It could result in substantial time savings in searching the file. A major application of MIRET will be implemented in the near future. The system will be interfaced to data-logging equipment directly coupled to an infrared spectrometer, thus performing automatic data recording and substance identification. Distribution of MIRET will be limited until further testing and developments are completed. This system is quite similar to the program previously described for powder diffraction identification (11, 12). This system is now in Version 12 and in use by over 150 companies and universities in house and is available on national time sharing. In this system, a single unknown pattern (7 components) can be solved (determination of each (11) G. G. Johnson, Jr., and V. Vand, Ind. Eng. Chem., 59 (8), 18 (1967). (12) G.G. Johnson, Jr., ibid., 61 ( 5 ) , 79 (1969).
component) in less than one minute on an IBM 360/67. In five years of development, this system has reached the degree of confidence and reliability that the infrared system will when it has been given the same amount of reprogramming. ACKNOWLEDGMENT
The authors thank ASTM (Richard Sherwood in particular) for allowing the use of the magnetic tape of reference spectra used as the data base for this project. We must also thank ASTM for the use of the Serial Number List and the Alphabetic List of published infrared spectra. We acknowledge Mr. Duncan Erley for his initial comments on the MIRET system. Finally, a note of appreciation to Dr. Mary Barnes of the Materials Research Laboratory who made the infrared spectra we have used in testing MIRET. RECEIVED for review October 7, 1970. Accepted August 26, 1971.
Signal-to-Noise Ratio in Fourier Transform Spectrometry Herbert M. Pickett’ and Herbert L. Strauss Department of Chemistry, University of California, Berkeley, Calif. 94720 The effect of interferogram length or of measurement time in Fourier transform spectrometry is discussed. Both signal-dependent noise and signal-independent noise are considered. We conclude that better spectra can be obtained with shorter interferograms when the spectral features are broad. This result is compared to results obtained using filtering and smoothing in conventional spectrometry. Both optical and nuclear magnetic resonance Fourier transform spectrometry are discussed and sample spectra are presented.
FOURIER TRANSFORM METHODS of spectrometry are becoming more popular rapidly, especially for measurements in the infrared (IR)region of the spectrum (1-4) and in nuclear magnetic resonance spectrometry (NMR) (5, 6). The spectra are recorded as a function of a parameter other than frequency, and this has a number of advantages. In NMR a single scan of a spectrum can be obtained in a time comparable to Tiwith none of the ringing associated with fast passage in the frequency domain. In IR and other optical spectrometry, the apparatus used to obtain the spectrum usually allows more light throughout and is often simpler than a conventional spectrometer ( I , 7 , 8). The Fourier transform methods usually also benefit from Fellgett’s ad1 Present address, Department of Chemistry, Harvard University, Cambridge, Mass. 02138.
-
(1) K. D. Moller and W. G. Rothschild, “Far-Infrared Spectroscopy,” John Wiley and Sons, New York, N.Y., 1971. 41, (6), 97A (1969). (2) M. J. D. Low, ANAL.CHEM. (3) G. Horlick and H. V. Malmstadt, ihid., 42, 1361 (7970). (4) G. Horlick, ibid., 43 (8), 61A (1971). (5) R. R. Ernst, AdGan. Magn. Resonance, 2, 1 (1969). (6) R. R. Ernst and W. A. Anderson, Rev. Sci. Instrum., 31, 93 (1966). (7) H. A. Gebbie and R. Q. Twiss, Rep. Prog. Phys., 29, 729 ( 1966). (8) J. Connes, Rev. Opt., 40, 45, 116, 171, 231 (1961); Translation by C. A. Flanagan, Navweps Report 8099, U.S. Naval Ordinance Test Station, China Lake, Calif., 1963.
vantage (7-9), sometimes called the multiplex advantagethat is, the signal-to-noise ratio is improved over the signalto-noise ratio of a frequency recorded spectrum by a factor which is proportional to the square root of the total frequency range covered divided by the resolution. Many of the theoretical and practical problems associated with achieving the optimum signal-to-noise ratio have been extensively discussed in the literature ( 5 , 6, 8-10). The optimum conditions for running a Fourier transform spectrometer are a bit different from those for frequency recording spectrometers. In this paper we discuss a number of ways to deal with the situation of fairly low resolution spectrometry in the presence of noise. In our discussion we will emphasize emission and absorption in optical spectrometry, although the same ideas are relevant to magnetic resonance spectrometry also. We present some of our far infrared results and some NMR spectra taken by M. P. Klein which show the same effects. In the infrared, measurements are made with an interferometer, and the intensity of the transmitted signal is recorded as a function of the path difference between the two interferometer mirrors. The measurement is fundamentally one of wavelength and only incidentally is it a function of time. Nevertheless, we will speak of the infrared measurement as being “in the time domain” so that we may follow the language Ernst uses for the NMR case (4, 5). In NMR, the free induction decay oscillations are at the absolute frequencies of the lines (in a fixed frame). However, it is important to note that in most NMR instruments the measured waveform is the beat signal between the free induction decay and a reference oscillator. In the discussion that follows, we assume that the noise has no strong dependence on frequency in the range of (9) P. Fellgett, J. Phys. Radium, 19, 187 (1958). (10) D. R. Lumb and G. C. Augason, Mem. SOC.Roy. Sci.Liege, 9,132 (1964).
ANALYTICAL CHEMISTRY, VOL. 44, NO. 2, FEBRUARY 1972
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interest, or equivalently that the noise in a time domain data point is uncorrelated with noise in other such data points (see Appendix). We will wish to distinguish between signal independent and signal dependent noise. A familiar source of signal independey noise is the Johnson or thermal noise of an amplifier. Another source of signal independent noise is that due to truncation error involved in the digitization of data. Signal dependent noise may come from any form of noise which tends to modulate the signal, such as a fluctuating source in optical spectrometry or a fluctuating magnetic field in NMR. In these cases of signal dependent noise, the noise power is proportional to the square of the signal voltage. Shot noise in photomultiplier tubes (11) is an example of signal dependent noise for which the noise power is proportional to the signal voltage. We will assume that any other sources of signal dependent noise that may be present also have the character that the noise power increases with an increase in signal. CHOICE OF NOMINAL RESOLUTION-THEORY In all of the time domain measurements, the nominal resolution of the spectrum is determined by the maximum excursion of the interferogram from the time origin. The interferogram of a spectral element of infinitesimal width (a Dirac delta function) will be a cosine function (12). The transformation of a finite length of this interferogram will produce a “spectrum” with a line shape whose width is inversely proportional to the length of the interferogram. The height of the spectral ljne is proportional to this length. The exact shape of the line is also determined by the method of apodization which is used (4,but the height and width of the line depend on the length of the interferogram once the apodization function is fixed. The situation is very different if a spectral element has a finite width. The interferogram then is a cosine wave of diminishing amplitude which vanishes at times much gieater than 1/15’, where 6 is the half width of the element. When the maximum time T for which the interferogram i s recorded is small compared to l/6, the spectral signal derived from the interferogram will still increase proportionally to T since for T > l/S, an increase in T will lead to no increase in the spectral signal. The situation is even more extreme if we include the effect of noise. We first consider the signal independent noise and, for this part of the noise, the root-mean-square noise in the transformed interferogram will increase as T1’2. Therefore the signalto-noise ratio (S/N) will be proportional to T + 1 ’ 2only when T > l/6, will lead to no increase in the noise level, and S/N will approach a constant value for large T. While this implies that no signal-to-noise ratio will be lost in going to large T , it is not apparent that any will be gained either. It would be more efficient to spend the extra time (Le,, time for which T >> l/6) recording duplicate data points in the high noise (i.e., high signal) regions of the interferogram (points at T
+ J(v +
vo)
(A. 12)
and
s:fi s-ffim
C(t) = 6(t, 0)
F(v) =
(-4.2)
Note that the choice of the range o f t and v in the fundamental transforms in (A.l) and (A.2) is somewhat arbitrary. We have chosen the form we feel is most natural for IR Fourier transform spectrometry. From the Wiener-Khintchine Theorem ( I I ) , the mean square noise voltage, [N(v)12,in the frequency domain is the Fourier transform of the time domain noise auto-correlation function, C(t).
[No32 =
Our Lorentzian line is given by
(4).
and the inverse transformation is given by G(t) = 2
and when the mean square noise is constant over the interferogram,
-
[W12dt
(A.6)
Le., the noise is white noise over the frequency range of interest. For a finite interferogram,
e-zs6t cos 2irxt dt
J(X) = s ST:
(-) + 1
f
=
2ir
x2
{e-z*6To(6cos 2nxT0 - x sin ZirxT,) 62
- ,-ZdT
(6 cos 2irxT
- x sin 27rxT)) (A.13)
Finally, S
~ ( v ,E ) ~ ( 0 )= - [e-zraro
2as
- e-2*6Tl
(A. 14)
and this with Equation A.8 gives Equation 1. The solid line in Figure 1 is J(x) for the case when To= 0 and T = 0.2/6. ACKNOWLEDGMENT
We thank Dr. M. P. Klein for providing sample NMR spectra. These spectra are part of a set which will appear in his forthcoming article on the practice of NMR Fourier transform spectrometry.
RECEIVED for review May 27, 1971. Accepted August 30, 1971. This work was supported in part by the National Science Foundation. The first author (HMP) was a National Science Foundation Predoctoral Fellow, and the second author (HLS) is an Alfred P. Sloan Fellow. (14) C . H. Townes and A. L. Schawlow, "Microwave Spectros-
copy," McGraw-Hill, New York, N.Y., 1955, Chap. 13.
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