DETECTION DEVICE AND METHOD

Abstract: The invention is a device for the multiplex detection of a plurality of characteristic signals from a plurality of samples. Previous multiplex systems using Hadamard transform multiplexing were often limited to 4n-1 samples (n = 1, 2, 3 etc.). The present invention overcomes this limitations by encoding signals after they had been modulated by a sine function or a cosine function (or a combination of sine and cosine functions). Such modulation of signals using the "sine/cosine transform" is easier to implement than Hadamard transforms and there is no limitation imposed on the number of samples.

Description: DETECTION DEVICE AND METHOD The present invention relates to a device for and method of multiplex detection of a plurality of signals characteristic of a plurality of samples. More particularly, the invention relates to techniques for the multiplex detection of signals by a spectrometer. A specific use which the present invention may be put to is in the technical area of Fourier Transform Infra Red Spectroscopy for process control, for the simultaneous acquisition of multiple sample spectra. The invention might, for example, be utilised in the control of mixing processes such as the gelatinisation of starch. Conventionally, spectrometers receive signals from one radiation source and one sample, and display the results as a single spectrum of the sample. In many applications for sensing and process control it is desirable to use a single spectrometer to detect and analyze the signals from many samples, because spectrometers are typically expensive and vulnerable in an industrial environment. One way of achieving this is to present each signal sequentially to the spectrometer by some appropriate switching mechanism, and detect and analyze the results separately. Alternatively, it is known from International Patent Application No. PCT/GB91/010T9 (assigned to British Technology Group Limited), whose disclosure is incorporated herein by reference, to employ a multiplexing technique in which several signals from different samples are presented and analysed simultaneously. With this technique, the total signal arriving at the detector was greater than if the signals were observed sequentially. The overall signal-to-noise ratio was increased relative to that which could be obtained by sequential sampling. In more detail, in this known multiplexing technique the signals from each sample (carrying spectral information characteristic of that sample) were combined and then detected in a number of experiments. In order to be able to determine the individual signals from the individual samples after detection of the combined (composite) signal, each individual signal was modulated using an encoding scheme. The combined signal was then decoded to reveal the individual signals. In this known technique, the encoding and decoding was carried out using Hadamard and inverse Hadamard transforms. Whilst the known Hadamard transform multiplexing technique has proven successful in practice, its implementation presents some difficulties. A particular problem with the Hadamard transform multiplexing technique is that it is preferably limited to there being 4Λ-1 samples (for n=l,2,3...), since for other numbers of samples the necessary number of experiments is not at a minimum. The present invention seeks to overcome these problems. According to the present invention, there is provided a device for the multiplex detection of a plurality of signals characteristic of a plurality of samples, comprising: encoding means for causing the respective signal characteristic of each sample to be modulated according to a respective different sine function, or a respective different cosine function, or a respective different mixed sine and cosine function; and means for combining and detecting the signals, whereby signals characteristic of a plurality of samples are detected simultaneously as a combined signal. By modulating the various signals according to different sine/cosine functions, the present "sine/cosine transform" technique invention can afford the advantage over the Hadamard transform technique described earlier of being easy to implement. Further, as opposed to the Hadamard transform technique, there are no real limitations to the number of samples which can be accommodated. As with the Hadamard transform invention described above, the present invention can enable the simultaneous acquisition of an arbitrary number of signals using one spectrometer or at least a reduced number of spectrometers, thus achieving a considerable cost saving in situations where several instruments might formerly have been required if a sequential sample acquisition scheme were used. Whilst a major advantage afforded by the present invention is the ability to operate with a reduced number of spectrometers, in many circumstances the invention can also provide an improved standard deviation of noise as compared with sequential sample acquisition schemes. These circumstances are where the noise is generated in the combining and detecting means. For instance, in the case of FT-NMR, the noise is generated mainly in the radio-frequency detection coils and in the detection preamplifier. As used herein, the terms "sine function", "cosine function" and "mixed sine and cosine function" include, for example, sine squared functions and the like. The different, say, sine functions according to which the respective signals are modulated may differ solely according to their frequency. Also, as used herein, the term "mixed sine and cosine" function includes such functions as might be encountered for example with Fourier and Hartley transforms. Again, as used herein, the term "samples" includes both a number of separate, distinct specimens and also a number of portions of the same specimen. Preferably, the encoding means is adapted to cause each signal to be modulated such that the magnitude of the modulated signal does not fall below zero. This feature is of importance if the invention is being used to detect signal intensities of any kind, such as radiation. Intensities by their very nature can not fall below zero; it is important that the sample multiplexing scheme can cope with this limitation. Hence, in one preferred embodiment, the encoding means is adapted to cause each modulated signal to be centred on a baseline level having a magnitude greater than zero. This can be a particularly simple way of implementing the invention. In this event, the encoding means may be adapted to cause each signal to be modulated according both to a sine squared and to a cosine squared function. Expressed another way, each signal may be caused to be modulated according to a sine, cosine or mixed sine and cosine function, with a baseline offset from the zero intensity level. In another preferred embodiment, the encoding means is adapted to cause each signal to be modulated such that negative half-cycles of the function are inverted, whereby the magnitude of the modulated signal does not fall below zero. This solution is generally not preferred over the solution previously described, except perhaps where there are only small (say, two or three) numbers of samples. Although the signals could be in any appropriate form, such as acoustical or electrical, typically, each signal is a radiation signal, in which event the device suitably includes at least one source of radiation for irradiating the samples to generate said signals. The radiation may be electromagnetic, and may for example be light, whether at its visible or invisible wavelengths. In one particularly preferred variant of the invention, suitable for use with radiation signals, and capable of modulating the respective signals with a baseline offset, the encoding means comprises a respective pair of polarising filters for each sample, the filters of each pair being movable relative to each other to modulate their respective signal. This can be a cheap and effective way of putting the invention into practice, since it may involve the use of readily available components. In another variant, the encoding means may comprise a respective graded filter device for each sample, each filter device being movable to modulate its respective signal. Whilst this variant could produce accurate modulations of the different signals, it is generally not preferred because it may require the manufacture of bespoke filters of the requisite densities. This could involve considerable expense. In a further preferred variant, the encoding means comprises a respective modulating member for each sample, each modulating member being movable and being shaped such that movement of the member can modulate its respective signal. The modulating member may, for example, involve a specially shaped slit on a rotatable disc or on a translatable plate, or it may involve a disc with a series of apertures of various sizes. Preferably, given an encoding matrix which represents in its rows and columns the modulation effected on each signal for each of a plurality of sampling instants, the encoding means is arranged such that the matrix is unitary. This can ensure that each of the decoded signals includes noise with the same standard deviation. Preferably, again, given an encoding matrix which represents in its rows and columns the modulation effected on each signal for each of a plurality of sampling instants, the encoding means is arranged such that the matrix is purely real. This can lead to the invention being easier to implement than in cases, such as where a Fourier transform is used, where the matrix is complex. The number of sampling instants is preferably no less than the number of samples, since this is usually necessary for the successful performance of the test. By "sampling instants" is meant preferably the instants at which the combining and detecting means detects the combined signal or signals. In one preferred embodiment (the "real time" embodiment), such sampling instants may represent uniformly spaced instants during a single experiment; in another preferred embodiment (the "pseudo-time" embodiment), such sampling instants may represent separate experiments spaced by an arbitrary (but nevertheless typically regular) time interval. Preferably, the encoding means is adapted to cause each signal to be modulated at a respective different frequency; in other words, the functions by which the respective samples are modulated differ as to frequency. This is a particularly effective way of putting the invention into practice. More preferably, relative to the modulation having the lowest frequency, the remaining modulations have frequencies which are integral multiples of such lowest frequency modulation. It is particularly preferred that the encoding means is adapted to cause each signal to be modulated according to the transform kernels of a discrete cosine, discrete sine or discrete Hartley transform, since these can be the easiest to implement, especially if the symmetric versions of the first two of these transforms are used. The discrete cosine and Hartley transforms are described in detail later; it will be understood that the discrete sine transform is simply the sine analogy of the cosine transform. A discrete Fourier transform may also be used to advantage. The detection device of the present invention may include decoding means for decoding the output of the combining and detecting means into a plurality of signals, such that each signal is characteristic only of its respective sample. The decoding means would normally operate according to the inverse transform of that used in the encoding means. In the case of a spectrometer, the inverse transform would recover spectra from each sample when used in conjunction with the further standard inverse transform conventionally required to calculate spectra from the observed signals. The device may also include a further encoding means for causing the signal from each sample to be encoded with information characteristic of that sample. In the case of spectrometers, the encoding would be wavelength encoding, to ensure that each wavelength arriving at the combining and detecting means is encoded in some way, to make it distinguishable from all the other wavelengths which may arrive. Such wavelength encoding might be achieved by a diffraction grating, by a simple filter system, or by a Fourier transform device such as a Michelson interferometer or a Nuclear Magnetic Resonance pulse generator. An analogous further decoding means may also be provided. The device may also include sample receiving means such as a series of sample mounts. The invention extends to a method of multiplex detecting a plurality of signals characteristic of a plurality of samples, comprising: causing the respective signal characteristic of each sample to be modulated according to a respective different sine function, or a respective different cosine function, or a respective different mixed sine and cosine function; and combining and detecting the signals, whereby signals characteristic of a plurality of samples are detected simultaneously as a combined signal. Preferred features of the invention, together with the theory underlying the invention, are now described with reference to the accompanying drawings, in which :- Figure 1 is a schematic diagram of a first preferred embodiment of the invention; Figure 2 is a schematic diagram of a second preferred embodiment of the invention; Figure 3 is a schematic diagram of a third preferred embodiment of the invention; Figure 4 is a schematic diagram of a fourth preferred embodiment of the invention; Figure 5 illustrates the shape of a modulating member for use in any of the preferred embodiments; Figure 6 is a flow diagram illustrating the operation of a set up phase in the invention; Figure 7 is a flow diagram illustrating the operation of a multiplexing phase in the invention using a first multiplexing scheme; Figure 8 is a flow diagram illustrating the operation of a multiplexing phase in the invention using a second multiplexing scheme; Figure 9 is a flow diagram illustrating the operation of a multiplexing phase in the invention using a third multiplexing scheme; Figure 10 is a flow diagram illustrating the operation of a multiplexing phase in the invention using a fourth multiplexing scheme; Figure 11 is a flow diagram illustrating the operation of a processing phase in the invention; Figure 12 is a schedule of phase angles d jk (deg.) to be used in a Discrete Fourier Transform multiplexing scheme with N=5; Figure 13a is a schedule of coefficients cos ,*) to be used in a Discrete Fourier Transform multiplexing scheme with N=5; Figure 13b is a schedule of coefficients sin θ,*) to be used in a Discrete Fourier Transform multiplexing scheme with N=5; Figure 13c is a schedule of coefficients cos 2 (θ 2) to be used in a Discrete Fourier Transform multiplexing scheme with N=5; Figure 13d is a schedule of coefficients cos 2 (π/4-θ, t /2) to be used in a Discrete Fourier Transform multiplexing scheme with N=5; Figures 14a to 14e show five synthetic spectra illustrating the signals characteristic respectively of five samples; Figures 15a to 15e illustrate the results of multiplexing the spectra shown in Figures 14a to 14e respectively in five different multiplexing experiments to yield five experimental spectra; and Figures 16a to 16e illustrate the results of demultiplexing the waveforms shown in Figures 15a to 15e to recover the original spectra shown respectively in Figures 14a to 14e. The construction of the detection device of the present invention is first described. In broad terms, the first three embodiments of the present invention are similar to the three embodiments disclosed in International Patent Application No. PCT/GB91/01019 referred to earlier, the main differences relating to the manner in which the sample multiplexing encoding is effected and the manner in which the signals detected by the detector are decoded. Referring first to Figure 1, the first embodiment of detection device is a multiplexing Fourier Transform Infra-Red (FT-IR) spectrometer. It comprises generally, in succession, a source ("Source") of radiation, a primary encoder ("Primary Encoder"), a splitter (not explicitly shown in this figure), a sample multiplexing encoder (A or B), a plurality of samples ("Samples"), a combiner (not shown explicitly in this figure) and a detector ("Detector"). The paths of the radiation from the source to the detector are denoted by arrows. In more detail, the source is a source of infra-red radiation, such as a flash lamp or laser. For other types of spectrometer, the source might be a microwave generator. The primary encoder is a Michelson interferometer, by which wavelength information (that is, radiation intensities at various wavelengths) is encoded into a series of intensities sorted as a function of the displacement of a moveable mirror in one arm of the interferometer. The splitter serves to split the signal from the primary encoder and to transmit this on to the samples or the sample multiplexing encoder, dependent upon whether the "A" or "B" configuration is adopted. The splitter might be some form of lens or fibre optic device or a partly-silvered mirror set at an angle to the radiation path.. The sample multiplexing encoder is placed in the path of the radiation from the radiation source, either before the samples (as designated at A), or after the samples (as designated at B). Encoding is accomplished by one of a variety of means described shortly. The signal characteristic of each sample is encoded independently of the other such signals. In the most preferred embodiment, the encoder comprises a respective pair of close- coupled linear polarising filters for each sample, the filters of each pair being rotatable relative to each other to modulate their respective signal. In the embodiment illustrated in Figure 1, there are three samples, and hence there would be three separate pairs of filters. The attenuation of the signals obtained by each pair is proportional to cos 2 θ, where θ is the angle between the planes of polarization. An attenuation of sin 2 θ can be achieved by utilising the fact that sin 2 θ = cos 2 (θ - π/2, mod 2π). In an alternative preferred embodiment, the encoder comprises a respective graded filter device for each sample, each such device being movable to modulate its respective signal. The filter device might, for example, comprise a series of neutral density filters of variable thickness and/or variable optical density selectively interposable in the radiation path. These filters are mounted on a sliding bar or a rotating filter wheel, so that the filter appropriate to each sample and experiment number can be selected. Alternatively, the filter device might comprise a single filter with varying optical density. The density could vary in discrete steps, or it could vary continuously if the filter is only moved in discrete steps. In a further alternative preferred embodiment, the encoder comprises a respective modulating member for each sample, each modulating member being movable and being shaped such that movement of the member can modulate its respective signal. An appropriately shaped member is shown in Figure 5. This consists of a rectangular flat plate 1 having a cut-out 2 in the shape shown in the figure, the plate being slidable across the path of the radiation to vary the occlusion of the radiation beam from the source. The two sectors of the cut-out are each shaped according to a sine (or cosine) function, so that equal increments of displacement of the plate would cause the radiation signal from the relevant sample to be modulated according to a sine (or cosine) function. In a variant of this embodiment, the modulating member could be a suitably shaped rotatable disc. In yet a further preferred embodiment, the encoder comprises a wheel with holes of the appropriate sizes. The wheel is rotated to select the appropriate hole before a new experiment is conducted. Finally, for example if the same number of sources are used as samples, it would be possible to vary the power supply to the sources to effect the requisite modulation. The samples themselves are physical (material) samples, and might be of any type, provided (in this particular embodiment) they are suitable for infra-red spectroscopy. They might, as just one example, be food samples. Although three samples are shown in Figure 1 , any number of samples (greater than one) could be used. Five or six might be a typical number. The combiner serves to combine the radiation signals from the individual samples so that a single composite signal is detected by the detector. The combiner might be some form of lens or fibre optic device. If the combiner were a fibre optic device, it could include optical fibres or other light guides extending from each individual sample to carry the various signals to a point at which they are all combined, and a single further optical fibre or other light guide to transmit the combined signal on to the detector. It will be understood that for many uses the samples may be physically separated from each other. It is important that the combiner (and also the splitter) is mounted in a rigid fashion so as to prevent changes in the relative phase of the individual signals due to variation during the course of the experiment. The detector is of a standard form used in FT-IR spectroscopy such as can convert the detected signal to digital form for subsequent analysis. It includes not only a sensor for sensing the composite signal, but also, in the preferred embodiment, means for decoding and analysing the signal to produce information relating to each of the individual samples. The decoding and analysing means would usually incorporate implementing software (for more details of which see Figures 6 to 11 and the description relating thereto). Referring to Figure 2, the second embodiment of detection device is similar to the first embodiment. The only difference resides in the position in the radiation path of the primary encoder. It will be understood that, for an FT-IR spectrometer, it is largely immaterial whether the primary encoder is placed before or after the samples. Referring to Figure 3, the third embodiment of detection device is similar to the second embodiment, the difference residing in the number of sources provided. Whereas in the second embodiment only one source is provided, in the third embodiment the same number of sources are provided as samples. This might be advantageous if, for example, the sample multiplexing encoder A were to operate directly on each source, say by varying the input voltage to the flash lamp so as to modulate its output. Further advantages of using multiple sources are firstly that they can ensure that a larger signal arrives at the detector and secondly that failure of a single source need not result in total signal loss but only in loss of signal from one sample. Referring to Figure 4, the fourth embodiment of detection device is similar to the second embodiment, the main difference residing again in the number of sources which are provided. For this embodiment, the device is configured as a Fourier Transform Raman Spectrometer such as detects the effects of Raman scattering. The sample itself could be the source of radiation (although this is not necessarily the case), so that no independent source need be provided at all. In the succeeding numbered sections, the theory underlying the invention and the various multiplexing schemes of practical interest are described. 1. OVERVIEW AND INTRODUCTION In overview, the encoding and decoding performed in the present invention may be defined in terms of an "encoding" matrix C having matrix coefficients C jk such that the experimental observations a which measure the combined signal presented to the detector (data-handling system) are related to the unknown individual signals s k from each sample, k, as follows: The unknowns s t are recovered by solving Equation (1) using the inverse of matrix C. Equation (1) is in fact quite general, and defines not only the sample multiplexing encoding but also where appropriate the encoding of the signals from the samples with wavelength information (that is, using the primary encoder). If this double level of encoding takes place, each of the elements of the vectors in Equation (1) is itself a vector and each of the element of the matrix C is itself a matrix. Equation (1) would then be solved by double inversion of the matrix. In the preferred embodiments of the present invention, as regards the wavelength multiplexing encoding, and ignoring for one moment the double level of encoding created by the sample multiplexing encoding, the relevant elements in the encoding matrix are determined by the physical technique used and the construction of the spectrometer. For example, a simple case of wavelength multiplexing is that of a dispersive instrument, such as a conventional grating Infra-Red (IR) spectrometer or continuous wave nuclear magnetic resonance (NMR) spectrometer. The measured values d j are radiation intensities at various wavelengths, distinguished by a wavelength sorting device such a diffraction grating. These intensities are recorded by the instrument and output in the form of a spectrum s k , k = 1 ,2, ... , N. Thus, in this simple case, the coefficients C jk for the wavelength encoding comprise a unit matrix. More complicated, and closely related, examples are provided by Fourier transform infrared (FT-IR) and pulsed Fourier transform nuclear magnetic resonance (FT-ΝMR) spectrometers. In the former case, wavelength information (that is, radiation intensities s k at various wavelengths) is encoded by a Michelson interferometer into a series of intensities d j sorted as a function of the position of a moveable mirror in one arm of the interferometer. In the FT-ΝMR case the encoding of wavelength information is achieved by the application of excitation pulses and by the consequent precession and dephasing of the nuclear magnetic moments within the sample. Intensity values s k as a function of frequency are encoded and recorded as a series of induced voltages d j (the so-called "free induction decays") measured at successive time intervals after the excitation of the sample. In both these more complicated cases, still ignoring the double level of encoding, the matrix C is a unitary matrix of Fourier coefficients. Equation (1) can be solved by carrying out an inverse Fourier transform of the data d } , with or without additional data processing such as apodisation. As regards the sample multiplexing encoding, and ignoring now in turn the wavelength multiplexing encoding, the coefficients C jk are the transform kernels of any appropriate sine, cosine or other transform, capable of modulating the respective signal characteristic of each sample according to a respective different sine, cosine or mixed sine and cosine function. A large family of suitable such transforms (including various discrete cosine and discrete sine transforms) is disclosed in Table 1 of a paper by Jain, A. K. entitled "Fundamentals of digital image processing", Prentice Hall, 1989, p.150. Other such transforms include the discrete Hartley transform and the discrete Fourier transform. References to the discrete Hartley transform are firstly a paper by Bracewell, R.N., J.Opt. Soc. Am., 22. 1832-1835 (1983), and secondly a paper by Williams, C.P. et al., Anal. Chem., £1* 428-431 (1989). For the present purposes, the discrete Fourier transform may in many circumstances be needlessly complicated, since generally the quantities to be transformed are real rather than complex quantities. It is to be noted that the sine or cosine (or Hartley) transform is a real transform, whereas the Fourier transform is a complex transform. The theoretical analysis below mainly concerns two related real transforms, the discrete cosine transform, and (a variant of this) the symmetric discrete cosine transform, but it is understood that it can be applied to other such transforms, such as what is termed herein the "discrete sine transform". In Table 1 of the Jain paper, the discrete cosine transform is referred to as an "Even Cosine" transform of order "-1" ("EDCT-l "), the symmetric discrete cosine transform is referred to as an "Even Cosine" transform of order "-2" ("EDCT-2"), and the discrete sine transform as an "Even Sine" transform of order "-2". In a simple case, as described later, for the sample multiplexing encoding the C jk are cosine terms, each of which corresponds to a phase angle θ jk . The values θ,* must be chosen such that each source signal s k is modulated by a "frequency" which can be distinguished from the frequencies of the other sources. In other words, the data d } must be sampled with steps in Q jk small enough to satisfy the Nyquist sampling condition for all the modulation frequencies. Thus, in brief, the detection device of the present invention preferably operates by encoding the signal from each sample with a separate characteristic frequency. 2. RESTRAINTS ON THE ENCODING MATRIX As regards the wavelength or similar encoding, as mentioned previously, the relevant coefficients of the matrix C are fixed by the physical processes involved. However, as regards the sample multiplexing encoding, the coefficients are chosen directly by the user. It has been determined pursuant to the present invention that there are four important limitations on the possible choices: 1. It must be possible to solve Equation (1). The basic requirement is that the determinant of C should be non-zero. In order to obtain accurate numerical solutions, it is also desirable that C should not be ill-conditioned. 2. It must be possible to realise the coefficients C Jk physically in some way. 3. It is desirable that the experimental errors or signal-to-noise ratios of the decoded s k values should be at least as good as, or better than, the errors obtained when measuring the s k values directly by some sequential technique. Further, the change in the signal-to-noise ratio should preferably be the same for all the s k . The latter requirement imposes a tight constraint on the matrix C; it must be unitary. It is shown later that the encoding matrix is preferably not only unitary but also real, that is, its inverse is equal to its transpose, when the quantities being transformed are real numbers. More generally, however, if the transformed quantities are complex (that is, there are two quantities measured by a quadrature detection scheme) the transformation matrix must be complex unitary. The unitary property also guarantees that the transformation is invertible. 4. It is desirable (though not essential) that the matrix C can be constructed and implemented for any order N. Changing N should not require radical redesign of the spectrometer. In the immediately succeeding sections, it is shown, amongst other things, that transforms based on the sine or cosine function fulfil the above mentioned restraints. Various such transforms are defined and discussed in Sections 5 to 7 below. In the final sections of the description, various aspects of the implementation of these transforms are discussed. ■ 1 j- 3. THE GENERAL LINEAR TRANSFORM AND ITS PROPERTIES In this section, it is shown that the encoding matrix must be real and unitary if the signal-to-noise properties are not to be adversely affected by the encoding/decoding scheme. A general linear transformation of an array f(x) of N real numbers, labelled by the index x = 0, 1,..., N - 1, to a real array F(u), u = 0, 1,..., N - 1, and back again, may be written in the form _v-ι F(u) - Σ ∑ g(u,x)f(x) (2) _c-0 where the functions g(u, x) and h(x, u) are called the forward and inverse transformation kernel, respectively. Substituting (2) into (3) we find that fix) - g(u, χ ' )f( χ ') (4) for x = 0, 1 ,..., N - 1. For this to be true generally, we require that for all x, x ':- _v-ι ∑ h(x,u)g(u,x' ) - δ χ (5) u- 0 where the Kronecker delta δ^. = 1 if* = x' and 0 otherwise. Also substituting (3) into (2) we find that F(u) - for u = 0, 1,..., N- l. For this to be true generally, we require that for all u, u ':- N- l Σ ∑ g(u,x)h(x,u') - δ t (6) E- O where δ^. = 1 if u = u' and 0 otherwise. The orthornormality conditions (5) and (6) are the necessary and sufficient conditions for the forward and inverse transforms (2) and (3) to hold. Now suppose that f(x) is an array of experimental data subject to gaussian measurement errors, in other words f(x) is a realisation of an array of gaussian random variables /(x with expectation values ( (x)) and standard deviations σ x). If we perform the generalized transformation (2) on the data /ft), it follows from the Central Limit Theorem (Kendall, M.G., Stuart, A. and Ord, J.K.; Advanced Theory of Statistics; Vol. 1 ; Edward Arnold - London, Melbourne and Auckland, 1991; ch. 7, pp. 198ff) that the transformed array F(u) will also be a realisation of an array of gaussian random variables F(u) with expectation values (F(u)) given by ΛM (F(u)) - ∑ g(u,x) (f(x)) (7) i- O and standard deviations a F (u) given by the Pythagorean rule for the addition of noise variances, o 2 F (u) = N ∑ ~ l g 2 (u,x) σ 2 f (x) r χ _t.- 00 J N- l ∑ g(u,x)h(x,u)σ f (x) x- 0 J provided that the transformation satisfies the condition g(u, x) = h(x, u), that is the forward and inverse transforms have the same kernel. If this condition holds and also σ x) = σ f for all x, then from (6) it follows that o F ^(u) - * b uu a f 2 - o f 2 for all u. In other words we may write σ F (u) = σ F for all u, where ° F ' ° f ( 8 ) General linear transformations satisfying the above condition (that g(u, x) = h(x, u)) transform a real array of gaussian random variables with the same standard deviation σ^into another array of gaussian random variables with the same standard deviation o F = o f . This invariance of the standard deviation of gaussian noise during transformations is a desirable property. In what follows we will be concerned exclusively with this class of transformations. The functions g(u, x) may be regarded as the elements of a transformation encoding matrix g, and likewise the functions h(x, u) are the elements of a matrix h which by definition is the inverse of g:h =g "1 . The condition g(u, x) = h(x, u) specifies that the inverse ofg equals its transpose, that isg is a real unitary matrix. (The generalization of this result to linear transformations of complex arrays is that the transformation kernels should be complex and satisfy g(u, x) = Λ * (x, u), that is the matrix is complex unitary.) In summary, for a linear transformation to transform a real array of gaussian random variables with the same standard deviation into another array of gaussian random variables with the same standard deviation, the transformation matrix must be a real unitary matrix. 4. USEFUL TRIGONOMETRIC SERIES We now derive some results for the sums of trigonometric series which will be employed later. First we define for r = 0,1,..., 2N- 1, f = -1 and N ≥ 2. Hence, When r = 0, S e (r) = Ν from (9). When r is even, S e (r) must be 0 to satisfy (10), since exp (iitr) = 1 and so exp (iπr/N) S e (r) = S e (r). When r is odd, exp (iπr) = -1 and so from (10), 1 ♦ t cotl — 2N We can summarise the above results as follows: Taking real parts. S(r) - R (S(r)) (12) and taking imaginary parts, N-l S e (r) - I m (S e (r)) - ∑ sinl im (13) u-0 From (12) and (13) we find that and 0 when r = 0 or even, |sin(— )♦ cos( — )cot(— ) - c ( ^) whenrisodd. Summarising, we have that for r = 0, 1 , ... , 2N - 1 N- l N when r = 0, cos ' ±r(u ) (14) u- 0 2' 0 otherwise, and 5. THE DISCRETE COSINE TRANSFORM røCT^ In this section, the Discrete Cosine Transform (DCT) is first defined; it is then demonstrated that this transformation has a real and unitary transformation matrix. We define the transformation functions g(u,x) and h(x,u) for the Discrete Cosine Transform as follows, where x = 0, 1 ,..., N - 1 : 1 g(0,x) - h(x,0) - N ' g(u,x) = h(x,u) u > 0. (16) It can be seen that the DCT has an encoding matrix such that, in the present invention, the signals characteristic of each sample can be modulated according to a respective different cosine function. It is to be noted that the first array element of the transformation matrix has a different form from the other array elements. It follows from the definition in (16) that ∑ h(x,u)g(u,x') - h(x,0)g(0s') + ∑ h(x,u)g(umx') u- 0 u- 1 From (12), the first term in the summation on the right-hand side is N when x + x' + 1 is zero, and, since 0*_, jc, j ' __ N- l, this never happens; zero when x + x' + 1 is even and > 0, i.e. x + x' is odd and > 0; and 1 when x + x' + 1 is odd and > 0, i.e. x + x' is even and ≥ 0. The second term in the summation is N when x = x'; zero when \x - x'\ is even and > 0, i.e. x + _-_ ■ ' is even and > 0; and 1 when \x - x'\ is odd and > 0, i.e. x + x' is odd and > 0. Combining all the alternatives, we find that N- l 1 when x = x', ∑ h(x,u)g(u,x') - b_ u- 0 0 otherwise. (17) It also follows from the definition (16) and the sum (14) that 0, 0, (18) The last of the four cases in (18) may be written as From (14), the first term in the summation is zero since u +u' > 0; the second term is N when u = u' and 0 otherwise. Summarising all the possible cases, (19) The functions g(u,x) and Λfx.u, ) defined in (16) therefore satisfy the necessary and sufficient conditions (5) and (6) for forward and inverse transformation kernels. g(u,x) is the kernel of the Discrete Cosine Transform (DCT). 6. THE SYMMETRIC DISCRETE COSINE TRANSFORM fSDCT In this section, the Symmetric Discrete Cosine Transform (SDCT) is first defined; it is then, again, demonstrated that this transform has a real and unitary transformation matrix. We define the transformation functions g(u,x) and h(x, u) for the Symmetric Discrete Cosine Transform as follows, where x = 0,1,..., N- 1 and w = 0,l,..., N - 1 : g(u,x) = h(x,u) - (20) As with the DCT, it can be seen that the SDCT has an encoding matrix such that, in the present invention, the signals characteristic of each sample can be modulated according to a respective different cosine function. It follows from the definition of the SDCT in (20) that ΛM cos — (χ + χ' + l)(« + ) cos -(x - x')(t/ * -) N u- 0 N 2 N 2 From (14), the first term in the summation of the right-hand side is zero except when x + x'+ 1 is zero; since 0 ≤ x.x'≤ N - 1 , this never happens. The second term is zero except when x = x'; then the sum is N. Thus, h(x,u)g(u,x ) - δ^, - < u - 0 *-' 0 otherwise. (21) It also follows from (20) that The first term in the summation is zero except when u + u' +1 is zero, which never happens. The second term is zero except when u = u'; then the sum is N. Thus, N- l 1 wΛen w = u' *(ιι,x)A(*,« ) - δ^ = 0 othenvise . (22) x- 0 The functions g(u,x) and Λ(x,w defined in (20) therefore satisfy the necessary and sufficient conditions (5) and (6) for forward and inverse transformation kernels. A comparison between ( 16) and (20) shows that g(u,x) is the kernel of a modified version of the Discrete Cosine Transform. All the array elements of the forward and inverse transformation matrices have the same form and the array is symmetric. Hence, as mentioned above this transformation is termed the Symmetric Discrete Cosine Transform (SDCT). 7. THE DISCRETE HARTLEY TRANSFORM AND THE DISCRETE FOURIER TRANSFORM In this section, the Discrete Hartley Transform (DHT) and the Discrete Fourier Transform (DFT) are defined and discussed. The Discrete Hartley Transform may be defined, using the foregoing notation, as a transform with the symmetric kernel g(u,x) - h(x,u) - (23) It follows that ∑ h(x,u)g(u,x') - I∑ <cos(2Ξ ) ♦ sin(^)} (2) u-0 Nu-0 N N N-l l*όι ,2πxu. ,2πx'u. . ,2πxu. ■ , 2πx'u = — 2^ »cos( )cos( ) + sιn( )sm( ) Nuu--0 N N N N ,2πxu. . ,2πx'u. . ,2πxu ,2πx'« λ , cos( )sιn( ) ♦ sιn( )cos( )/ N N J N ' N IT { ∞ S(2 -(X - x')) ♦ έm( (x ♦ x'))} N ι u=o N iV N The sums of these cosine and sine terms, which are standard results of Fourier Transform theory, are easily found by the method of Section 4 above. It turns out that N-l ∑ 2zππ«u,, ,. λ ,N when x = x cos( (x - x))=L ., u-0 N 0 otherwise (24) and that ΛΓ-I .2πw ∑ sin(— (x ♦ x')) = 0 for all x,x', (25) u-0 N and hence Thus the kernel (23) satisfies the condition (5) for the forward and inverse transformation kernels, and may similarly be shown to satisfy (6). The kernel (23) may be rewritten in a simpler form by employing the identity, ,. π,. . , π Λ 1 cos(— ) = sιn(— ) = — . (26) 4 4 v/2 Hence 2 , ,2iixu. ,π Ν . / 2πxu. . , π x ^ g(u,x) = h(x,u) = — {cos(— — )cos(— ) sιn(— — - )sm(— )} (27) N N 4 N 4 The closely related kernel for the Discrete Fourier Transform may be written (as is well- known) g(u,x) - h -(x,u) - -^{ C os(-^) ♦ sin(^)}, (28) fit N N where r = -1 and the asterisk denotes the complex conjugate. This kernel is complex, and in general the arrays F(u) and/fø) will be complex. It is, however, possible to separate the real and imaginary parts. Let F R (u) and F(u) be the real and imaginary parts of F(u) respectively, and similarly for f(x) and/fx). The transform (2) can be separated into transforms yielding F*(u) and F'fu) respectively: (29) These equations involve only real quantities. The inverse transform may be separated in a similar way. It will be understood from the foregoing that the Discrete Hartley Transform is a mixed sine and cosine transform, like the Discrete Fourier Transform; however, unlike the DFT but like the Discrete Cosine and Symmetric Discrete Cosine Transforms, the DHT is a purely real transform. It is therefore easier than the DFT to implement. In practice, the DCT, SDCT and DHT have been found to be equally easy to implement, and have produced results of approximately equal quality. 8. THE MULTIPLEX ADVANTAGE In this section we examine how the multiplex advantage (signal-to-noise improvement) comes about in the sample multiplexing spectrometer. We consider N samples with corresponding signals, s h k= 1,2,..., N; these are the signals which would be measured by N sequential experiments on the samples if a conventional spectrometer were employed. Using a multiplexing spectrometer, we measure N data d J t j = 1 ,2,..., N, which are weighted sums of the s k . Following the notation of (1), we can write N *j - ∑ C jk s k (30) k- l We can relate these quantities to the arrays f(x) and F(u) defined above with the following equivalences: u -j - 1, x = k - \, d F(u), - fix), C jk - g(u,x)- (31) If the transform kernel satisfies the orthonormality conditions (5) and (6), (30) can be inverted numerically to recover the desired signals s k where C kj ~ ' = h(x,u). From the discussion in Section 3 it follows that if the transform is real and unitary, and if the data d } are measured with a detector D 0 which gives rise to additive gaussian measurement errors (noise) with identical standard deviations σ for all j, independently of the signals, then the errors in the computed signals s k will also be gaussian with standard deviations σ for all k. So far it does not appear that anything has been gained by using a multiplexing scheme. The origin of a multiplex advantage may be seen more clearly if a specific form for the transformation kernel is substituted. In what follows, the SDCT is used as an example. The DCT (or indeed any other suitable transform) could be used, but the symmetry of the SDCT makes the algebra particularly simple. From (20), (30) and (31), 4 - T Nr or, equivalently, where d = V(N/2)d j . We now choose to measure the quantities d with the detector £>„, which will give us measurement errors with standard deviations σ. The d j can be computed from d j = /(2/N)d j ' , with standard deviations ^(2/Njσ. The signals s k can be recovered by or, equivalently, The computed s* will have the same standard deviations as the d } , that is J(2IN)o. But if we had measured the s k directly using the same detector D a , the measured values would have had standard deviations σ. Therefore the multiplexing scheme gives a reduction of * (2 N) in the noise level, or equivalently an enhancement of <J(NI2) in the signal-to-noise ratio. Thus the origin of a multiplexing advantage in this instance lies in the choice of what is actually measured, the quantities d defined by (33). In summary, it has been shown for the SDCT, and can be shown for other sine, cosine, Hartley and Fourier transforms, that use of these transforms to multiplex the signals from a number of individual samples can give rise to the multiplex advantage of enhanced signal to noise ratio. 9. PRACTICAL MULTIPLEXING SCHEMES Four possible practical multiplexing schemes are now outlined. Scheme / modulates the sample signals by a cosine function. This scheme is not applicable to cases where the signals are intrinsically non-negative, such as radiation intensities. The other three multiplexing schemes are designed to circumvent this problem, with some loss of efficiency in the signal-to-noise enhancement which is achieved by multiplexing. Schemes /// and IV use cosine-squared functions to modulate the signal intensities. All of these schemes use the Symmetric Discrete Cosine Transform; other like transforms may also be used. 9.1 Multiplexing Scheme / For simplicity in what follows, we choose to omit the primes in (33) and (34). The basic multiplexing scheme (which will be called Scheme I) using the SDCT is then N d s k ∞sβ Jk , (35) k- l where θ ikk < - )(* - b (36) N 2 2 ' in radians, or equivalently in degrees, fory = 1,2,..., N, and the inversion is given by s. = — ∑ d. ∞s θ,. (38) * N j T\ J Jk If the measured d } have standard errors σ, then it can be seen that the computed s k have standard errors V(2/N)a, that is, the signal-to-noise enhancement using the multiplexing scheme is In other words, either (36) or (37) provides the relevant values of the phase angle Q jk for each sample k and each experiment/. The signals from each sample are modulated according to the value of cosθ^ for each experiment so that they will each be of the form of a sine (or cosine) wave having a particular frequency. It will be apparent that, relative to the modulation having the lowest frequency (the modulation for the sample with k = 1), the remaining modulations have frequencies which are integral multiples of such lowest frequency modulation. This is a characteristic of the Symmetric Discrete Cosine Transform. As an example, taking the relatively simple case in Equation (37) where N = 3, for the first sample (k=\), the values of θ Jk for the necessary three experiments ( =1,2,3) would be 15, 45 and 75°; for the second sample the values would be 45, 135 and 225°; for the third they would be 75, 225 and 375°. The frequency of modulation of the signal from the third sample would be five times as great as that from the first sample. The signal-to-noise enhancement can be derived in a different way as follows. From (20) and (21), AM N- l ∑ h(x,u)g(ujc') = a , = — ∑ cos θ^ cos θ χ , u U - 0 N U-0 where θ xuu = - /y(* + - 2 )/(V« + - 2 )_. • Then by setting x = x' and using the equivalences (31 ), N - 1 2 for k - 1 ,2,...N. By the Pythagorean theorem for the addition of noise standard deviations, the s k computed by (38) have standard deviations given by N — cosθ 2σ ∑ cos 2 θ . = ajQJN), Σ N for all k, as was required to be proven. One problem with Scheme /above is that it is necessary to modulate the signals s k by a cosine function which is negative for some values of/ and k. If the signals are, for example, electrical in nature, this presents no problems; however, if the signals are intrinsically positive, as is the case with any form of radiation such as light intensity, the scheme must be modified, since it is intrinsically impossible to modulate a non-negative radiation signal by a negative number. In general terms, the modified scheme is required to encode each signal such that the magnitude of the modulated signal does not fall below zero. Broadly speaking, this can be achieved in one of two ways. Either, the negative portion of the modulation may be inverted, the inverted portions of the modulated signal either being caused to be presented to the detector at a different time from the positive, non-inverted portions, or being caused to be presented to a separate detector. Multiplexing Scheme // below represents such an approach. Alternatively, this may be achieved by centring the sine (or cosine) function on a baseline signal having a magnitude greater than zero, that is, by imposing a sinusoidally oscillating modulation of the signal upon some constant background level. The zero level of the modulation thus corresponds to the background level, and the negative modulation can be represented by a positive signal which is nevertheless less than the background level. The problem of recovering the signals from the individual samples is then simply one of baseline correction followed by inverse transformation. Two specific schemes to achieve this are presented as Schemes III and IV below. These schemes rely on the fact that a modulation of the form cosθ superimposed on a constant background can be regarded, by a standard trigonometric identity, as equivalent to a modulation of the form cos 2 (θ/2) with the background removed. 92 Multiplexing Scheme // In this scheme, in a first variant, 2N experiments are performed using one detector, or, in a second variant, N experiments are performed using two identical detectors. In the first variant, a first set of N experiments is performed in which the N measured data ct } are weighted sums of the signals s k as in (35), but, for values of/, k for which cos θ Jk < 0, the modulation function is set to zero, that is, the contribution of s k to d * } is zero when cos θ jk < 0. We may write where c+ _ f cos θ yλ when cos θ Jk > 0, jk 1 0 otherwise. In a second set of N experiments, using the same detector Z>„ , the N measured data dj are again weighted sums of the s h but, for values of/, k for which cos θ jk > 0, the modulation function is set to zero, that is, the contribution of s k to d } ~ is zero when cos θ^ > 0. For values of k with cos d jk < 0, the modulation function is taken to be - cos θ,*. We may write N d ~ . J k- l C jk S * (41) where In all cases, the signals are modulated by a positive factor in the range 0 to 1. This is physically realizable for signals such as light intensities which are intrinsically non- negative. For each set of N experiments, the modulated signals have the appearance of a half-wave rectified waveform. In the second variant of the scheme, N experiments are performed with the N quantities d* j being measured by the first detector, and another N quantities d j being measured in parallel by the second (identical) detector. In both variants, the difference of the two datasets is calculated to derive the dy. where C J* = C jk - C j ~ k = COs θ J' 44) for all k. As in Scheme /, the signals are then computed by S k - ∑ d cosθ Jk . (45) If the measured data c } , d each have standard deviations σ due to noise, then by the Pythagorean theorem for the addition of noise standard deviations, the differences d } have standard deviations */2σ. Following the argument of the previous section, the computed s k will have standard deviations * /2σ, that is, the signal-to-noise enhancement appears to be However, it must be remembered that 2N measurements were made. If the experiments had been performed sequentially, i.e. one sample at a time, then 2 measurements could have been made on each sample and averaged with standard errors of σ/</2. Therefore the net signal-to-noise enhancement of the multiplexing scheme relative to a sequential acquisition scheme is only (l/2)-/(N/2), a factor of 2 worse than Scheme /. This loss of efficiency occurs because twice as much noisy data is acquired as in Scheme /, but almost half the contributions to the measured data by the sample signals s k are blocked (when C + jk and C jk are zero). 9.3. Multiplexing Scheme /// As in the previous scheme, 2N experiments are performed using one detector, or N experiments using two identical detectors D a . The first N measurements are of data a*. , which are sums of the s k weighted by cosine-squared functions. The phase angles are Θ. 2, where the θ jk are given by (36):- d° - ∑ s k ∞s 2 (Θ 2). (46) J k- l In the second N measurements, the s k are modulated by sine-squared functions:- d S . = ∑ ^ sin 2 (θ 2) . (47) J k- l In all cases, the signals are modulated by a positive factor in the range 0 to 1 , which is physically realizable. The difference of the two datasets is next calculated, employing a standard trigonometric identity:- d - d c d - d S = ∑ s k cos * jk . (48) J k- l The s k can be recovered by the transform (45). By arguments analogous to those used for Scheme //, the net signal-to-noise enhancement relative to a sequential acquisition scheme works out to be 9.4 Multiplexing Scheme TV This multiplexing scheme is particularly preferred, since it has a requirement for fewer experiments to be carried out, or for fewer spectrometers. In this scheme, it is necessary to perform N + 1 experiments using one detector £)„ . The first N measurements are of data d c j which are sums of the s k weighted by cosine- squared functions, as in (46) above. The (N+ l)th. measurement is a quantity d° N+l which is just the sum of the unweighted signals:- ι - Σ S k . (49) k. \ We calculate N quantities d, defined by N N 2d c d n C A - ∑ s k (2 cos^ Jk /2)- l) - ∑ s k ∞sθ Jk (50) and then the s k can be recovered by the transform (45). d° N+ , acts as a 'baseline correction' to the d r If it and all the d c J have noise standard deviation σ, then the d j have standard deviations given by + σ 2 ) = * 5σ. It follows that the computed s k have standard deviations -/(2/N) x 5σ, that is, the signal-to-noise enhancement is -/(N/10) relative to a sequential acquisition scheme using single measurements of each signal. In Schemes I to IV above, the Symmetric Discrete Cosine Transform was used as just one example of a possible transform. However, other cosine, sine or mixed sine/cosine transforms may be used in a similar way. For instance, the implementation of an encoding scheme for the Discrete Hartley Transform (23) proceeds identically, except that the formula for the phase angles Q jk is modified in accordance with (27): V^ (/ - i) <*- i) - ( 51 ) The implementation of the Discrete Fourier Transform is conceptually more difficult. Because all the quantities involved in the actual implementation are intrinsically real, it is necessary to use the form (29) expressed in terms of real arrays f(x),f(x). (29) can be rewritten in terms of signal arrays s k R , s k and encoded data arrays df, dj as follows: ,, N ■ Σ (s k R cosβ Jk sinθ Jk,}' ; k- l N ∑ k k cosθ. k * s k sinθ t l. ( 52 ) k-l J The relevant phase angles here are: θ,* = ^- - 1)(* - 1)* ( 53 ) From here, one can proceed in either of two ways. Firstly, one can consider the Ν quantities s k R as the original signals to be encoded, and the Ν quantities s k as zero. Ν experiments are then performed to measure the Ν quantities d , and a further Ν experiments to measure the dj. The signals s k R can be recovered from the df and dj by the appropriate inverse Fourier Transform. Alternatively, and preferably, the number of samples is redefined as being 2N. The first N signals are denoted by sj. (If the number of samples is odd, s N l is set to zero, which is tantamount to using a "dummy" sample.) 2N experiments are then performed to measure the df and dj. In effect, this scheme packs the signals into both the real and imaginary components of the Fourier Transform. In circumstances where it is not possible to modulate the signal by negative quantities, it would be necessary to supplement the modulation scheme by baseline offset corrections such as those described under Multiplexing Schemes II to IV above. 10. REAL TIME OR VIRTUAL TIME OPERATION References herein to the modulation of the signals from the various samples according to different sine, cosine or mixed sine and cosine functions are to be understood to include references both to real time and to virtual time operation. Also, where reference is made to each signal being modulated at a different frequency, the frequency is to be understood as either a real frequency or a "pseudo-frequency". Hence, in real time modulation, the spectral elements, comprising the signals from every sample, are monitored by the spectrometer for a series of time intervals Δt during which time-dependent modulation of the signal takes place. For example, a number of optical signals could be attenuated by rotating sectors or moving graduated filters in the light path, the modulation of each signal being according to a different, say, cosine function. In this example, the combined signal from all the sources is an interferogram, which is collected (digitized) in an appropriate manner. Inverse Fourier or cosine transformation of the time domain signals then gives a set of spectrometer responses separated by the frequency differences of the intensity modulations. For dispersive instruments, this corresponds to a set of spectra, while for multiplexing instructions a second set of transforms is required before spectra are obtained. In the case of systems where whole spectra are required, several thousand spectral elements may be involved and therefore in principle several thousand Fourier or cosine transforms. However, the size of the transforms depends on the number of cycles of modulation which are used, and this may be reduced to a minimum level in which the number of samples in the time domain, for each spectral element, is no more than the number of sources used. It will be apparent that with real time modulation the signals from each sample are modulated according to sine or cosine functions each having a different "real" frequency, the modulation taking place actually through the various sampling instants. With virtual time modulation, on the other hand, the signals from each sample are modulated according to sine or cosine functions each having a different pseudo-frequency, the modulation taking place between the various sampling instants, with the sampling instants representing different experiments. Virtual time modulation may be important in circumstances where real time modulation might involve difficulties with synchronising the signals from the various samples. In virtual time modulation a number of separate experiments are conducted, the time separation between which may be arbitrary, hence giving rise to the concept of a "pseudo" frequency. Whilst the actual time separation between the experiments can be arbitrary, it will be appreciated from the analysis presented below that the modulation employed for each sample will progress from experiment to experiment with uniform phase shift increments, and, further, that the phase shift increments will differ from sample to sample. For example, consider a system consisting of N samples. During a series of separate experiments, the signal intensities from the N samples are modulated by the coefficients C Jk = cos θ Jk , where the phase angles θ Jk are functions of the sample number k and the experiment number /. Thus each sample is modulated by an amount corresponding to a separate phase shift. During each experiment, the N modulated sample signals are combined into one interferogram d y This process continues through the necessary minimum of N experiments and the effect is to modulate the intensity of each sample throughout the cycle of N experiments by a cosine wave with a discrete pseudo frequency. The pseudo-frequency is understood to vary according to the experiment number. The operation of the invention is now described with reference to Figures 6 to 13. The operation of the invention based on the Symmetric Discrete Cosine Transform is described with reference to Figures 6 to 11, whilst the operation of the invention based on the Fourier Transform is described with reference to Figures 12 and 13. Referring at first particularly to Figure 6, the detection device is set up according to the steps outlined in this figure. After the Start Step 10, the desired sample multiplexing encoding transform is selected in Step 12. This might be the Symmetric Discrete Cosine Transform, as described in detail above. Then in Step 14 values are computed of the phase angles d jk for j=l,...Jf and Λ=1,...,N, where Nis the number of samples. These phase angles might, for example, be computed from Equation (37), appropriate to Multiplexing Scheme /. Next (Step 16) the spectrometer configuration is selected. Specifically, the number of sources and the order of the primary and sample multiplexing encoders and samples is selected, in accordance, for example, with the principles discussed in relation to the preferred embodiments of the invention described above. Then in Step 18 the desired multiplexing scheme (for instance one of Schemes I to TV described above) is selected. It is noted in passing that Scheme / would only be suitable in situations where the signal to be modulated has negative as well as positive values, such as would be the case with electrical signals. Finally, in Step 20, the appropriate multiplexing scheme is executed, as described next. In the ensuing description of the various multiplexing schemes, like reference numerals refer to like steps. Referring now to Figure 7, Multiplexing Scheme / is executed as follows. After the Start Step 100, theNsamples numbered A=l,...Nare installed in the detection device, ready for their characteristics to be ascertained by whatever technique (such as FT-IR) has been chosen (Step 102). In the implementing software of the device, the experiment number,/, is next set to 1 (Step 104). At Question Step 106, the implementing software then enquires whether the value of/ is equal to N + 1. If the answer is "no", then Steps 108 to 112 are proceeded with. If "yes", then the experiment is finished and the processing phase begins (Step 114, for details of which see later). If the answer is "no", then one progresses to Step 108, in which, for each sample k, the relevant signal is modulated by cosθ j *, where θ Jk is given by Equation (37). Then (Step 110) the signals from each sample are combined (added together) and the result is recorded as d j (see Equation (35)). Finally (Step 112), the experiment number y is incremented (that is, a new experiment is embarked upon), and the procedure reverts to question Step 106 until the requisite number of experiments has been completed. Referring now to Figure 8, Multiplexing Scheme // is executed as follows. After the Start Step 200, Steps 202, 204 and 206 are embarked upon, exactly as with the corresponding steps in Scheme /, except that the destinations from the Question Step 206 are different. In the present case, if the answer to the question is "no", then Steps 208 to 212 are proceeded with. If "yes", then Steps 214 to 232 are proceeded with. The extra steps in Scheme Hare to allow for the "negative" part of the modulation of the signals. Following the "no" path first, one progresses to Step 208, in which, for each sample k, the relevant signal is modulated by cosθ,*, where θ jk is given by Equation (37), if cosθ. λ is greater than 0, and otherwise by zero (see Equation (40)). Then (Step 210) the signals from each sample are combined (added together) and the result is recorded as d + j (see Equation (39)). Finally (Step 212), the experiment number/ is incremented (that is, a new experiment is embarked upon), and the procedure reverts to Question Step 206 until the requisite first N experiments have been completed. Once the first N experiments have been completed (so that/=N+l), the answer to Question Step 206 is "yes", and processing proceeds to Step 214, where y is again set to 1. One then progresses to a further Question Step 216, where again it is enquired whether the second set of N experiments has been completed, this time for the "negative" part of the modulation. If "no", then processing proceeds to Steps 218 to 222. If "yes", then processing proceeds to Steps 224 to 232. Hence, if the answer is "no", one progresses first to Step 218, in which, for each sample k, the relevant signal is modulated by -cosθ,*, where θ^ is given by Equation (37), if cosθ y * is less than 0, and otherwise by zero (see Equation (42)). Then (Step 220) the signals from each sample are combined (added together) and the result is recorded as d } (see Equation (41)). Finally (Step 222), the experiment number/ is incremented (that is, a new experiment is embarked upon), and the procedure reverts to Question Step 216 until the requisite second N experiments have been completed. Once these have been completed (that is, the answer to Question Step 216 is "yes"), then processing proceeds to Steps 224 to 232. In Step 224, the implementing software sets the value of the experiment number/ to 1. Next, in Question Step 226, it is enquired whether the value of/ is greater than N. If the answer is "yes", then the experiment is finished and the processing phase begins (Step 232, for details of which see later). If the answer is "no", then the total signal d i for each experiment is computed as the sum of d* j and d } (see Equation (43)) in Step 228. This process is repeated for all the values of/ (see Step 230). Referring now to Figure 9, Multiplexing Scheme III is executed similarly to Multiplexing Scheme // described above and hence like steps are denoted by like reference numerals. As illustrated in Figure 9, Scheme /// differs from Scheme // in the following respects. Firstly, in Step 308, for each sample k, the relevant signal is modulated by cos 2 (θ /lS 2), where θ jk is given by Equation (37). Then (Step 310) the signals from each sample are combined (added together) and the result is recorded as tf j (see Equation (46)). Secondly, and likewise, in Step 318, for each sample k, the relevant signal is modulated by sin^θ ^), where β jk is again given by Equation (37). Then (Step 320) the signals from each sample are combined (added together) and the result is recorded as _ . (see Equation (47)). Thirdly, and finally, in Step 328 the total signal d for each experiment is computed as the sum of d° j and d } (see Equation (48)), analogously to the procedure adopted in Step 228. Referring now to Figure 10, Multiplexing Scheme TVx ' s executed as follows. After the Start Step 400, the N samples numbered installed in the detection device, ready for their characteristics to be ascertained by whatever technique has been chosen (Step 402). In the implementing software of the device, the experiment number,/, is next set to 1 (Step 404). At Question Step 406, the implementing software then enquires whether the value of/ is equal to N+ 1. If the answer is "no", then Steps 408 to 412 are proceeded with. If "yes" , then Steps 414 to 426 are proceeded with. If the answer is "no", then one progresses to Step 408, in which, for each sample k, the relevant signal is modulated by cos 2 (θ A /2), where Q Jk is given by Equation (37). Then (Step 410) the signals from each sample are combined (added together) and the result is recorded as <f, (see Equation (46)). Finally (Step 412), the experiment number / is incremented (that is, a new experiment is embarked upon), and the procedure reverts to Question Step 406 until N experiments have been completed. Once the N experiments have been completed, so that the answer at Question Step 406 is "yes", then a further baseline correction experiment is conducted in Steps 414 and 416. Specifically, in Step 414, a final experiment is conducted in which for each sample k, Q jk is set to zero, so that cos (θ, 2) is equal to 1; in other words, none of the signals are modulated. Then, for this one experiment, all of the (un-modulated) signals are combined (added together) and the result is recorded as df N+l (see Equation (49)). Once the final baseline correction experiment has also been completed, the processing proceeds to determine the value of d } for each experiment. At Step 418, the implementing software sets the value of the experiment number, /, to 1. Then question Step 420 is reached, in which an enquiry is made as to whether/ is greater than N. If the answer is "yes", then the experiment is finished and the processing phase begins (Step 426, for details of which see later). If the answer is "no", then the value of d } is computed as 2d c J - d ff ^, (see Equation (50) - Step 422). Step 424 ensures that this is carried out for each of the N experiments. Referring now particularly to Figure 11 , the processing phase is carried out in the analysis means of the detection device as follows. After the Start Step 500, the sample number k is first set to 1 (Step 502). Next, at the Question Step 504, enquiry is made as to whether the sample number in the processing phase is equal to N+l. If the answer is "no", the actual signals s k for each sample are decoded from the derived values of d j using Equation (38) (Step 506). Step 508 ensures that this process is carried out for each of the N samples. If the answer is "yes" (that is, when the actual signals s k for each sample have been evaluated), one proceeds to Step 510, at which any additional processing which might be required is carried out. Such processing might, for example, be apodization or Fourier transformation of the signals s k . Then (Step 512), the processed signals are displayed, possibly as spectra, and/or further analysed. Processing is concluded at the Stop Step 514. Referring now to Figures 12 and 13a to 13d, the salient features of the operation of the invention using the Discrete Fourier Transform are now described, where these differ markedly from the features described in relation to Figures 6 to 11. In Figure 12 are provided, by way of example, the values of the phase angles θ^ to be used in a test with five or ten samples, as derived from Equation (53) with N = 5. In Figures 13a to 13d are provided the corresponding values of cos(θ,,*.), sin θ,*), cos 2 (θ.*/2) and cos 2 (π/4 - θ Jk /2), respectively. Figures 13a and 13b cover the case where the signals may be negative as well as positive, whilst Figures 13c and 13d cover the case where the signals are intrinsically non-negative. In the case of five samples, the coefficients in Figures 13a and 13b are used to modulate the signals s" k (k = 1, ..., 5) in a series of ten experiments to measure the quantities d R ] and _/,, according to Equation (52) (with the quantities ^ taken as zero). For the case often samples, the coefficients in Figures 13a and 13b are used to modulate the signals s^ (assigned to the first five samples) and s k ' (assigned to the second five samples) in a series of ten experiments to measure the quantities dj and dj, according to Equation (52). In the event that the signals cannot be modulated by negative coefficients, five samples can be handled by performing five experiments in which the modulation is of the form cos 2 (β 2) and the quantities d = ∑ k _ os (θ i 2) are measured, followed by a further five experiments in which the modulation is of the form cos 2 (π/4 - θ j /2) and the quantities df = ∑ k s*. cos 2 (π/4 - β jk /2) are measured. The relevant coefficients are tabulated in Figures 13c and 13d respectively. An additional experiment is required to measure d m = Σ k s R k (for the baseline correction). Using the facts that 2cos 2 (θ J 2) - 1 = cosθ.* and 2cos 2 (π/4 - θ jk /2) - 1 = sinθ^ the eleven encoded data arrays f. j Q ^ l, ..., 5) and d > , can be combined in a manner similar to that employed in Multiplexing Scheme TV, Equation (50). In this way ten arrays equivalent to dj = ∑ k s*. cosθ.* and d J = Σ k s R k sinθ,-. can be computed. An inverse Fourier transformation then recovers the original signals s".. Modulation (attenuation) of the signals by factors of cos 2 (θ 2) and cos 2 (π/4 - d jk /2) can, as usual, be achieved by rotating a pair of cross-polarising filters relative to each other by the appropriate angles. In all cases, the angles should be normalised to the range 0 - 90° since this is the effective range of cross-polarising filters. Although the test can be performed with a Discrete Fourier Transform as just described, it has no particular advantage over the Discrete Cosine, Symmetric Discrete Cosine or Discrete Hartley Transforms, and it has the disadvantage of added complexity. The latter three transforms are therefore preferred. Finally, results derived using the Symmetric Discrete Cosine Transform with the present invention are given in Figures 14 to 16. In Figures 14 are given five synthetic spectra for each of five samples, together with added noise. The standard deviation of the noise in each spectrum was the same for each sample, but the signal-to-noise ratios were different; the spectra are plotted with the same maximum vertical heights for each sample. In Figures 15a to 15e are given the results of combining the five spectra shown in Figures 14a to 14e using the Multiplexing Scheme / defined in Equation (35), with angles θ jk appropriate to five channels (as given by Equation (36)). In Figures 16, corresponding to the respective Figures 14, are given the original spectra as recovered using Equation (38). It can be seen that the original spectra have been recovered to a good degree of accuracy. Indeed, a comparison of Figures 14 and 16 reveals a noticeable improvement in signal-to- noise ratio in Figures 16, that is, using the multiplexing scheme. It will be understood that the present invention has been described above purely by way of example, and modifications of detail can be made within the scope of the invention.

Claims: CLAIMS 1. Device for the multiplex detection of a plurality of signals characteristic of a plurality of samples, comprising: encoding means for causing the respective signal characteristic of each sample to be modulated according to a respective different sine function, or a respective different cosine function, or a respective different mixed sine and cosine function; and means for combining and detecting the signals, whereby signals characteristic of a plurality of samples are detected simultaneously as a combined signal. 2. A device according to Claim 1 wherein the encoding means is adapted to cause each signal to be modulated such that the magnitude of the modulated signal does not fall below zero. 3. A device according to Claim 2 wherein the encoding means is adapted to cause each modulated signal to be centred on a baseline level having a magnitude greater than zero. 4. A device according to Claim 2 or 3 wherein the encoding means is adapted to cause each signal to be modulated according both to a sine squared and to a cosine squared function. 5. A device according to Claim 2 wherein the encoding means is adapted to cause each signal to be modulated such that negative half-cycles of the sine function are inverted, whereby the magnitude of the modulated signal does not fall below zero. 6. A device according to any of the preceding claims wherein the combining and detecting means is adapted to detect radiation. 7. A device according to any of the preceding claims further including at least one source of radiation for irradiating the samples to generate said signals. 8. A device according to any of the preceding claims wherein the encoding means comprises a respective pair of polarising filters for each sample, the filters of each pair being movable relative to each other to modulate their respective signal. 9. A device according to any of the preceding claims wherein the encoding means comprises a respective graded filter device for each sample, each filter device being movable to modulate its respective signal. 10. A device according to any of the preceding claims wherein the encoding means comprises a respective modulating member for each sample, each modulating member being movable and being shaped such that movement of the member can modulate its respective signal. 11. A device according to any of the preceding claims wherein, given an encoding matrix which represents in its rows and columns the modulation effected on each signal for each of a plurality of sampling instants, the encoding means is arranged such that the matrix is unitary. 12. A device according to any of the preceding claims wherein, given an encoding matrix which represents in its rows and columns the modulation effected on each signal for each of a plurality of sampling instants, the encoding means is arranged such that the matrix is purely real. 13. A device according to any of the preceding claims wherein the number of sampling instants is no less than the number of samples. 14. A device according to any of the preceding claims wherein the encoding means is adapted to cause each signal to be modulated at a respective different frequency. 15. A device according to Claim 14 wherein, relative to the modulation having the lowest frequency, the remaining modulations have frequencies which are integral multiples of such lowest frequency modulation. 16. A device according to any of the preceding claims wherein the encoding means is adapted to cause each signal to be modulated according to the transform kernels of a discrete cosine, discrete sine, discrete Hartley or discrete Fourier transform. 17. A device according to any of the preceding claims including decoding means for decoding the output of the combining and detecting means into a plurality of signals, such that each signal is characteristic only of its respective sample. 18. A device according to any of the preceding claims including a further encoding means for causing the signal from each sample to be encoded with information characteristic of that sample. 19. A method of multiplex detecting a plurality of signals characteristic of a plurality of samples, comprising: causing the respective signal characteristic of each sample to be modulated according to a respective different sine function, or a respective different cosine function, or a respective different mixed sine and cosine function; and combining and detecting the signals, whereby signals characteristic of a plurality of samples are detected simultaneously as a combined signal. 20. A method according to Claim 19 wherein each signal is caused to be modulated such that the magnitude of the modulated signal does not fall below zero. 21. A method according to Claim 20 wherein each modulated signal is centred on a baseline level having a magnitude greater than zero. 22. A method according to Claim 20 or 21 wherein each signal is caused to be modulated according both to a sine squared and to a cosine squared function. 23. A method according to Claim 20 wherein each signal is caused to be modulated such that negative half-cycles of the function are inverted, whereby the magnitude of the modulated signal does not fall below zero. 24. A method according to any of Claims 19 to 23 wherein each signal is a radiation signal. 25. A method according to any of Claims 19 to 24 wherein, given an encoding matrix which represents in its rows and columns the modulation effected on each signal for each of a plurality of sampling instants, this matrix is unitary. 26. A method according to any of Claims 19 to 25 wherein, given an encoding matrix which represents in its rows and columns the modulation effected on each signal for each of a plurality of sampling instants, this matrix is purely real. 27. A method according to any of Claims 19 to 26 wherein the number of sampling instants is no less than the number of samples. 28. A method according to any of Claims 19 to 27 wherein each signal is caused to be modulated at a respective different frequency. 29. A method according to Claim 28 wherein, relative to the modulation having the lowest frequency, the remaining modulations have frequencies which are integral multiples of such lowest frequency modulation. 30. A method according to any of Claims 19 to 29 wherein each signal is caused to be modulated according to the transform kernels of a discrete cosine, discrete sine, discrete Hartley or discrete Fourier transform. 31. A method according to any of Claims 19 to 30 wherein the output of the combining and detecting means is decoded into a plurality of signals, each signal being characteristic only of its respective sample. 32. A method according to any of Claims 19 to 31 wherein the signal from each sample is caused to be encoded with information characteristic of that sample. 33. A device for the multiplex detection of a plurality of signals substantially as herein described with reference to and as illustrated in Figures 1 - 13. 34. A method of multiplex detecting a plurality of signals substantially as herein described.

Inventors: BELTON PETER STANLEY, WRIGHT KEVIN MICHAEL, BELTON, PETER, STANLEY, WRIGHT, KEVIN, MICHAEL
Applicants: BELTON PETER STANLEY, BRITISH TECH GROUP, WRIGHT KEVIN MICHAEL, BELTON, PETER, STANLEY, BRITISH TECHNOLOGY GROUP LIMITED, WRIGHT, KEVIN, MICHAEL
Application year: 1995
Main classification: G01J3
IPC codes: G01J3