SIGNAL PROCESSING

Abstract: Data processing methods and apparatus embodying an improved version of a gradiometer transform of the type disclosed in EP-A-0849565, which provides a method of extracting a signal from noisy data, comprising generating a plurality of components of a gradiometer transform by correlating a normalised data-set with a plurality of gradiometer step functions of differing orders, to obtain a transform vector F, and wherein an N<th> order fit to the data is obtained from data APPROX d0 + d1T1(t) + D2T2(t) + ... DnTn(t) -1 </= t </= 1 where T0, T1, T2, ..., TN are Chebychev polynomials of the first kind and the coefficients d0, d1, d2, ..., dN are obtained from the matrix equation (A) and where JT(N) is the (N+1)x(N+1) submatrix of a generic matrix JT.

Description: "Signal Processing" The present invention relates to signal processing methods, apparatus and systems. The invention is useful in a diverse range of fields including, but not limited to, electrophysiology, non-destructive testing (NDT) and geoscience . More particularly, the invention relates to a signal processing technique based on the use of a "gradiometer transform" which is analogous to the operation of a physical gradiometer device, but is of wider application and does not suffer the limitations imposed by the physical structure of such devices. The technique relates especially to the extraction of signals and other information from data subject to noise and/or background drift. More particularly, the invention relates to an improved gradiometer transform technique for signal processing. The underlying theory and use of gradiometer transform techniques is disclosed in EP-A-0849565, the disclosure whereof is incorporated herein by reference. The Gradiometer Transform was originally defined as follows. Any analogue or digital data stream is correlated with each of a series of Gradiometer step functions in turn. Correlating the data set with the (m+l) th Gradiometer step function h m (t) produces a single real number F m the (in+1) th transform component (see Fig. 1). Evaluating N+l of these transform components generates a transform vector F = Fx F„ which is used to obtain an N th order polynomial fit to the data, namely data = c 0 + c x t + c 2 t 2 + ... c n t n where the coefficients c 0 , c 1 , ..., c N are obtained from the vector F using the matrix equation and where J ~l is the (N+l) x (N+l) submatrix of a generic matrix J "1 defined in EP-A-0849565. The Gradiometer Transform can be used to extract an underlying signal from a data set which has been corrupted by noise and/or drift. The process by which the technique is used to extract the desired signal is illustrated in Fig. 2. Correlating the data set with each of the Gradiometer step functions in turn will yield a unique transform vector F. This process (which may be hard-wired) is referred to herein as the forward gradiometer transform. Depending on which features of the underlying signal one wishes to retain and which features of the data set one wishes to reject, this transform vector F_ is replaced with a modified vector F new , which is always of finite length and which may, if so desired, have the first m of its components replaced by zeros. This process is referred to herein as SEAR-filtering. Depending on the particular application, the process of SEAR-filtering may be pre-defined, or may be automated, both of which may be hard-wired. In certain applications however, it may be desirable to leave this process open for the user to define on a case-by-case basis. The final step is to take the inverse gradiometer transform of the new transform vector F new . This will produce a polynomial fit to the data as described above . It has been found that there is a numerical instability inherent in the preferred embodiment of the Gradiometer Transform as disclosed in EP-A-0849565. When it is desirable to use a transform vector F new consisting of a large number (more than about 50) transform components, the transform is unable to accurately reconstruct the underlying signal over its entire length. This problem is illustrated in Figs. 3 and 4. In Figure 3, the forward transform of the data shown top left is calculated. The pictures at the bottom of Figure 3, from left to right, show the result of taking the inverse gradiometer transform of transform vectors F new consisting of the first 60, 75, 100 or 150 components of the transform vector F of the data. It is evident that as the number of transform components is increased, the range over which the transform can reliably reconstruct the underlying signal is decreased. The pictures at the bottom of Figure 4 show the result of taking the inverse gradiometer transform of the first 55, 75, 90 and 100 transform components of the transform vector of the data shown top left. While it can again be seen that numerical instabilities reduce the range over which the signal can be reconstructed, it is clear that in this case it is necessary to include at least 100 transform components in order to capture the high frequency components present in the original data. The present invention relates to an improved version of the inverse gradiometer transform which overcomes the stated problem of high numbers of transform components resulting in unreliable signal reconstruction over the desired time interval. In accordance with a first aspect of the invention, there is provided a data processing method comprising generating a plurality of components of a gradiometer transform by correlating a normalised data-set with a plurality of gradiometer step functions of differing orders, to obtain a transform vector F, and wherein an N th order fit to the data is obtained from data « d 0 + d/T^t) + d 2 T 2 (t) + ... c^T t) -1 < t < 1 where T 0 , T 1# T 2 , ..., T N are Chebychev polynomials of the first kind and the coefficients d 0 , d x , d 2 , ..., are obtained from the matrix equation and where J lN) is the (N+l)x (N+l) submatrix of a generic matrix JT. Preferably, said data-set comprises sampled data and said correlation of the data-set with said step functions is performed using a modified data set in which each data sample is replaced by a piecewise continuous rectangular step function. The components of the gradiometer transform can be used to obtain useful information regarding the data-set, such as information regarding the background drift of a time-varying signal. It is particularly preferred that the data-set comprises a signal and the method is employed to remove background drift from said signal. In accordance with a second aspect of the invention there is provided data processing apparatus comprising data processing means for applying the method in accordance with the first aspect of the invention to a data-set from a data source. Preferably, said apparatus includes output means for generating an output data-set resulting from the application of said method. In certain embodiments, the apparatus includes input means for generating an input signal in response to variations in a parameter to be monitored by the apparatus. Preferably also, the apparatus includes sampling means for dividing said signal into discrete samples . The data-processing means may comprise a suitably programmed computer or a dedicated unit in which the method is implemented by means of a program encoded in hardware or firmware. The data-source may be a sensor for monitoring a physical parameter or a store of previously obtained or synthesised data. In describing embodiments of the present invention, reference is made to the accompanying drawings, in which: Fig. 1 illustrates the basic gradiometer forward transform process applied to source data; Fig. 2 illustrates the basic gradiometer forward and reverse transform process by which a signal is reconstructed from source data; Figs. 3 and 4 illustrate the limitations of the prior art gradiometer transform when extended to a large number of transform components; Fig. 5 shows a comparison of results obtained using the prior art gradiometer transform and the improved gradiometer transform in accordance with the present invention; Fig. 6 illustrates the difference in results obtained when the prior art gradiometer transform and the improved gradiometer transform in accordance with the present invention are applied to speech data; Fig. 7 illustrates a typical implementation of the invention using a general purpose computer; and Fig. 8 illustrates a typical implementation of the invention in a dedicated hardware unit. The initial steps of the gradiometer transform method are described in detail in EP-A-0849565 and will not be repeated here. For a given data set, the transform vector F is calculated as before. A modified transform vector F new is also obtained as before. In accordance with the present invention, however, an N ch order fit to the data is obtained from data - d 0 + ά T x { ) + d 2 T 2 (t) + ... d n T n (t) -1 ≤ t ≤ 1 where T 0 , T 17 T 2 , ..., T N are Chebychev polynomials of the first kind (Abramowitz and Stegun, 1970) and the coefficients d 0 , d l r d 2 , ..., d N are obtained from the matrix equation and where JT (N! is the (N+l) x (N+l) submatrix of a new generic matrix JT . The matrix JT is a generic matrix of infinite size, and is the matrix product of three matrices which are easily calculated. The matrix JT is given by JT = B Δ D where the matrix B is the infinite upper-triangular matrix whose elements b ± j are such that . [j if j is odd "M = i [0 otherwise b f 2j if / ' > 2, j ≥ / ' , and / ' and / are both even or both odd l [0 otherwise , the matrix Δ is the inverse of the sparse matrix whose elements in the m eh row are 1 in those columns whose column number is an odd multiple of m and zero elsewhere, and the matrix D is the diagonal matrix D = diag[l, , 1/3, , ]. Fig. 5 illustrates the performance of the improved gradiometer transform compared to that of the previously disclosed gradiometer transform when each is used to reconstruct the data shown in Figure 4. It is clear that the improved transform allows accurate reconstruction of the underlying signal over its entire length even in those cases where it is necessary to include a high number of transform components in order to capture key features of the signal. Fig. 6 shows an application from the field of speech processing. The data shown in the top picture represent a digital recording of a section of speech. It can be seen that taking 100 transform components is insufficient to capture the high-frequency speech components. In order to accurately represent the signal, it is necessary to take approximately 300 transform components before signal reconstruction - at which point the original transform is unable to reconstruct any but the smallest section of the original data. The scatter plot shown bottom left indicates that the new transform has adequately captured the original signal . The present invention thus provides an improved implementation of the gradiometer transform as disclosed in EP-A-0849565. The invention may be implemented as a software program running on a general purpose computer. Such a program may operate on "live" or stored data from any applicable source, such as a sensor. An example of this is illustrated in Fig. 7, which shows a computer 10, such as a personal computer (PC) , for running a program to perform the Gradiometer Transform, connected to a data source 12, which may be a sensor adapted to generate a signal in response to variations in a physical parameter of interest or a source of stored data. Where the data source 12 is a sensor or the like which outputs an analogue signal, or an analogue data store, then the complete signal processing unit will include analogue to digital converter (ADC) means 14 for sampling the analogue signal or data prior to processing by the computer 10. It will be understood that the ADC means might comprise a separate unit connected between the data source 12 and the computer 10, as illustrated, or may be incorporated into either the data source 12 or the computer 10. It will also be understood that, where the data source comprises a store of digital data, this store might be external to the computer 10 or may comprise the computer's own data store (typically the hard disk of the computer) . The digital data may originate from any applicable source and may be transferred to the data store by any suitable means. The computer 10 may be adapted to generate an external output 16 from the Gradiometer Transform program which may be employed for any required purpose . Alternatively, the method may be implemented by a program stored in hardware or firmware in a dedicated data processing unit 16, as is illustrated schematically in Figure 8. Again, the program operates on "live" or stored data from an internal or external data source 20 (sensor, data store or the like, as above) , the complete system including ADC means 22 as required. The data processing unit 18 generates an output 24 which may be employed for any required purpose . Where the system is intended for a specific purpose, then the components of the system may be integrated into a single unit comprising, for example, a sensor, ADC means and data processing means. A general purpose unit might comprise the data processing means alone or the ADC and data processing means, with provision for connection to other components as required. In any of the cases mentioned above, the program implementing the Gradiometer Transform, and other system components, may be dedicated to a particular application or may be user-configurable to suit specific requirements. The detailed design of software, firmware and hardware components of data processing systems to implement the Gradiometer Transform, as discussed above, is a matter of routine engineering and will vary widely according to the requirements of particular applications. Accordingly, these aspects are not described in detail herein. Improvements and modifications may be incorporated without departing from the scope of the invention. References M.Abramowitz, I.A.Stegun, Handbook of Mathematical Functions, (Diver, New York, 1970) p. 795.

Claims: Claims 1. A data processing method comprising generating a plurality of components of a gradiometer transform by correlating a normalised data-set with a plurality of gradiometer step functions of differing orders, to obtain a transform vector F, and wherein an N th order fit to the data is obtained from data - d 0 + d-Tjtt) + d 2 T 2 (t) + ... d„T n (t) -1 ≤ t ≤ 1 where T 0 , T x , T 2 , ..., T N are Chebychev polynomials of the first kind and the coefficients d 0 , d l f d 2 , ..., d N are obtained from the matrix equation and where JT m is the (N+l) x (N+l) submatrix of a generic matrix JT. 2. The method of Claim 1, wherein said data-set comprises sampled data and said correlation of the data-set with said step functions is performed using a modified data set in which each data sample is replaced by a piecewise continuous rectangular step function. 3. The method of Claim 1 or Claim 2 , wherein said data-set comprises a signal and the method is employed to remove background drift from said signal. 4. Data processing apparatus comprising data processing means adapted for applying the method as claimed in Claims 1 to 3 to a data-set from a data source . 5. Data processing apparatus as claimed in Claim 4, said apparatus including output means for generating an output data-set resulting from the application of said method . 6. Data processing apparatus as claimed in Claim 4 or Claim 5, wherein the apparatus includes input means for generating an input signal in response to variations in a parameter to be monitored by the apparatus. 7. Data processing apparatus as claimed in Claim 6, wherein the apparatus includes sampling means for dividing said signal into discrete samples . 8. Data processing apparatus as claimed in any one of Claims 4 to 7, comprising a suitably programmed computer in which the method is implemented by means of a program encoded in hardware or firmware. 9. Data processing apparatus as claimed in any one of Claims 4 to 7, comprising a dedicated unit in which the method is implemented by means of a program encoded in hardware or firmware. 10. Data processing appartus as claimed in any one of Claims 4 to 9, wherein said data-source comprises a sensor for monitoring a physical parameter. 11. Data processing appartus as claimed in any one of Claims 4 to 9, wherein said data-source comprises a store of previously obtained or synthesised data.

Inventors: JONES ELIZABETH ANN, JONES, ELIZABETH, ANN
Applicants: JONES ELIZABETH ANN, UNIV PAISLEY, JONES, ELIZABETH, ANN, THE UNIVERSITY COURT OF THE UNIVERSITY OF PAISLEY
Application year: 2000
Main classification: G06F17
IPC codes: G06F17