equation}
|
||
P\_q = \int\_{-q/2}^{q/2} e^2 p(e) de = \frac{q^2}{12}
|
||
\end{equation}
|
||
|
||
The other important parameter of a noise source is the power spectral density (PSD), which indicates how the noise power spreads in different frequency bands.
|
||
To find the power spectral density, we need to calculate the Fourier transform of the autocorrelation function of the noise.
|
||
|
||
Assuming that the noise samples are not correlated with one another, we can approximate the autocorrelation function with a delta function in the time domain.
|
||
Since the Fourier transform of a delta function is equal to one, the **power spectral density will be frequency independent**.
|
||
Therefore, the quantization noise is white noise with total power equal to \\(P\_q = \frac{q^2}{12}\\).
|
||
|
||
Thus, the two-sided PSD (from \\(\frac{-f\_s}{2}\\) to \\(\frac{f\_s}{2}\\)), we should divide the noise power \\(P\_q\\) by \\(f\_s\\):
|
||
|
||
\begin{equation}
|
||
\int\_{-f\_s/2}^{f\_s/2} \Gamma(f) d f = f\_s \Gamma = \frac{q^2}{12}
|
||
\end{equation}
|
||
|
||
<div class="important">
|
||
<div></div>
|
||
|
||
Finally, the Power Spectral Density of the quantization noise of an ADC is equal to:
|
||
|
||
\begin{equation}
|
||
\begin{aligned}
|
||
\Gamma &= \frac{q^2}{12 f\_s} \\\\\\
|
||
&= \frac{\left(\frac{\Delta V}{2^n}\right)^2}{12 f\_s} \text{ in } \left[ \frac{V^2}{Hz} \right]
|
||
\end{aligned}
|
||
\end{equation}
|
||
|
||
</div>
|
||
|
||
<div class="exampl">
|
||
<div></div>
|
||
|
||
Let's take a 18bits ADC with a range of +/-10V and a sample frequency of 10kHz.
|
||
|
||
The quantization is:
|
||
\\[ q = \frac{20}{2^{18}} = 0.000076 \ [V] = 76 \ [\mu V] \\]
|
||
|
||
\\[ \Gamma\_Q = \frac{q^2}{12 f\_N} = 4.85 \cdot 10^{-14} \quad [V^2/Hz] \\]
|
||
|
||
</div>
|
||
|
||
{{< youtube b9lxtOJj3yU >}}
|
||
|
||
|
||
|
||
## Bibliography {#bibliography}
|
||
|
||
<a id="org60f0e22"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7.
|