The HADAMARD TRANSFROM A Thousand Points of Light:
in Chemical Analysis and Instrumentation Patrick J. Treado and Michael D. Morris Department of Chemistry The University of Michigan Ann Arbor, Ml 48109-1055
Most chemists appreciate the impact of the Fourier transform on chemical analysis. IR spectroscopy and NMR spectroscopy were revolutionized by the introduction of multiplexing instrumentation that produced the Fourier transform of the desired spectra. Somewhat less familiar, but no less important, are image- and signal-processing techniques that are based on the Fourier transform. Not all chemists are aware that the Fourier transform is just one of several related linear transforms. The Fourier transform is based on sines and cosines; the other transforms are based on other periodic functions. The Hadamard transform is the most common of these alternative transforms. Based on square waves rather than on trigonometric functions, this transform can be applied in three different areas in analytical chemistry. First, it is used to multiplex or encode analytical signals so that many signals can be measured simultaneously. In Hadamard techniques the multiplexer turns elements on and off or makes them switch between two values. The same formalism describes multiplexing schemes in which pseudorandom sequences are employed to excite a sys0003-2700/89/0361-723A/$01.50/0 © 1989 American Chemical Society
tern. In general, Fellgett's advantage, the signal/noise (S/N) ratio improvement from multiplexing many signals, is sought. Fellgett's advantage can be realized if the detector is noisy, because one unit of detector noise is distributed among all the measured signals. Second, the transform is used in multiplexed imaging. Here, the goal is usually to obtain spatially resolved signals without focusing a high-intensity source on a sample. Fellgett's advantage may be absent. A power distribution advantage is available; source powers can be 2 or 3 orders of magnitude higher than for tightly focused radiation. Finally, the Hadamard transform is used in data transformations. Information is redistributed so that signal compression (i.e., truncation of the trans-
ists are most familiar with the use of Hadamard multiplexing schemes for IR spectroscopy (i). Hadamard encoding of the output of dispersive spectrometers proved less attractive than multiplexing with a Michelson interferometer, which requires a Fourier transform to recover a spectrum. Investigation of other applications of the Hadamard transform has continued in many laboratories worldwide. Elegant technical advances in multiplexing technology have suddenly revived interest in spectroscopic applications. Increased emphasis on spectroscopic microscopy with laser and other intense sources has spurred interest in Hadamard imaging. Early developments in Hadamard techniques were reviewed by Decker (2) and by Marshall and Comisarow
INSTRUMENTATION form) can occur without loss of essential features. The transform is also used before transmission of signals so that the effects of corruption of one or more data words are distributed over the entire signal, rather than causing severe localized loss of information. Here, workers often emphasize the computational efficiency of t h e Hadamard transform compared with that of the Fourier transform. These areas of application were defined by about 1970. Analytical chem-
(3). There is extensive coverage of Hadamard transform spectroscopy, imaging, and signal processing in Griffith's 1978 review volume (4). The 1979 Harwit and Sloane monograph (1) on spectroscopic and imaging applications of Hadamard transforms remains the standard comprehensive introduction to both theory and practice in this field. In this article we will emphasize recent developments. Hadamard multiplexing can be understood from the simple three-ele-
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989 · 723 A
INSTRUMENTATION ment case shown in Figure 1. We can illuminate the three resolution ele ments xi, X2, and χ 3 one at a time, as in Figure la. Three measurements are re quired to define the signals at X\, X2, and X3. However, we can also illuminate the elements two at a time, as shown in Figure lb. In this case, we can com pletely define the system by a set of
three equations. These equations can be written in matrix form, as shown. The matrix inversion that recovers X\, X2, and X3 from the measured yi, yz, and yz is called an inverse Hadamard trans form. The operation connecting the X; with the yi is the Hadamard transform itself. In a practical multiplexing system, hundreds or even thousands of ele ments are multiplexed. The principles remain the same whatever the number of multiplexed elements. What is im portant is that the multiplexing in volves switching some elements on (i.e., multiplication by +1) while leaving others off (i.e., multiplication by 0). The multiplexing sequence is a system atic set of combinations of on/off ele ments. Physically, this combination can be realized with mechanical masks, electronic or electrooptical switches, or any appropriate kind of on/off switch ing system. In practice, cyclic systems are used. Each combination is generated from the previous one by shifting its ele ments one position to the left (or right) and placing the overflow in the vacant position. In Figure lc, shifting the fiveelement mask to the left one unit at a time generates the three combinations of Figure lb. We can generalize the system of Fig ure 1. Any Hadamard code generates the sum of signals from each element weighted by 0 or 1.
>',Ί
= Σ 8» Χ
(i)
In Equation 1, y;- is the signal from the j-th combination of on/off ele ments, and Xi is the signal generated at the i-th element. The vector s ; = (sy, S2j, · · · , Snj) has values 1 for each onelement and 0 for each off-element. To recover all η signals, we need η measurements. These can be defined by the η linear independent equations that completely define the system.
Y = SX
Sample
*-Σ'
(2.1)
*1*2
y,-5/aXi Figure 1. Hadamard multiplexing.
;=i
(a) A point source illuminates element x< to gen erate a signal. Elements x2 and x3 can be illumi nated by moving the source to each sequentially or by moving each element into the source path. (b) Hadamard encoding. The three elements are systematically illuminated in groups of two to generate signals that are the sum of signals from both illuminated elements. The illumination at each element is found by solving the system of equations, shown in matrix form, (c) A cyclic ap proach to Hadamard encoding. The cyclic mask allows a single long mask to be used to generate the required combinations.
;=i
(2-2)
In matrix notation, Equations 2.1-2.η may be written as Equation 3. Y = S X
724 A · ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
(3)
The system can be solved by calculat ing the inverse of matrix X, according to Equation 4. X = S" (4) The matrix S l is computationally easy to generate and is given by Equation 5. S" x =
W (5) n + 1 Here W is a matrix that has — l's where S T has 0's and + l ' s where ST has + l ' s . Matrix S, called a Sylvester matrix, is derived from a Hadamard matrix, which contains elements + 1 and - 1 only. Despite this, the operation is called a Hadamard transform. Rules and algorithms for generating suitable combinations of elements sy have been described by Harwit and Sloane (2). The Hadamard transform may be viewed as describing functions using a basis set of square waves. In contrast, Fourier transforms use sine waves for the basis set. The Hadamard variable corresponding to frequency is termed sequency. Many of the familiar proper ties of Fourier transforms are observed with Hadamard transforms. The gen eral theory of Hadamard transforms and the uses of the transform and se quency in image and signal processing have been discussed by Harmuth (5). As with the Fourier transform, there is a fast Hadamard transform (FHT). In both cases, the matrix defining the transform must be factorable. In the case of the Sylvester matrices, the F H T can be defined if the number of ele ments is equal to 2k - 1, where k is an integer. For the Hadamard matrix, containing + l ' s and - l ' s , the F H T is defined if the number of elements is some 2*. Because the FHT requires only addition and subtraction, it exe cutes about 8-10 times faster than the fast Fourier transform (FFT) on a gen eral-purpose computer. In general, Hadamard transform multiplexing has the same kinds of ad vantages and limitations as multiplex ing using Fourier encoding. Fellgett's advantage, for example, can be ob tained in a spectroscopic system where detector noise is dominant, but not in a shot noise limited system. Fellgett's advantage is about the same order of magnitude for both multiplexing schemes. In each case, the instrument analog-to-digital converter dynamic range must be increased by -Jn, where η is the number of channels or signals multiplexed relative to the require ments of a single-channel measure ment. The choice between Fourier and Ha damard encoding depends largely on the kind of experiment. A Michelson interferometer automatically produces
INSTRUMENTATION a Fourier encoding. For imaging appli cations, the simplest spatial encoding is a system of open and closed aper tures, which automatically produces a Hadamard encoding. Arguments about small differences in multiplex advan tage or even about computation time are overshadowed by the requirements of the experiment. How Hadamard spectroscopy and imaging are done Both spectroscopic and imaging appli cations require a sequence of Hada mard masks that encode the informa tion. Either one-dimensional or two-di mensional encodings are possible, as shown in Figure 2. The most common mask technology is a series of apertures that are fabricated on a transparent substrate. Metal films on glass, quartz, or silicon have been used. Slots milled in thin sheets of metal have also been employed. These systems all use cyclic encodings so that the mask is translat ed or rotated by one aperture width to generate sequential encodings. This principle is illustrated in Figure 2a for a one-dimensional encoding. The active aperture area is defined by a framing mask. After collection of a complete set of data points, an inverse Hadamard transform is calculated to recover the desired spectrum or image. Spectroscopy is performed by using Hadamard masks to encode the dis persed output of a grating spectrome ter for presentation to a single detector. Imaging is performed either by encod ing the source radiation or by encoding the image itself. The same principle can be used to
(a)
generate two-dimensional mask se quences, as shown in Figure 2b. Here, the mask is folded. For the 15-element sequence shown, the first five elements form one row, the next five form the second, and the last five form the third row. The mask is translated one unit aperture width, as with the one-dimen sional sequence. The mathematical op erations are the same as for the onedimensional sequence. If a plot of the data is required, it must be rearranged from a linear sequence into the appro priate (x, y) coordinates. It is also pos sible to fold the sequence twice, as shown in Figure 2c. This results in a more compact mask but requires trans lation in two dimensions. The Fateley group {6-9) has demon strated that optical shutter arrays (OSAs) can be used as Hadamard mask systems. An OSA, which is also called a spatial light modulator, contains a ma trix of electronically addressable, switchable elements that can be made transparent or opaque. A transparent element represents a 1 in a Hadamard sequence; an opaque element is a 0. An OSA can generate a complete set of Ha damard sequences but requires no me chanical movement. The OSA can be used to construct multiplexed spectrometers or imaging devices with no moving parts. Such de vices are more compact than mechani cal mask systems, provide excellent spectral subtraction capability, and may be more rugged. In addition, the OSA allows selective filtering by leav ing certain elements always off, that is, opaque (9). Here, important opportu nities for S/N ratio improvement exist.
Stationary framing mask of length π = 15 -Slit width Linear cyclic mask of length 2n - 1 = 29 stepped horizontally past the framing mask
(b)
Slit area
Stationary framing masks of area η = 3 χ 5
2D array stepped horizontally past the framing mask
ο)
2D array stepped horizontally and vertically past the framing mask
Figure 2. One- and two-dimensional Hadamard encodings. (a) A 15-element linear system of encodings. Each encoding is formed by shifting the mask sequence one element to the left, (b) A two-dimensional encoding is formed by folding the one-dimensional sequence into three rows of five elements each. The entire array is shifted one element to the left to generate the next encoding in the sequence, (c) Two foldings generate a more compact mechanical implementation. The first five encodings are generated by horizontal shifts only. The next five encodings are generated by shifting one unit down and following the same horizontal shift sequence. The final five encodings are gen erated by shirting one more unit down and following the same horizontal shift sequence.
OSA technology is currently limited to certain wavelength ranges. For exam ple, liquid crystal devices work in the visible and near-IR but not in the UV or through most of the fingerprint re gion of the IR. Hadamard encoding is not limited to optical or beam spectroscopies. Hada mard sequences can be used to define pseudorandom excitation sequences. For example, Hadamard sequences of potential pulses can be used to make electrode impedance measurements. Hadamard sequences of rf pulses can be used for NMR spectroscopy. Of course, any data set can be sub jected to Hadamard transformation. Image transmission from the early days of the space program employed this technique. It is used in pattern recogni tion, image compression, and other postprocessing applications. Recent spectroscopic applications Sugimoto (10) has proposed a slitless low-resolution near-IR (1-1.8 μπι) Ha damard spectrometer using a Hadamard-encoded 16-element array of light-emitting diodes (LEDs) as the source and a 32-element array of ger manium diodes as the detector system. The encoded LEDs replace the slits of a conventional spectrometer, thus im proving the throughput. Here, the mul tiplexing provides a Jacquinot or throughput advantage rather than the Fellgett's advantage obtained by mul tiplexing the dispersed output onto a single detector. The output of the LED array is collimated so that a remote 0.25-m spectrometer with a 1-km path length can be used. With this system, the minimum detectable absorbance is 2 X 10~5 with a 1-s integration time. Sugimoto calculates that an absor bance of 5 X 10~7 could be reached with a 30-min integration time. As config ured for environmental measurements, the system is capable of monitoring NH 3 , CH 4 , C 2 H 4 , C 3 H 8 , and the H 0 2 radical with parts-per-billion detection limits and 1-s time resolution. The first grating instrument to em ploy stationary electrooptical encoding masks at the output was an IR spec trometer described by Hammaker et al. (6). They fabricated a 63-element va nadium dioxide thermodiachromatic array, which was operated in the 7501000-cm -1 region with 3-cm _1 resolu tion. They demonstrated the expected Fellgett's advantage and described the merits of stationary mask systems. Be cause of limitations on the size of the mask, the entire fingerprint region could not be sampled at once. It may prove difficult to design a spectrograph with a sufficiently flat field or to find a material for mask fabrication to allow
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989 · 727 A
INSTRUMENTATION 1.00 0.83 0.67 0.50 0.33 0.17 ^ 0.00 I 1.00= 0.83-
I s 0.671 0.50 3. .·& % %
(a)
(b)
0.330.170.001.000.83 0.67 0.50 0.33
0.17 0.00
200 600 10001400 1800 2200 Wavenumbers (cm -1 )
Figure 3. The effect of Rayleigh line re jection on Hadamard transform Raman spectra. (a) Spectrum of 2-nitropropane with no significant Rayleigh scattered light passed by the multi plexer, (b) and (c) The same spectrum, but with increasingly large amounts of Rayleigh scatter passed by the multiplexer. (Adapted with permis sion from Reference 9.)
acquisition of a complete spectrum. The limited spectral coverage of many electrooptical masks can be an advantage in routine monitoring. Here, only a narrow spectral region is usually needed. In contrast, an interferometer, without prior spectral filtering, records information from the entire spectrum at once, generating both excess noise and some additional computational burden to recover only the region sought. In this area the Hadamard in strument may be both theoretically su perior and cost-effective. In visible and near-IR spectrosco pies, Fateley and co-workers (6-9, 11) have used liquid crystal optical shutter arrays (LC-OSAs) as stationary Hada mard mask systems. Current commer cial devices are useful at wavelengths above ~530 nm, but future versions may operate further in the blue. With an LC-OSA this group has demonstrat ed Hadamard Raman and atomic emis sion spectroscopies. The Fateley group has encoded the output of a 1-m subtractive double spectrometer equipped with 1180 groove-per-millimeter grat ings. Because the LC-OSA used has a 0.6-mm aperture, the available resolu tion was about 30 cm - 1 . Raman spectroscopy is a difficult ex periment for any multiplex technique, as practitioners of FT-Raman have pointed out (12). The source noise on
the intense Rayleigh line is redistribut ed across the entire spectrum by either Hadamard or Fourier multiplexing. This problem is exacerbated with the LC-OSA, because the device has appre ciable off-state transmission (7,8). Ha damard encoding allows rejection of much of this noise by placement of an extra mask in front of the OSA ele ments, which encode the line. The ef fect can be seen in the 2-nitropropane Raman spectra shown in Figure 3. Mul tiplexing allows acquisition of Raman spectra using visible or near-IR excita tion. Fateley and co-workers have dem onstrated that multiplexing generates excellent S/N ratio Raman spectra with uncooled silicon photodiodes as detectors. As with IR spectroscopy, a real ad vantage of Hadamard multiplexing over Fourier multiplexing is that the entire spectrum need not be encoded at once. Hadamard Raman spectroscopy may prove superior to FT-Raman for routine monitoring over a restricted wavenumber range. Using current technology at least, it is easier to gener ate a complete multiplexed Raman spectrum interferometrically than with a Hadamard mask system. There is an exact analog of dispersive Hadamard spectrometry available for mass spectrometry (13, 14). The ions dispersed by a conventional magnetic sector instrument are passed through a sequence of Hadamard masks con structed as slits in metal and mechani cally translated and then detected with a single-channel detector. Fellgett's advantage is obtained.
Recent imaging applications
Hadamard multiplexing is well suited to multiplexed imaging. Two types of imaging are possible: signal-encoded imaging and source-encoded imaging. In source encoding, the excitation source is encoded with a sequence of Hadamard masks. In signal encoding, the signal image is multiplexed. The operation of a two-dimensional signalencoded imager is depicted in Figure 4. The image is projected through a series of Hadamard masks, each containing 63 elements folded into 9 X 7 arrays. The encoded signals are then projected onto a single-element detector. Upon reverse Hadamard transform of the 63 independent signals, the spatially re solved image can be recovered. Image (signal) encoding offers ad vantages with optical spectroscopies involving reflected, scattered, or emit ted light. The image flux is usually far less intense than the source flux, so the mask does not have to be rugged. Also, diffraction from the mask apertures of the incoherent image is less severe than that of the coherent laser source. Where imaging detectors do not ex ist, source-encoded multiplexing, which is generally applicable to any lin ear spectroscopy, is often the most practical form of imaging. The alterna tive to source encoding is tight focusing or other generation of a small-area source and raster scanning of the source across the image. Even when there is theoretically little or no classi cal multiplex advantage, encoding al lows use of unfocused beams. In that
Encoded image η
Mask η
Projected image
Mask 2
2
M.
S î-1 Encoded image 2
Mask 1
Recovered image
Encoded image 1
Figure 4. Operation of a 63-element Hadamard imaging system. The 63 encodings of the image of the numeral " 2 " are generated by 63 sequences, which are represented by the 63 rows of the Sylvester matrix, S. The encodings on average represent about half the intensity in the image. The inverse Sylvester matrix, S~\ is multiplied by the vector of the 63 encoded signals to recover the image of the numeral.
728 A · ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
case the local source flux or power den sity can be reduced by at least 2 or 3 orders of magnitude. High signal inten sities can be maintained, and the possi bility of sample damage is minimized. Our group has called this a power dis tribution advantage, which perhaps is as commonly sought as classical multi plex advantages in Hadamard imaging. We have investigated the imaging properties of Hadamard systems gen erally (15-17). We have shown that for a system of uniform mask elements, resolution is ideally determined by convolution with a unit aperture. In an optical system, an encoded source beam can often be condensed to im prove resolution or an image can be magnified before encoding. In these ways diffraction-limited Hadamard imaging, in principle, is possible. IR and visible Hadamard imaging was demonstrated by Harwit and co workers in the early 1970s. All of this work used dispersive spectrometers to achieve spectral resolution. However, Kraenz and Kunath (18) coupled a two-dimensional Hadamard imaging system to an FT-IR microscope. By taking a complete interferogram at each step of the Hadamard sequence, they obtained complete spatial/spec tral images at spatial resolution up to 64 X 65 square 50-μπι pixels. These workers were seeking a spatial multi plex advantage by the use of defocused IR radiation. The photoacoustic experiments of Coufal and co-workers (19) were the first demonstration of source-encoded Hadamard imaging for an important laboratory spectroscopy that has no imaging detectors. They employed 7and 15-element linear masks placed di rectly over the sample being imaged to obtain spatially resolved signals. They also obtained Fellgett's advantage, be cause photoacoustic spectroscopy with an electret microphone is detector noise limited. The depth-resolving ca pabilities of photoacoustic spectrosco py were maintained in the multiplexed system. Two-dimensional Hadamard masks can also be used. The Nanjing University group (20) has shown that local power density in two-dimensional photoacoustic imaging can be reduced by a factor of 1000. Our own group has used similar sys tems to generate transverse photother mal deflection images. Like photo acoustic spectroscopy, transverse photothermal deflection has no available im aging detectors. Source-encoded Hada mard multiplexing is necessary. Excimer lasers can be used as sources for UV imaging if the encoding masks are able to withstand high peak power laser intensities (21). Spatial distribution of
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ANALYTICAL CHEMISTRY, VOL. 6 1 , NO. 11, JUNE 1, 1989 · 729 A
INSTRUMENTATION
Right angle prism
Eyepiece
Aperture
Hadamard mask
Monochromator Photomultiplier tube
Collimating lens
Translation stage control
Microscope objective Sample
Laser excitation beam
Computer
Microscope stage (x,y,z translation)
Figure 5. An image-encoded Hadamard Raman microscope. The sample is illuminated with an unfocused laser beam. The Raman scatter is collected through a con ventional compound microscope and presented to a Hadamard mask sequence before passage through a spectrometer. Spatial resolution depends on the relation between total magnification in the system and the size of a unit aperture in the mask. (Adapted with permission from Reference 16.)
the laser light is an attractive alterna tive to attenuation of expensive laser power by 2 orders of magnitude. Masks containing hundreds of elements are easily generated and employed in these multiplexing systems (15,22). There is little degradation of S/N ratio beyond that caused by the factor of two attenu ation of average power that occurs when masking is used. Most source-encoded imaging has been near-field imaging. The mask is placed as close to the sample as physi cally possible, and the sampled area is equal to the aperture area. This ap proach offers limited resolution. For the most part, minimum aperture di mensions are limited by fabrication technologies and have been 50 μπι on a side, or larger. With any optical spec troscopy, it is possible to use coarse masks to encode the beams and then to condense the encoded beams, similar to the way photolithography is done. This approach has been applied to pho tothermal detection (16) and Raman imaging (23). Masks with 150-400-μπι unit aperture dimensions have been used to generate images with 13-μπι resolution (16). There is no reason that diffraction-limited images with resolu tion of 0.5 μία or less cannot be generat ed by this technique, so long as proper ly designed imaging optics are used. Source-encoded Hadamard Raman imaging (23) and signal-encoded Hada mard Raman microscopy (24) have been demonstrated. With a signal-en
coded instrument, Hadamard images have been obtained with nearly diffrac tion-limited resolution. Figure 5 is a diagram of a Hadamard transform Raman microscope con structed in our laboratories. Unfocused laser light illuminates the sample. The scattered signal is collected, magnified, and projected to the encoding mask by the microscope optics. Spectral resolu tion is provided by the monochroma-
tor. Figure 6 is an image of polycrystalline benzoic acid obtained with this in strument. Multiplexing can be advantageous in combination with multichannel Ra man detectors. Two-dimensional (CCD) detectors can provide complete spectra that are spatially resolved in one direction but not in two. Hadamard multiplexing may provide an in expensive route to two-dimensional spectral/spatial Raman imaging with these devices. Several groups (25-28), primarily as tronomers, have used signal-encoded Hadamard X-ray telescopes to im prove the S/N ratios in observations of nebulae, solar flares, black holes in space, and plasmas in nuclear reactors. The masks are fabricated as holes in metal foils, and the radiation is direct ed to counting detectors. Hadamard excitation sequence applications Hadamard sequences are pseudoran dom pulse sequences. Optical and Xray spectroscopy and imaging use masks based on these sequences. How-, ever, they can be used to define pseu dorandom excitations of a system. An efficient form of ac polarography (29-31) uses Hadamard sequences of small-amplitude sine-wave harmonics to exercise a dropping mercury elec trode. The dc response and the inphase and quadrature components can be obtained simultaneously, allowing complete specification of the cell im pedance. The technique is equivalent to conventional ac polarography, as
Figure 6. The image of a benzoic acid crystal (992 cm -1 ) obtained with the instru ment of Figure 5. (Adapted with permission from Reference 17.)
ANALYTICAL CHEMISTRY, VOL. 6 1 , NO. 11, JUNE 1, 1989 · 731 A
INSTRUMENTATION Conclusions
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
•1.1
Potential (V)
Figure 7. Quadrature (top) and in-phase (bottom) ac polarograms of 0.1 mM CdCI2 in 0.1 M KCI, shown at 114.6 Hz. Excitation wave form amplitude, Vc = 50 mV. (Adapted from Reference 30.)
shown in Figure 7, but is more efficient. As with any Hadamard technique, the required computations involve no multiplication or division. A laboratory microcomputer can perform them in real time or near real time. Hadamard pseudorandom excitation has been proposed to improve the efficiency of multichannel tandem Fourier transform mass spectrometry (FTMS/FTMS). McLafferty and coworkers (32, 33) suggest the use of Hadamard weighted excitation functions to dissociate differing combinations of precursor ions. In each step, a different set of half of the precursor ions is dissociated, while product ion masses are monitored. The FTMS/FTMS spectra for the product ions are recovered by inverse Hadamard transform of the multiplexed spectra. Hadamard transform FTMS/FTMS should provide a Fellgett's advantage over single precursor ion measurements. Other tandem MS experiments can be similarly enhanced. The Hadamard transform is a linear transform. The usual inverse Hadamard transformation fails if the system does not respond linearly. Blumich and Ziessow describe techniques for treating nonlinear saturation effects in Hadamard NMR spectrometry (34). Hadamard pulse sequences are useful in NMR because they allow distribution of the excitation power over the entire data acquisition time. Both Hadamard phase change and amplitude change se-
quences can be used. However, if high pulse energies are used, it is necessary to solve the Bloch equations numerically and introduce saturation correction terms into the data treatment for recovery of the spectral line shapes.
Hadamard transformation remains a field with proven areas of application and many awaiting exploration. Technologies for fabrication of moving or stationary masks are now available for most of the electromagnetic spectrum. Computer addressing of stationary mask sequences (OSAs) is straightforward. Computer control of stepping motors and piezoelectric translators is routine. These advances make oncedifficult imaging and spectroscopic applications quite practical. Hybrid multichannel/multiplex experiments may become increasingly important as ways to use multichannel detectors to obtain complete spectral/ spatial data on samples. Fourier transform spectral encoding and Hadamard spatial encoding have been demonstrated in IR spectral microscopy. A similar approach could prove useful with Raman spectroscopy. Much Hadamard imaging has been near-field imaging. It is now known that near-field visible imaging can be used to resolve features 2 orders of magnitude smaller than the Rayleigh diffraction limit (37, 38). Yet sub-Ray leigh Hadamard imaging has not yet been explored in any region of the electromagnetic spectrum. With such exciting prospects we can expect future progress in Hadamard transform applications to be rapid and far from random.
Data transformations Hadamard encoding of data sets for image processing and image compression was one of the earliest applications of the transform (35). The computational efficiency of Hadamard transforms was a major force behind much of this work. Computational considerations have driven most applications to chemical data transformations. It was recognized early that the Hadamard and Fourier transforms allowed about the same amount of data compression by truncation of high-frequency or sequency terms. Recently Zupan and co-workers (36) demonstrated that Hadamard and Fourier transformation are about equally effective in compression of IR spectra. Of course, calculation of the Hadamard transform is about eight times faster than calculation of the Fourier transform. Figure 8 shows that truncation of the high-sequency terms in the benzenesulfonamide spectra leads to broadened recovered spectra with loss of detail. Compressed spectra recovered after Hadamard transformation have the high-intensity narrow portions removed from the top of peaks before broadening is noticeable.
732 A · ANALYTICAL CHEMISTRY, VOL. 6 1 , NO. 11, JUNE 1, 1989
Figure 8. Effect of truncation of the Fourier transform (FFT) and Hadamard transform (FHT) on the recovered 512point IR spectrum of benzenesulfonamide. Truncation of the FFT broadens the spectrum; truncation of the FHT attenuates high-intensity features. (Adapted with permission from Reference 36.)
INSTRUMENTATION References
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ACE GLASS 131 ALLTECH 62 BLUE M 181 CHEMICAL DYNAMICS 197 CONOSTAN 27-28 ELDEX 143 FLUKA CHEMICAL 205 GILMONT 111 GRAPHIC CONTROLS 147 HAMMOND DRIERITE 205 HOTPACK 99 ISCO 63 LABCONCO 145 MILTON ROY/APD 37 NEYTECH/J.M. NEY . . . . 101-175 OMEGA IFC PHARMACIA LKB 65 RUDOLPH RESEARCH 113 SHIMADZU 53 SWAGELOK 76-77 UICINC 101 VALCO 141 WHEATON 99,119 The LabGuide is your true onestop buying source for scientific products.
Patrick J. Treado {right) is a third-year graduate student at the University of Michigan pursuing a Ph.D. in analytical chemistry. He received his B.S. degree in chemistry from Georgetown University in 1985. His research interests are in Raman spectroscopy, photothermal spectroscopy, and microscopy. Michael D. Morris (left) is professor of chemistry at the University of Michigan. He received his B.A. degree in chemistry from Reed College in 1960 and his Ph.D. in chemistry from Harvard University in 1964. After several years on the chemis try faculty at Penn State, he joined the University of Michigan in 1969. His research interests are in analytical laser spectroscopy, with special emphasis on Raman spectroscopy.
734 A · ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
