# Old-school Boxcar

The boxcar averager is a well-known class of instruments that help to recover signals buried in noise. Publications older than half a century are easy to find [1], [2]. The employed measurement technique is particularly suited to signals with a small duty cycle, meaning that the interesting part of the signal is only a small portion of the time domain series. This is, for instance, the case for signals derived from a pulsed source for time-resolved measurements. When start events trigger such signals it is possible to finely define a time window where the signal is measured, to discard the signal outside of that window and therefore also discard the noise contained there.

The origin of the word boxcar is somewhat obscure but it is thought to be derived from the similarity in outline of an American boxcar train and a train of pulses viewed on an oscilloscope [3]. The mathematical boxcar function is any function which is zero over the entire timeline except for a single interval where it is equal to a constant [4].

## Gating, Integration and Averaging

Boxcar averagers generally consist of two components; a gated integrator and a signal averager. The gate function acts on the time domain and requires a start point and an end point for the measurement. When the gate is open the input signal gets integrated and when it is closed no signal is integrated. The integrated result corresponds to the area underneath the pulse. The integration can also be delayed compared to the start event. Assuming the input unit is Volts, the output unit of the boxcar integrator is V * s.

The signal averager takes the result of a number of integrations and performs averaging. Averaging acts as a low-pass filter, reducing the dynamic of the signal of interest but improving the signal-to-noise proportionally to the square root of the number of averaged samples. For many pulsed experiments this is the main operation mode with a large number of pulses taken to retrieve the information of interest. Two averaging algorithms are usually provided: linear moving and exponential weighted average. These algorithms are reasonably simple to implement with analog technology. Linear averaging treats all integrated pulses with the same weight (sum of samples divided by the number of samples) and has a memory limited to the examined window (boxcar), while exponential weighted average weights newer samples more than older ones in the time series and has theoretically an unlimited memory of samples taken in the past.

Traditional boxcars implement a static measurement mode by generating a voltage that is proportional to the measurement result of the averager on some physical output connector. This method is static as the control parameters of the boxcar (gating time, gating delay, and gating length) are not changed over time.

A second common measurement method is dynamic, as a sweep of the gating delay is performed in combination with a short integration time (smaller than the length of the pulse) and can be used to depict the waveform of the pulse instead.

## Analog vs. Digital Boxcars

Commercial boxcars averagers have been around for a long time but most commercial instruments are of an older design. Often they consist of PCI cards or NIM modules that are combined within a mainframe rack to provide one or more channels. These instruments mostly rely on analog electronic components and therefore the setting ranges of the configuration parameters are limited. Having served thousands of scientists for decades, such analog boxcars are part of the established old-school landscape of signal recovery instruments. For today's high-end applications the functionality of these boxcars is not sufficient anymore and a new range of digital instruments is necessary.

Digital boxcar averagers follow a different implementation philosophy. As a consequence, the comparison to analog instruments is not straightforward. Even if digital instruments provide the equivalent function of their analog counterparts, some specification parameters are provided just for the sake of comparison.

Numerous advantages are provided by digital boxcars including superior specifications, a wider range of settings and unique features that cannot be practically offered by analog instruments. Digital boxcars are only suited to recover signals from periodic sources. In this respect the capability is marginally lower compared to analog instruments that can easily cope with asynchronous pulsed events. For most applications, however, this limitation is not considered problematic.

The table below details how specifications of analog and digital boxcars compare and where digital boxcars are superior to their analog counterparts.

static boxcar mode consists of signal gating, integration, averaging - equivalent behavior between digital and analog boxcars equivalent to waveform reconstruction, or peak form analyzer in the pulsed laser community - whereas analog instruments require an external ramp generator, digital instruments will perform the reconstruction instantaneously without the need for external equipment digital boxcars shall define an input bandwidth that is given by the anti-aliasing filters in front of the analog-to-digital converters - a typical specification could be one third of the sampling frequency parameter defining the minimum and maximum trigger rate - instruments might have different limits when the trigger is generated internally or externally - the trigger rate is equivalent to laser repetition rate when using a pulsed laser typical analog specification due to the limited capability of an analog integrator to discharge - this parameter can be very low with digital boxcars, providing support for a much higher repetition rate typical analog specification indicating the analog gain that can be applied to the signal before integration - in the digital world a similar but not equivalent specification exists, as the gain is applied to the input signal before the analog-to-digital converter typical analog specification defining the overall gain provided from input to output, gain = Vout - Vin, this parameter can be arbitrarily large in digital boxcars specification defining the time window in which the instrument can perform integration - it should not come as a surprise that digital boxcars can exceed the resolution of the sampling rate thanks to a suitable internal implementation typical analog specification used for the dynamic boxcar mode - in the digital domain this parameter is a legacy, and can be easily given in 360 degrees full range - no specific limitation may apply digital implementations are far superior to analog instruments the update frequency of the boxcar output

## References

[1] Boxcar Integrator with Long Holding Times, R. J. Blume, Rev. Sci. Instrum. 32, 1016 (1961), doi: 10.1063/1.1717602

[2] High Stability Boxcar Integrator for Fast NMR Transients in Solids, D. Ware and P. Mansfield, Rev. Sci. Instrum. 37, 1167 (1966), doi: 10.1063/1.1720449

[3] The Boxcar Detector, A synchronous detector that is used to recover waveforms buries in noise, J.D.W. Abernethy, Wireless World (December 1970)

[4] Wikipedia, Boxcar Function