110 lines
4.1 KiB
Markdown
110 lines
4.1 KiB
Markdown
+++
|
||
title = "Signal to Noise Ratio"
|
||
author = ["Thomas Dehaeze"]
|
||
draft = false
|
||
+++
|
||
|
||
### Backlinks {#backlinks}
|
||
|
||
- [Power Spectral Density]({{< relref "power_spectral_density" >}})
|
||
- [Voltage Amplifier]({{< relref "voltage_amplifier" >}})
|
||
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
|
||
- [Position Sensors]({{< relref "position_sensors" >}})
|
||
|
||
Tags
|
||
: [Electronics]({{< relref "electronics" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
|
||
|
||
|
||
## SNR to Noise PSD {#snr-to-noise-psd}
|
||
|
||
From ([Jabben 2007](#org620f0ec)) (Section 3.3.2):
|
||
|
||
> Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
|
||
> In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
|
||
> The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
|
||
> The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.
|
||
>
|
||
> Usually, the SNR is specified up to a certain cut-off frequency.
|
||
> If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
|
||
> \\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
|
||
> with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
|
||
|
||
<div class="examp">
|
||
<div></div>
|
||
|
||
Let's take an example.
|
||
|
||
- \\(x\_{fr} = 170 V\\)
|
||
- \\(C\_{snr} = 85 dB\\)
|
||
- \\(f\_c = 200 Hz\\)
|
||
|
||
The Power Spectral Density of the output voltage is:
|
||
\\[ S\_{snr} = \frac{170^2}{8 \cdot 200 \cdot {10^{\frac{2 \cdot 85}{20}}}} = 5.7 \cdot 10^{-8}\ V^2/Hz \\]
|
||
|
||
And the RMS of that noise up to \\(f\_c\\) is:
|
||
\\[ S\_{rms} = \sqrt{S\_{snr} \cdot f\_c} \approx 3.4\ mV \\]
|
||
|
||
</div>
|
||
|
||
|
||
## Convert SNR to Noise RMS value {#convert-snr-to-noise-rms-value}
|
||
|
||
The RMS value of the noise can be computed from:
|
||
\\[ N\_\text{rms} = 10^{-\frac{S\_{snr}}{20}} S\_\text{rms} \\]
|
||
where \\(S\_{snr}\\) is the SNR in dB and \\(S\_\text{rms}\\) is the RMS value of a sinus taking the full range.
|
||
|
||
If the full range is \\(\Delta V\\), then:
|
||
\\[ S\_\text{rms} = \frac{\Delta V/2}{\sqrt{2}} \\]
|
||
|
||
<div class="examp">
|
||
<div></div>
|
||
|
||
As an example, let's take a voltage amplifier with a full range of \\(\Delta V = 20V\\) and a SNR of 85dB.
|
||
The RMS value of the noise is then:
|
||
\\[ n\_\text{rms} = 10^{-\frac{S\_{nrs}}{20}} s\_\text{rms} \\]
|
||
|
||
\\[ n\_\text{rms} = 10^{-\frac{85}{20}} \frac{10}{\sqrt{2}} \approx 0.4 mV\_\text{rms} \\]
|
||
|
||
</div>
|
||
|
||
|
||
## Convert wanted Noise RMS value to required SNR {#convert-wanted-noise-rms-value-to-required-snr}
|
||
|
||
If the wanted full range and RMS value of the noise are defined, the required SNR can be computed from:
|
||
\\[ S\_{snr} = 20 \log \frac{\text{Signal, rms}}{\text{Noise, rms}} \\]
|
||
|
||
<div class="examp">
|
||
<div></div>
|
||
|
||
Let's say the wanted noise is \\(1 mV, \text{rms}\\) for a full range of \\(20 V\\), the corresponding SNR is:
|
||
\\[ S\_{snr} = 20 \log \frac{\frac{20/2}{\sqrt{2}}}{10^{-3}} \approx 77dB \\]
|
||
|
||
</div>
|
||
|
||
|
||
## Noise Density to RMS noise {#noise-density-to-rms-noise}
|
||
|
||
From ([Fleming 2010](#org094853a)):
|
||
\\[ \text{RMS noise} = \sqrt{2 \times \text{bandwidth}} \times \text{noise density} \\]
|
||
|
||
If the noise is normally distributed, the RMS value is also the standard deviation \\(\sigma\\).
|
||
The peak to peak amplitude is then approximatively \\(6 \sigma\\).
|
||
|
||
<div class="examp">
|
||
<div></div>
|
||
|
||
- noise density = \\(20 pm/\sqrt{Hz}\\)
|
||
- bandwidth = 100Hz
|
||
|
||
\\[ \sigma = \sqrt{2 \times 100} \times 20 = 0.28nm RMS \\]
|
||
The peak-to-peak noise will be approximately \\(6 \sigma = 1.7 nm\\)
|
||
|
||
</div>
|
||
|
||
|
||
## Bibliography {#bibliography}
|
||
|
||
<a id="org094853a"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
|
||
|
||
<a id="org620f0ec"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
|