WO1981001464A1

WO1981001464A1 - Method and apparatus for obtaining derivatives of photometric signals

Info

Publication number: WO1981001464A1
Application number: PCT/AU1980/000092
Authority: WO
Inventors: M Hammer
Original assignee: Varian Techtron Pty Ltd; M Hammer
Priority date: 1979-11-19
Filing date: 1980-11-19
Publication date: 1981-05-28
Also published as: GB2074725B; JPS56501582A; GB2074725A

Abstract

Method and apparatus for obtaining derivatives of photometric signals in spectroscopy wherein an analogue signal characteristic of a sample (12) is converted to a digital signal which is differentiated using an approximation algorithm in a non-recursive digital filter (18) operating at predetermined differentiating intervals. The derivative is then low pass filtered in a recursive digital filter (23) at filtering intervals shorter than the differentiating intervals to reconstruct an interpolated derivative.

Description

METHOD AND APPARATUS FOR OBTAINING DERIVATIVES OF PHOTOMETRIC SIGNALS.

This invention relates to a method and apparatus for obtaining derivatives of photometric signals in a spectroscopic apparatus. The invention will be described in relation to obtaining derivatives of absorbance but it should be understood that derivatives of other photometric signals such as transmittance and emission may also be obtained and that the invention is described in relation to absorbance by way of example only.

In using a spectroscopic apparatus it is sometimes desirable to determine derivatives of absorbance, both when the instrument is in its scanning mode and when it is not. Generally, both the first and second derivatives in the instrument scanning and non-scanning modes are required although higher derivatives may also be required. In the past these derivatives have been obtained using analogue techniques which are relatively complex and difficult to implement. Furthermore, the technique most commonly used in the past has the tendency to amplify noise in the system with the result that a compromise is usually made between accuracy of answer and noise/quantization affects superimposed on the answer. In fact derivatives of an order higher than the first derivative have been extremely difficult because of noise effects.

It is an object of the present invention to provide a method and apparatus for obtaining derivatives of photometric signals which minimize the problems of the prior art.

According to the present invention there is provided spectroscopic apparatus, for analysing a, sample and including: means for generating a photometric signal characteristic of a sample being analysed, differentiating means operative to receive said photometric signal and to generate a derivative of said photometric signal to predetermined differentiating intervals, and, low pass output filter means operative to receive said derivative from said differentiating means and to reconstruct an interpolated derivative output by filtering said derivative at filtering intervals shorter than said differentiating intervals.

Preferably the photometric signal is first generated in analogue form and is converted to digital form before the generation of the derivative. The digital photometric signals may be generated at input intervals shorter than the differentiating intervals.

The derivatives may be generated by an approximation algorithm in a two stage non-recursive digital filter for a first derivative and in a three stage non-recursive digital filter for a second derivative.

The output filter means preferably comprises a two pole recursive digital filter for the first derivative and a four pole recursive digital filter for the second derivative.

Alternatively for the second derivative a two pole recursive digital filter may be used to which the second derivative is applied every second filtering interval to obtain a first filtered output which is reapplied to the output filter at every alternate filtering interval. The output filter means preferably has. filtering intervals, about fifteen times shorter than the differentiating intervals and preferably has a feedback factor of about 0.97,and cut-off frequency between 1 and 4 seconds but other values are possible.

The method of spectroscopic analysis according to the present invention includes the steps of: generating a photometric signal characteristic of the sample being analysed, generating derivative of said photometric signal at predetermined differentiating intervals, and filtering said derivative so as to remove unwanted higher frequency components and so as to simultaneously reconstruct an interpolated derivative output by filtering said derivative at filtering intervals shorter than said differentiating intervals.

A preferred embodiment of the invention will now be described with reference to the accompanying drawings wherein:

Figure 1 shows a schematic diagram of spectroscopic apparatus according to the present invention;

Figure 2 shows a two pole recursive digital filter for use as the output filter in Figure 1; and,

Figure 3 shows a four pole recursive digital filter for use as the output filter in Figure 1.

The spectroscopic apparatus shown in Figure 1 includes means 9 for generating a photometric signal characteristic of a sample 12 being analysed and-which includes a radiant energy source 10 for generating a radiant energy beam 11 to which a sample 12 is exposed, in this case by interposing the sample 12 in the beam path. Detection means 13 receive the beam 11. after encoun-tering the. sample 12 and generates an analogue photometric signal characteristic of the sample 12, this signal in the illustrated apparatus being percentage of the beam transmitted (%T).

Amongst the various modes of operation of a spectroscopic apparatus, it may be desirable to determine the first and second derivatives of a photometric signal when the apparatus is in its non- scanning mode (dPS/dt and d ²PS/dt²

respectively) and the first and second derivatives when scanning (dPS/dλ and d ²ps/dλ² respectively).

Derivatives of signals may be obtained by an approximation technique for obtaining derivatives of a photometric signal (PS) representative of the sample involves use of the following approximation:-

for the first derivative and

for higher order derivatives. Substituting ABS for PS gives the formulae for derivaties of absorbance (ABS). Derivatives with, respect to wavelength (.derivatives when scanning! may be obtained from the time derivative by suitable scaling; the scale factor varying both with the order of the derivative and with scan rate employed in the apparatus.

In the illustrated form of the invention the signal representative of the percentage transmission (%T) is quantized or converted into a stepwise signal to make it more convenience for later processing so that derivatives of ABS may be obtained. This conversion is achieved in an analogue to digital converter (A/D converter) 14 which may be a 12 bit device having a unipolar output. It will be appreciated however, that other types of converters 14 may be used. When a 12 bit A/D converter 14 is used, resolution may not be sufficient to adequately cover the range occupied by the signal being converted, and if that is the case a companding circuit 15 to expand low level signals is provided. This companding circuit 15 is to be used whenever the input is less that 1/16 full scale. The decision of whether to use the compander 15 or not for any particular conversion is to be made by carrying out a single test on the input. If the compander 15 is used the result of the A/D conversion must be divided by 16.

After the %T signal is quantized pre-differentiation filtering is carried out to remove higher frequency components from the stepwise output of the A/D converter 14 and thereby smooth the signal. Such filtering is carried out by a digital filter 16 such as a recursive digital filter. In spectroscopic apparatus having a chopper interrupted beam 11, it has been found that a filter 16 having two poles is usually adequate, although a filter 16. having more than two poles could be used if desired.. When a two pole recursive filter is used, it is prefered that the filter 16 has a cut-off frequency of about 7 samples Ce.g. about 0.2 seconds) - the time for a sample being the chopper interrupt period which may be about 25 - 30 ms.

To obtain absorbance (ABS) from the smoothed %T signal the logarithm to the base 10 is taken and the logarithm may be derived by logarithm computation means 17 or from a programmable read-only memory (PROM) used as a look-up-table.

The apparatus includes differentiating means 18 operative to receive the absorbance signal and to generate a derivative of the absorbance signal at predetermined differentiating intervals. The differentiating intervals are longer than the input intervals at which the digital %T signal is generated by the A/D converter 14, the input intervals being the same as the chopper interrupt intervals.

The derivatives either when scanning or when not scanning are obtained from the ABS signal in the differentiating means 18 using the approximation techniques mentioned above.

The accuracy of this technique increases as Δt approaches zero, but as Δt is reduced the term ABS_{t -Δt} tends towards ABS_t and hence ABS_t - ABS_t-Δt becomes a very small difference between two potentially large numbers. This has the effect of greatly amplifying noise. The accuracy of this technique also depends on the ratio between the differentiating frequency and the highest signal frequency component. Generally a low differentiating frequency is desirable and it has been found that suitable results are obtainable, with, a differentiating to input interval ratio about 6:1 for qualitative analysis or 8:1 or higher for quantitative analysis.

In the preferred form of the invention, the first derivative of ABS in either the scanning r non-

scannin mode, is obtained through the use of a

digital filter 19 which is capable of implementing the approximation algorithm for the first derivative. This includes the calculation of the change in absorbance between successive differentiating intervals, i.e. PS_t - PS_t_₁, where PS_t_₁ is a first absorbance amplitude sampled and PS_t is a second and the latest absorbance amplitude sampled one differentiating interval after the first sampling.

It has been found that a non-recursive filter 19 having two stages is adequate, but a filter 19 having more than two stages may be used if desired. In general the nth derivative will require n+1 filter stages. The differentiating interval for obtaining the first derivative with respect to wavelength using a non-recursive digital filter 19 may be about 0.5 second, but a sample period of 0.69 second is preferred. For differentiating with respect to time, the sampling interval may be chosen within a wide range such as between 2 to 9999 seconds.

A differentiating interval of above magnitude tends to minimise amplification of noise, but results in a stepwise or bar-graph type output of the derivative since the output data only change every differentiating interval.

To obtain the second derivative of ABS in either the scanning or non-scannin mode, use is made

of a digirtal filter 2a similar to th.e filter 19. used for obtaining the first derivative. That filter 20 is of the non-recursive type and isa three stage filter with the same differentiating interval as that of the two stage filter 19 used to obtain the first derivative. The differentiating filter 20 may be operative to calculate the second derivative of absorbance by the approximation algorithm outlined above which includes the calculation of

PS_t - 2PS_t-1 + PS_t-2, where PS_t-2 is a first absorbance amplitude sampled, PS_t-1 is a second absorbance amplitude sampled one differentiating interval after the first sampling, and PS is a third and the latest absorbance amplitude sampled one differentiating interval after the second sampling.

According to the illustrated form of the invention, only one o£ the first or second derivatives is determined at any one time. The spectroscopic apparatus therefore may have a mode selecting means 21 which is operative to select the required one of the two derivative modes.

The apparatus also includes means 22 for level scaling of the output of the differentiating means 18.

The output of the scaling means 22 is connected to low pass output filter means 23 which is operative to reconstruct an interpolated derivative output by filtering the derivative at filtering intervals shorter than the differentiating intervals. The process of digital differentiation introduces considerable noise and this unwanted noise component of the derivative is removed by the low pass function of the filter means 23.

In the case of first derivatives when linear interpolation between two successive derivatives is required a non- recursive digital filter 23 may be used as the output filter. Alternatively, an exponential interpolation between successive derivatives may be obtained using a recursive digital filter 23. When the recursive filter 23 is used the filter 23 should have a time constant of sufficient duration to provide a substantially linear interpolation.

It is preferred that the post-differentiation filtering be carried out using a recursive digital filter 23 shown in Figure 2. This filter 23 has two poles 30,31 although a higher number of poles may be used. The filter 23 is clocked at a frequency fifteen times higher than the differentiating frequency. The filter 23 may have a feedback factor A of about 0.97 and a cut-off frequency between 1 and 4 seconds.

The filter 23 includes summers 32, 33 and delaying means 34,35 each providing a signal delayequal to one. filtering interval. The feedback means 36,37 provide the feedback weighting factor of about 0.97.

In the case of the second derivative, the noise generated by the differentiating means 18 is inherently larger than with the first derivative and it is therefore preferred to have a filter 23 with more poles and a lower cut-off frequency than that used for the first derivative. The second derivative post differentiator filter 23 is preferably a four pole recursive digital filter and having a cut-off frequency between 1 and 4 seconds.

As shown in Figure 3, the output filter 23 has four poles 38-41 with summers 42-45, delay means 46-49 and feedback means 50-53 providing the feedback weighting factor A of about 0.97. Because the nett input factor including scaling is the square of the value required for the first derivative, it is possible to use for second derivative filtering the same output filtering means used for the first derivative twice in cascade. This may be achieved by applying the second derivative to the output filter 23 every second filtering interval to obtain a first filtered output and the first filtered output is reapplied to the output filter 23 at every alternate interval to obtain the interpolated derivative output.

This has the effect of implementing a filter wi.th. four poles and at th.e same time increasing the effective filtering interval from 1/30 seconds to 1/15 seconds. The nett result is a filter with four pole roll off, a cut-off frequency of about 4 seconds and requiring no more processing time than the first derivative case.

In general the output filter 23 has a progressively high number of poles and a progressively lower cut-off frequency for successively higher order derivatives.

Apparatus according to the invention also includes output means 24 including a digital to analogue converter 25 (D/A converter) for converting the output from the output filter 23 to analogue form. This D/A converter 25 may be the same device as the analogue to digital converter 14, the device being operative in alternating opposite modes. For example, the combined A/D and D/A device may be operative in its D/A mode at all times when the A/D mode is not being used, i.e. about 30ms in every 33ms.

A sample and hold circuit 26 (S/H circuit) as is known in the art may be used to store successive outputs from the D/A converter 25 for the time interval of one conversion. The high frequency components present in the analogue output may be removed by a low pass filtering using an analogue filter 27 with a time constant of about 150 milliseconds.

The logarithmic conversion, approximation alogorithms and post-differentiation filtering may be carried out by use of dedicated discrete digital circuitry. Alternatively, a microprocessor may be used to perform the logarithmic conversion, alogorithms and filtering, and that is generally the preferred approach because of its convenience. It will be appreciated that the co-efficients for the digital filters together wi:th the number of stages will determine the type of filter characteristic to be employed. The techniques for determining the co-efficients and the choice of the number of stages will be clear to those skilled in the art.

Also it will be appreciated that the chopper interrupt intervals, input intervals, differentiating intervals, output filtering intervals, filter feedback factors and cut-off frequencies are chosen to suit the particular method and apparatus in which the invention is used and that values for all these variables other than those mentioned herein may be chosen without departing from the scope of the invention.

Claims

1. Spectroscopic apparatus, fpr analysing a sample and including; means for generating a photometric signal characteristic of a sample being analysed, differentiating means operative to receive said photometric signal and to generate a derivative of said photometric signal at predetermined differentiating intervals, and, low pass output filter means operative to receive said derivative from said differentiating means and to reconstruct an interpolated derivative output by filterning said derivative at filtering intervals shorter than said differentiating intervals.

2. Spectroscopic apparatus according to Claim 1, wherein said photometric signal generating means includes: a radiant energy source for generating a radiant energy beam to which the sample can be exposed, detection means for receiving said beam after encountering the sample and for generating an analogue photometric signal characteristic of said sample, analogue to digital conversion means for converting said analogue photometric signal to digital photometric signals constituting said photometric signal, and pre-differentiating digital filter means operative to receive said digital photometric signal and to remove higher frequency components thereof and smooth the digital photometric signal.

3. Spectroscopic apparatus according to Claim 2, wherein said digital photometric signals are generated at input intervals shorter than said differentiating intervals.

4. Spectroscopic apparatus according to Claim 3 wherein said pre-differentiating digital filter means comprises a recursive digital filter having a cut-off frequency of about 7 input intervals.

5. Spectroscopic apparatus according to Claim 3, wherein said differentiating means is operative to calculate the first derivative of said photometric signal by an approximation algorithm which includes the calculation of the change in said photometric signal between successive differentiating intervals.

6... Spectroscopic apparatus according to Claim 5 wherein said differentiating means comprises a two stage non-recursive digital filter.

7. Spectroscopic apparatus according to Claim 3 wherein said differentiating means calculates the second derivative of said photometric signal by an approximation algorithm which includes the calculation of:

PS_t - 2PS_t-1 +PS_t-2 , where PS_t-2 is a first photometric signal amplitude sampled, PS_t-1 is a second photometric signal amplitude sampled, one differentiating interval after the first sampling, and PS_t is a third and the latest photometric signal amplitude sampled one differentiating interval after the second sampling.

8. Spectroscopic apparatus according to Claim 7 wherein said differentiating means comprises a three stage non- recursive digital filter.

9. Spectroscopic apparatus according to Claim 7 wherein said output filter means comprises a two pole recursive digital filter to which the second derivative is applied at every second input interval to obtain a first filtered output and said first filtered output is reapplied to the output filter at every alternate input interval to obtain said interpolated derivative output.

10. Spectroscopic apparatus according to Claim 1 wherein said output filter means comprises a recursive digital filter.

11. Spectroscopic apparatus according to Claim 10 wherein said filtering intervals are about 15 times shorter than said differentiating intervals.

12. Spectroscopic apparatus according to Claims 10 or 11 wherein said filter has a feedback factor of about 0.97.

13. Spectroscopic apparatus according to Claims 10,11 or 12 wherein said filter has a cut-off frequency between 1 and 4 seconds.

14. Spectroscopic apparatus according to any one of Claims 10 to 13 wherein said filter is two pole recursive digital filter,

15. Spectroscopic apparatus, according to any one of Claims 10 to 13 wherein said filter is a four pole recursive digital filter.

16. Spectroscopic apparatus according to any one of

Claims 10 to 13 wherein said filter has a progressively higher number of poles and a progressively lower cut-off frequency for successively higher order derivatives.

17. Spectroscopic apparatus according to Claim 1 wherein said output filter means comprises a non-recursive digital filter.

18. Spectroscopic apparatus according to Claim 2 wherein said output means includes digital to analogue conversion means for converting said derivative output to an analogue derivative output.

19. Spectroscopic apparatus according to Claim 18 wherein said analogue to digital conversion means and said digital to analogue conversion means are comprised by the same device being operative in alternating opposite modes.

20. A method of spectroscopic analysis of a sample including the steps of: generating a photometric signal characteristic of the sample being analysed, generating derivative of said photometric signal at predetermined differentiating intervals, and filtering said derivative so as to remove unwanted higher frequency components and so as to simultaneously reconstruct an interpolated derivative output by filtering said derivative at filtering intervals shorter than said differentiating intervals.

21. A. method of spectroscopic analysis according to Claim 20 and including: generating a radiant energy beam to which the sample can be exposed, receiving said beam after encountering the sample and generating an analogue photometric signal characteristic of said sample, converting said analogue photometric signal to digital photometric signals constituting said photometric signal, and filtering said digital photometric signal to remove higher frequency components thereof and to smooth the digital photometric signal_*

22. A method of spectroscopic analysis according to Claim 21 wherein said digital photometric signals are generated at input intervals shorter than said differentiating intervals.

23. A method of spectroscopic analysis according to Claim 20 wherein the step of derivative generation comprises calculating the first derivative of said photometric signal by an approximation algorithm which includes calculation of the change in said photometric signal between successive differentiating intervals.

24. A method of spectroscopic analysis according to Claim 20 wherein the step of derivative generation comprises calculating the second derivative of said photometric signal by an approximation algorithm which includes calculating:

PS_t - 2PS_t_₁ + PS_t_₂, where PS_t-2 is a first photometric signal amplitude sampled, PS_t_₁ is a second photometric signal amplitude sampled one differentiating interval after the first sampling, and

PS is a third and the latest photometric signal amplitude sampled one differentiating interval after the second sampling.

25. A method of spectroscopic analysis according to Claim 20 wherein said filtering intervals are about fifteen times shorter than said differentiating intervals.