Method and apparatus for reducing coupling between signals in a measurement system

Abstract

A method and an apparatus for separating a composite signal into a plurality of signals is described. A signal processor receives a composite signal and separates a composite signal in to separate output signals. Pre-demodulation signal values are used to adjust the demodulation scheme.

Claims

1. A system which measures one or more physiological characteristics of a living subject non-invasively by measuring pulsing light of two or more wavelengths after attenuation by body tissue, the system comprising: at least one emitter configured to emit light of at least two or more different wavelengths using a modulation scheme that cycles the at least two or more different wavelengths, the at least one emitter positioned to emit light in the direction of a living subject; at least one detector configured to detect the light of the at least two or more different wavelengths emitted from the at least one emitter after the light has been attenuated by tissue of the living subject, the detector further configured to generate a detector signal representative of the emitted light; a demodulation system configured to demodulate the detector signal, the demodulation system comprising an optimizing demodulator configured to reduce crosstalk by calculating updated coefficients that minimize a predetermined error using baseline coefficients, wherein the demodulation system is configured to use the updated coefficients to determine a plurality of demodulation functions used to demodulate the detector signal.

2. The system of claim 1, the demodulation system is further configured to optimize the demodulation based on previously demodulated signal values.

3. The system of claim 2, wherein the previously demodulated signal values comprises a plurality of demodulated signal values corresponding respectively to the at least two or more different wavelengths.

4. The system of claim 3, wherein each of the plurality of demodulated signal values is determined using a demodulation function.

5. The system of claim 1, wherein the demodulation system is further configured to optimize during an initial calibration period.

6. The system of claim 5, wherein the demodulation system is further configured to optimize during periodic calibration periods during measurement.

7. The system of claim 1, wherein at least one demodulation function is used for each of the at least two or more different wavelengths.

8. A method of demodulating a composite signal generated by applying at least first and second periodic pulses of electromagnetic energy to a system having a physiological parameter to be measured and by non-invasively receiving signals responsive to said electromagnetic energy after attenuation by body tissue of a living subject, said signals received as a composite signal having at least first and second components responsive to said at least first and second pulses respectively, said method comprising: receiving, at one or more signal processors, a composite signal responsive to electromagnetic energy of at least first and second periodic pulses of electromagnetic energy after attenuation by body tissue; applying, using the one or more signal processors, a first demodulation scheme to said composite signal to generate at least a first demodulated signal; and automatically optimizing, using an optimizing demodulator to reduce crosstalk, the first demodulation scheme by determining a plurality of updated coefficients that minimizes a predetermined error using baseline coefficients, the plurality of updated coefficients used to determine a plurality of demodulation functions which are used to demodulate the composite signal.

9. The method of claim 8, wherein the optimizing reduces crosstalk.

10. The method of claim 9, wherein the optimizing occurs continuously.

11. The method of claim 9, wherein the optimizing occurs interspersed with measurements.

12. The method of claim 8, further comprising applying a second demodulation scheme to said composite signal to generate a second demodulated signal and optimizing the second demodulation scheme based at least in part on the second demodulated signal.

13. The method of claim 12, wherein said signal processor is further configured to optimize at least one of the first and second demodulation schemes in the absence of at least one of the first and second periodic pulses of electromagnetic energy.

14. The method of claim 12, wherein said signal processor is further configured to optimize at least one of the first and second demodulation schemes based on the demodulated signals in order to further reduce crosstalk interspersed with measurements.

15. A system for demodulating a composite signal generated by applying at least first and second periodic pulses of electromagnetic energy to a system having a physiological parameter to be measured and by non-invasively receiving signals responsive to said electromagnetic energy after attenuation by body tissue of a living subject, said signals received as a composite signal having at least first and second components responsive to said at least first and second pulses respectively, said system comprising: means for receiving a composite signal responsive to electromagnetic energy of at least first and second periodic pulses of electromagnetic energy after attenuation by body tissue; means for applying a first demodulation scheme to said composite signal to generate at least a first demodulated signal; and means for automatically optimizing the first demodulation scheme by determining a plurality of updated coefficients that minimizes a predetermined error using baseline coefficients, the plurality of updated coefficients used to determine a plurality of demodulation functions which are used to demodulated the composite signal.

16. The system of claim 15, wherein the optimizing reduces crosstalk.

17. The system of claim 15, further comprising applying a second demodulation scheme to said composite signal to generate a second demodulated signal and optimizing the second demodulation scheme based at least in part on the second demodulated signal.

18. The system of claim 17, wherein said signal processor is further configured to optimize at least one of the first and second demodulation schemes in the absence of at least one of the first and second periodic pulses of electromagnetic energy.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present disclosure will be described below in connection with the accompanying figures.

(2) FIG. 1 is a block diagram of a multi-channel processing system that uses feedback from one or more outputs to configure the operation of a signal separator that separates a composite signal into a plurality of output signals.

(3) FIG. 2 is a block diagram of a two-channel signal processing system to determine blood oxygen saturation in a subject, wherein illumination is provided by back-to-back Light-Emitting Diodes (LEDs).

(4) FIG. 3 is a block diagram of a multi-channel signal processing system to determine blood constituents (e.g., oxygen saturation) in a subject, wherein illumination is provided by N diodes or illumination sources.

(5) FIG. 4 is a block diagram of a specific embodiment of the multi-channel processing system of FIG. 1.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

(6) FIG. 1 shows a topology of a multi-channel measurement or communication system 100. The system 100 has a signal combiner 103 for combining one or more input signals S.sub.1 . . . S.sub.N into a composite signal and a signal separator 104 for separating the composite signal into one or more output signals Ŝ.sub.1 . . . Ŝ.sub.M The output signals Ŝ.sub.1 . . . Ŝ.sub.M can include estimates of the input signals S.sub.1 . . . S.sub.N. The input signals S.sub.1 . . . S.sub.N are corrupted by pre-combination distortion 101-102 respectively, and, optionally, by combination distortion in the signal combiner 103. The combiner 103 combines the N input signals into a composite signal (or composite signals). The combiner 401 can combine signals by addition, subtraction, multiplication, division, modulation, non-linear processes, linear processes, estimation, combinations thereof, etc. The composite signal is provided through a communication channel to the separator 104. The composite signal is distorted by communication channel distortion 110. The separator 104 separates the composite signal into M output signals, where M can be less than N, greater than N, or equal to N. In one embodiment, the separator also provides one or more additional output signals {circumflex over (n)}.sub.0 . . . {circumflex over (n)}.sub.K corresponding to estimates of other signals, such as, for example, noise signals, error signals, etc.

(7) Due to errors in the system 100, the output signals Ŝ.sub.1 . . . Ŝ.sub.M are typically not exact copies of the input signals, but rather are estimates of the input signals. The accuracy of these estimates is a measure of system performance. The pre-combination distortion 101-102, the combiner distortion, and/or the channel distortion 110 tend to introduce crosstalk between the channels and thereby corrupt the output signals. The pre-combination distortion 101-102, combiner distortion, and the channel distortion 110 can be caused by variations in manufacturing tolerances, delay, movement, temperature effects, degradation of components due to age or other factors, noise, etc.

(8) A module 105 is provided to configure the separator 104 to improve the quality of the separation function and thereby improve the quality of the output signals. One or more of the output signals from the separator are provided to the module 105 to provide feedback regarding the quality of the output signals and/or feedback regarding the operation of the separator 104. The module 105 uses feedback from one or more of the output signals Ŝ.sub.1 . . . Ŝ.sub.M (and, optionally, the output signals {circumflex over (n)}.sub.0 . . . {circumflex over (n)}.sub.K) to monitor the quality of the separation function and to provide control information to control the operation of the separator. In one embodiment, the module 105 is configured by using configuration data obtained from the combiner 103. Such configuration data can be obtained by calibration procedures that test the operation of the combiner 103 before or during system use.

(9) In one embodiment, the module 105 configures demodulators in the signal separator 104 using, at least in part, calibration data obtained during a calibration period. For example, in one embodiment involving a two channel system, the calibration data includes first and second calibration data corresponding to the first and second output signals during a first time period, and third and fourth calibration data corresponding to the first and second demodulated output signals during a second time period. In one embodiment, the second transmitter is turned off during the first time period, and the first transmitter is turned off during the second time period. In one embodiment, the module 105 configures the first demodulation signal and the second demodulation signal by adjusting initial parameters that define the first demodulation signal and the second demodulation signal. The configuration module adjusts the initial parameters using, at least in part, the calibration data obtained during a calibration period.

(10) FIG. 2 is a block diagram of a two-channel signal processing system 200 that fits the general topology shown in FIG. 1. The system 200 is configured to determine one or more blood constituents in a subject, such as, for example, a human subject. In the example presented, the measurements are performed on a portion of the subject, such as a finger 202 illustrated in FIG. 2A. An LED modulation circuit 204 drives a pair of back-to-back light emitting diodes (LEDs) 206, 208 by applying a periodic signal to the two light emitting diodes 206, 208. Light from the diodes 206, 208 passes through the finger 202 and is detected by a detector 250. An output from the detector 250 is provided to a signal processing block 270. A control output from the signal processing block 270 is provided to the LED modulation circuit 204. The signal processing block 270 also provides outputs Ŝ.sub.1 and Ŝ.sub.2 corresponding to the light detected from the diodes 206, 208, and, optionally, output signals {circumflex over (n)}.sub.0 . . . {circumflex over (n)}.sub.K corresponding to estimates of noise or other signals.

(11) In one embodiment, the LED 206 is selected to emit electromagnetic energy in the red visible light range, and has a wavelength of, for example, approximately 660 nanometers. The LED 208 is selected to emit electromagnetic energy in the infrared range, and has a wavelength of, for example, approximately 905 nanometers. The LED modulation circuit 204 supplies current to activate the LEDs 206 and 208. Each LED is activated for a time period τ which can be different for the different LEDs. The pulses from the LEDs 206 and 208 repeat with a periodicity T.

(12) FIG. 3 is a block diagram of a multi-channel signal processing system 300 that also fits the topology shown in FIG. 1. Like the system 200, the system 300 is configured to determine blood oxygen saturation or other blood constituents in a subject, such as, for example, a human subject. In FIG. 3, an LED modulation circuit 314 drives N diodes, where N is two or greater, thus allowing greater flexibility than the system 200. In FIG. 3, the diodes 206 and 208 are shown, along with an N'th diode 309, it being understood that the diode 309 is omitted if N=2. The LED modulation circuit 314 is configured to allow the diodes 206, 208, and 309 to be driven independently such that the diodes can be driven at separate times or in overlapping time periods if desired. Light from the diodes 206, 208, 309 passes through the finger 202 and is detected by the detector 250. The output from the detector 250 is provided to a signal processing block 371. A control output from the signal processing block 371 is provided to the LED modulation circuit 314. The signal processing block 371 also provides outputs S.sub.1 through S.sub.M, where M is greater than or equal to one, but need not be equal to N.

(13) FIG. 4 shows one embodiment of an adjustable multi-channel modulator/demodulator system 400. The system 400 is based on the topology of the system 100 and can be used in a system for measuring blood constituents (e.g., pulse oximetry, carboxyhemoglobin, etc.) as shown in FIGS. 2 and 3. In the system 400, an input S.sub.1(t) and a modulation input M.sub.1(t) are provided to a first modulator 491. A signal input S.sub.N(t) and a modulation input M.sub.N(t) are provided to an N.sup.th modulator 402.

(14) The photodetector 250 is modeled as an adder 405. The outputs of the modulators 491 and 402 are added together in the adder 405, in the presence of noise n(t) to generate a composite signal M(t) where:
S(t)=S.sub.1(t)M.sub.1(t)+ . . . +S.sub.N(t)M.sub.N(t)+n(t)  (1)

(15) The S(t) signal output of the adder 405 (i.e., the output of the detector 250) is applied to the input of a signal-processing block 410. Within the signal-processing block 410, the signal S(t) is passed through an amplifier 497 and through an analog bandpass filter 498. The analog bandpass filter 498 provides anti-aliasing and removal of low frequency noise and DC. The desired signal components in the signals S.sub.i(t) are frequency shifted by the operation of the modulation signals M.sub.i(t) and are passed by the analog bandpass filter 498.

(16) The output of the analog bandpass filter 498 is sampled by an analog-to-digital converter 499 and converted therein to digital signals and provided to an input of an optional decimation block 420.

(17) The filtered (and, optionally, decimated) signal S(t) is sampled to produce a sampled-data signal S(k) that is provided to the first input of the first mixer 424, to the first input of the N'th mixer 412, and to the first input of a noise channel mixer 413. A first demodulating signal D.sub.1(k) is provided to a second input of a first mixer 424 from a signal generator 431. The N.sup.th demodulating signal D.sub.N(k) is provided to an N.sup.th mixer 412 from an output of a signal generator 432. The noise demodulating signal D.sub.0(k) is provided to the noise channel mixer 413 from an output of a signal generator 441. A control input to each of the signal generators 431, 432, and 441 is provided by the output of the adjuster algorithm 450. In yet another embodiment, the adjuster algorithm 450 may also be controlled by other signal processing elements downstream of the signal processor 400.

(18) The outputs of the mixers 413, 424, and 412 are provided as respective inputs to decimation blocks 440, 430, and 434 respectively. Each of the decimation blocks 440, 430, and 434 has a control input provided by the output of the adjuster algorithm block 450. The output of the decimation block 440 is an estimate of the signal n(t) and it is provided to an input of the adjuster algorithm block 450. In an alternate embodiment, the signal estimates Ŝ.sub.i(k) are also provided to the adjuster algorithm block 450.

(19) An output of the decimator 430 is a signal Ŝ.sub.1(k), which, as discussed above, is an estimate of the signal S.sub.1(k) (where S.sub.1(k) corresponds to a sampled-data representation of S.sub.1(t)). Likewise, the output of the decimation block 434 is an estimate of the signal S.sub.N(t). As shown above, the selection of the demodulating signals D.sub.i(t) for i=0 . . . N in accordance with the present disclosure substantially reduces or eliminates the effects of noise in the output signals Ŝ.sub.i(k) and n(k), and also substantially reduces or eliminates crosstalk between the signals.

(20) When the system 400 is used in connection with a blood constituent measurement system as shown in FIGS. 2 and 3 the red LED 206 provides a light intensity represented as I.sub.RD, and the infrared LED 208 provides a light intensity represented as I.sub.IR. The effects of turning the LEDs 206, 208 on and off on a periodic bases are modeled by the first multiplier or modulator 290 which applies a first modulation signal M.sub.1(t) to the red light intensity to generate a modulated red signal I.sub.RDMOD(t) and by a second multiplier the modulator 292 which applies a second modulation signal M.sub.2(t) to the infrared light intensity to generate a modulated infrared signal I.sub.IRMOD(t). The modulated light red signal and the modulated infrared signal are applied to the finger 202, or other body portion, as described above. The blood in the finger 202 has a volume and scattering components, which vary throughout each cardiac cycle. The blood carries oxygen and other materials therein. The oxygen content is a function of both the blood volume and the concentration of the oxygen in the blood volume. The concentration of the oxygen in the blood volume is generally measured as blood oxygen saturation for reasons which are described in full in U.S. Pat. Nos. 5,482,036 and 5,490,505, both of which are hereby incorporated by reference in their entirety. As further described in the two referenced patents, the blood oxygen saturation is determined by comparing the relative absorption of the red light and the infrared light in the finger 202. The comparison is complicated by the noise caused by movement, ambient light, light scattering, and other factors. The signals S.sub.1(t) and S.sub.2(t) represent the effect of the time-varying volume and scattering components of the blood in the finger 202 on the red light and the infrared light, respectively, passing through the finger 202 from the LEDs 206, 208 to the detector 250.

(21) As shown in FIG. 4, a set of N+1 signals S.sub.i[k] i=1 . . . N, and n(k) are sampled at a desired sample rate. The signals are combined according to the formula:
S(k)=M.sub.1(k)S.sub.1(k)+ . . . +M.sub.N(k)S.sub.N(k)+n(k)  (2)

(22) In one embodiment, each of the decimators 420, 440, 430, and 434 includes a digital lowpass filter and a sample rate compressor. In one embodiment, the characteristics of the digital lowpass filters (e.g., the number of filter coefficients and values of the filter coefficients) and the sample rate compression factor of each decimator are fixed. In one embodiment, the characteristics of the digital lowpass filters (e.g., the number of filter coefficients or values of the filter coefficients) and the sample rate compression factor of each decimator are provided by the adjustment algorithm 450. The signal generators 431, 432 and 441 generate the demodulation sequences for the demodulators 424, 412, and 413 respectively. The demodulation sequences produced by the signal generators 431, 432 and 441 are controlled by the adjuster algorithm 450.

(23) In one embodiment, the adjuster algorithm 450 adjusts the pre-demodulation decimation rate R.sub.1 (in the demodulator 420), and the post-demodulation decimation rate R.sub.2 (in the demodulators 430, 434 and 440) according to the noise in the noise estimate {circumflex over (n)}(k) and (optionally) according to the signals Ŝ.sub.i(k). The product R.sub.1R.sub.2 is the total decimation rate from the signal S(k) at the output of the A/D converter 499 to the signals Ŝ.sub.i(k) at the output of the signal processing block 400. The adjuster algorithm may adjust R.sub.1 and R.sub.2 such that the product R.sub.1R.sub.2 varies, or the adjuster algorithm may adjust R.sub.1 and R.sub.2 such that the product R.sub.1R.sub.2 is substantially constant. Typically, the adjuster algorithm will keep the R.sub.1R.sub.2 product constant so that the signal processing blocks downstream of the signal processor 400 will operate at a substantially constant sample rate.

(24) In one embodiment, the adjuster algorithm 450 adjusts the demodulation signals D.sub.i(k) to reduce or eliminate crosstalk. In one embodiment, the adjuster algorithm 450 reduces crosstalk by configuring the demodulators, as discussed in more detail below.

(25) One skilled in the art will recognize that the lowpass filters provided in connection with the decimation blocks can provide other filter functions in addition to lowpass filtering. Thus, for example, the lowpass filters 420, 430, 440, and 450, and the decimators 420, 430, 434, and 440 can provide other filter functions (in addition to lowpass filtering) such as, for example, bandpass filtering, bandstop filtering, etc. Moreover, the post-demodulation decimation rate R.sub.2 need not be the same for each output channel. Thus, for example, in FIG. 4, the decimator 440 can have a first decimation rate R.sub.2=r.sub.1 while the decimators 430 and 434 have a second decimation rate R.sub.2=r.sub.2.

(26) The demodulators above are described in terms of digital signal processing on sampled data. Thus, the demodulator signals are written D.sub.i(k). The demodulators and the filtering associated with the demodulators can be done in using analog processing (using time-domain demodulator signals D.sub.i(t)) or on sampled data signals (using digital-domain demodulator signals D.sub.i(k)). For convenience, the following development describes the demodulator signals primarily in the time domain, with the understanding that the modulators can be implemented using digital signal processing or analog processing.

(27) The characteristics of the demodulation signals D.sub.1(t) and D.sub.2(t) affect how much crosstalk is seen in the output signals. In an diagonal system, that is, when the demodulator has been diagonalized, there is, ideally, no crosstalk. The first output signal Ŝ.sub.1(t) is an estimate (or approximation) to the signal S.sub.1(t). Similarly, the second output signal Ŝ.sub.2 (t) is an estimate (or approximation) to the signal S.sub.2(t). When the composite signal S(t) is a linear combination of the signals S.sub.i(t), then the relationship between the signals S.sub.1(t) and the signals Ŝ.sub.i(t). When M.sub.1=cos ωt, M.sub.2=sin ωt, n(t)=0, and there is no distortion (e.g., no pre-combination, combiner, or channel distortion) then:

(28) $\begin{array}{cc}S\left(t\right)=S_1\left(t\right)\cos \omega t+S_2\left(t\right)\sin \omega t\mathrm{Then}\text{:}& \left(3\right)\\ {\hat{S}}_1\left(t\right)=\mathrm{LP}\left[D_1\left(t\right)S\left(t\right)\right]& \left(4\right)\\ {\hat{S}}_2\left(t\right)=\mathrm{LP}\left[D_2\left(t\right)S\left(t\right)\right]\mathrm{If}\text{:}& \left(5\right)\\ D_1\left(t\right)=2\cos \omega t& \left(6\right)\\ D_2\left(t\right)=2\sin \omega t\mathrm{then}& \left(7\right)\\ \begin{array}{c}{D}_1\left(t\right)S\left(t\right)=2S_1\left(t\right){\cos }^2\omega t+2S_2\left(t\right)\sin \omega t\cos \omega t\\ =S_1\left(t\right)-S_1\left(t\right)\cos 2\omega t+S_2\left(t\right)\sin 2\omega t\end{array}& \left(8\right)\end{array}$

(29) After lowpass filtering to remove the terms with a frequency of 2ωt and higher
Ŝ.sub.1(t)=S.sub.1(t)  (9)

(30) Similarly for

(31) $\begin{array}{cc}{\hat{S}}_2\left(t\right)=\mathrm{LP}\left[D_2\left(t\right)S\left(t\right)\right]\mathrm{then}& \left(10\right)\\ \begin{array}{c}{D}_2\left(t\right)S\left(t\right)=2S_1\left(t\right)\sin \omega t\cos \omega t+2S_2\left(t\right){\sin }^2\omega t\\ =S_2\left(t\right)-S_2\left(t\right)\cos 2\omega t+S_1\left(t\right)\sin 2\omega t\end{array}& \left(11\right)\end{array}$

(32) After lowpass filtering to remove the terms with a frequency of 2ωt
Ŝ.sub.2(t)=S.sub.2(t)  (12)

(33) In the above analysis, it was assumed that there are no time delays or phase shifts in the signal S(t), and thus, configuration is relatively straightforward

(34) When an unknown delay (or phase error) is introduced, then the signals are no longer diagonal. Consider, for example, the situation when a delay Δ is introduced into the composite signal. Then:
S(t)=cos ω(t−Δ)S.sub.1(t−Δ)+sin ω(t−Δ)S.sub.2(t−Δ)

(35) It then follows that:

(36) $\begin{array}{c}{\hat{S}}_1\left(t\right)=\mathrm{LP}\left[2\cos \omega t\left(\cos \omega \left(t-\Delta \right)S_1\left(t-\Delta \right)+\sin \omega \left(t-\Delta \right)S_2\left(t-\Delta \right)\right)\right]\\ =\mathrm{LP}\left[2\cos \omega t\left(\cos \omega t\cos \mathrm{\omega \Delta }+\sin \omega t\sin \mathrm{\omega \Delta }\right)S_1\left(t-\Delta \right)\right]+\\ \mathrm{LP}\left[2\cos \omega t\left(\sin \omega t\cos \mathrm{\omega \Delta }-\cos \omega t\sin \mathrm{\omega \Delta }\right)S_2\left(t-\Delta \right)\right]\\ =\cos \mathrm{\omega \Delta }{S}_1\left(t-\Delta \right)-\sin \mathrm{\omega \Delta }{S}_2\left(t-\Delta \right)\end{array}$

(37) The above equations can be expressed in matrix form as:

(38) $\begin{array}{cc}\left[\begin{array}{c}{\hat{S}}_1\left(t\right)\\ {\hat{S}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\omega \Delta }& -\sin \mathrm{\omega \Delta }\\ \sin \mathrm{\omega \Delta }& \cos \mathrm{\omega \Delta }\end{array}\right]\left[\begin{array}{c}{S}_1\left(t-\Delta \right)\\ S_2\left(t-\Delta \right)\end{array}\right]\mathrm{Then}& \left(13\right)\\ \left[\begin{array}{c}{S}_1\left(t-\Delta \right)\\ S_2\left(t-\Delta \right)\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\omega \Delta }& \sin \mathrm{\omega \Delta }\\ -\sin \mathrm{\omega \Delta }& \cos \mathrm{\omega \Delta }\end{array}\right]\left[\begin{array}{c}{\hat{S}}_1\left(t\right)\\ {\hat{S}}_2\left(t\right)\end{array}\right]& \left(14\right)\end{array}$

(39) The above equation can be expressed as

(40) $\begin{array}{cc}\begin{array}{c}\left[\begin{array}{c}{S}_1\left(t-\Delta \right)\\ S_2\left(t-\Delta \right)\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\omega \Delta }& \sin \mathrm{\omega \Delta }\\ -\sin \mathrm{\omega \Delta }& \cos \mathrm{\omega \Delta }\end{array}\right].Math.\mathrm{LP}\left[\begin{array}{c}{D}_1\left(t\right)S\left(t\right)\\ D_2\left(t\right)S\left(t\right)\end{array}\right]\\ =\mathrm{LP}\left[\left[\begin{array}{cc}\cos \mathrm{\omega \Delta }& \sin \mathrm{\omega \Delta }\\ -\sin \mathrm{\omega \Delta }& \cos \mathrm{\omega \Delta }\end{array}\right]\left[\begin{array}{c}{D}_1\left(t\right)\\ D_2\left(t\right)\end{array}\right]S\left(t\right)\right]\\ =\mathrm{LP}\left[\left[\begin{array}{c}{\stackrel{\_}{D}}_1\left(t\right)\\ {\stackrel{\_}{D}}_2\left(t\right)\end{array}\right]S\left(t\right)\right]\end{array}\mathrm{where}\begin{array}{c}{\stackrel{\_}{D}}_1\left(t\right)=\cos \mathrm{\omega \Delta }{D}_1\left(t\right)+\sin \mathrm{\omega \Delta }{D}_2\left(t\right)\\ =2\cos \mathrm{\omega \Delta }\cos \omega t+2\sin \mathrm{\omega \Delta }\sin \omega t\end{array}& \left(15\right)\end{array}$

(41) and similarly for D.sub.2(t). Thus the modified demodulation functions D.sub.1(t) and D.sub.2(t) can be expressed as a linear combination of basis functions. If the time delay Δ can be predicted, then the demodulator functions can be calculated and programmed into the communication system. However, in many cases the time delay Δ is not known or changes over time. As described below, the demodulator functions can be determined by system calibration procedures.

(42) When an unknown phase shift (or phase error) is introduced, then there may be crosstalk in the system. Consider, for example, the situation when a phase error φ.sub.1 occurs in the signal S.sub.1(t) and a phase error φ.sub.2 occurs in the signal S.sub.2(t). The phase errors can be caused by intrinsic properties of the components, intrinsic properties of the system, component variations, time delays, etc. In the presence of the phase errors:

(43) $\begin{array}{cc}\begin{array}{c}{D}_1\left(t\right)S\left(t\right)=2A_1{S}_1\left(t\right)\sin \left(\omega t+{\varphi }_1\right)\sin \omega t+2A_2{S}_2\left(t\right)\cos \left(\omega t+{\varphi }_2\right)\sin \omega t\\ =2A_1{S}_1\left(t\right)\sin \omega t\left[\sin \omega t\cos {\varphi }_1+\cos \omega t\sin {\varphi }_1\right]+\\ 2A_2{S}_2\left(t\right)\sin \omega t\left[\cos \omega t\cos {\varphi }_2+\sin \omega t\sin {\varphi }_2\right]\\ =A_1{S}_1\left(t\right)\cos {\varphi }_1-A_1{S}_1\left(t\right)\cos {\varphi }_1\cos 2\omega t+\\ A_1{S}_1\left(t\right)\sin {\varphi }_1\sin 2\omega t+A_2{S}_2\left(t\right)\cos {\varphi }_2\sin 2\omega t-\\ A_2{S}_2\left(t\right)\sin {\varphi }_2+A_2{S}_2\left(t\right)\sin {\varphi }_2\cos 2\omega t\end{array}& \left(16\right)\end{array}$

(44) Thus, after lowpass filtering
Ŝ.sub.1(t)=A.sub.1S.sub.1(t)cos φ.sub.1−A.sub.2S.sub.2(t)sin φ.sub.2  (17)

(45) The above equation shows crosstalk because Ŝ.sub.1(t) depends in part on components of S.sub.2(t) when A.sub.2≠0 and φ.sub.2≠nπ where n=0, ±1, ±2 . . . .

(46) Similarly,
Ŝ.sub.2(t)=A.sub.1S.sub.1(t)sin φ.sub.1+A.sub.2S.sub.2(t)cos φ.sub.2  (18)

(47) The above equations can be expressed in matrix form as:

(48) $\begin{array}{cc}\left[\begin{array}{c}{\hat{S}}_1\left(t\right)\\ {\hat{S}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}{A}_1\cos {\varphi }_1& -A_2\sin {\varphi }_2\\ A_1\sin {\varphi }_1& A_2\cos {\varphi }_2\end{array}\right]\left[\begin{array}{c}{S}_1\left(t\right)\\ S_2\left(t\right)\end{array}\right]& \left(19\right)\end{array}$

(49) After inversion

(50) $\begin{array}{cc}\left[\begin{array}{c}{S}_1\left(t\right)\\ S_2\left(t\right)\end{array}\right]=\frac{1}{\cos \left({\varphi }_1-{\varphi }_2\right)}\left[\begin{array}{cc}\frac{\cos {\varphi }_2}{A_1}& \frac{\sin {\varphi }_1}{A_1}\\ -\frac{\sin {\varphi }_2}{A_2}& \frac{\cos {\varphi }_1}{A_2}\end{array}\right]\left[\begin{array}{c}{\hat{S}}_1\left(t\right)\\ {\hat{S}}_2\left(t\right)\end{array}\right]& \left(20\right)\\ \left[\begin{array}{c}{S}_1\left(t\right)\\ S_2\left(t\right)\end{array}\right]=\frac{1}{\cos \left({\varphi }_1-{\varphi }_2\right)}\left[\begin{array}{cc}\frac{\cos {\varphi }_2}{A_1}& \frac{\sin {\varphi }_1}{A_1}\\ -\frac{\sin {\varphi }_2}{A_2}& \frac{\cos {\varphi }_1}{A_2}\end{array}\right]\left[\begin{array}{c}{D}_1\left(t\right)\\ D_2\left(t\right)\end{array}\right]S\left(t\right)\mathrm{Then}& \left(21\right)\\ \left[\begin{array}{c}{S}_1\left(t\right)\\ S_2\left(t\right)\end{array}\right]=\left[\begin{array}{c}{\stackrel{\_}{D}}_1\left(t\right)\\ {\stackrel{\_}{D}}_2\left(t\right)\end{array}\right]S\left(t\right)& \left(22\right)\end{array}$

(51) Where D.sub.i(t) are modified demodulation functions, given by:

(52) $\begin{array}{cc}\left[\begin{array}{c}{\stackrel{\_}{D}}_1\left(t\right)\\ {\stackrel{\_}{D}}_2\left(t\right)\end{array}\right]=\frac{1}{\cos \left({\varphi }_1-{\varphi }_2\right)}\left[\begin{array}{c}\frac{\cos {\varphi }_2\sin \omega t+\sin {\varphi }_1\cos \omega t}{A_1}\\ \frac{-\sin {\varphi }_2\sin \omega t+\cos {\varphi }_1\cos \omega t}{A_s}\end{array}\right]& \left(23\right)\end{array}$

(53) The modified demodulation functions have the form

(54) 0$\begin{array}{cc}{\stackrel{\_}{D}}_i\left(t\right)=\underset{j=1}{\overset{N}{.Math.}}{\alpha }_j{\Phi }_j\left(t\right)& \left(24\right)\end{array}$

(55) Choosing the coefficients α.sub.j to eliminate crosstalk configures the modulator.

(56) In one embodiment, the coefficients α.sub.j can be fixed coefficients computed using known properties of the system. However, as system properties change over time, such fixed coefficients may lead to unacceptably high levels of crosstalk. Moreover, variations from device to device may cause the fixed coefficients to give unacceptably high levels of crosstalk.

(57) Higher performance (that is, lower crosstalk) can be obtained by computing the coefficients as part of a system calibration or initialization procedure. Such calibration can be performed during system startup (e.g., when the system is turned on, or when the system begins processing data, etc.) and/or at regular intervals. In one embodiment, the adjuster algorithm 450 computes the coefficients α.sub.j from the calibration data. In one embodiment, the coefficients α.sub.j are chosen by making four calibration-type measurements to measure four parameters ξ.sub.11, ξ.sub.12, ξ.sub.21, and ξ.sub.22, where:
ξ.sub.11=Ŝ.sub.1|.sub.S.sub.1.sub.=A,S.sub.2.sub.=0=Aa.sub.1 cos φ.sub.1  (25)
ξ.sub.12=Ŝ.sub.1|.sub.S.sub.1.sub.=0,S.sub.2.sub.=B=−Ba.sub.2 sin φ.sub.2  (26)
ξ.sub.21=Ŝ.sub.2|.sub.S.sub.1.sub.=A,S.sub.2.sub.=0=Aa.sub.1 sin φ.sub.1  (27)
ξ.sub.22=Ŝ.sub.2|.sub.S.sub.1.sub.=0,S.sub.2.sub.=B=Ba.sub.2 cos φ.sub.2  (28)

(58) where A and B are amplitudes. Then

(59) $\begin{array}{cc}{a}_1=\frac{\sqrt{{\xi }_{11}^2+{\xi }_{21}^2}}{A}& \left(29\right)\\ a_2=\frac{\sqrt{{\xi }_{12}^2+{\xi }_{22}^2}}{B}& \left(30\right)\\ \tan {\varphi }_1=\frac{{\xi }_{21}}{{\xi }_{11}}& \left(31\right)\\ \tan {\varphi }_2=-\frac{{\xi }_{12}}{{\xi }_{22}}& \left(32\right)\end{array}$

(60) From the above equations, it is evident that crosstalk depends only on φ.sub.1 and φ.sub.2. Moreover, φ.sub.1 and φ.sub.2 can be chosen to eliminate crosstalk without knowing A or B. This is useful for systems such as pulse oximetry systems where absolute measurements of a channel are difficult or impractical, but where relative measurements (e.g., channel-to-channel measurements) are practical.

(61) The demodulation signals D.sub.i(t) can be generated using the values of φ.sub.1 and φ.sub.2 from the above equations. Alternatively, the demodulation signals D.sub.i(t) can be generated from quadrature components as:
D.sub.1(t)=b.sub.11 sin ωt+b.sub.12 cos ωt  (33)
D.sub.2(t)=b.sub.21 sin ωt+b.sub.22 cos ωt  (34)

(62) where the coefficients b.sub.ij are computed from φ.sub.1 and φ.sub.2.

(63) In one embodiment, the demodulation functions are adapted from baseline coefficients, which are then improved through a calibration or initialization procedure to produce actual coefficients. The baseline coefficients are typically obtained from known properties of the system. The actual coefficients are usually relatively close in value to the baseline coefficients. This provides one way to assess the operational status of the system and to evaluate the calibration procedure. In one embodiment, if the actual coefficients are too different from the baseline parameters then it is assumed that the calibration procedure failed in some manner or that the equipment has failed in some manner, and appropriate measures can be taken (e.g., alert the operator, sound a warning, etc.)

(64) To find the actual coefficients, the demodulation functions are initially given by:
D.sub.1(t)=α.sub.11 sin ωt+α.sub.12 cos ωt  (35)
D.sub.2(t)=α.sub.21 sin ωt+α.sub.22 cos ωt  (36)

(65) Where the coefficients α.sub.ij are the baseline coefficients determined from known or assumed properties of the signal S(t). For example, in one embodiment α.sub.ij=δ.sub.ij. In one embodiment, where initial estimates are available for φ.sub.1 and φ.sub.2, then the values of α.sub.ij can be computed as discussed above.

(66) The crosstalk reduction obtained using demodulation functions based on the coefficients α.sub.ij can often be improved by computing new coefficients α.sub.ij and corresponding new demodulation functions D.sub.i(t) where:
D.sub.1(t)=α.sub.11 sin ωt+α.sub.12 cos ωt  (37)
D.sub.2(t)=α.sub.21 sin ωt+α.sub.22 cos ωt  (38)

(67) The process of finding the coefficients α.sub.ij begins by measuring two data sets, x.sub.1(t) and x.sub.2(t), as follows:
x.sub.1(t)=S(t)|.sub.S.sub.1.sub.=A,S.sub.2.sub.=0  (39)
x.sub.2(t)=S(t)|.sub.S.sub.1.sub.=0,S.sub.2.sub.=B  (40)

(68) The data sets x.sub.1(t) and x.sub.2(t) are used to enforce the following constraint:

(69) $\begin{array}{cc}{\int }_0^{\mathrm{nT}}{x}_i\left(t\right){\stackrel{\_}{D}}_i\left(t\right)d=0& \left(41\right)\end{array}$

(70) where i=1, 2, n=1, 2, 3 . . . , and T is a time period corresponding to one complete modulation cycle. From the above constraint and the definitions of the demodulation functions, it follows that:

(71) $\begin{array}{cc}{\stackrel{\_}{\alpha }}_{11}{\int }_0^{\mathrm{nT}}{x}_1\left(t\right)\sin \omega \mathrm{tdt}+{\stackrel{\_}{\alpha }}_{12}{\int }_0^{\mathrm{nT}}{x}_1\left(t\right)\cos \omega \mathrm{tdt}=0& \left(42\right)\\ {\stackrel{\_}{\alpha }}_{21}{\int }_0^{\mathrm{nT}}{x}_2\left(t\right)\sin \omega \mathrm{tdt}+{\stackrel{\_}{\alpha }}_{22}{\int }_0^{\mathrm{nT}}{x}_2\left(t\right)\cos \omega \mathrm{tdt}=0& \left(43\right)\end{array}$

(72) It is convenient to define

(73) $\begin{array}{cc}{\gamma }_{11}={\int }_0^{\mathrm{nT}}{x}_1\left(t\right)\sin \omega \mathrm{tdt}& \left(44\right)\\ {\gamma }_{12}={\int }_0^{\mathrm{nT}}{x}_1\left(t\right)\cos \omega \mathrm{tdt}& \left(45\right)\\ {\gamma }_{21}={\int }_0^{\mathrm{nT}}{x}_2\left(t\right)\sin \omega \mathrm{tdt}& \left(46\right)\\ {\gamma }_{22}={\int }_0^{\mathrm{nT}}{x}_2\left(t\right)\cos \omega \mathrm{tdt}& \left(47\right)\end{array}$

(74) and to define

(75) $\begin{array}{cc}{\beta }_{\mathrm{ij}}=\frac{{\gamma }_{\mathrm{ij}}}{{\left[\underset{k}{.Math.}{\gamma }_{\mathrm{ik}}^2\right]}^{1/2}}& \left(48\right)\\ \mathrm{where}& \\ \underset{k}{.Math.}{\beta }_{\mathrm{ik}}^2=1& \left(49\right)\end{array}$

(76) Then
α.sub.11β.sub.11+α.sub.12β.sub.12=0  (50)
α.sub.21β.sub.21+α.sub.22β.sub.22=0  (51)

(77) In one embodiment, to reduce crosstalk, it is desired to find the coefficients α.sub.ij closest (in the sense of minimizing some specified error, such as, for example, a least squared error) to the coefficients α.sub.ij such that the above constraints are satisfied. One solution, obtained by minimizing the least squared error is:

(78) $\begin{array}{cc}{\stackrel{\_}{\alpha }}_{\mathrm{ij}}={\alpha }_{\mathrm{ij}}-\left(\underset{k}{.Math.}{\alpha }_{\mathrm{ik}}{\beta }_{\mathrm{ik}}\right){\beta }_{\mathrm{ij}}& \left(52\right)\end{array}$

(79) The term in parentheses can be described as the baseline crosstalk.

(80) One of ordinary skill in the art will recognize that optimization methods other than least squares can be used. The solution methods for configuration are, for simplicity, described above in terms of a two-channel system. Using the above teachings, the extension to multi-channel systems is straightforward.

(81) Although described above in connection with a particular embodiment of the present disclosure, it should be understood the description of the embodiment is illustrative of the disclosure and are not intended to be limiting. Although described above in connection with a pulse oximetry system wherein a parameter to be measured is the attenuation of red and infrared light passing through a portion of a subject's body, it should be understood that the method and apparatus described herein can also be used for other measurements where two or more signals are passed through a system to be analyzed. In particular, the present disclosure can be used to demodulate two combined parametric signals responsive to the system to be analyzed where the two parametric signals have a predetermined timing relationship between them, as described herein. The disclosure can be used in connection with various physiological parameter measurement systems, such as, for example, systems that measure blood constituents, blood oxygen carboxyhemoglobin, methemoglobin, glucose, etc. Various modifications and applications may occur to those skilled in the art without departing from the true spirit and scope of the invention as defined in the appended claims.