Update Content - 2020-09-04

content/zettels/analog_to_digital_converters.md

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+title = "Analog to Digital Converters"
+author = ["Thomas Dehaeze"]
+draft = false
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+
+Tags
+: [Electronics]({{< relref "electronics" >}})
+
+
+## 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] (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](#org5158d30)).
+
+<a id="org5158d30"></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="examp">
+  <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>
+
+<./biblio/references.bib>