Signal Processing for Interferometry
This work was presented at the
Society of America meeting on
Fourier Transform Spectroscopy in Santa Barbara,
CA on June 25, 1999.
A new vision of Fourier
transform-infrared (FT-IR) signal processing is presented. The elements
of the signal processing, taken individually, are not novel; the novel
combination produces unexpected and useful results. First, the transfer
functions of the spectrometer electronics including the detector elements,
preamps and analog-to-digital converters (ADCs) are measured and recorded.
Then, simultaneously-sampled pairs of data points are recorded from both
the infrared and reference laser channels. The interferometer mirror may
be scanned with any combination of velocities from constant to sinusoidal
to random. The data pairs are recorded at evenly spaced intervals of time
without regard to the reference laser modulation frequency or phase. The
data are then post-processed by a variety of means that produce equivalent
results. The first step in the data processing is to remove the effects,
inasmuch as possible, of the detector, preamp, ADCs and noise to generate
accurate representations of the pure optical signals. This step removes
the frequency-dependent delays (caused by the electro-optic and electronic
components) so that the phase and amplitude relationships of the original
optical signals are restored. The next step is to convert the data, from
points spaced by equal intervals of time to points spaced by equal intervals
of retardation, via a difficult interpolation. Several approaches are described
in the literature, but these are computationally demanding.1,2,3,4,5,6,7
The interpolation step is extremely powerful because, in combination with
purification of the optical signals, it converts data sampled with any
combination of mirror velocities to interferometric data which is independent
of mirror velocity.
There are three convergent historical
roots of this work. The first was the desire to use sigma-delta analog-to-digital
converters in FT-IR spectrometry.1,2 The second was the desire
to correct for velocity errors in conventional FT-IR spectrometers, particularly
ones which are operated in harsh field conditions.8,9 The third
was the fact that almost all very rapid-scan FT-IR spectrometers produce
a sinusoidal profile of retardation vs. time.6 The common goal
of these three areas is to correct the effects of mirror velocity variation.
The signal processing approaches described here are successful in reaching
the goal in all three cases.
Two new methods of interpolation
are examined and compared to previous methods. The first new process is
the use of wavelet transforms to extract the frequency and phase of the
laser signal. Together with the use of wavelet and inverse wavelet transforms
to interpolate the infrared signal, this is equivalent to the second approach.
The second approach uses the inverse of Kawata's method10 to
extract the phases of the laser and infrared signals, thereby converting
them to zero-based representations. An amplitude scale factor for the infrared
signal must be preserved, but this is a fairly simple operation. With the
zero-based representations, interpolation, to account for the relationship
between time and retardation, is very straightforward. Kawata's method
can then be used to reconstruct an amplitude representation of the infrared
signal, but with data points evenly spaced in retardation. These two methods
produce results equivalent to previous approaches, but offer much better
computational efficiency. It is thought that, together with the use of
new digital signal processors, these approaches will be capable of real-time
processing of more than one thousand interferograms per second.6
In particular, Texas Instruments has scheduled for 1998 (and rescheduled
for 1999) the release of the TMS320C67 processor which is capable of more
than 1 billion floating point operations per second (FLOPS). Boards combining
multiple processors of this type will be suitable for real-time signal
processing for a variety of applications such as fast process control,
quality control and spectral imaging.
The transfer functions of the laser
and infrared channels are measured by excitation with fast solid state
emitters. In the proof-of-feasibility stage, light-emitting-diodes (LEDs)
have been used11,12 It has been found that inexpensive (<
US$2 in 1999) LEDs can reach frequencies higher than 2 MHz. Much higher
frequencies may also be available with these inexpensive components. A
variety of faster and more expensive solid state emitters are also available
and might be appropriate for spectrometer construction. For example, superluminescent
diodes used with optical fibers for communication rates in the range of
hundreds of MHz, and laser diodes, for the same purpose, with frequencies
well into the GHz range, are available for less than US$500. While semiconductor
emitters are notoriously unstable with temperature, it is thought that
the wavelength drift will have minimal effect on measurement of the phase
and magnitude response of signal channels. This is partly because the effects
of wavelength drift with wideband detectors are small. Further, the temperature
drifts can be made small and slow by the use of insulation, thermal mass,
and active control. The crucial issue is that the emitters be stable in
their temporal response, and preferably in their total emitted energy.
Since their temporal response is on the order of nanoseconds to microseconds,
small changes of speed over temperature will have minimal effect on the
measurement of infrared and laser channel transfer functions with time
constants in the milliseconds to tens of microseconds range. There are
several approaches which might be used to measure the transfer functions.
In general, the transfer functions of the detector channels would be measured
at mirror turnaround points during rapid-scan operation, or during mirror
settling time in step-scan operation. The simplest approach to extracting
transfer functions is the use of sinusoidal excitation to drive the emitters,
and consequently the detector channels. The phase and magnitude of the
detector response can be readily measured by recording and demodulating
with the same DSP core that operates the spectrometer. The advantages of
this approach are conceptual simplicity and consequently straightforward
implementation. Other approaches might include pulse or pseudorandom modulation.
The general approach of shifting
the complexity of spectrometer operation and signal processing from electronic
hardware to electronic software has roots in the computing revolution that
made FT-IR spectrometry possible and then practical. In the end, this revolution
will make FT-IR spectrometry cheap. The vision of shifting signal processing
from conventional electronic hardware to computer hardware and software
was first expounded for step-scan in 199313,14 and for rapid-scan
in 1994.15 This vision seeks to lower the cost and improve the
performance of interferometric measurements. One ironic aspect of the trend
is that the complexity of the hardware of a digital signal processor or
computer central processor is really many orders of magnitude greater than
the complexity of the conventional electronic circuits that it might replace.
Further, the capital cost of the integrated circuit fabrication facilities
is far greater than the cost of facilities for fabrication of conventional
analog components. Because the marginal cost of each additional integrated
circuit is very low and because the marginal cost of each additional unit
of software is even lower, the tradeoffs become more attractive with each
passing year. The fundamental tradeoff of shifting electronic functionality
from analog components to digital signal processing is slower execution
time. However, the cost of computing is dropping at a compounded rate of
30% per year as the capability increases at a compounded 30% per year.
Speed is no longer an issue. In 1960, a unit of computing power equal to
1 thousand FLOPS cost about US$5 per hour. This year (1999), the same performance
will cost about US $0.00001 per hour (measured with constant 1960 dollars).
This is just short of 6 orders of magnitude change over four decades. As
certain companies have foreseen, value has shifted significantly from hardware
to software. A convergence of market forces is once again working a revolution
in FT-IR spectrometry.
J. W. Brault, lecture
L14 at 10th International Conference on Fourier Transform Spectroscopy,
August 27 - September 1, 1995, Budapest, Hungary.
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