Signal processing apparatus

[0001] This application is a continuation of application Ser. No. 09/195,791, filed Nov. 17, 1998, now U.S. Pat. No. 7,328,053 which is a continuation of application Ser. No. 08/859,837, filed May 16, 1997 (now U.S. Pat. No. 6,157,850), which is a continuation of application Ser. No. 08/320,154, filed Oct. 7, 1994 (now U.S. Pat. No. 5,632,272 issued May 27, 1997), which is a continuation-in-part of application Ser. No. 08/132,812, filed Oct. 6, 1993 (now U.S. Pat. No. 5,490,505).

[0002] 1. Field of the Invention

[0003] The present invention relates to the field of signal processing. More specifically, the present invention relates to the processing of measured signals, containing a primary signal portion and a secondary signal portion, for the removal or derivation of either the primary or secondary signal portion when little is known about either of these components. More particularly, the present invention relates to modeling the measured signals in a novel way which facilitates minimizing the correlation between the primary signal portion and the secondary signal portion in order to produce a primary and/or secondary signal. The present invention is especially useful for physiological monitoring systems including blood oxygen saturation systems.

[0004] 2. Description of the Related Art

[0005] Signal processors are typically employed to remove or derive either the primary or secondary signal portion from a composite measured signal including a primary signal portion and a secondary signal portion. For example, a composite signal may contain noise and desirable portions. If the secondary signal portion occupies a different frequency spectrum than the primary signal portion, then conventional filtering techniques such as low pass, band pass, and high pass filtering are available to remove or derive either the primary or the secondary signal portion from the total signal. Fixed single or multiple notch filters could also be employed if the primary and/or secondary signal portion(s) exist at a fixed frequency(s).

[0006] It is often the case that an overlap in frequency spectrum between the primary and secondary signal portions exists. Complicating matters further, the statistical properties of one or both of the primary and secondary signal portions change with time. In such cases, conventional filtering techniques are ineffective in extracting either the primary or secondary signal. If, however, a description of either the primary or secondary signal portion can be derived, correlation canceling, such as adaptive noise canceling, can be employed to remove either the primary or secondary signal portion of the signal isolating the other portion. In other words, given sufficient information about one of the signal portions, that signal portion can be extracted.

[0007] Conventional correlation cancelers, such as adaptive noise cancelers, dynamically change their transfer function to adapt to and remove portions of a composite signal. However, correlation cancelers require either a secondary reference or a primary reference which correlates to either the secondary signal portion only or the primary signal portion only. For instance, for a measured signal containing noise and desirable signal, the noise can be removed with a correlation canceler if a noise reference is available. This is often the case. Although the amplitude of the reference signals are not necessarily the same as the amplitude of the corresponding primary or secondary signal portions, they have a frequency spectrum which is similar to that of the primary or secondary signal portions.

[0008] In many cases, nothing or very little is known about the secondary and/or primary signal portions. One area where measured signals comprising a primary signal portion and a secondary signal portion about which no information can easily be determined is physiological monitoring. Physiological monitoring generally involves measured signals derived from a physiological system, such as the human body. Measurements which are typically taken with physiological monitoring systems include electrocardiographs, blood pressure, blood gas saturation (such as oxygen saturation), capnographs, other blood constituent monitoring, heart rate, respiration rate, electro-encephalograph (EEG) and depth of anesthesia, for example. Other types of measurements include those which measure the pressure and quantity of a substance within the body such as cardiac output, venous oxygen saturation, arterial oxygen saturation, bilirubin, total hemoglobin, breathalyzer testing, drug testing, cholesterol testing, glucose testing, extra vasation, and carbon dioxide testing, protein testing, carbon monoxide testing, and other in-vivo measurements, for example. Complications arising in these measurements are often due to motion of the patient, both external and internal (muscle movement, vessel movement, and probe movement, for example), during the measurement process.

[0009] Many types of physiological measurements can be made by using the known properties of energy attenuation as a selected form of energy passes through a medium.

[0010] A blood gas monitor is one example of a physiological monitoring system which is based upon the measurement of energy attenuated by biological tissues or substances. Blood gas monitors transmit light into the test medium and measure the attenuation of the light as a function of time. The output signal of a blood gas monitor which is sensitive to the arterial blood flow contains a component which is a waveform representative of the patient's arterial pulse. This type of signal, which contains a component related to the patient's pulse, is called a plethysmographic wave, and is shown in FIG. 1 as curve s. Plethysmographic waveforms are used in blood gas saturation measurements. As the heart beats, the amount of blood in the arteries increases and decreases, causing increases and decreases in energy attenuation, illustrated by the cyclic wave s in FIG. 1.

[0011] Typically, a digit such as a finger, an ear lobe, or other portion of the body where blood flows close to the skin, is employed as the medium through which light energy is transmitted for blood gas attenuation measurements. The finger comprises skin, fat, bone, muscle, etc., shown schematically in FIG. 2, each of which attenuates energy incident on the finger in a generally predictable and constant manner. However, when fleshy portions of the finger are compressed erratically, for example by motion of the finger, energy attenuation becomes erratic.

[0012] An example of a more realistic measured waveform S is shown in FIG. 3, illustrating the effect of motion. The primary plethysmographic waveform portion of the signal s is the waveform representative of the pulse, corresponding to the sawtooth-like pattern wave in FIG. 1. The large, secondary motion-induced excursions in signal amplitude obscure the primary plethysmographic signal s. Even small variations in amplitude make it difficult to distinguish the primary signal component s in the presence of a secondary signal component n.

[0013] A pulse oximeter is a type of blood gas monitor which non-invasively measures the arterial saturation of oxygen in the blood. The pumping of the heart forces freshly oxygenated blood into the arteries causing greater energy attenuation. As well understood in the art, the arterial saturation of oxygenated blood may be determined from the depth of the valleys relative to the peaks of two plethysmographic waveforms measured at separate wavelengths. Patient movement introduces motion artifacts to the composite signal as illustrated in the plethysmographic waveform illustrated in FIG. 3. These motion artifacts distort the measured signal.

[0014] This invention provides improvements upon the methods and apparatus disclosed in U.S. patent application Ser. No. 08/132,812, filed Oct. 6, 1993, entitled Signal Processing Apparatus, which earlier application has been assigned to the assignee of the instant application. The present invention involves several different embodiments using the novel signal model in accordance with the present invention to isolate either a primary signal portion or a secondary signal portion of a composite measured signal. In one embodiment, a signal processor acquires a first measured signal and a second measured signal that is correlated to the first measured signal. The first signal comprises a first primary signal portion and a first secondary signal portion. The second signal comprises a second primary signal portion and a second secondary signal portion. The signals may be acquired by propagating energy through a medium and measuring an attenuated signal after transmission or reflection. Alternatively, the signals may be acquired by measuring energy generated by the medium.

[0015] In one embodiment, the first and second measured signals are processed to generate a secondary reference which does not contain the primary signal portions from either of the first or second measured signals. This secondary reference is correlated to the secondary signal portion of each of the first and second measured signals. The secondary reference is used to remove the secondary portion of each of the first and second measured signals via a correlation canceler, such as an adaptive noise canceler. The correlation canceler is a device which takes a first and second input and removes from the first input all signal components which are correlated to the second input. Any unit which performs or nearly performs this function is herein considered to be a correlation canceler.

[0016] An adaptive correlation canceler can be described by analogy to a dynamic multiple notch filter which dynamically changes its transfer function in response to a reference signal and the measured signals to remove frequencies from the measured signals that are also present in the reference signal. Thus, a typical adaptive correlation canceler receives the signal from which it is desired to remove a component and receives a reference signal of the undesired portion. The output of the correlation canceler is a good approximation to the desired signal with the undesired component removed.

[0017] Alternatively, the first and second measured signals may be processed to generate a primary reference which does not contain the secondary signal portions from either of the first or second measured signals. The primary reference may then be used to remove the primary portion of each of the first and second measured signals via a correlation canceler. The output of the correlation canceler is a good approximation to the secondary signal with the primary signal removed and may be used for subsequent processing in the same instrument or an auxiliary instrument. In this capacity, the approximation to the secondary signal may be used as a reference signal for input to a second correlation canceler together with either the first or second measured signals for computation of, respectively, either the first or second primary signal portions.

[0018] Physiological monitors can benefit from signal processors of the present invention. Often in physiological measurements a first signal comprising a first primary portion and a first secondary portion and a second signal comprising a second primary portion and a second secondary portion are acquired. The signals may be acquired by propagating energy through a patient's body (or a material which is derived from the body, such as breath, blood, or tissue, for example) or inside a vessel and measuring an attenuated signal after transmission or reflection. Alternatively, the signal may be acquired by measuring energy generated by a patient's body, such as in electrocardiography. The signals are processed via the signal processor of the present invention to acquire either a secondary reference or a primary reference which is input to a correlation canceler, such as an adaptive noise canceler.

[0019] One physiological monitoring apparatus which benefits from the present invention is a monitoring system which determines a signal which is representative of the arterial pulse, called a plethysmographic wave. This signal can be used in blood pressure calculations, blood constituent measurements, etc. A specific example of such a use is in pulse oximetry. Pulse oximetry involves determining the saturation of oxygen in the blood. In this configuration, the primary portion of the signal is the arterial blood contribution to attenuation of energy as it passes through a portion of the body where blood flows close to the skin. The pumping of the heart causes blood flow to increase and decrease in the arteries in a periodic fashion, causing periodic attenuation wherein the periodic waveform is the plethysmographic waveform representative of the arterial pulse. The secondary portion is noise. In accordance with the present invention, the measured signals are modeled such that this secondary portion of the signal is related to the venous blood contribution to attenuation of energy as it passes through the body. The secondary portion also includes artifacts due to patient movement which causes the venous blood to flow in an unpredictable manner, causing unpredictable attenuation and corrupting the otherwise periodic plethysmographic waveform. Respiration also causes the secondary or noise portion to vary, although typically at a lower frequency than the patients pulse rate. Accordingly, the measured signal which forms a plethysmographic waveform is modeled in accordance with the present invention such that the primary portion of the signal is representative of arterial blood contribution to attenuation and the secondary portion is due to several other parameters.

[0020] A physiological monitor particularly adapted to pulse oximetry oxygen saturation measurement comprises two light emitting diodes (LED's) which emit light at different wavelengths to produce first and second signals. A detector registers the attenuation of the two different energy signals after each passes through an absorptive media, for example a digit such as a finger, or an earlobe. The attenuated signals generally comprise both primary (arterial attenuator) and secondary (noise) signal portions. A static filtering system, such as a bandpass filter, removes a portion of the secondary signal which is outside of a known bandwidth of interest, leaving an erratic or random secondary signal portion, often caused by motion and often difficult to remove, along with the primary signal portion.

[0021] A processor in accordance with one embodiment of the present invention removes the primary signal portions from the measured signals yielding a secondary reference which is a combination of the remaining secondary signal portions. The secondary reference is correlated to both of the secondary signal portions. The secondary reference and at least one of the measured signals are input to a correlation canceler, such as an adaptive noise canceler, which removes the random or erratic portion of the secondary signal. This yields a good approximation to a primary plethysmographic signal as measured at one of the measured signal wavelengths. As is known in the art, quantitative measurements of the amount of oxygenated arterial blood in the body can be determined from the plethysmographic signal in a variety of ways.

[0022] The processor of the present invention may also remove the secondary signal portions from the measured signals yielding a primary reference which is a combination of the remaining primary signal portions. The primary reference is correlated to both of the primary signal portions. The primary reference and at least one of the measured signals are input to a correlation canceler which removes the primary portions of the measured signals. This yields a good approximation to the secondary signal at one of the measured signal wavelengths. This signal may be useful for removing secondary signals from an auxiliary instrument as well as determining venous blood oxygen saturation.

[0023] In accordance with the signal model of the present invention, the two measured signals each having primary and secondary signal portions can be related by coefficients. By relating the two equations with respect to coefficients defined in accordance with the present invention, the coefficients provide information about the arterial oxygen saturation and about the noise (the venous oxygen saturation and other parameters). In accordance with this aspect of the present invention, the coefficients can be determined by minimizing the correlation between the primary and secondary signal portions as defined in the model. Accordingly, the signal model of the present invention can be utilized in many ways in order to obtain information about the measured signals as will be further apparent in the detailed description of the preferred embodiments.

[0024] One aspect of the present invention is a method for use in a signal processor in a signal processor for processing at least two measured signals S₁ and S₂ each containing a primary signal portion s and a secondary signal portion n, the signals S₁ and S₂ being in accordance with the following relationship:
S₁=s₁+n₁
S₂=s₂+n₂

[0025] where s₁ and s₂, and n₁ and n₂ are related by:
s₁=r_as₂ and n₁=r_vn₂

[0026] and where r_a and r_v are coefficients.

[0027] The method comprises a number of steps. A value of coefficient r_a is determined which minimize correlation between s₁ and n₁. Then, at least one of the first and second signals is processed using the determined value for r_a to significantly reduce n from at least one of the first or second measured signal to form a clean signal.

[0028] In one embodiment, the clean signal is displayed on a display. In another embodiment, wherein the first and second signals are physiological signals, the method further comprises the step of processing the clean signal to determine a physiological parameter from the first or second measured signals. In one embodiment, the parameter is arterial oxygen saturation. In another embodiment, the parameter is an ECG signal. In yet another embodiment, wherein the first portion of the measured signals is indicative of a heart plethysmograph, the method further comprises the step of calculating the pulse rate.

[0029] Another aspect of the present invention involves a physiological monitor. The monitor has a first input configured to receive a first measured signal S₁ having a primary portion, s₁, and a secondary portion n₁. The monitor also has a second input configured to received a second measured signal S₂ having a primary portion s₂ and a secondary portion n₂. Advantageously, the first and the second measured signals S₁ and S₂ are in accordance with the following relationship:
S₁=s₁+n₁
S₂=s₂+n₂

[0030] where s₁ and s₂, and n₁ and n₂ are related by:
S₁=r_as₂ and n₁=r_vn₂

[0031] and where r_a and r_v are coefficients.

[0032] The monitor further has a scan reference processor, the scan reference processor responds to a plurality of possible values for r_a to multiply the second measured signal by each of the possible values for r_a and for each of the resulting values, to subtract the resulting values from the first measured signal to provide a plurality of output signals. A correlation canceler having a first input configured to receive the first measured signal, and having a second input configured to receive the plurality of output signals from the saturation scan reference processor, provides a plurality of output vectors corresponding to the correlation cancellation between the plurality of output signals and the first measured signal. An integrator having an input configured to receive the plurality of output vectors from the correlation canceler is responsive to the plurality of output vectors to determine a corresponding power for each output vector. An extremum detector is coupled at its input to the output of the integrator. The extremum detector is responsive to the corresponding power for each output vector to detect a selected power.

[0033] In one embodiment, the plurality of possible values correspond to a plurality of possible values for a selected blood constituent. In one embodiment the, the selected blood constituent is arterial blood oxygen saturation. In another embodiment, the selected blood constituent is venous blood oxygen saturation. In yet another embodiment, the selected blood constituent is carbon monoxide.

[0034] Another aspect of the present invention involves a physiological monitor. The monitor has a first input configured to receive a first measured signal S₁ having a primary portion, s₁, and a secondary portion, n₁. The monitor also has a second input configured to received a second measured signal S₂ having a primary portion S₂ and a secondary portion n₂. The first and the second measured signals S₁ and S₂ are in accordance with the following relationship:
S₁=s₁+n₁
S₂=s₂+n₂

[0035] where s₁ and s₂, and n₁ and n₂ are related by:
S₁=r_as₂ and n₁=r_vn_s

[0036] and where r_a and r_v are coefficients.

[0037] A transform module is responsive to the first and the second measured signals and responsive to a plurality of possible values for r_a to provide at least one power curve as an output. An extremum calculation module is responsive to the at least one power curve to select a value for r_a which minimizes the correlation between s and n, and to calculate from the value for r_a a corresponding saturation value as an output. A display module is responsive to the output of saturation calculation to display the saturation value.

[0081] The present invention involves a system which utilizes first and second measured signals that each contain a primary signal portion and a secondary signal portion. In other words, given a first and second composite signals S₁(t)=s₁(t) +n₁(t) and S₂(t)=s₂(t)+n₂(t), the system of the present invention can be used to isolate either the primary signal portion s(t) or the secondary signal portion n(t). Following processing, the output of the system provides a good approximation n″(t) to the secondary signal portion n(t) or a good approximation s″(t) to the primary signal portion s(t).

[0082] The system of the present invention is particularly useful where the primary and/or secondary signal portion n(t) may contain one or more of a constant portion, a predictable portion, an erratic portion, a random portion, etc. The primary signal approximation s″(t) or secondary signal approximation n″(t) is derived by removing as many of the secondary signal portions n(t) or primary signal portions s(t) from the composite signal S(t) as possible. The remaining signal forms either the primary signal approximation s″(t) or secondary signal approximation n″(t), respectively. The constant portion and predictable portion of the secondary signal n(t) are easily removed with traditional filtering techniques, such as simple subtraction, low pass, band pass, and high pass filtering. The erratic portion is more difficult to remove due to its unpredictable nature. If something is known about the erratic signal, even statistically, it could be removed, at least partially, from the measured signal via traditional filtering techniques. However, often no information is known about the erratic portion of the secondary signal n(t). In this case, traditional filtering techniques are usually insufficient.

[0083] In order to remove the secondary signal n(t), a signal model in accordance with the present invention is defined as follows for the first and second measured signals S₁ and S₂:

[0084] $\begin{array}{c}{S}_1=s_1+n_1\\ S_2=s_2+n_2\\ \mathrm{with}\\ s_1=r_a{s}_2\mathrm{and}{n}_1=r_v{n}_2\\ \mathrm{or}\\ r_a=\frac{s_1}{s_2}\mathrm{and}{n}_1=\frac{n_1}{n_2}\end{array}$

[0085] where s₁ and n₁ are at least somewhat (preferably substantially) uncorrelated and S₂ and n₂ are at least somewhat (preferably substantially) uncorrelated. The first and second measured signals S₁ and S₂ are related by correlation coefficients r_a and r_v as defined above. The use and selection of these coefficients is described in further detail below.

[0086] In accordance with one aspect of the present invention, this signal model is used in combination with a correlation canceler, such as an adaptive noise canceler, to remove or derive the erratic portion of the measured signals.

[0087] Generally, a correlation canceler has two signal inputs and one output. One of the inputs is either the secondary reference n′(t) or the primary reference s′(t) which are correlated, respectively, to the secondary signal portions n(t) and the primary signal portions s(t) present in the composite signal S(t). The other input is for the composite signal S(t). Ideally, the output of the correlation canceler s″(t) or n″(t) corresponds, respectively, to the primary signal s(t) or the secondary signal n(t) portions only. Often, the most difficult task in the application of correlation cancelers is determining the reference signals n′(t) and s′(t) which are correlated to the secondary n(t) and primary s(t) portions, respectively, of the measured signal S(t) since, as discussed above, these portions are quite difficult to isolate from the measured signal S(t). In the signal processor of the present invention, either a secondary reference n′(t) or a primary reference s′(t) is determined from two composite signals measured simultaneously, or nearly simultaneously, at two different wavelengths, λa and λb.

[0088] A block diagram of a generic monitor incorporating a signal processor according to the present invention, and a correlation canceler is shown in FIGS. 4a and 4b. Two measured signals, S_λa(t) and S_λb(t), are acquired by a detector 20. One skilled in the art will realize that for some physiological measurements, more than one detector may be advantageous. Each signal is conditioned by a signal conditioner 22a and 22b. Conditioning includes, but is not limited to, such procedures as filtering the signals to remove constant portions and amplifying the signals for ease of manipulation. The signals are then converted to digital data by an analog-to-digital converter 24a and 24b. The first measured signal Sa(t) comprises a first primary signal portion, labeled herein s_λa(t), and a first secondary signal portion, labeled herein n_λa(t). The second measured signal S_λb(t) is at least partially correlated to the first measured signal S_λa(t) and comprises a second primary signal portion, labeled herein s_λb(t), and a second secondary signal portion, labeled herein n_λb(t). Typically the first and second secondary signal portions, n_λa(t) and n_λb(t), are uncorrelated and/or erratic with respect to the primary signal portions s_λa(t) and s_λb(t). The secondary signal portions n_λa(t) and n_λb(t) are often caused by motion of a patient in physiological measurements.

[0089] The signals S_λa(t) and S_λb(t) are input to a reference processor 26. The reference processor 26 multiplies the second measured signal S_λb(t) by either a factor r_a=s_λa(t)/s_λb(t) or a factor r_v=n_λa(t)/n_λb(t) and then subtracts the second measured signal S_λb(t) from the first measured signal S_λa(t). The signal coefficient factors r_a and r_v are determined to cause either the primary signal portions s_λa(t) and s_λb(t) or the secondary signal portions n_λa(t) and n_λb(t) to cancel, respectively, when the two signals S_λa(t) and S_λb(t) are subtracted. Thus, the output of the reference processor 26 is either a secondary reference signal n′(t)=n_λa(t)−r_an_λb(t), in FIG. 4a, which is correlated to both of the secondary signal portions n_λa(t) and n_λb(t) or a primary reference signal s′(t)=s_λa(t)−r_vs_λb(t), in FIG. 4b, which is correlated to both of the primary signal portions s_λa(t) and s_λb(t). A reference signal n′(t) or s′(t) is input, along with one of the measured signals S_λa(t) or S_λb(t), to a correlation canceler 27 which uses the reference signal n′(t) or s′(t) to remove either the secondary signal portions n_λa(t) or n_λb(t) or the primary signal portions s_λa(t) or s_λb(t) from the measured signal S_λa(t) or S_λb(t). The output of the correlation canceler 27 is a good primary signal approximation s″(t) or secondary signal approximation n″(t). In one embodiment, the approximation s″(t) or n″(t) is displayed on a display 28.

[0090] In one embodiment, an adaptive noise canceler 30, an example of which is shown in block diagram form in FIG. 5a, is employed as the correlation canceler 27, to remove either one of the erratic, secondary signal portions n_λa(t) and n_λb(t) from the first and second signals S_λa(t) and S_λb(t). The adaptive noise canceler 30 in FIG. 5a has as one input a sample of the secondary reference n′(t) which is correlated to the secondary signal portions n_λa(t) and n_λb(t). The secondary reference n′(t) is determined from the two measured signals S_λa(t) and S_λb(t) by the processor 26 of the present invention as described herein. A second input to the adaptive noise canceler, is a sample of either the first or second composite measured signals S_λa(t)=S_λa(t)+n_λa(t) or S_λb(t)=s_λb(t)+n_λb(t).

[0091] The adaptive noise canceler 30, in FIG. 5b, may also be employed to remove either one of primary signal portions s_λa(t) and s_λb(t) from the first and second measured signals S_λa(t) and S_λb(t). The adaptive noise canceler 30 has as one input a sample of the primary reference s′(t) which is correlated to the primary signal portions s_λa(t) and s_λb(t). The primary reference s′(t) is determined from the two measured signals S_λa(t) and S_λb(t) by the processor 26 of the present invention as described herein. A second input to the adaptive noise canceler 30 is a sample of either the first or second measured signals S_λa(t)=s_λa(t)+n_λa(t) or S_λb(t)=s_λb(t)+n_λb(t).

[0092] The adaptive noise canceler 30 functions to remove frequencies common to both the reference n′(t) or s′(t) and the measured signal S_λa(t) or S_λb(t). Since the reference signals are correlated to either the secondary signal portions n_λa(t) and n_λb(t) or the primary signal portions s_λa(t) and s_λb(t), the reference signals will be correspondingly erratic or well behaved. The adaptive noise canceler 30 acts in a manner which may be analogized to a dynamic multiple notch filter based on the spectral distribution of the reference signal n′(t) or s′(t).

[0093] FIG. 5c illustrates an exemplary transfer function of a multiple notch filter. The notches, or dips in the amplitude of the transfer function, indicate frequencies which are attenuated or removed when a signal passes through the notch filter. The output of the notch filter is the composite signal having frequencies at which a notch is present removed. In the analogy to an adaptive noise canceler 30, the frequencies at which notches are present change continuously based upon the inputs to the adaptive noise canceler 30.

[0094] The adaptive noise canceler 30 (FIGS. 5a and 5b) produces an output signal, labeled herein as s″_λa(t), s″_λb(t), n″_λa(t) or n″_λb(t) which is fed back to an internal processor 32 within the adaptive noise canceler 30. The internal processor 32 automatically adjusts its own transfer function according to a predetermined algorithm such that the output of the internal processor 32 labeled b_λ(t) in FIG. 5a and c_λ(t) in FIG. 5b, closely resembles either the secondary signal portion n_λa(t) or n_λb(t) or the primary signal portion s_λa(t) or s_λb(t). The output b_λ(t) of the internal processor 32 in FIG. 5a is subtracted from the measured signal, s_λa(t) or s_λb(t), yielding a signal output s″_λa(t)=s_λa(t)+n_λa(t)−b_λa(t) or a signal output s″_λb(t)=s_λb(t)+n_λb(t)−b_λb(t). The internal processor optimizes s″_λa(t) or s″_λb(t) such that s″_λa(t) or s″_λb(t) is approximately equal to the primary signal s_λa(t) or s_λb(t), respectively. The output c_λ(t) of the internal processor 32 in FIG. 5b is subtracted from the measured signal, S_λa(t) or S_λb(t), yielding a signal output given by n″_λa(t)=s_λa(t)+n_λa(t)−c_λa(t) or a signal output given by n″_λb(t)=s_λb(t)+n_λb(t)−c_λb(t). The internal processor optimizes n″_λa(t) or n″_λb(t) such that n″_λa(t) or n″_λb(t) is approximately equal to the secondary signal portion n_λa(t) or n_λb(t), respectively.

[0095] One algorithm which may be used for the adjustment of the transfer function of the internal processor 32 is a least-squares algorithm, as described in Chapter 6 and Chapter 12 of the book Adaptive Signal Processing by Bernard Widrow and Samuel Steams, published by Prentice Hall, copyright 1985. This entire book, including Chapters 6 and 12, is hereby incorporated herein by reference.

[0096] Adaptive processors 30 in FIGS. 5a and 5b have been successfully applied to a number of problems including antenna sidelobe canceling, pattern recognition, the elimination of periodic interference in general, and the elimination of echoes on long distance telephone transmission lines. However, considerable ingenuity is often required to find a suitable reference signal n′(t) or s′(t) since the portions n_λa(t), n_λb(t), s_λa(t) and s_λb(t) cannot easily be separated from the measured composite signals S_λa(t) and S_λb(t). If either the actual secondary portion n_λa(t) or n_λb(t) or the primary signal portion s_λa(t) or s_λb(t) were a priori available, techniques such as correlation cancellation would not be necessary.

[0097] Generalized Determination of Primary and Secondary Reference Signals

[0098] An explanation which describes how the reference signals n′(t) and s′(t) may be determined follows. A first signal is measured at, for example, a wavelength λa, by a detector yielding a signal S_λa(t):
S_λa(t)=S_λa(t)+n_λa(t)   (1)

[0099] where s_λa(t) is the primary signal portion and n_λa(t) is the secondary signal portion.

[0100] A similar measurement is taken simultaneously, or nearly simultaneously, at a different wavelength, λb, yielding:
S_λb(t)=s_λb(t)+n_λb(t)   (2)

[0101] Note that as long as the measurements, S_λa(t) and S_λb(t), are taken substantially simultaneously, the secondary signal components, n_λa(t) and n_λb(t), are correlated because any random or erratic functions affect each measurement in nearly the same fashion. The substantially predictable primary signal components, s_λa(t) and s_λb(t), are also correlated to one another.

[0102] To obtain the reference signals n′(t) and s′(t), the measured signals S_λa(t) and S_λb(t) are transformed to eliminate, respectively, the primary or secondary signal components. In accordance with the present invention one way of doing this is to find proportionality constants, r_a and r_v, between the primary signal portions s_λa(t) and sλ_b(t) and the secondary signal portions n_λa(t) and n_λb(t) such that the signals can be modeled as follows:
s_λa(t)=r_as_λb(t) n_λa(t)=r_vn_λb(t).   (3)

[0103] In accordance with the inventive signal model of the present invention, these proportionality relationships can be satisfied in many measurements, including but not limited to absorption measurements and physiological measurements. Additionally, in accordance with the signal model of the present invention, in most measurements, the proportionality constants r_a and r_v can be determined such that:
n_λa(t)≠r_an_λb(t) s_λa(t)≠r_vs_λb(t).   (4)

[0104] Multiplying equation (2) by r_a and then subtracting equation (2) from equation (1) results in a single equation wherein the primary signal terms s_λa(t) and s_λb(t) cancel:
n′(t)=S_λa(t)−r_aS_λb(t)=n_λa(t)−r_an_λb(t);   (5a)

[0105] a non-zero signal which is correlated to each secondary signal portion n_λa(t) and n_λb(t) and can be used as the secondary reference n′(t) in a correlation canceler such as an adaptive noise canceler.

[0106] Multiplying equation (2) by r_v and then subtracting equation (2) from equation (1) results in a single equation wherein the secondary signal terms n_λa(t) and n_λb(t) cancel, leaving:
s′(t)=S_λa(t)−r_vS_λb(t)=s_λa(t)−r_vs_λb(t);   (5b)

[0107] a non-zero signal which is correlated to each of the primary signal portions s_λa(t) and s_λb(t) and can be used as the signal reference s′(t) in a correlation canceler such as an adaptive noise canceler.

[0108] Example Of Determination Of Primary And Secondary Reference Signals In An Absortive System

[0109] Correlation canceling is particularly useful in a large number of measurements generally described as absorption measurements. An example of an absorption type monitor which can advantageously employ correlation canceling, such as adaptive noise canceling, based upon a reference n′(t) or s′(t) determined by a processor of the present invention is one which determines the concentration of an energy absorbing constituent within an absorbing material when the material is subject to change. Such changes can be caused by forces about which information is desired or primary, or alternatively, by random or erratic secondary forces such as a mechanical force on the material. Random or erratic interference, such as motion, generates secondary components in the measured signal. These secondary components can be removed or derived by the correlation canceler if a suitable secondary reference n′(t) or primary reference s′(t) is known.

[0110] A schematic N constituent absorbing material comprising a container 42 having N different absorbing constituents, labeled A₁, A₂, A₃, . . . A_N, is shown in FIG. 6a. The constituents A₁ through A_N in FIG. 6a are arranged in a generally orderly, layered fashion within the container 42. An example of a particular type of absorptive system is one in which light energy passes through the container 42 and is absorbed according to the generalized Beer-Lambert Law of light absorption. For light of wavelength λa, this attenuation may be approximated by:

[0111] $\begin{array}{cc}I=I_o\exp \left(-\sum _{i=1}^N{\epsilon }_{i,\lambda a}{c}_i{x}_i\right)& \left(6\right)\end{array}$

[0112] Initially transforming the signal by taking the natural logarithm of both sides and manipulating terms, the signal is transformed such that the signal components are combined by addition rather than multiplication, i.e.:

[0113] $\begin{array}{cc}{S}_{\pi }=1n\left(I_o/I\right)=\sum _{i=1}^N{\epsilon }_{i,\lambda a}{c}_i{x}_i& \left(7\right)\end{array}$

[0114] where I₀ is the incident light energy intensity; I is the transmitted light energy intensity; ε_i,λa is the absorption coefficient of the i^th constituent at the wavelength λa; x_i(t) is the optical path length of i^th layer, i.e., the thickness of material of the i^th layer through which optical energy passes; and c_i(t) is the concentration of the i^th constituent in the volume associated with the thickness x_i(t). The absorption coefficients ε₁ through ε_N are known values which are constant at each wavelength. Most concentrations c₁(t) through c_N(t) are typically unknown, as are most of the optical path lengths x_i(t) of each layer. The total optical path length is the sum of each of the individual optical path lengths x_i(t) of each layer.

[0115] When the material is not subject to any forces which cause change in the thicknesses of the layers, the optical path length of each layer, x_i(t), is generally constant. This results in generally constant attenuation of the optical energy and thus, a generally constant offset in the measured signal. Typically, this offset portion of the signal is of little interest since knowledge about a force which perturbs the material is usually desired. Any signal portion outside of a known bandwidth of interest, including the constant undesired signal portion resulting from the generally constant absorption of the constituents when not subject to change, is removed. This is easily accomplished by traditional band pass filtering techniques. However, when the material is subject to forces, each layer of constituents may be affected by the perturbation differently than other layers. Some perturbations of the optical path lengths of each layer x_i(t) may result in excursions in the measured signal which represent desired or primary information. Other perturbations of the optical path length of each layer x_i(t) cause undesired or secondary excursions which mask primary information in the measured signal. Secondary signal components associated with secondary excursions must also be removed to obtain primary information from the measured signal. Similarly, the ability to compute secondary signal components caused by secondary excursions directly allows one to obtain primary signal components from the measured signal via simple subtraction, or correlation cancellation techniques.

[0116] The correlation canceler may selectively remove from the composite signal, measured after being transmitted through or reflected from the absorbing material, either the secondary or the primary signal components caused by forces which perturb or change the material differently from the forces which perturbed or changed the material to cause respectively, either the primary or secondary signal component. For the purposes of illustration, it will be assumed that the portion of the measured signal which is deemed to be the primary signal s_λa(t) is the attenuation term ε₅c₅x₅(t) associated with a constituent of interest, namely A₅, and that the layer of constituent A₅ is affected by perturbations different than each of the layers of other constituents A₁ through A₄ and A₆ through A_N. An example of such a situation is when layer A₅ is subject to forces about which information is deemed to be primary and, additionally, the entire material is subject to forces which affect each of the layers. In this case, since the total force affecting the layer of constituent A₅ is different than the total forces affecting each of the other layers and information is deemed to be primary about the forces and resultant perturbation of the layer of constituent A₅, attenuation terms due to constituents A₁ through A₄ and A₆ through A_N make up the secondary signal portion n_λa(t). Even if the additional forces which affect the entire material cause the same perturbation in each layer, including the layer of A₅, the total forces on the layer of constituent A₅ cause it to have different total perturbation than each of the other layers of constituents A₁ through A₄ and A₆ through A_N.

[0117] It is often the case that the total perturbation affecting the layers associated with the secondary signal components is caused by random or erratic forces. This causes the thickness of layers to change erratically and the optical path length of each layer, x_i(t), to change erratically, thereby producing a random or erratic secondary signal component n_λa(t). However, regardless of whether or not the secondary signal portion n_λa(t) is erratic, the secondary signal component n_λa(t) can be either removed or derived via a correlation canceler, such as an adaptive noise canceler, having as one input, respectively, a secondary reference n′(t) or a primary reference s′(t) determined by a processor of the present invention as long as the perturbation on layers other than the layer of constituent A₅ is different than the perturbation on the layer of constituent A₅. The correlation canceler yields a good approximation to either the primary signal s_λa(t) or the secondary signal n_λa(t). In the event that an approximation to the primary signal is obtained, the concentration of the constituent of interest, c₅(t), can often be determined since in some physiological measurements, the thickness of the primary signal component, x₅(t) in this example, is known or can be determined.

[0118] The correlation canceler utilizes either the secondary reference n′(t) or the primary reference s′(t) determined from two substantially simultaneously measured signals S_λa(t) and S_λb(t). S_λa(t) is determined as above in equation (7). S_λb(t) is determined similarly at a different wavelength λb. To find either the secondary reference n′(t) or the primary reference s′(t), attenuated transmitted energy is measured at the two different wavelengths λa and λb and transformed via logarithmic conversion. The signals S_λa(t) and S_λb(t) can then be written (logarithm converted) as:

[0119] $\begin{array}{cc}{S}_{\lambda a}\left(t\right)={\epsilon }_{5,\lambda a}{c}_5{x}_5\left(t\right)+\sum _{i=1}^4{\epsilon }_{i,\lambda a}{c}_i{x}_i+\sum _{i=6}^N{\epsilon }_{i,\lambda a}{c}_i{x}_i& \left(8\right)\\ S_{\lambda a\left(t\right)}={\varepsilon }_{5,\lambda a}{c}_5{x}_5\left(t\right)+n_{\lambda a}\left(t\right)& \left(9\right)\\ S_{\lambda b\left(t\right)}={\epsilon }_{5,\lambda b}{c}_5{x}_5\left(t\right)+\sum _{i=1}^4{\epsilon }_{i,\lambda b}{c}_i{x}_i+\sum _{i=6}^N{\epsilon }_{i,\lambda b}{c}_i{x}_i& \left(10\right)\\ S_{\lambda b\left(t\right)}={\varepsilon }_{5,\lambda b}{c}_5{x}_5\left(t\right)+n_{\lambda b}\left(t\right)& \left(11\right)\end{array}$

[0120] Further transformations of the signals are the proportionality relationships in accordance with the signal model of the present invention defining r_a and r_v, similar to equation (3), which allows determination of a noise reference n′(t) and a primary reference s′(t). These are:
ε_5,λa=r_aε_5,λb   (12a)
n_λa=r_vn_λb   (12b)
where
n_λa≠r_an_λb   (13a)
ε_5,λa≠r_vε_5,λb   (13b)

[0121] It is often the case that both equations (12) and (13) can be simultaneously satisfied. Multiplying equation (11) by r_a and subtracting the result from equation (9) yields a non-zero secondary reference which is a linear sum of secondary signal components:

[0122] $\begin{array}{cc}{n}^{\prime }\left(t\right)=S_{\lambda a}\left(t\right)-r_a{S}_{\lambda b}\left(t\right)=n_{\lambda a}\left(t\right)-r_a{n}_{\lambda b}\left(t\right)& \left(14a\right)\\ =\sum _{i=1}^4{\epsilon }_{i,\lambda a}{c}_i{x}_i\left(t\right)+\sum _{i+6}^N{\epsilon }_{i,\lambda a}{c}_i{x}_i\left(t\right)-\sum _{i=1}^4{r}_a{\epsilon }_{i,\lambda b}{c}_i{x}_i\left(t\right)+\sum _{i=6}^N{r}_a{\epsilon }_{i,\lambda b}{c}_i{x}_i\left(t\right)& \left(15a\right)\\ =\sum _{i=1}^4{c}_i{x}_i\left(t\right)\left[{\epsilon }_{i,\lambda a}-r_a{\epsilon }_{i,\lambda a}-r_a{\epsilon }_{i,\lambda b}\right]+\sum _{i=6}^N{c}_i{x}_i\left(t\right)\left[{\epsilon }_{1,\lambda a}-r_a{\epsilon }_{i,\lambda b}\right]& \left(16a\right)\end{array}$

[0123] Multiplying equation (11) by r_v and subtracting the result from equation (9) yields a primary reference which is a linear sum of primary signal components:

[0124] $\begin{array}{cc}{s}^{\prime }\left(t\right)=S_{\lambda a}\left(t\right)-r_v{S}_{\lambda b}\left(t\right)=s_{\lambda a}\left(t\right)-r_v{s}_{\lambda b}\left(t\right)& \left(14b\right)\\ =c_5{x}_5\left(t\right){\varepsilon }_{5,\lambda a}-r_v{c}_5{x}_5\left(t\right){\varepsilon }_{5,\lambda b}& \left(15b\right)\\ =c_5{x}_5\left(t\right)\left[{\varepsilon }_{5,\lambda a}-r_v{\varepsilon }_{5,\lambda b}\right].& \left(16b\right)\end{array}$

[0125] A sample of either the secondary reference n′(t) or the primary reference s′(t), and a sample of either measured signal S_λa(t) or S_λb(t), are input to a correlation canceler 27, such as an adaptive noise canceler 30, an example of which is shown in FIGS. 5a and 5b and a preferred example of which is discussed herein under the heading PREFERRED CORRELATION CANCELER USING A JOINT PROCESS ESTIMATOR IMPLEMENTATION. The correlation canceler 27 removes either the secondary portion n_λa(t) or n_λb(t), or the primary portions, S_λa(t) or S_λb(t), of the measured signal yielding a good approximation to either the primary signals S″_λa(t)≈ε_5,λac₅x₅(t) or s″_λb(t)≈ε_5,λbc₅x₅(t) or the secondary signals n″_λa(t)≈n_λa(t) or n″b(t)≈n_λb(t). In the event that the primary signals are obtained, the concentration c₅(t) may then be determined from the approximation to the primary signal s″_λa(t) or s″_λb(t) according to:
c₅(t)≈s″_λa(t)/ε_5,λax₅(t)   (17a)
or
c₅(t)≈s″_λb(t)/ε_5,λbx₅(t)   (17b)

[0126] As discussed previously, the absorption coefficients are constant at each wavelength λa and λb and the thickness of the primary signal component, x₅(t) in this example, is often known or can be determined as a function of time, thereby allowing calculation of the concentration c₅(t) of constituent A₅.

[0127] Determination of Concentration or Saturation in a Volume Containing more than One Constituent

[0128] Referring to FIG. 6b, another material having N different constituents arranged in layers is shown. In this material, two constituents A₅ and A₆ are found within one layer having thickness x_5,6(t)=x₅(t)+x₆(t), located generally randomly within the layer. This is analogous to combining the layers of constituents A₅ and A₆ in FIG. 6a. A combination of layers, such as the combination of layers of constituents A₅ and A₆, is feasible when the two layers are under the same total forces which result in the same change of the optical path lengths x₅(t) and x₆(t) of the layers.

[0129] Often it is desirable to find the concentration or the saturation, i.e., a percent concentration, of one constituent within a given thickness which contains more than one constituent and is subject to unique forces. A determination of the concentration or the saturation of a constituent within a given volume may be made with any number of constituents in the volume subject to the same total forces and therefore under the same perturbation or change. To determine the saturation of one constituent in a volume comprising many constituents, as many measured signals as there are constituents which absorb incident light energy are necessary. It will be understood that constituents which do not absorb light energy are not consequential in the determination of saturation. To determine the concentration, as many signals as there are constituents which absorb incident light energy are necessary as well as information about the sum of concentrations.

[0130] It is often the case that a thickness under unique motion contains only two constituents. For example, it may be desirable to know the concentration or saturation of A₅ within a given volume which contains A₅ and A₆. In this case, the primary signals s_λa(t) and s_λb(t) comprise terms related to both A₅ and A₆ so that a determination of the concentration or saturation of A₅ or A₆ in the volume may be made. A determination of saturation is discussed herein. It will be understood that the concentration of A₅ in a volume containing both A₅ and A₆ could also be determined if it is known that A₅+A₆=1, i.e., that there are no constituents in the volume which do not absorb incident light energy at the particular measurement wavelengths chosen. The measured signals S_λa(t) and S_λb(t) can be written (logarithm converted) as:

[0131] $\begin{array}{cc}{S}_{\lambda a}\left(t\right)={\epsilon }_{5,\lambda a}{c}_5{x}_{5,6}\left(t\right)+{\epsilon }_{6,\lambda a}{c}_6{x}_{5,6}\left(t\right)+n_{\lambda a}\left(t\right)& \left(18a\right)\\ =s_{\lambda a}\left(t\right)+n_{\lambda a}\left(t\right)& \left(18b\right)\\ S_{\lambda b}\left(t\right)={\epsilon }_{5,\lambda b}{c}_5{x}_{5,6}\left(t\right)+{\epsilon }_{6,\lambda b}{c}_6{x}_{5,6}\left(t\right)+n_{\lambda b}\left(t\right)& \left(19a\right)\\ =s_{\lambda b}\left(t\right)+n_{\lambda b}\left(t\right).& \left(19b\right)\end{array}$

[0132] It is also often the case that there may be two or more thicknesses within a medium each containing the same two constituents but each experiencing a separate motion as in FIG. 6c. For example, it may be desirable to know the concentration or saturation of A₅ within a given volume which contains A₅ and A₆ as well as the concentration or saturation of A₃ within a given volume which contains A₃ and A₄, A₃ and A₄ having the same constituency as A₅ and A₆ respectively. In this case, the primary signals s_λa(t) and s_λb(t) again comprise terms related to both A₅ and A₆ and portions of the secondary signals n_λa(t) and n_λb(t) comprise terms related to both A₃ and A₄. The layers, A₃ and A₄, do not enter into the primary equation because they are assumed to be perturbed by a different frequency, or random or erratic secondary forces which are uncorrelated with the primary force. Since constituents 3 and 5 as well as constituents 4 and 6 are taken to be the same, they have the same absorption coefficients (i.e., ε_3,λa=ε_5,λa; ε_3,λb=ε_5,λb; ε_4,λa=ε_6,λa and ε_4,λb=ε_6,λb. Generally speaking, however, A₃ and A₄ will have different concentrations than A₅ and A₆ and will therefore have a different saturation. Consequently a single constituent within a medium may have one or more saturations associated with it. The primary and secondary signals according to this model may be written as:

[0133] $\begin{array}{cc}{s}_{\lambda a}\left(t\right)=\left[{\varepsilon }_{5,\lambda a}{c}_5+{\varepsilon }_{6,\lambda a}{c}_6\right]x_{5,6}\left(t\right)& \left(20a\right)\\ n_{\lambda a}\left(t\right)=\left[{\varepsilon }_{5,\lambda a}{c}_3+{\varepsilon }_{6,\lambda a}{c}_4\right]x_{3,4}\left(t\right)+\sum _{i=1}^2{\epsilon }_{i,\lambda a}{c}_i{x}_i\left(t\right)+\sum _{i=7}^n{\epsilon }_{i,\lambda a}{c}_i{x}_i& \\ n_{\lambda a}\left(t\right)=\left[{\varepsilon }_{5,\lambda a}{c}_3+{\varepsilon }_{6,\lambda a}{c}_4\right]x_{3,4}\left(t\right)+n_{\lambda a}\left(t\right)& \left(20c\right)\\ s_{\lambda b}\left(t\right)=\left[{\varepsilon }_{5,\lambda b}{c}_5+{\varepsilon }_{6,\lambda b}{c}_6\right]x_{5,6}\left(t\right)& \left(21a\right)\\ n_{\lambda b}\left(t\right)=\left[{\varepsilon }_{5,\lambda b}{c}_3+{\varepsilon }_{6,\lambda b}{c}_4\right]x_{3,4}\left(t\right)+\sum _{i=1}^2{\epsilon }_{i,\lambda b}{c}_i{x}_i\left(t\right)+\sum _{i=7}^N{\epsilon }_{i,\lambda b}{c}_i{x}_i\left(t\right).& \left(21b\right)\\ n_{\lambda b}\left(t\right)=\left[{\varepsilon }_{5,\lambda b}{c}_3+{\varepsilon }_{6,\lambda b}{c}_4\right]x_{3,4}\left(t\right)+n_{\lambda b}\left(t\right)& \left(21c\right)\\ & \end{array}$

[0134] where signals n_λa(t) and n_λb(t) are similar to the secondary signals n_λa(t) and n_λb(t) except for the omission of the 3, 4 layer.

[0135] Any signal portions whether primary or secondary, outside of a known bandwidth of interest, including the constant undesired secondary signal portion resulting from the generally constant absorption of the constituents when not under perturbation, should be removed to determine an approximation to either the primary signal or the secondary signal within the bandwidth of interest. This is easily accomplished by traditional band pass filtering techniques. As in the previous example, it is often the case that the total perturbation or change affecting the layers associated with the secondary signal components is caused by random or erratic forces, causing the thickness of each layer, or the optical path length of each layer, x_i(t), to change erratically, producing a random or erratic secondary signal component n_λa(t). Regardless of whether or not the secondary signal portion n_λa(t) is erratic, the secondary signal component n_λa(t) can be removed or derived via a correlation canceler, such as an adaptive noise canceler, having as one input a secondary reference n′(t) or a primary reference s′(t) determined by a processor of the present invention as long as the perturbation in layers other than the layer of constituents A₅ and A₆ is different than the perturbation in the layer of constituents A₅ and A₆. Either the erratic secondary signal components n_λa(t) and n_λb(t) or the primary components s_λa(t) and s_λb(t) may advantageously be removed from equations (18) and (19), or alternatively equations (20) and (21), by a correlation canceler. The correlation canceler, again, requires a sample of either the primary reference s′(t) or the secondary reference n′(t) and a sample of either of the composite signals S_λa(t) or S_λb(t) of equations (18) and (19).

[0136] Determination of Primary and Secondary Reference Signals for Saturation Measurements

[0137] One method for determining reference signals s′(t) or n′(t) from the measured signals S_λa(t) and S_λb(t) in accordance with one aspect of the invention is what will be referred to as the constant saturation approach. In this approach, it is assumed that the saturation of A₅ in the volume containing A₅ and A₆ and the saturation of A₃ in the volume containing A₃ and A₄ remains relatively constant over some period of time, i.e.:
Saturation(A₅(t))=c₅(t)/[c₅(t)+c₆(t)]  (22a)
Saturation(A₃(t))=c₃(t)/[c₃(t)+c₄(t)]  (22b)
Saturation(A₅(t))={1+[c₆(t)/c₅(t)]}⁻¹   (23a)
Saturation(A₃(t))={1+[c₄(t)/c₃(t)]}⁻¹   (23b)

[0138] are substantially constant over many samples of the measured signals S_λa and S_λb. This assumption is accurate over many samples since saturation generally changes relatively slowly in physiological systems.

[0139] The constant saturation assumption is equivalent to assuming that:
c₅(t)/c₆(t)=constant₁   (24a)
c₃(t)/c₄(t)=constant₂   (24b)

[0140] since the only other term in equations (23a) and (23b) is a constant, namely the numeral 1.

[0141] Using this assumption, the proportionality constants r_a and r_v which allow determination of the secondary reference signal n′(t) and the primary reference signal s′(t) in the constant saturation method are:

[0142] $\begin{array}{cc}{r}_a=\frac{{\varepsilon }_{5,\lambda a}{c}_{5-}{x}_{5,6}\left(t\right)+{\varepsilon }_{6,\lambda a}{c}_6{x}_{5,6}\left(t\right)}{{\varepsilon }_{5,\lambda b}{c}_5{x}_{5,6}\left(t\right)+{\varepsilon }_{6,\lambda b}{c}_6{x}_{5,6}\left(t\right)}& \left(25a\right)\\ =s_{\lambda a}\left(t\right)/s_{\lambda b}\left(t\right)& \left(26a\right)\\ =\frac{{\varepsilon }_{5,\lambda a}{c}_5+{\varepsilon }_{6,\lambda a}{c}_6}{{\varepsilon }_{5,\lambda b}{c}_5+{\varepsilon }_{6,\lambda b}{c}_6}& \left(27a\right)\\ =\frac{{\varepsilon }_{5,\lambda a}\left(c_5/c_6\right)+{\varepsilon }_{6,\lambda a}}{{\varepsilon }_{5,\lambda b}\left(c_5/c_6\right)+{\varepsilon }_{6,\lambda b}}& \left(28a\right)\\ \approx s_{\lambda a}^{″}\left(t\right)/s_{\lambda b}^{″}\left(t\right)={\mathrm{constant}}_3;& \left(29a\right)\\ \mathrm{where}& \\ n_{\lambda a}\left(t\right)\ne r_a\left(t\right)n_{\lambda b}\left(t\right)& \left(30a\right)\\ \mathrm{and}& \\ r_v=\frac{{\varepsilon }_{5,\lambda a}{c}_3{x}_{3,4}\left(t\right)+{\varepsilon }_{6,\lambda a}{c}_4{x}_{3,4}\left(t\right)}{{\varepsilon }_{5,\lambda b}{c}_3{x}_{3,4}\left(t\right)+{\varepsilon }_{6,\lambda b}{c}_4{x}_{3,4}\left(t\right))}& \left(25b\right)\\ =n_{\lambda a}\left(t\right)/n_{\lambda b}\left(t\right)& \left(26b\right)\\ =\frac{{\varepsilon }_{5,\lambda a}{c}_3+{\varepsilon }_{6,\lambda a}{c}_4}{{\varepsilon }_{5,\lambda b}{c}_3+{\varepsilon }_{6,\lambda b}{c}_4}& \left(27b\right)\\ =\frac{{\varepsilon }_{5,\lambda a}\left(c_3/c_4\right)+{\varepsilon }_{6,\lambda a}}{{\varepsilon }_{5,\lambda b}\left(c_3/c_4\right)+{\varepsilon }_{6,\lambda b}}& \left(28b\right)\\ \approx n_{\lambda a}^{″}\left(t\right)/n_{\lambda b}^{″}\left(t\right)={\mathrm{constant}}_4;& \left(29b\right)\\ \mathrm{where}& \\ s_{\lambda a}\left(t\right)\ne r_v\left(t\right)s_{\lambda b}\left(t\right).& \left(30b\right)\end{array}$

[0143] In accordance with the present invention, it is often the case that both equations (26) and (30) can be simultaneously satisfied to determine the proportionality constants r_a and r_v. Additionally, the absorption coefficients at each wavelength ε_5,λa, ε_6,λa, ε_5,λb, and ε_6,λb are constant and the central assumption of the constant saturation method is that c₅(t)/c₆(t) and c₃(t)/c₄(t) are constant over many sample periods. Thus, new proportionality constants r_a and r_v may be determined every few samples from new approximations to either the primary or secondary signal as output from the correlation canceler. Thus, the approximations to either the primary signals s_λa(t) and s_λb(t) or the secondary signals n_λa(t) and n_λb(t), found by the correlation canceler for a substantially immediately preceding set of samples of the measured signals S_λa(t) and S_λb(t) are used in a processor of the present invention for calculating the proportionality constants, r_a and r_v, for the next set of samples of the measured signals S_λa(t) and S_λb(t).

[0144] Multiplying equation (19) by r_a and subtracting the resulting equation from equation (18) yields a non-zero secondary reference signal:
n′(t)=S_λa(t)−r_aS_λb(t)=n_λa(t)−r_an_λb(t).   (31a)

[0145] Multiplying equation (19) by r_v and subtracting the resulting equation from equation (18) yields a non-zero primary reference signal:
s′(t)=S_λa(t)−r_vS_λb(t)=s_λa(t)−r_vs_λb(t).   (31b)

[0146] When using the constant saturation method in patient monitoring, initial proportionality coefficients can be determined as further explained below. It is not necessary for the patient to remain motionless even for an initialization period. With values for the proportionality coefficients r_a and r_v determined, a correlation canceler may be utilized with a secondary reference n′(t) or a primary reference s′(t).

[0147] Determination of Signal Coefficients for Primary and Secondary Reference Signals Using the Constant Saturation Method

[0148] In accordance with one aspect of the present invention, the reference processor 26 of FIGS. 4a and FIG. 4b of the present invention may be configured to multiply the second measured assumed signal S_λb(t)=s_λb(t)+n_λb(t) by each of a plurality of signal coefficients r₁, r₂, . . . r_n and then subtract each result from the first measured signal S_λa(t)=s_λa(t)+n_λa(t) to obtain a plurality of reference signals
R′(r, t)=s_λa(t)−r s_λb(t)+n_λa(t)−r n_λb(t)   (32)

[0149] for r=r₁, r₂, . . . r_n as shown in FIG. 7a. In other words, a plurality of signal coefficients are chosen to represent a cross section of possible signal coefficients.

[0150] In order to determine either the primary reference s′(t) or the secondary reference n′(t) from the above plurality of reference signals of equation (32), signal coefficients r_a and r_v are determined from the plurality of assumed signal coefficients r₁, r₂, . . . r_n. The coefficients r_a and r_v are selected such that they cause either the primary signal portions s_λa(t) and s_λb(t) or the secondary signal portions n_λa(t) and n_λb(t) to cancel or nearly cancel when they are substituted into the reference function R′(r, t), e. g.
s_λa(t)=r_as_λb(t)   (33a)
n_λa(t)=r_vn_λb(t)   (33b)
n′(t)=R′(r_a, t)=n_λa(t)−r_an_λb(t)   (33c)
s′(t)=R′(r_v, t)=s_λa(t)−r_vs_λb(t).   (33d)

[0151] In other words, coefficients r_a and r_v are selected at values which reflect the minimum of correlation between the primary signal portions and the secondary signal portions. In practice, one does not usually have significant prior information about either the primary signal portions s_λa(t) and s_λb(t) or the secondary signal portions n_λa(t) and n_λb(t) of the measured signals S_λa(t) and S_λb(t). The lack of this information makes it difficult to determine which of the plurality of coefficients r₁, r₂, . . . r_n correspond to the signal coefficients r_a=s_λa(t)/s_λb(t) and r_v=n_λa(t)/n_λb(t).

[0152] One approach to determine the signal coefficients r_a and r_v from the plurality of coefficients r₁, r₂, . . . r_n employs the use of a correlation canceler 27, such as an adaptive noise canceler, which takes a first input which corresponds to one of the measured signals S_λa(t) or S_λb(t) and takes a second input which corresponds to successively each one of the plurality of reference signals R′(r₁, t), R′(r₂, t), . . . , R′(r_n, t) as shown in FIG. 7a. For each of the reference signals R′(r₁, t), R′(r₂, t), . . . , R′(r_n, t) the corresponding output of the correlation canceler 27 is input to a “squares” operation 28 which squares the output of the correlation canceler 27. The output of the squares operation 28 is provided to an integrator 29 for forming a cumulative output signal (a summation of the squares). The cumulative output signal is subsequently input to an extremum detector 31. The purpose of the extremum detector 31 is to chose signal coefficients r_a and r_v from the set r₁, r₂, . . . r_n by observing which provide a maximum in the cumulative output signal as in FIGS. 7b and 7c. In other words, coefficients which provide a maximum integrated output, such as energy or power, from the correlation canceler 27 correspond to the signal coefficients r_a and r_v which relate to a minimum correlation between the primary signal portions and the secondary signal portions in accordance with the signal model of the present invention. One could also configure a system geometry which would require one to locate the coefficients from the set r₁, r₂, . . . r_n which provide a minimum or inflection in the cumulative output signal to identify the signal coefficients r_a and r_v.

[0153] Use of a plurality of coefficients in the processor of the present invention in conjunction with a correlation canceler 27 to determine the signal coefficients r_a and r_v may be demonstrated by using the properties of correlation cancellation. If x, y and z are taken to be any collection of three time varying signals, then the properties of some correlation cancelers C(x, y) may be defined as follows:
Property (1) C(x, y)=0 for x, y correlated   (34a)
Property (2) C(x, y)=x for x, y uncorrelated   (34b)
Property (3) C(x+y, z)=C(x, z)+C(y, z)   (34c)

[0154] With properties (1), (2) and (3) it is easy to demonstrate that the energy or power output of a correlation canceler with a first input which corresponds to one of the measured signals S_λa(t) or S_λb(t) and a second input which corresponds to successively each one of a plurality of reference signals R′(r₁, t), R′(r₂, t), . . . , R′(r_n, t) can determine the signal coefficients r_a and r_v needed to produce the primary reference s′(t) and secondary reference n′(t). If we take as a first input to the correlation canceler the measured signal S_λa(t) and as a second input the plurality of reference signals R′(r₁, t), R′(r₂, t), . . . , R′(r_n, t) then the outputs of the correlation canceler C(S_λa(t), R′(r_j,t)) for j=1, 2, . . . , n may be written as
C(s_λa(t)+n_λa(t), s_λa(t)−r_js_λb(t)+n_λa(t)−r_jn_λb(t))   (35)

[0155] where j=1, 2, . . . n and we have used the expressions
R′(r, t)=S_λa(t)−rS_λb(t)  (36)
S_λa(t)=s_λa(t)+n_λa(t)   (37a)
S_λb(t)=s_λb(t)+n_λb(t).   (37b)

[0156] The use of property (3) allows one to expand equation (35) into two terms
C(S_λa(t), R′(r,t))=C(s_λa(t), s_λa(t)−rs_λb(t)+n_λa(t)−rn_λb(t))+C(n_λa(t), s_λa(t)−rs_λb(t)+n_λa(t)−rn_λb(t))   (38)

[0157] so that upon use of properties (1) and (2) the correlation canceler output is given by
C (S_λa(t), R′(r_j,t))=s_λa(t) δ (r_j−r_a)+n_λa(t) δ (r_j−r_v)   (39)

[0158] where δ (x) is the unit impulse function
δ (x)=0 if x≠0
δ (x)=1 if x=0.   (40)

[0159] The time variable, t, of the correlation canceler output C(S_λa(t), R′(r_j, t)) may be eliminated by computing its energy or power. The energy of the correlation canceler output is given by

[0160] $\begin{array}{cc}\begin{array}{c}{E}_{\lambda a}\left(r_j\right)=\int C^2(S_{\lambda a}\left(t\right),R^{\prime }\left(r_j,t\right)dt\\ =\delta \left(r_j-r_a\right)\int s_{\lambda a}^2\left(t\right)dt+\delta \left(r_j-r_v\right)\int n_{\lambda a}^2\left(t\right)dt.\end{array}& \left(41a\right)\end{array}$

[0161] It should be understood that one could, equally well, have chosen the measured signal S_λb(t) as the first input to the correlation canceler and the plurality of reference signals R′(r₁, t), R′(r₂, t), . . . , R′(r_n, t) as the second input. In this event, the correlation canceler energy output is

[0162] $\begin{array}{cc}\begin{array}{c}{E}_{\lambda b}\left(r_j\right)=\int C^2(S_{\lambda b}\left(t\right),R^{\prime }\left(r,t\right)dt\\ =\delta \left(r_j-r_a\right)\int s_{\lambda b}^2\left(t\right)dt+\delta \left(r_j-r_v\right)\int n_{\lambda b}^2\left(t\right)dt.\end{array}& \left(41b\right)\end{array}$

[0163] It should also be understood that in practical situations the use of discrete time measurement signals may be employed as well as continuous time measurement signals. A system which performs a discrete transform (e.g., a saturation transform in the present example) in accordance with the present invention is described with reference to FIGS. 11-22. In the event that discrete time measurement signals are used integration approximation methods such as the trapezoid rule, midpoint rule, Tick's rule, Simpson's approximation or other techniques may be used to compute the correlation canceler energy or power output. In the discrete time measurement signal case, the energy output of the correlation canceler may be written, using the trapezoid rule, as

[0164] $\begin{array}{cc}{E}_{\lambda a}\left(r_j\right)=\delta \left(r_j-r_a\right)\Delta t\left\{\sum _{i=0}^n{s}_{\lambda a}^2\left(t_i\right)-0.5\left(s_{\lambda a}^2\left(t_0\right)+s_{\lambda a}^2\left(t_n\right)\right)\right\}+& \left(42a\right)\\ \delta \left(r_j-r_v\right)\Delta t\left\{\sum _{i=0}^n{n}_{\lambda a}^2\left(t_i\right)-0.5\left(n_{\lambda a}^2\left(t_0\right)+n_{\lambda a}^2\left(t_n\right)\right)\right\}& \\ E_{\lambda b}\left(r\right)=\delta \left(r_j-r_a\right)\Delta t\left\{\sum _{i=0}^n{s}_{\lambda b}^2\left(t_i\right)-0.5\left(s_{\lambda b}^2\left(t_0\right)+s_{\lambda b}^2\left(t_n\right)\right)\right\}+& \left(42b\right)\\ \delta \left(r_j-r_v\right)\Delta t\left\{\sum _{i=0}^n{n}_{\lambda b}^2\left(t_i\right)-0.5\left(n_{\lambda b}^2\left(t_0\right)+n_{\lambda b}^2\left(t_n\right)\right)\right\}& \end{array}$

[0165] where t_i is the i^th discrete time, t₀ is the initial time, t_n is the final time and Δt is the time between discrete time measurement samples.

[0166] The energy functions given above, and shown in FIG. 7b, indicate that the correlation canceler output is usually zero due to correlation between the measured signal S_λa(t) or S_λb(t) and many of the plurality of reference signals R′(r₁, t), R′(r₂, t), . . . , R′(r_n, t). However, the energy functions are non zero at values of r_j which correspond to cancellation of either the primary signal portions s_λa(t) and s_λb(t) or the secondary signal portions n_λa(t) and n_λb(t) in the reference signal R′(r_j, t). These values correspond to the signal coefficients r_a and r_v.

[0167] It should be understood that there may be instances in time when either the primary signal portions s_λa(t) and s_λb(t) or the secondary signal portions n_λa(t) and n_λb(t) are identically zero or nearly zero. In these cases, only one signal coefficient value will provide maximum energy or power output of the correlation canceler.

[0168] Since there may be more than one signal coefficient value which provides maximum correlation canceler energy or power output, an ambiguity may arise. It may not be immediately obvious which signal coefficient together with the reference function R′(r, t) provides either the primary or secondary reference. In such cases, it is necessary to consider the constraints of the physical system at hand. For example, in pulse oximetry, it is known that arterial blood, whose signature is the primary plethysmographic wave, has greater oxygen saturation than venous blood, whose signature is the secondary erratic or random signal. Consequently, in pulse oximetry, the ratio of the primary signals due to arterial pulsation r_a=s_λa(t)/s_λb(t) is the smaller of the two signal coefficient values while the ratio of the secondary signals due to mainly venous blood dynamics r_v=n_λa(t)/n_λb(t) is the larger of the two signal coefficient values, assuming λa=660 nm and λb=910 nm.

[0169] It should also be understood that in practical implementations of the plurality of reference signals and cross correlator technique, the ideal features listed as properties (1), (2) and (3) above will not be precisely satisfied but will be approximations thereof. Therefore, in practical implementations of this embodiment of the present invention, the correlation canceler energy curves depicted in FIG. 7b will not consist of infinitely narrow delta functions but will have finite width associated with them as depicted in FIG. 7c.

[0170] It should also be understood that it is possible to have more than two signal coefficient values which produce maximum energy or power output from a correlation canceler. This situation arises when the measured signals each contain more than two components each of which are related by a ratio as follows:

[0171] $\begin{array}{cc}\begin{array}{c}{s}_{\lambda a}\left(t\right)=\sum _{i=0}^n{f}_{\lambda a,i}\left(t\right)\\ s_{\lambda b}\left(t\right)=\sum _{i=0}^n{f}_{\lambda b,i}\left(t\right)\\ \mathrm{where}\\ \begin{array}{ccc}{f}_{\lambda a,i}\left(t\right)=r_i{f}_{\lambda b,i}\left(t\right)& & i=1,\dots ,n\end{array}\\ r_i\ne r_j.\end{array}& \left(43\right)\end{array}$

[0172] Thus, reference signal techniques together with a correlation cancellation, such as an adaptive noise canceler, can be employed to decompose a signal into two or more signal components each of which is related by a ratio.

[0173] Preferred Correlation Canceler Using a Joint Process Estimator Implementation

[0174] Once either the secondary reference n′(t) or the primary reference s′(t) is determined by the processor of the present invention, the correlation canceler can be implemented in either hardware or software. The preferred implementation of a correlation canceler is that of an adaptive noise canceler using a joint process estimator.

[0175] The least mean squares (LMS) implementation of the internal processor 32 described above in conjunction with the adaptive noise canceler of FIGS. 5a and FIG. 5b is relatively easy to implement, but lacks the speed of adaptation desirable for most physiological monitoring applications of the present invention. Thus, a faster approach for adaptive noise canceling, called a least-squares lattice joint process estimator model, is used in one embodiment. A joint process estimator 60 is shown diagrammatically in FIG. 8 and is described in detail in Chapter 9 of Adaptive Filter Theory by Simon Haykin, published by Prentice-Hall, copyright 1986. This entire book, including Chapter 9, is hereby incorporated herein by reference.

[0176] The function of the joint process estimator is to remove either the secondary signal portions n_λa(t) or n_λb(t) or the primary signal portions s_λa(t) or s_λb(t) from the measured signals S_λa(t) or S_λb(t), yielding either a primary signal approximation S″_λa(t) or S″_λb(t) or a secondary signal approximation n″_λa(t) or n″_λb(t). Thus, the joint process estimator estimates either the value of the primary signals s_λa(t) or s_λb(t) or the secondary signals n_λa(t) or n_λb(t). The inputs to the joint process estimator 60 are either the secondary reference n′(t) or the primary reference s′(t) and the composite measured signal S_λa(t) or S_λb(t). The output is a good approximation to the signal S_λa(t) or S_λb(t) with either the secondary signal or the primary signal removed, i.e. a good approximation to either s_λa(t), s_λb(t), n_λa(t) or n_λb(t).

[0177] The joint process estimator 60 of FIG. 8 utilizes, in conjunction, a least square lattice predictor 70 and a regression filter 80. Either the secondary reference n′(t) or the primary reference s′(t) is input to the least square lattice predictor 70 while the measured signal S_λa(t) or S_λb(t) is input to the regression filter 80. For simplicity in the following description, S_λa(t) will be the measured signal from which either the primary portion s_λa(t) or the secondary portion n_λa(t) will be estimated by the joint process estimator 60. However, it will be noted that S_λb(t) could also be input to the regression filter 80 and the primary portion s_λb(t) or the secondary portion n_λb(t) of this signal could be estimated.

[0178] The joint process estimator 60 removes all frequencies that are present in both the reference n′(t) or s′(t), and the measured signal S_λa(t). The secondary signal portion n_λa(t) usually comprises frequencies unrelated to those of the primary signal portion s_λa(t). It is improbable that the secondary signal portion n_λa(t) would be of exactly the same spectral content as the primary signal portion s_λa(t). However, in the unlikely event that the spectral content of s_λa(t) and n_λa(t) are similar, this approach will not yield accurate results. Functionally, the joint process estimator 60 compares the reference input signal n′(t) or s′(t), which is correlated to either the secondary signal portion n_λa(t) or the primary signal portion s_λa(t), and input signal S_λa(t) and removes all frequencies which are identical. Thus, the joint process estimator 60 acts as a dynamic multiple notch filter to remove those frequencies in the secondary signal component n_λa(t) as they change erratically with the motion of the patient or those frequencies in the primary signal component s_λa(t) as they change with the arterial pulsation of the patient. This yields a signal having substantially the same spectral content and amplitude as either the primary signal s_λa(t) or the secondary signal n_λa(t). Thus, the output S″_λa(t) or n″_λa(t) of the joint process estimator 60 is a very good approximation to either the primary signal s_λa(t) or the secondary signal n_λa(t).

[0179] The joint process estimator 60 can be divided into stages, beginning with a zero-stage and terminating in an m^th-stage, as shown in FIG. 8. Each stage, except for the zero-stage, is identical to every other stage. The zero-stage is an input stage for the joint process estimator 60. The first stage through the m^th-stage work on the signal produced in the immediately previous stage, i.e., the (m-1)^th-stage, such that a good primary signal approximation S″_λa(t) or secondary signal approximation n″_λa(t) is produced as output from the m^th-stage.

[0180] The least-squares lattice predictor 70 comprises registers 90 and 92, summing elements 100 and 102, and delay elements 110. The registers 90 and 92 contain multiplicative values of a forward reflection coefficient Γ_f,m(t) and a backward reflection coefficient Γ_b,m(t) which multiply the reference signal n′(t) or s′(t) and signals derived from the reference signal n′(t) or s′(t). Each stage of the least-squares lattice predictor outputs a forward prediction error f_m(t) and a backward prediction error b_m(t). The subscript m is indicative of the stage.

[0181] For each set of samples, i.e. one sample of the reference signal n′(t) or s′(t) derived substantially simultaneously with one sample of the measured signal S_λa(t), the sample of the reference signal n′(t) or s′(t) is input to the least-squares lattice predictor 70. The zero-stage forward prediction error f₀(t) and the zero-stage backward prediction error b₀(t) are set equal to the reference signal n′(t) or s′(t). The backward prediction error b₀(t) is delayed by one sample period by the delay element 110 in the first stage of the least-squares lattice predictor 70. Thus, the immediately previous value of the reference n′(t) or s′(t) is used in calculations involving the first-stage delay element 110. The zero-stage forward. prediction error is added to the negative of the delayed zero-stage backward prediction error b₀(t-1) multiplied by the forward reflection coefficient value Γ_f,l(t) register 90 value, to produce a first-stage forward prediction error f₁(t). Additionally, the zero-stage forward prediction error f₀(t) is multiplied by the backward reflection coefficient Γ_b,l(t) register 92 value and added to the delayed zero-stage backward prediction error b₀(t-1) to produce a first-stage backward prediction error b₁(t). In each subsequent stage, m, of the least square lattice predictor 70, the previous forward and backward prediction error values, f_m−1(t) and b_m−1(t-1), the backward prediction error being delayed by one sample period, are used to produce values of the forward and backward prediction errors for the present stage, f_m(t) and b_m(t).

[0182] The backward prediction error b_m(t) is fed to the concurrent stage, m, of the regression filter 80. There it is input to a register 96, which contains a multiplicative regression coefficient value κ_m,λa(t). For example, in the zero-stage of the regression filter 80, the zero-stage backward prediction error b₀(t) is multiplied by the zero-stage regression coefficient κ_0,λa(t) register 96 value and subtracted from the measured value of the signal S_λa(t) at a summing element 106 to produce a first stage estimation error signal e_1,λa(t). The first-stage estimation error signal e_1,λa(t) is a first approximation to either the primary signal or the secondary signal. This first-stage estimation error signal e_1,λa(t) is input to the first-stage of the regression filter 80. The first-stage backward prediction error b₁(t), multiplied by the first-stage regression coefficient κ_1,λa(t) register 96 value is subtracted from the first-stage estimation error signal e_1,λa(t) to produce the second-stage estimation error e_2,λa(t). The second-stage estimation error signal e_2,λa(t) is a second, somewhat better approximation to either the primary signal s_λa(t) or the secondary signal n_λa(t).

[0183] The same processes are repeated in the least-squares lattice predictor 70 and the regression filter 80 for each stage until a good approximation e_m,λa(t), to either the primary signal s_λa(t) or the secondary signal n_λa(t) is determined. Each of the signals discussed above, including the forward prediction error f_m(t), the backward prediction error b_m(t), the estimation error signal e_m,λa(t), is necessary to calculate the forward reflection coefficient Γ_f,m(t), the backward reflection coefficient Γ_b,m(t), and the regression coefficient κ_m,λa(t) register 90, 92, and 96 values in each stage, m. In addition to the forward prediction error f_m(t), the backward prediction error b_m(t), and the estimation error e_m,λa(t) signals, a number of intermediate variables, not shown in FIG. 8 but based on the values labeled in FIG. 8, are required to calculate the forward reflection coefficient Γ_f,m(t), the backward reflection coefficient Γ_b,m(t), and the regression coefficient κ_m,λa(t) register 90, 92, and 96 values.

[0184] Intermediate variables include a weighted sum of the forward prediction error squares ℑ_m(t), a weighted sum of the backward prediction error squares β_m(t), a scalar parameter Δ_m(t), a conversion factor γ_m(t), and another scalar parameter ρ_m,λa(t). The weighted sum of the forward prediction errors ℑ_m(t) is defined as:

[0185] $\begin{array}{cc}{𝔍}_m\left(t\right)=\sum _{i=1}^t{\lambda }^{t-i}{f_m\left(i\right)}^2;& \left(44\right)\end{array}$

[0186] where λ without a wavelength identifier, a or b, is a constant multiplicative value unrelated to wavelength and is typically less than or equal to one, i.e., λ≦1. The weighted sum of the backward prediction errors β_m(t) is defined as:

[0187] $\begin{array}{cc}{\beta }_m\left(t\right)=\sum _{i=1}^t{\lambda }^{t=i}{b_m\left(i\right)}^2& \left(45\right)\end{array}$

[0188] where, again, λ without a wavelength identifier, a or b, is a constant multiplicative value unrelated to wavelength and is typically less than or equal to one, i.e., λ≦1. These weighted sum intermediate error signals can be manipulated such that they are more easily solved for, as described in Chapter 9, §9.3 of the Haykin book referenced above and defined hereinafter in equations (59) and (60).

[0189] Description Of The Joint Process Estimator

[0190] The operation of the joint process estimator 60 is as follows. When the joint process estimator 60 is turned on, the initial values of intermediate variables and signals including the parameter Δ_m−1(t), the weighted sum of the forward prediction error signals ℑ_m−1(t), the weighted sum of the backward prediction error signals β_m−1(t), the parameter ρ_m,λa(t), and the zero-stage estimation error e_0,λa(t) are initialized, some to zero and some to a small positive number δ:
Δ_m−1(0)=0;   (46)
ℑ_m−1(0)=δ;   (47)
β_m−1(0)=δ;   (48)
ρ_m,λa(0)=0;   (49)
e_0,λa(t)=S_λa(t) for t≧0.   (50)

[0191] After initialization, a simultaneous sample of the measured signal S_λa(t) or S_λb(t) and either the secondary reference n′(t) or the primary reference s′(t) are input to the joint process estimator 60, as shown in FIG. 8. The forward and backward prediction error signals f₀(t) and b₀(t), and intermediate variables including the weighted sums of the forward and backward error signals ℑ₀(t) and β₀(t), and the conversion factor γ₀(t) are calculated for the zero-stage according to:
f₀(t)=b₀(t)=n′(t)   (51a)
ℑ₀(t)=β₀(t)=λℑ₀(t-1)+|n′(t)|²   (52a)
γ₀(t-1)=1   (53a)

[0192] if a secondary reference n′(t) is used or according to:
f₀(t)=b₀(t)=s′(t)   (51b)
ℑ₀(t)=β₀(t)=λℑ₀(t-1)+|s′(t)|²   (52b)
^Y0(t-1)=1   (53b)

[0193] if a primary reference s′(t) is used where, again, λ without a wavelength identifier, a or b, is a constant multiplicative value unrelated to wavelength.

[0194] Forward reflection coefficient Γ_f,m(t), backward reflection coefficient Γ_b,m(t), and regression coefficient κ_m,λa(t) register 90, 92 and 96 values in each stage thereafter are set according to the output of the previous stage. The forward reflection coefficient Γ_f,l(t), backward reflection coefficient Γ_b,l(t), and regression coefficient κ_1,λa(t) register 90, 92 and 96 values in the first stage are thus set according to the algorithm using values in the zero-stage of the joint process estimator 60. In each stage, m≧1, intermediate values and register values including the parameter Δ_m−1(t); the forward reflection coefficient Γ_f,m(t) register 90 value; the backward reflection coefficient Γ_b,m(t) register 92 value; the forward and backward error signals f_m(t) and b_m(t); the weighted sum of squared forward prediction errors ℑ_f,m(t), as manipulated in §9.3 of the Haykin book; the weighted sum of squared backward prediction errors β_b,m(t), as manipulated in §9.3 of the Haykin book; the conversion factor γ_m(t); the parameter ρ_m,λa(t); the regression coefficient κ_m,λa(t) register 96 value; and the estimation error e_m+1λa(t) value are set according to:
Δ_m−1(t)=λΔ_m−1(t-1)+{b_m−1(t-1)f*_m−1(t)/γ_m−1(t-1)}  (54)
Γ_f,m(t)={Δ_m−1(t)/β_m−1(t-1)}  (55)
Γ_b,m(t)=−{Δ*_m−1(t)/ℑ_m−1(t)}  (56)
f_m(t)=f_m−1(t)+Γ*_f,m(t)b_m−1(t-1)   (57)
b_m(t)=b_m−1(t-1)+Γ*_b,m(t)f_m−1(t)   (58)
ℑ_m(t)=ℑ_m−1(t)−{|Δ_m−1(t)|²/β_m−1(t-1)}  (59)
β_m(t)=β_m−1(t-1)−{|Δ_m−1(t)|²/β_m−1(t)}  (60)
γ_m(t-1)=γ_m−1(t-1)−{|b_m−1(t-1)|²/β_m−1(t-1)}  (61)
ρ_m,λa(t)=λρ_m,λa(t-1)+{b_m(t)ε*_m,λa(t)/γ_m(t)}  (62)
κ_m,λa(t)={ρ_m,λa(t)/β_m(t)}  (63)
ε_m+1,λa(t)=ε_m,λa(t)−κ*_m(t)b_m(t)   (64)

[0195] where a (*) denotes a complex conjugate.

[0196] These equations cause the error signals f_m(t), b_m(t), e_m,λa(t) to be squared or to be multiplied by one another, in effect squaring the errors, and creating new intermediate error values, such as Δ_m−1(t). The error signals and the intermediate error values are recursively tied together, as shown in the above equations (54) through (64). They interact to minimize the error signals in the next stage.

[0197] After a good approximation to either the primary signal s_λa(t) or the secondary signal n_λa(t) has been determined by the joint process estimator 60, a next set of samples, including a sample of the measured signal S_λa(t) and a sample of either the secondary reference n′(t) or the primary reference s′(t), are input to the joint process estimator 60. The re-initialization process does not re-occur, such that the forward and backward reflection coefficient Γ_f,m(t) and Γ_b,m(t) register 90, 92 values and the regression coefficient κ_m,λa(t) register 96 value reflect the multiplicative values required to estimate either the primary signal portion s_λa(t) or the secondary signal portion n_λa(t) of the sample of S_λa(t) input previously. Thus, information from previous samples is used to estimate either the primary or secondary signal portion of a present set of samples in each stage.

[0198] In a more numerically stable and preferred embodiment of the above described joint process estimator, a normalized joint process estimator is used. This version of the joint process estimator normalizes several variables of the above-described joint process estimator such that the normalized variables fall between −1 and 1. The derivation of the normalized joint process estimator is motivated in the Haykin text as problem 12 on page 640 by redefining the variables defined according to the following conditions:

[0199] $\begin{array}{c}{\stackrel{\_}{f}}_m\left(t\right)=\frac{f_m\left(t\right)}{\sqrt{{𝔍}_m\left(t\right){\gamma }_m\left(t-1\right)}}\\ \stackrel{\_}{b}\left(t\right)=\frac{b_m\left(t\right)}{\sqrt{{\beta }_m\left(t\right){\gamma }_m\left(t\right)}}\\ {\stackrel{\_}{\Delta }}_m\left(t\right)=\frac{{\Delta }_m\left(t\right)}{\sqrt{{𝔍}_m\left(t\right){\beta }_m\left(t-1\right)}}\end{array}$

[0200] This transformation allows the conversion of Equations (54)-(64) to the following normalized equations:

[0201] $\begin{array}{c}{\stackrel{\_}{\Delta }}_{m-1}\left(t\right)={{{\stackrel{\_}{\Delta }}_{m-1}\left(t-1\right)\left[1-{f_{m-1}\left(t\right)}^2\right]}^{1/2}\left[1-{{\stackrel{\_}{b}}_{m-1}\left(t-1\right)}^2\right]}^{1/2}+\\ {\stackrel{\_}{b}}_{m-1}\left(t-1\right){\stackrel{\_}{f}}_{m-1}\left(t\right)\\ {\stackrel{\_}{b}}_m\left(t\right)=\frac{\left[{\stackrel{\_}{b}}_{m-1}\left(t-1\right)-{\stackrel{\_}{\Delta }}_{m-1}\left(t\right){\stackrel{\_}{f}}_{m-1}\left(t\right)\right]}{{{\left[1-{{\stackrel{\_}{\Delta }}_{m-1}\left(t\right)}^2\right]}^{1/2}\left[1-{{\stackrel{\_}{f}}_{m-1}\left(t\right)}^2\right]}^{1/2}}\\ {\stackrel{\_}{f}}_m\left(t\right)=\frac{\left[{\stackrel{\_}{f}}_{m-1}\left(t\right)-{\stackrel{\_}{\Delta }}_{m-1}\left(t\right){\stackrel{\_}{b}}_{m-1}\left(t-1\right)\right]}{{{\left[1-{{\stackrel{\_}{\Delta }}_{m-1}\left(t\right)}^2\right]}^{1/2}\left[1-{{\stackrel{\_}{b}}_{m-1}\left(t-1\right)}^2\right]}^{1/2}}\\ {\beta }_m\left(t\right)=\left[1-{{\stackrel{\_}{\Delta }}_{m-1}\left(t\right)}^2\right]{\beta }_{m-1}\left(t-1\right)\\ {\gamma }_m\left(t\right)={\gamma }_{m-1}\left(t\right)\left[1-{{\stackrel{\_}{b}}_{m-1}\left(t\right)}^2\right]\\ {\rho }_m\left(t\right)=\lambda ·{\left[\frac{{\gamma }_m\left(t\right){\beta }_m\left(t-1\right)}{{\gamma }_m\left(t-1\right){\beta }_m\left(t\right)}\right]}^{1/2}{\rho }_m\left(t-1\right)+{\stackrel{\_}{b}}_m\left(t\right){\varepsilon }_m\left(t\right)\\ {\varepsilon }_{m+1,\lambda a}\left(t\right)={\varepsilon }_{m,.\lambda a}\left(t\right)-{\rho }_m\left(t\right){\stackrel{\_}{b}}_m\left(t\right)\end{array}$

[0202] Initialization of Normalized Joint Process Estimator

[0203] Let N(t) be defined as the reference noise input at time index n and U(t) be defined as combined signal plus noise input at time index t the following equations apply (see Haykin, p. 619):

[0204] 1. To initialize the algorithm, at time t=0 set
Δ_m−1(0)=0
β_m−1(0)=δ=10⁻⁶
γ₀(0)=1

[0205] 2. At each instant t≧1, generate the various zeroth-order variables as follows:

[0206] $\begin{array}{c}{\gamma }_0\left(t-1\right)=1\\ {\beta }_0\left(t\right)=\lambda {\beta }_0\left(t-1\right)+N^2\left(t\right)\\ {\stackrel{\_}{b}}_0\left(t\right)={\stackrel{\_}{f}}_0\left(t\right)=\frac{N\left(t\right)}{\sqrt{{\beta }_0\left(t\right)}}\end{array}$