4.1 KiB
+++ title = "Signal to Noise Ratio" author = ["Thomas Dehaeze"] draft = false +++
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
From (Jabben 2007) (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.
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: \[ S_{rms} = \sqrt{S_{snr} \cdot f_c} \approx 3.4\ mV \]
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: \[ S_\text{rms} = \frac{\Delta V/2}{\sqrt{2}} \]
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: \[ n_\text{rms} = 10^{-\frac{S_{nrs}}{20}} s_\text{rms} \]
\[ n_\text{rms} = 10^{-\frac{85}{20}} \frac{10}{\sqrt{2}} \approx 0.4 mV_\text{rms} \]
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: \[ S_{snr} = 20 \log \frac{\text{Signal, rms}}{\text{Noise, rms}} \]
Let's say the wanted noise is \(1 mV, \text{rms}\) for a full range of \(20 V\), the corresponding SNR is: \[ S_{snr} = 20 \log \frac{\frac{20/2}{\sqrt{2}}}{10^{-3}} \approx 77dB \]
Noise Density to RMS noise
From (Fleming 2010): \[ \text{RMS noise} = \sqrt{2 \times \text{bandwidth}} \times \text{noise density} \]
If the noise is normally distributed, the RMS value is also the standard deviation \(\sigma\). The peak to peak amplitude is then approximately \(6 \sigma\).
- 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 \(6 \sigma = 1.7 nm\)
Bibliography
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.
Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.