# UHF Arithmetic Unit

Zurich Instruments introduces the first functional unit for arithmetic operations on the results of lock-in amplifier and Boxcar measurements. The UHF Arithmetic Unit is included with every UHF Instrument but of course the functionality increases with further upgrades like UHF-BOX and UHF-PID installed.

## 4 Arithmetic Units

- More than 50 input parameters
- Add, subtract, multiply and divide demodulator samples (X, Y, R, Θ)
- Add, subtract, multiply and divide demodulator samples Boxcar output samples (requires UHF-BOX option)
- Polar conversion of arbitrary Cartesian demodulator outputs
- Fixed coefficients and auxiliary inputs as scaling factors
- Results available on auxiliary outputs connectors and as PID inputs (requires UHF-PID option)
- Fast streaming to host computer

The four arithmetic units (AU) allow real-time operations using the measurement results of demodulators, Boxcar units and auxiliary inputs connectors as input parameters. As an example the UHFLI, with its two signal inputs and a total of eight dual phase demodulators provides the chance to combine more than 50 different input parameters. The possible operations include addition, subtraction, multiplication, division, scaling, plus absolute value and phase angle calculation of complex numbers (e.g. X_{1} + i*Y_{2}).

## LabOne Integration

A dedicated tab inside the LabOne user interface accommodates the four AU units. This unique integration enables quick graphical definition of the arithmetic operations and real-time monitoring of the results. These results are indeed streamed at a configurable rate to the host computer where they can be saved to the hard disk or analyzed in one of the time domain and frequency domain tools of LabOne:

- Plotter
- Sweeper
- Numerical
- Software Trigger

The use of the AU is particularly attractive for applications where the result of the arithmetic operation is directly used to provide feedback to an experimental setup; the results can be directly used as an input for the PID controllers (requires UHF-PID option) and are also directly available on any of the auxiliary outputs. Since they are available on the auxiliary outputs, they can also be selected as demodulator inputs, permitting a second lock-in step such as tandem demodulation.

## Wide Range of Applications

### Balanced detection: noise suppression by differential measurement

\(c_0\cdot\{R_2,X_2,Y_2,\Theta_2\} - c_1\cdot\{R_1,X_1,Y_1,\Theta_1\}\)

Differential signal measurement is a powerful method for cancelling out certain noise components in order to improve signal to noise ratio. The method is quite universal and applied in a wide range of situations, for instance in the optical domain where laser spectroscopy and imaging setups are limited by laser amplitude noise. In order to overcome such limitations one can split the laser beam into a probe beam, which passes through the actual setup, and a reference beam which does not, the beams being captured by separate photodiodes (PD). Subtracting the electrical PD signals cancels out the noise components which are common to both beams, and for measurements of periodic signals (pulsed lasers, heterodyne, etc.) also removes unwanted DC components which can help improve further signal analysis down stream.

In situations where it is not possible to place the two PDs close enough to directly subtract their photo currents, one can connect them to the two inputs of the lock-in amplifier where they are measured with two demodulators with identical settings, in particular referenced to the same oscillator and with the same filter settings. Subtracting the demodulator outputs reduces the coherent noise from the signal of interest and increases the SNR.

One limitation of this scheme is that the intensity of the two light beams needs to be carefully matched (assuming a symmetric detection setup) in order to maximize noise suppression. This process can be tedious, and for setups where the transmission of the probe beam experiences significant changes an auto-balancing approach can be useful in order maintain noise suppression over the course of the measurement. This can be achieved by introducing a slowly varying scaling parameter with lower bandwidth than the actual signal (LP, low-pass filtered), for instance g = LP(Rsig) / LP(Rref). The resulting signal would then be derived as c0 * Rsig - g * Rref, c0 being an adjustable scaling factor to maximize performance.

### Normalization and relative measurement

\(\frac{R_1}{R_2}\)

For measurements where tiny differences between two samples need to be detected while the signals themselves can change over many orders of magnitude - not uncommon for instance for impedance and optical transmission measurements - a relative measurement by dividing the two signals can help to properly keep track of the relevant measurement quantity. This also easily allows boosting of the signal by numerical scaling independent of the actual signal levels, which comes in handy when the signal is further processed with a PID controller to provide feedback to an experimental setup. In such situations the controller parameters will not have to be readjusted every time the signal levels change. This could be used, for example, in a spectroscopy setup where the frequency of a laser is adjusted to obtain a defined optical transmission of a gas cell.

### Modulation parameter output for AM and FM signals

- \(R_m=\frac{\sqrt{(X_2+X_3)^2+(Y_2+Y_3)^2}}{R_1}\) (AM modulation with normalization)
- \(R_m=\frac{\sqrt{(X_2-X_3)^2+(Y_2-Y_3)^2}}{R_1}\) (FM modulation with normalization)
- \(h_{AM}=\frac{X_1}{X_2}\) (AM modulation index)

The analysis of signals with multiple frequencies, such as amplitude, frequency or phase modulated signals, can conveniently be performed with direct sideband extraction. The UHF-MOD option provides the demodulated outputs X,Y for the carrier (demodulator 1) and the sidebands independently (demodulator 2 and 3). Whenever an experiment requires the entire signal contained in the two first sidebands this can now be easily determined, even normalized to the carrier. The sum (for AM), or respectively the difference (for FM) of the sidebands provides a factor √2 SNR improvement compared to measurement of one sideband only.

### Dual Frequency Resonance Tracking (DFRT)

\(R_1-R_2 \)

Reliable resonance frequency tracking often relies on phase-locked loops that exploit the phase information to provide fast feedback. However, there are case where the SNR ratio of the phase signal is insufficient for the feedback or ambiguous due to physical properties, for example in near-field scanning of polar materials where the phase flips by 180 degrees at the domain boundaries.

Instead of relying on the phase signal one can probe the resonance with two frequencies detuned in order to hit the steepest slopes right and left of the resonance center frequency. The difference of the two amplitude signals provides an excellent means of providing an error signal to a PID controller (requires UHF-PID option). The DFRT Method blog article provides more details of the actual implementation.