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Basics of Boxcar Averaging

Boxcar averagers help to recover signals buried in noise. This well-known class of instruments features in publications older than half a century [1], [2]. The 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 the case for signals derived from a pulsed source for time-resolved measurements, for example. When start events trigger such signals, it is possible to define a time window where the signal is measured, discarding the signal outside of that window and, consequently, discard the noise present 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 that is zero over the entire timeline except for a single interval where it takes a constant value [4].


Gating, Integration, and Averaging

Boxcar averagers generally consist of two components, namely 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; when it is closed, no signal is integrated. The integrated result corresponds to the area underneath the pulse. The integration can also be delayed with respect to the start event. Assuming that 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 while improving the signal-to-noise ratio 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. There exist two main averaging algorithms: linear moving and exponential weighted average, both reasonably simple to implement with analog technology. Linear averaging assigns the same weight to all integrated pulses (sum of samples divided by the number of samples), and has a memory limited to the examined window (boxcar); exponential weighted average assigns larger weights to newer samples than it does to older ones in the time series, and theoretically has an unlimited memory of past samples.

Traditional boxcar averagers 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) do not change over time.

Boxcar static mode

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

Boxcar dynamic mode

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Analog vs Digital Boxcars

Commercial boxcars averagers are a mature technology, but most commercial instruments are characterized by an older design. They often consist of PCI cards or NIM modules combined within a mainframe rack to provide one or more channels. These instruments mostly rely on analog electronic components, 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; a new range of digital instruments is needed.

Digital boxcar averagers follow an entirely different strategy. As a consequence, the comparison with analog instruments is not straightforward. Even if digital instruments perform tasks equivalent to those of their analog counterparts, some of the parameters included in the table below are provided for the sake of comparison only.

Digital boxcar averagers come with numerous advantages that include superior specifications, a wider range of settings, and unique features that cannot be offered by analog instruments in practice. Digital boxcar averagers are only suitable for the recovery of signals from periodic sources. In this respect, their capability is marginally lower compared to analog instruments that can easily cope with asynchronous pulsed events. For most applications, however, this limitation is not viewed as problematic.

This table presents the specifications of analog and digital boxcars, highlighting cases where digital boxcars are superior to their analog counterparts.

Static boxcar mode Consists of signal gating, integration, averaging – the behavior is equivalent for digital and analog boxcars
Dynamic boxcar mode Equivalent to waveform reconstruction, or peak form analyzer in the pulsed laser community – analog instruments require an external ramp generator, whereas digital instruments perform the reconstruction instantaneously without the need for external equipment
Signal input bandwidth Digital boxcars define an input bandwidth given by the anti-aliasing filters in front of the analog-to-digital converters – a typical specification could be 1/3 of the sampling frequency
Repetition rate Parameter defining the minimum and maximum trigger rate – instruments might have different limits when the trigger is generated internally or externally, and the trigger rate is equivalent to the laser repetition rate when using a pulsed laser
Integrator dead time 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
Boxcar sensitivity range Typical analog specification indicating the analog gain that can be applied to the signal before integration – in the digital world there exists a similar specification, with the gain applied to the input signal before the analog-to-digital converter
Boxcar gain Typical analog specification defining the overall gain provided from input to output, gain = Vout - Vin – this parameter can be arbitrarily large in digital boxcars
Integrator gating time Parameter defining the time window over which the instrument can perform integration – digital boxcars can exceed the resolution of the sampling rate thanks to a suitable internal implementation
Integrator gating delay 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 as no specific limitation applies
Boxcar averaging length Digital instruments outperform analog ones
Boxcar output Parameter defining the update frequency of the boxcar output


Further Reading

[1] Blume, R.J. Boxcar integrator with long holding times. Rev. Sci. Instrum. 32, 1016 (1961).

[2] Ware, D. & Mansfield, P. High stability boxcar integrator for fast NMR transients in solids. Rev. Sci. Instrum. 37, 1167 (1966).

[3] Abernethy, J.D.W. The boxcar detector – a synchronous detector that is used to recover waveforms buried in noise, Wireless World (December 1970).

[4] Boxcar function, Wikipedia.

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