# Noise Measurement¶

Lock-in amplifiers can be used to measure the noise on a signal. By quantifying the noise of a system one can estimate the maximum achievable performance.

## How Does a Lock-in Measure Noise?¶

A lock-in amplifier measures the signal amplitude close to a given reference frequency with a defined bandwidth around this reference frequency. The noise in an input signal near the reference frequency appears as noise in the lock-in amplifier signal output.

The noise is the standard deviation of the measured X or Y value and is
measured by first calculating the average, X_{avg}, over a
defined period of time. Then, this signal, X_{avg}, is
subtracted from the X value to get the deviation. Finally, the RMS
(root-mean-square) is calculated, corresponding to the total noise power
of the input signal within a defined bandwidth around the reference
frequency. The value is correct for input noise with Gaussian
distribution of the noise power, which is normally the case.

Most of the times the noise spectral density is of interest, which is
the normalization of the X_{noise} made independent of the
filter bandwidth. To calculate the noise spectral density from the
calculated RMS noise, one needs to divide the measured value by the
square root of the bandwidth √BW. The noise spectral density has the
units V/√Hz.

The related equations are X_{noise} = RMS(X -
X_{avg})/√BW, and Y_{noise} = RMS(Y -
Y_{avg})/√BW respectively. The X and Y noise are expected to be
identical.

## Measuring the Noise of the HF2LI/HF2IS¶

A LabVIEW example (ziExample-HF2-Noise.vi) is available to measure the noise in an input signal. To measure the equivalent input noise of the HF2, remove all BNC connectors from the input of the device and apply the following settings in the LabOne UI.

Signal Input 1 range / AC / Diff / 50 | 0.01 V / ON / OFF / ON |

Demodulator 1 Low-Pass Filter | BW 3dB = 100 Hz, Order = 4 |

Oscillator 1 Frequency | 1 MHz |

Signal Output 1 switch | OFF |

Run the example, ziExample-HF2-Noise.vi. Make sure that the correct Demodulator is selected. The noise spectral density should now show a value close to 5 nV/√Hz. By changing the settings in the user interface, the noise behavior of the device can be analyzed in more detail. For example, changing the reference frequency to 10 kHz will slightly increase the spectral noise density, because of flicker noise that is larger at lower frequencies and generally present in all electronic circuits.