Chapter 10
Signal Processing for Time-of-Flight Applications Ihor A. Lys
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When compact disc players werefirstintroduced, their cost, and the cost of the discs themselves, was astronomical. Economists will maintain that as volume grew, prices dropped. This is not the entire picture, however. The high cost was due to technological complexity, not just small volume. Today every CD player contains a significant amount of signal processing circuitry, yet because of the advances made in the early eighties, costs only a fraction of what comparable technology cost then. Music reproduction faced the same constraints in the eighties that signal processing for TOF faces today. This chapter examines the mathematical and practical aspects of analog and digital signal processing as applied to TOF application, but it should be noted that most of the material is quite generalized. Time to Digital Conversion Time to digital converters, or TDCs, are the most basic of signal acquisition elements. The basic block diagram is shown in Fig 1. A TDC consists of a counter and two discriminators. The counter is started by a pulsefromthe start discriminator, and stopped by a pulsefromthe stop discriminator. The time interval can then be read out and stored. The TDC is then reset and is ready for another event. This process is usually repeated thousands of times, and an output is presented to the user in histogram form. TDCs have several properties which make them attractive for some forms of TOF work. They are inexpensive, and quite fast, with count rates of 6 GHz being common. Theyfindmany applications in high energy physics experiments, and pulse counting applications, where the number of pulses received is small (usually less than 5) and there is a long time between successive events. These applications were the original impetus for the development of TDCs. TDCs properties may also preclude their use in many applications. A stop event terminates a TDCs cycle, and it is therefore required that there be a stop pulse, and that it occur before the next start pulse. TDCs need to have some maximum time
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In Time-of-Flight Mass Spectrometry; Cotter, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1993.
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TIME-OF-FLIGHT MASS S P E C T R O M E T R Y
to record in the case that no matching stop pulse is received. Until multi-stop TDCs were developed, there had to be a 1 to 1 correspondence between start and stop pulses. Multi-stop TDCs solve the problems associated with the 1 to many nature of many ionization events, but at a price in complexity and cost. A multi-stop TDC will record events separated by some minimum time, called the dead time, but will not record anything which happened in-between. Start iscriminator
$ Start
Index
Histogram Memory 0
Waveform Display
1
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Stop
Counter iscriminator
ËLOCK
2
ί| Stop
M[n]=M[n]+l' *
Clock
3
\
... Ν
V
Figure 1. Block diagram of a TDC. Signals of varying heights produce the same output, including the time value. This is not usually desired, because the height of the received signal may cause a slope-induced error from the leading edge. The recorded time always leads the time at which the peak of the signal occurred in a TDC, but the amount of this lead is dependent on the signal amplitude. TDCs do not quantify the size of the input signals, and the only information which is actually recovered is that there was some signal strength at the time recorded. Constant-fraction discriminators can be used to overcome the time error introduced by signal height variations, but the height of the signal is still not recorded. TDCs do not have any ability to separate signals from noise. Signals with poor signal to noise ratio (SNR) will prematurely trigger the stop discriminator, and will therefore generate no usable data. There has been much discussion about the SNR required for good operation of a TDC. The SNR value needed can be expressed in terms of an arbitrary "falsing rate" which expresses the percentage of false stop pulses to true stop pulses through the use of noise distribution analysis. Obviously a lower falsing rate is better, but because of the nature of the analysis, some reasonable rate needs to be chosen. In any event, the signal must be larger than the noise level. There has been interest in "windowed" TDCs, which impose some dead time after the start pulse is received, to remove early noise events. This can be quite useful in TOF applications where many of the stop bins in a multi-stop TDC can be wasted on useless low-mass ions, preventing the acquisition of high-mass ions. At this point however TDCs become more and more difficult to use, require much more careful setup, and often require experiments to be repeated in order to generate good data. After adding all of the improvements that TDCs have undergone, they become complex, expensive devices, and Analog to Digital converters should be a better choice. Transient Recorders Transient Recorders are the other option available today. Analog to digital conversion is the basis of much of this technology. In the mid 1980's several companies introduced the first commercially viable digital oscilloscopes. These devices quickly revolutionized the transient recorder field, and have replaced both In Time-of-Flight Mass Spectrometry; Cotter, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1993.
Downloaded by PENNSYLVANIA STATE UNIV on August 2, 2012 | http://pubs.acs.org Publication Date: December 21, 1993 | doi: 10.1021/bk-1994-0549.ch010
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Signal Processing for Time-of-Flight Applications
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analog scopes and more traditional transient recorders. Their initially high cost and poor performance limits have been overcome. Today one can buy hand-held digital oscilloscopes for under $1000 that have performance specifications comparable to those of expensive rack mount instruments of the early 1980's. A Transient Recorder is composed of several components, as shown in Fig 2. The core is a Sample and Hold unit(S/H), Analog-to-Digital Converter (ADC), and a memory. When packaged like a traditional oscilloscope, a transient recorder is referred to as a Digital Storage Oscilloscope, or DSO. When a trigger is received by the unit, sampling begins at a rate determined by the clock, called the Sample Rate, and the digital representation of the voltage present at the input is stored in successive memory locations. It is important to note that each start (trigger) pulse generates a complete representation of the entire signal. This is significantly more data than a TDC recovers. Each memory location corresponds to one minuscule segment in time. Its value represents the voltage of the input signal during that period. Each trigger can produce as much as 200000 data values. $ Counter
Clock
I Sample And Hold
^ Quantizer
Index,
Memory
Waveform Display
M|n]
Figure 2. Block diagram of a transient recorder. At first this may appear to be the complete solution to the TOF signal problem. Several issues remain however. TOF applications have constantly demanded more performance from DSOs. All of the general specifications of a DSO have been pushed by TOF needs. In many respects the fields of radar processing (the original impetus for the development of transient recorders) and TOF are similar in their demands. The signals have wide dynamic range, very wide bandwidth, poor signal to noise ratio, poor repeatability, long signal length, and low repetition rates. In addition, there is a desire to recover all of the signal produced at each point in time. These characteristics demand high sampling rates, long memories, high resolution A/D converters, and significant signal processing abilities. Furthermore, these characteristics must be met in single shot acquisitions. In order to further understand the limiting factors affecting DSOs, an in depth discussion of signals and the operations performed on them is necessary. Continuous Signals All real world signals are continuous in nature. At each infinitesimal point in time the signal has some absolute amplitude. The signal can be analyzed, or processed to enhance certain aspects or to remove others. A mathematical representation of some continuous time signals would be: Sin(t) x(t)= ASin(a>t), y(t)= z(t)= A · Sign[Sin(œt)l
v(t)= A
In Time-of-Flight Mass Spectrometry; Cotter, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1993.
(1)
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TIME-OF-FLIGHT MASS SPECTROMETRY
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In this list, x(t) is just a sine wave of amplitude A and offrequencyω; y(t) is commonly referred to as the sine function, essentially a rounded impulse at t=0; z(t) is a square wave of amplitude A and offrequencyω; v(t) is a DC signal of amplitude A. Each of these signals has a value at every point in time,from-oo to +00, and x(t), y(t) and v(t) are also continuous. Continuous time signal manipulation operations have effects on these signals, often changing them in many subtle ways. For example a signal x(t) could be amplified, producing x'(t)=Ax(t). It could also be delayed, producing x"(t)=Ax(t-to). Two signals can be added together, producing x'"(t)=Ax(t)+By(t). Many other operations are possible. Fourier Transforms Fourier analysis is a method of decomposing a signal into a series of sines and cosines, with differing amplitudes and frequencies. It can be shown that any signal that meets certain criteria, called the Dirchlet conditions, will have a Fourier representation. The Fourier transformation of a signal x(t) is Χ(ω), as given by the Fourier Integral: +00
(2)
Χ(ω)= jxity-^dt -00
Note that this integral is an infinite integral, and that each evaluation of it ie
produces one value of Χ(ω). The analytic notation e~ * with j = V - ϊ is used to represent cos cot, the different frequency sinewaves over which the integral is evaluated. In mathematics the variable / is commonly used for this value, but in electrical engineering, / is reserved to symbolize current, and so j is used instead. When a periodic signal is transformed, the limits of integration can be relaxed, and the signal need only be integrated over a single period. The importance of eq. (2) cannot be overemphasized. It transforms the time-domain to the frequency domain. A function such as x(t) = A · cos(o? /) domain corresponds to two impulses of amplitude A · π in thefrequencydomain at ω = ±ω . Χ(ω) is zero at all other points. Note that there two impulses. Their relative amplitudes are proportional to the phase of the input sinusoid. This is graphically represented in Fig 3. m
m
e
t m i e
0
0
"A'Pi
1 -wO1 1
1
wO
1
Figure 3. Fourier representation of x(t) = A' cos(w 00
Many functions will have more complex Fourier transforms. A square pulse in time will transform to an odd looking function shown in Fig 4. This transform relation is:
In Time-of-Flight Mass Spectrometry; Cotter, R.; ACS Symposium Series; American Chemical Society: Washington, DC, 1993.
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Signal Processing for Time-of-Flight Applications
Χ(ω) =
*(
