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+++
title = "Signal to Noise Ratio"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Electronics ]({{< relref "electronics" >}} ), [Dynamic Error Budgeting ]({{< relref "dynamic_error_budgeting" >}} )
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## SNR to Noise PSD {#snr-to-noise-psd}
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From ([Jabben 2007](#org05d266b)) (Section 3.3.2):
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> 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.
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< 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}
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From ([Fleming 2010](#org8235840)):
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\\[ \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\\).
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< div class = "examp" >
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< 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}
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< a id = "org8235840" ></ 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– < https: // doi . org / 10 . 1109 / tmech . 2009 . 2028422 > .
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< a id = "org05d266b" > < / a > Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
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## Backlinks {#backlinks}
- [Piezoelectric Actuators ]({{< relref "piezoelectric_actuators" >}} )
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- [Power Spectral Density ]({{< relref "power_spectral_density" >}} )
- [Position Sensors ]({{< relref "position_sensors" >}} )
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- [Voltage Amplifier ]({{< relref "voltage_amplifier" >}} )
- [Voltage Amplifier ]({{< relref "voltage_amplifier" >}} )
