# Patent

## Title
Method and apparatus for estimating respiratory impedance

## Abstract

The acoustic impedance of the respiratory system can be inferred from oscillations that are generated in an airway of a subject. The impedance describes the frequency-dependent relation between the resulting oscillations in flow and pressure. When the impedance varies from inspiration to expiration, it has to be estimated with a high time resolution. A method is provided that reliably estimates the impedance in time intervals that are short enough for physiological purposes. A simple version of the uncertainty principle has been derived for discrete time and frequency. A discrete time-frequency transform has been developed that gives an optimal time-frequency resolution according to this principle. The transform is orthonormal, which permits an analysis of variance in the discrete time-frequency domain. The impedance follows from bivariate least-squares analysis in the time-frequency domain, under the assumption that noise is present in both flow and pressure.

## Background
TECHNICAL FIELD OF THE INVENTION
The invention relates to a method and apparatus for estimating respiratory impedance from forced pressure oscillations.
BACKGROUND TO THE INVENTION
The respiratory or acoustic impedance of a human respiratory system can be measured to obtain information concerning the resistance, compliance and inertia of the airways, lungs and chest wall. This information is useful in diagnosing the nature and severity of a variety of respiratory diseases such as chronic obstructive pulmonary disease (COPD), asthma and bronchitis.
In a forced oscillation technique (FOT), such as that described in“Expiratory Flow Limitation Detected by Forced Oscillation and Negative Expiratory Pressure” by Dellacà et al., pages 363-374 European Respiratory Journal Vol. 29 No. 2 (reference 1), acoustic waves are directed into the respiratory system while the person breathes normally, and the response is measured to determine the respiratory impedance.
The respiratory impedance describes the frequency-dependent relation in the oscillations resulting from the acoustic waves in terms of flow and pressure. Where the respiratory impedance varies from inspiration to expiration (as in some diseases and other medical conditions), the respiratory impedance must be estimated with a fine time resolution. However, in the prior art, little consideration has been given to the reliability of techniques for estimating the respiratory impedance in time intervals that are short enough for physiological purposes (i.e. shorter than the duration of an inhalation or exhalation).
Therefore, there is a need for a method and apparatus for reliably estimating the respiratory impedance in short time intervals.
SUMMARY OF THE INVENTION
According to a first aspect of the invention, there is provided a method of estimating respiratory impedance, the method comprising generating pressure waves in a patient interface device in communication with an airway of a subject; determining the flow and pressure of the gas in a pneumatic system that includes the patient interface device and an airway of such a subject to produce respective time series representing the flow and pressure; transforming the respective time series to the time-frequency domain to create a transformed time series; estimating the power of the flow and pressure from the respective transformed time series; estimating respective cross spectra of the flow and pressure based on the transformed time series; and estimating the respiratory impedance of the subject from the estimated power and cross spectra.
According to a second aspect of the invention, there is provided an apparatus for estimating respiratory impedance, the apparatus comprising a patient interface device; an exitation source for generating oscillating pressure, flow, or volume of gas in such an airway of such a subject; means for determining the flow and pressure of gas in a pneumatic circuit defined by the patient interface device and an airway of such a subject and for outputting respective time series of values representing the flow and pressure; a processor configured to transform the respective time series to the time-frequency domain; estimate a power of the flow and the pressure from the respective transformed time series; estimate respective cross spectra of the flow and pressure based on the respective transformed time series; and estimate the respiratory impedance of the subject from the estimated power and cross spectra.
According to a third aspect of the invention, there is provided a computer program product comprising a computer readable medium with computer readable code embodied therein, the computer readable code being configured such that, on execution by a suitable processor or computer, the processor or computer performs the method described above.
The estimate of respiratory impedance may be made by a diagnostic tool that uses the estimate to assess obstruction of the airways or to estimate the severity of disease. The diagnostic tool may therefore also be used to assess the effectiveness of treatments (whether pharmacological or otherwise) that should affect the respiratory impedance.
Thus, according to a fourth aspect of the invention, there is provided a method of diagnosing a physiological condition, the method comprising measuring respiratory impedance as described above, and diagnosing a physiological condition on the basis of the measured respiratory impedance.
The estimates of respiratory impedance may also or alternatively be used to adapt the settings of machine used in the treatment of a medical condition, for example a non-invasive ventilator that is used for counteracting airway obstruction, or in determining a treatment regimen (for example which particular medication or device to use, the medication dosage, etc.) for the physiological condition.
Thus, according to a fifth aspect of the invention, there is provided a method of treatment of a physiological condition, the method comprising measuring respiratory impedance as described above, and determining and/or administering a treatment for the physiological condition on the basis of the measured respiratory impedance.



BRIEF DESCRIPTION OF THE DRAWINGS
Preferred embodiments of the invention will now be described in detail, by way of example only, with reference to the following drawings, in which:
FIG. 1 is a block diagram of an apparatus for estimating the respiratory impedance from forced pressure oscillations in accordance with the invention;
FIG. 2 is a schematic diagram of the linear model used to estimate the respiratory impedance in accordance with the invention;
FIGS. 3A and 3B illustrate the circularity assumption and the uncertainty in the discrete time domain respectively;
FIGS. 4A and 4B illustrate time series for which Δt2+N2Δf2 is minimal;
FIG. 5 illustrates the bivariate least squares in the time-frequency domain;
FIG. 6 illustrates confidence region in the complex plane;
FIG. 7 illustrates the respiratory impedance in a healthy patient as a function of time and frequency with confidence limits for forced pressure oscillations at five different frequencies;
FIG. 8 illustrates the respiratory impedance in a patient with COPD as a function of time and frequency with confidence limits for forced pressure oscillations at five different frequencies;
FIG. 9 is a flow chart illustrating the steps in the method of estimating the respiratory impedance in accordance with the invention;
FIG. 10 is a flow chart illustrating the processing steps performed by the apparatus according to the invention in more detail;
FIG. 11 shows the minimal and maximal singular values of the matrix C as a function of the number of events N in a time series;
FIG. 12 illustrates the relation between uncertainty in time Δt and the centers of gravity of the eigenvectors of CHC for N=16;
FIG. 13 illustrates the relation between uncertainty in time Δt and uncertainty in frequency Δf for N=16;
FIG. 14 is a graph plotting νmin,n against fn;
FIG. 15 illustrates the uncertainty in the plot of NΔf vs. Δt for N=16;
FIG. 16 illustrates the estimation of 100(1−α) % confidence limits for {circumflex over (Z)}t,m in the complex plane;
FIG. 17 illustrates the squared coherence as a function of time and frequency for a healthy subject; and
FIG. 18 illustrates the squared coherence as a function of time and frequency for a subject with COPD.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
As described above, the acoustic impedance of the respiratory system can be inferred from forced pressure oscillations in a mouthpiece or facemask (forced oscillation technique, FOT). Since this ‘respiratory impedance’ depends on resistance, compliance and inertia of the airways, lungs and chest wall, it provides insight into the nature and severity of a variety of respiratory diseases. The frequency-dependent impedance is often variable in time. In chronic obstructive pulmonary disease, airway resistance usually increases during expiration. In sleep apnea or snoring, resistance may increase during inspiration. This asks for a reliable estimation of impedance in short time intervals, shorter than the duration of a breath.
Respiratory impedance is defined as the complex-valued transfer function that describes a linear relation between flow q and pressure p as a function of frequency f,

p(f)=q(f)z(f),  (1)

where the impedance is denoted by z(f). This equation only holds if z(f) is constant and time is infinite. If the system is temporarily stable, z(f) may still be estimated in a short time interval. This is limited, however, by the uncertainty principle, which says that a signal cannot be sharply localized in both the time and frequency domains. In addition, the use of short time intervals makes the estimation increasingly dependent on chance events. The question is how this would affect the estimation of a transfer function like z(f), which may only be stable in a part of the respiratory cycle.

In the following, the transfer function of a temporarily stationary process is estimated by linear regression in the discrete time-frequency domain. An uncertainty relation is derived for discrete time and frequency, which is used to obtain an optimal time-frequency resolution. Analysis of the statistical properties of the used impedance estimator yields a frequency-dependent parameter of respiratory mechanics, of which both the estimated values and the confidence limits can be followed in time.
A table (Table 1) is included at the end of the Detailed Description which provides a glossary of symbols and abbreviations used in the following explanation of the invention.
Methods
This section of the Detailed Description sets out the mathematical basis for the method and apparatus according to the invention. A less mathematical explanation of the invention is provided in the “Discussion” section below.
A schematic diagram of an apparatus 2 according to the invention is shown in FIG. 1. Briefly, the used FOT device 2 consists of a patient interface device 4 connected to an airway of a subject to create a pneumatic system that includes an airway of such a subject. Patient interface device 4 is any device suitable to provide a pneumatic connection with the airway of a subject, such a mask or mouthpiece. Patient interface device 4 is operatively coupled to an excitation source 6 that generates pressure waves, flow changes, or volume changes with relatively low frequencies, for example frequencies from 8 to 24 Hz. In an exemplary embodiment of the present invention, patient interface device 4 is a loudspeaker.
In the illustrated embodiment, a patient breathes air through a mesh-wired resistance 9 that is near to pneumotach head 8, which is located between the patient interface device 4 and exitation source 6. The forced waves are partly reflected in the airways and lungs of the patient and lead to oscillations in airflow and pressure which are measured in the FOT device 2. The airflow through the pneumotach head 8 is measured using a differential pressure transducer 10 and the pressure is measured using a second pressure transducer 12 near to the mouthpiece 4. An analog-to-digital convertor 12 yields respective time series of flow and pressure, {qt} and {pt}, where the integer t is the discrete time index, which are provided to a processor or computer 16 for analysis. The computer or processor 16 can also provide the signals for controlling the frequency of the acoustic waves generated by the loudspeaker 6 via the analog-to-digital convertor 14 and an amplifier 18. Further details of a specific embodiment of the apparatus 2 are provided in the section below entitled “The measurement apparatus”.
The present invention also contemplates that other gas flow delivery devices can be coupled to the airway of the subject in addition to excitation source 6. For example, a pressure support system or ventilator can be coupled to patient interface device 4 to provide a flow of gas, for example, while excitation source 6 oscillates the airway of the subject. It should also be noted that the flow and pressure measurements can be made at any location along the pneumatic system, including measuring the pressure and flow within the pressure support system or ventilator. The measured pressure and flow can then be used to determine the flow and pressure of the gas in the pneumatic system, such as at the airway of the patient, using any conventional technique.
The present invention also contemplates that excitation source 6 can be a components of the pressure support system or the ventilator rather than a stand-alone apparatus. For example, a pressure support system or ventilator may include a valve to control the flow/pressure of gas delivered to the patient. The present invention contemplates oscillating such a valve to produce the low frequency pressure waves, flow changes, or volume changes. Of course, any other device or system that is capable of generating the low frequency pressure waves, flow changes, or volume changes, such as a piston, can be used as excitation source 6.
Rather than measuring pressure and/or flow using a pressure and/or flow sensors, the present invention also contemplates that the flow or the pressure can be set or controlled, such as by the pressure support system or ventilator. In which case, the set pressure or the set flow is used as the flow of pressure for present purposes rather than the output of differential pressure transducer 10 or second pressure transducer 12.
The interval between two samples (in seconds) is the reciprocal of the sample rate (in Hz). In this section, the time- and frequency-dependent impedance is derived from these time series, using relatively simple linear algebra. The reader who is not familiar with linear algebra is referred to the Discussion section below, where the main results are summarized in words.
A Simple Linear Model.
To account for chance events, assume that the bivariate time series {qt, pt} is a realization of the bivariate stochastic process {Qt, Pt}. Deterministic variables are written in lowercase and random variables (RVs) in uppercase. Further assume that time is infinitely long (t= . . . , −1, 0, 1, . . . ). Referring now to FIG. 2, the excitation source input xt is filtered through linear time-invariant filters (2 and 3) with impulse response sequences {hqx,l} and {hpx,l}. This gives flow and pressure as predicted by the linear model,




  
    
      
        
          q
          
            p
            ,
            t
          
        
        ≡
        
          
            ∑
            
              l
              =
              0
            
            ∞
          
          ⁢
          
            
              h
              
                qx
                ,
                l
              
            
            ⁢
            
              x
              
                t
                -
                l
              
            
            ⁢
            
              
            
            ⁢
            and
            ⁢
            
              
            
            ⁢
            
              p
              
                p
                ,
                t
              
            
          
        
        ≡
        
          
            ∑
            
              l
              =
              0
            
            ∞
          
          ⁢
          
            
              h
              
                px
                ,
                l
              
            
            ⁢
            
              
                x
                
                  t
                  -
                  l
                
              
              .
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




Suppose that these ‘true’ values are disturbed by two independent sources of zero mean white noise with respective variances σQ2 and σP2. These sources of noise are also passed through linear time-invariant filters (1 and 4) to give the ‘error’ terms Qe,t and Pe,t. The complete model is then written as




  
    
      
        {
        
          
            
              
                
                  Q
                  t
                
                =
                
                  
                    μ
                    Q
                  
                  +
                  
                    q
                    
                      p
                      ,
                      t
                    
                  
                  +
                  
                    Q
                    
                      e
                      ,
                      t
                    
                  
                
              
            
          
          
            
              
                
                  P
                  t
                
                =
                
                  
                    μ
                    P
                  
                  +
                  
                    p
                    
                      p
                      ,
                      t
                    
                  
                  +
                  
                    P
                    
                      e
                      ,
                      t
                    
                  
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




where the mean values μQ and μP are constants. Since the resulting Qt and Pt are normally distributed, the supposed bivariate process {Qt, Pt} is completely stationary.

The Circularity Assumption.
While the described stationary process is infinitely long, it is an aim of the invention to describe transient phenomena, where stationary can only be assumed for a finite sequence of N events. This can be solved by assuming that time is circular, in the sense that the last event precedes the first. The resulting time series may then be represented by placing the N events at regular intervals along a circle, like a clock. One can endlessly pursue the circle in the same direction without leaving it. The resulting infinite time series is periodic in N. That is, for any integer j,

xt=xt+jN.  (4)

Substituting u=t−l+j N, the first convolution in Eq. 2 can be rewritten as




  
    q
    
      p
      ,
      t
    
  
  =
  
    
      ∑
      
        u
        =
        0
      
      
        N
        -
        1
      
    
    ⁢
    
      
        ∑
        
          j
          =
          
            -
            ∞
          
        
        ∞
      
      ⁢
      
        
          h
          
            qx
            ,
            
              t
              -
              u
              +
              jN
            
          
        
        ⁢
        
          
            x
            
              u
              -
              jN
            
          
          .
        
      
    
  




Due to the supposed periodic character of xt (Eq. 4), xu−jN=xu. The infinite sum in the above equation can be defined as hqx,t−u ‘periodized to length N’, denoted by




  
    
      
        
          h
          
            qx
            ,
            
              t
              -
              u
            
          
          o
        
        ≡
        
          
            ∑
            
              j
              =
              
                -
                ∞
              
            
            ∞
          
          ⁢
          
            h
            
              qx
              ,
              
                t
                -
                u
                +
                
                  jN
                  .
                
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




As a result,





  
    
      
        
          q
          
            p
            ,
            t
          
        
        =
        
          
            ∑
            
              u
              =
              0
            
            
              N
              -
              1
            
          
          ⁢
          
            
              h
              
                qx
                ,
                
                  t
                  -
                  u
                
              
              o
            
            ⁢
            
              
                x
                u
              
              .
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




Since hqx,t−uo is also periodic in N, the index t−u can be augmented with N if t−u is negative (so that the index always falls in the range from zero to N−1, which is convenient). When the sequences are represented by column vectors, Eq. 6 can also be written as (if N=4)




  
    
      
        
          [
          
            
              
                
                  q
                  
                    p
                    ,
                    0
                  
                
              
            
            
              
                
                  q
                  
                    p
                    ,
                    1
                  
                
              
            
            
              
                
                  q
                  
                    p
                    ,
                    2
                  
                
              
            
            
              
                
                  q
                  
                    p
                    ,
                    3
                  
                
              
            
          
          ]
        
        =
        
          
            
              [
              
                
                  
                    
                      h
                      
                        qx
                        ,
                        0
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        3
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        2
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        1
                      
                      o
                    
                  
                
                
                  
                    
                      h
                      
                        qx
                        ,
                        1
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        0
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        3
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        2
                      
                      o
                    
                  
                
                
                  
                    
                      h
                      
                        qx
                        ,
                        2
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        1
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        0
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        3
                      
                      o
                    
                  
                
                
                  
                    
                      h
                      
                        qx
                        ,
                        3
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        2
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        1
                      
                      o
                    
                  
                  
                    
                      h
                      
                        qx
                        ,
                        0
                      
                      o
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    
                      x
                      0
                    
                  
                
                
                  
                    
                      x
                      1
                    
                  
                
                
                  
                    
                      x
                      2
                    
                  
                
                
                  
                    
                      x
                      3
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        7
        )
      
    
  




The N×N matrix in this equation is a circulant (a square matrix whose rows are right-shifted versions of the previous row, with wraparound at the edges). When the first row is denoted by hqxH, where the superscript H stands for the Hermitian transpose, the circulant may also be written as circ{hqxH}. Let x≡{x0, . . . , xN−1}, Cqx≡circ{hqxH} and qp≡{qp,0, . . . , qp,N−1}. Then qp=Cqxx. Similarly, pp=Cpxx, so that Eq. 3 becomes




  
    
      
        {
        
          
            
              
                
                  Q
                  =
                  
                    
                      μ
                      Q
                    
                    +
                    
                      q
                      p
                    
                    +
                    
                      Q
                      e
                    
                  
                
              
            
            
              
                
                  P
                  =
                  
                    
                      μ
                      P
                    
                    +
                    
                      p
                      p
                    
                    +
                    
                      P
                      e
                    
                  
                
              
            
          
          ,
          
            or
            ⁢
            
              

            
            ⁢
            
              {
              
                
                  
                    
                      Q
                      =
                      
                        
                          μ
                          Q
                        
                        +
                        
                          
                            C
                            qx
                          
                          ⁢
                          x
                        
                        +
                        
                          Q
                          e
                        
                      
                    
                  
                
                
                  
                    
                      P
                      =
                      
                        
                          μ
                          P
                        
                        +
                        
                          
                            C
                            px
                          
                          ⁢
                          x
                        
                        +
                        
                          
                            P
                            e
                          
                          .
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




The Discrete Frequency Domain.
To derive the frequency-dependent impedance, the model of Eq. 8 has to be transformed to the discrete frequency domain. As a step-up to time-frequency analysis, this is briefly summarized here. Transformation of a given time series x≡{x0, . . . , xN−1} to the discrete frequency domain partitions the series into a set of harmonic oscillations with discrete frequencies fn≡n/N, for n=0, . . . , N−1. Such an oscillation is described by the N-vector

fn≡{1,ωn,ωn·2, . . . ,ωnt, . . . ,ωn(N−1)},  (9)

where ω≡exp(2πi/N) and i2=−1. Due to Euler's theorem,

exp(2πint/N)=cos(2πnt/N)+i sin(2πnt/N),  (10)

so that the vector fn actually describes a combined oscillation along the real and imaginary axis with frequency fn. It is readily shown that





  
    
      
        
          
            f
            n
            H
          
          ⁢
          
            f
            
              n
              ′
            
          
        
        =
        
          {
          
            
              
                
                  N
                  ,
                
              
              
                
                  
                    n
                    =
                    
                      n
                      ′
                    
                  
                  ;
                
              
            
            
              
                
                  0
                  ,
                
              
              
                
                  n
                  ≠
                  
                    
                      n
                      ′
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




As a result, the squared norm ∥fn∥2=fnHfn equals N, while two oscillations at different harmonic frequencies are orthogonal.
Transformation of x to the discrete frequency domain is performed by the ‘orthonormal discrete Fourier transform’ (ODFT) matrix F, defined as




  
    
      
        
          
            F
            ≡
            
              
                1
                
                  N
                
              
              ⁢
              
                {
                
                  ω
                  
                    -
                    nt
                  
                
                }
              
            
          
          =
          
            
              1
              
                N
              
            
            ⁡
            
              [
              
                
                  
                    
                      f
                      0
                      H
                    
                  
                
                
                  
                    ⋮
                  
                
                
                  
                    
                      f
                      
                        N
                        -
                        1
                      
                      H
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        12
        )
      
    
  




where n,t=0, . . . , N−1. This transform yields the N-vector F x, whose components are the inner products fnHx/√{square root over (N)}. From Eq. 11, it follows that the rows (and columns) of F are orthonormal so that the matrix is unitary,

FFH=FHF=I,  (13)

where I is the N×N identity matrix. This permits a synthesis of x from the N oscillations,





  
    
      
        x
        =
        
          Ix
          =
          
            
              
                F
                H
              
              ⁢
              Fx
            
            =
            
              
                
                  
                    
                      1
                      N
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              f
                              0
                            
                          
                          
                            ⋯
                          
                          
                            
                              f
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            f
                            0
                            H
                          
                        
                      
                      
                        
                          ⋮
                        
                      
                      
                        
                          
                            f
                            
                              N
                              -
                              1
                            
                            H
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                x
              
              =
              
                
                  1
                  N
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      0
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        f
                        n
                      
                      ⁡
                      
                        (
                        
                          
                            f
                            n
                            H
                          
                          ⁢
                          x
                        
                        )
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Accordingly, x is written as the weighted sum of N harmonic oscillations with frequencies fn.
The vectors fn are eigenvectors of circulant matrices, so that we can write

Cqxfn=fnHqx,n.  (15)

The complex number Hqx,n is the transfer function from x to q for frequency fn, defined as

Hqx,n≡hqxHfn.  (16)

When Hpx,n is defined in similar manner, the impedance for frequency fn is

zn≡Hpx,n/Hqx,n.  (17)

When the eigenvalues are put into the diagonal matrix Λqx≡diag{Hqx,0, . . . , Hqx,N−1} and the corresponding eigenvectors as columns into the N×N matrix FH, Eq. 15 becomes

CqxFH=FHΛqx.  (18)

Circulants are normal matrices (they commute with their Hermitian transpose). Eq. 18 is therefore in agreement with the spectral theorem, which says that an N×N normal matrix has N orthogonal eigenvectors (in this case, the N orthogonal column vectors fn).
The Uncertainty Principle for Discrete Time and Frequency.
The components of x≡{xt} are sharply localized in the time domain, but uncertain in the frequency domain. Conversely, the components of the Fourier transform F x are sharply localized in the frequency domain, but uncertain in the time domain. To obtain a signal with small uncertainty in both domains, a measure of uncertainty has to be defined.
Assuming that time is circular, the approach of Forbes and Alonso seems most appropriate (see reference 3). Referring now to FIG. 3A, in the time domain, the squared components |xt|2 are placed as point masses at regular intervals on a circle with radius rt in the complex plane, centered at the origin, with the size of each point mass in FIG. 3A corresponding to the weight of each event. Time is represented by the arc length between two events. The circumference of the circle is the total period N (where N=12 in this illustration), so that the radius equals rt≡N/(2π). A given event at time t is located at point rtωt in the complex plane. Forbes and Alonso (reference 3) proposed several measures of uncertainty. Here a slightly different measure is used, defined as the weighted mean of the squared distances between each event at time t and the event at a reference time tR,




  
    
      
        
          Δ
          t
          2
        
        ≡
        
          
            ∑
            
              t
              =
              0
            
            
              N
              -
              1
            
          
          ⁢
          
            
              
                
                
                  
                    
                      r
                      t
                    
                    ⁢
                    
                      ω
                      t
                    
                  
                  -
                  
                    
                      r
                      t
                    
                    ⁢
                    
                      ω
                      
                        t
                        R
                      
                    
                  
                
                
              
              2
            
            ⁢
            
              
                
                  
                  
                    x
                    t
                  
                  
                
                2
              
              /
              
                
                  ∑
                  
                    t
                    =
                    0
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      
                      
                        x
                        t
                      
                      
                    
                    2
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




This measure of uncertainty has the advantage that it can easily be expressed as a ratio of quadratic forms. Without loss of generality, it can be assumed that tR=0 (as shown in FIG. 3A). Then the squared uncertainty Δt2 is

Δt2≡rt2xH(Ω−I)H(Ω−I)x/∥x∥2,  (20)

where the diagonal matrix Ω≡diag{1, ω1, . . . , ωN−1}. It is possible to define

Δt≡rt∥Ax∥/∥x∥,  (21)

where A≡Ω−I. Physically, Δt2 is the second moment of inertia of the set of point masses about an axis (perpendicular to the plane of the circle) through the reference point R (corresponding to time tR). As depicted in FIG. 3B, this moment of inertia is directly related to the center of gravity. Geometrically, Δt equals the distance from R to one of the two points where the vertical line through the center of gravity intersects with the circle. So, Δt is fully determined by the horizontal distance from the center of gravity to R. If the center of gravity coincides with R, then the weight of x is entirely concentrated at time tR and Δt is zero. If the center of gravity lies at the opposite point of the circle, then the weight is entirely concentrated at that point and Δt is maximal, equal to 2rt. Thus, in FIG. 3B, the vertical line through the center of gravity B crosses the circle at point C. The uncertainty Δt equals the distance CR.

In the frequency domain, the components of F x are associated with frequencies fn≡n/N. The same components would be obtained for frequencies n/N+j, where j is any integer (see Eqs. 9, 10). The components of F x are therefore periodic in the frequency domain, with unit period. A comparable measure of uncertainty can thus be defined in the frequency domain, relative to a reference frequency fR. The squared magnitudes of the components of F x are again placed as point masses on a circle in the complex plane. The radius now equals rf≡1/(2π) so that the circumference equals unity. If fR=0, then

Δf≡rf∥(Ω−I)Fx∥/∥x∥.  (22)

The diagonal matrix Ω is related to the ‘time shift operator’ T≡circ{0, . . . , 0, 1}. Premultiplication of x by T circularly shifts the components of x one place downward. It readily follows from Eq. 18 that F T−1=ΩF (the ‘shift theorem’), so that

Δf=rf∥(ΩF−F)x∥/∥x∥=rf∥F(T−1−I)x∥/∥x∥. 

Since F is unitary, ∥F(T−1−I)x∥=∥(T−1−I)x∥. Letting B≡T−1−I,

Δf≡rf∥Bx∥/∥x∥.  (23)

Now consider the matrix




  C
  ≡
  
    
      
        r
        t
      
      ⁡
      
        [
        
          
            
              A
            
          
          
            
              B
            
          
        
        ]
      
    
    .
  




The total uncertainty in time and frequency may be expressed by the ratio

xHCHCx/∥x∥2=rt2xH(AHA+BHB)x/∥x∥2=Δt2+N2Δf2.  (24)

This ratio (a Rayleigh quotient) can take values in the range from the minimal to the maximal squared singular value of C. This yields the corresponding uncertainty principle for discrete time and frequency,

σmin2≦Δt2+N2Δf2≦σmax2.  (25)

Accordingly, it is impossible that Δt and Δf are both very small, since the total uncertainty Δt2+N2Δf2 should at least be equal to σmin2. The total uncertainty is minimal if x is an eigenvector of CHC that corresponds to Δmin2. The eigenvectors of CHC can be regarded as discrete Mathieu functions. An interesting property of CHC is that it commutes with F, so that CHC and F share the same eigenvectors. The eigenvalue of F that corresponds to vmin equals unity, so that F vmin=vmin. As is depicted in FIG. 4A, this means that Δt=NΔf if x equals vmin. FIG. 4 shows the time series for which the sum Δt2+N2Δf2 is minimal. In FIG. 4A, the eigenvector is derived for reference time tR=0 and frequency fR=0 (upper graph). This is also an eigenvector of the Fourier matrix F (with unit eigenvalue), so that the time series remains unchanged after transformation to the discrete frequency domain (lower graph). In these graphs, both of the time and frequency axes are circularly shifted for visual reasons. FIG. 4B corresponds to FIG. 4A but with a reference frequency fR=1⁄4 In FIG. 4B, closed circles represent real values, open circles represent imaginary values, and crosses indicates the absolute values.
See the section entitled“Derivation of the uncertainty principle” below for a more detailed analysis of the uncertainty principle.
The Discrete Time-Frequency Domain.
The ODFT partitions a time series of N events into a set of oscillations with N different frequencies. A simple way to perform a time-frequency analysis is to compute the ODFT repeatedly for short-lasting sequences of M successive events (with M<N). Accordingly, the series is partitioned into M oscillations with frequencies fm≡m/M (with m=0, . . . , M−1), which amounts to a discrete ‘short-time Fourier transform’. The resulting uncertainty in time and frequency is minimal if each short-lasting oscillation is ‘windowed’ with the components of vmin.
For a sequence of M events, such a windowed oscillation is described by the M-vector

vmin(m)=Ωmvmin.  (26)

An example is shown in FIG. 4B. Note that vmin(m) is in fact a circularly shifted version of vmin in the frequency domain. As is illustrated in FIG. 4 and proven in the “Derivation of the uncertainty principle” section below (see Eq. B19), it has the same uncertainty in time and frequency as the eigenvector vmin (if fm is taken as reference frequency). Let the components of vmin(m) be denoted by νmin,t(m). An N-vector that describes a short-lasting oscillation is then obtained by inserting zeros in the middle of vmin(m) (so that the unimodal structure of the vector remains intact). This gives

hm,0≡{νmin,0(m), . . . ,νmin,K(m),0, . . . ,0,νmin,K+1(m), . . . ,νmin,M−1(m)}/∥vmin∥,  (27)

where K is the smallest integer less than or equal to M/2. This oscillation is centered about time t=0 and frequency fm=m/M. In general, the oscillation in

hm,t≡Tthm,0  (28)

is centered about time t and frequency fm.

The best-fitting linear relation between x and hm,t in the least-squares sense follows from the orthogonal projection of x onto the (complex) one-dimensional subspace spanned by hm,t. Taking into account that hm,t is defined as a unit vector (see Eq. 27), this projection is

hm,thm,tHx/∥hm,t∥2=hm,thm,tHx.  (29)

For a given frequency fm, let the N×N matrix Mm be defined as

Mm≡circ{hm,0H}.  (30)

This is a circulant that acts a band-pass filter with resonant frequency fm. Premultiplication of x by Mm gives an N-vector that may be denoted by a tilde or ‘wave sign’,

{tilde over (x)}m≡Mmx.  (31)

The components of {tilde over (x)}m are the inner products {tilde over (x)}m,t≡hm,tHx.
The full time frequency transform (TFT), encompasses all frequencies fm for m=0, . . . , M−1. Let the MN×N matrix M be defined as




  
    
      
        M
        ≡
        
          
            
              1
              
                M
              
            
            ⁡
            
              [
              
                
                  
                    
                      M
                      0
                    
                  
                
                
                  
                    ⋮
                  
                
                
                  
                    
                      M
                      
                        M
                        -
                        1
                      
                    
                  
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        32
        )
      
    
  




Transformation to ‘the’ discrete time-frequency domain (the time-frequency domain) is performed by premultiplication of x by M (specific for a given M). This gives the MN-vector {tilde over (x)}≡M x. Back-transformation means multiplication again, by MH, giving MH{tilde over (x)}=MHM x. In the section entitled “Derivation of the time-frequency transform” below, it is shown that the columns of M are orthonormal, so that

MHM=I.  (33)

Back-transformation thus recovers the original time series, since MHM x=I x=x. This makes a time-frequency synthesis of x possible according to




  
    
      
        x
        =
        
          
            1
            M
          
          ⁢
          
            
              ∑
              
                m
                =
                0
              
              
                M
                -
                1
              
            
            ⁢
            
              
                ∑
                
                  t
                  =
                  0
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
                  h
                  
                    m
                    ,
                    t
                  
                
                ⁢
                
                  
                    
                      x
                      ~
                    
                    
                      m
                      ,
                      t
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




The time series is thus written as the weighted sum of MN short-lasting oscillations, each centered about time t and frequency fm (cf. Eq. 14). For a given M, these oscillations give an optimal time-frequency resolution according to the uncertainty principle of Eq. 25. As shown in the “Derivation of the uncertainty principle” section below, the oscillations can be approximated by a truncated Gaussian with M′=10·Δt nonzero elements.
Input to the Excitation Source.
It is convenient to use a set of harmonic oscillations as excitation source input x. To analyze the data, the scale M of the TFT can then be chosen so that the frequencies of these oscillations coincide with the TFT resonant frequencies fm. First consider a single sinusoidal input with frequency fn and amplitude am. Due to Euler's theorem (Eq. 10), a real-valued harmonic oscillation is the sum of a complex-valued oscillation and its complex conjugate. The excitation source input can thus be written

x=1⁄2(amfn+am*fn*),  (35)

where the asterisk denotes the complex conjugate. From Eqs. 9 and 10, it follows that fn*=fN−n. The frequency fn≡n/N can be matched with a TFT resonant frequency fm≡m/M if N is an integer multiple of M. If in is subsequently chosen so that fn=fm,

{tilde over (x)}m=Mm(1⁄2amfn+1⁄2amfN−n)≈1⁄2amfn.  (36)

The latter approximation is valid if M is large enough and the corresponding Δf small enough to suppress the oscillation at frequency fN−n by the TFT filter with resonant frequency fn=fm. A full representation of the excitation source input in the time-frequency domain is obtained if the component at the ‘complementary frequency’ fN−n (here equal to fM-m) is also included by adding MM-m to Mm in Eq. 36.
The model of Eq. 8 is transformed to the time-frequency domain by premultiplying both the equations by M. Concentrating on the frequencies of the interest (e.g., the frequency of the forced oscillation described by Eq. 35), multiplying the upper part of Eq. 8 by Mm yields

MmQ=MmμQ+MmCqxx+MmQe  (37)

It is readily shown that MmμQ is zero if m≠0. Thus,

{tilde over (Q)}m=MmCqxx+{tilde over (Q)}e,m  (38)

Since circulants all commute, it follows that MmCqxx=CqxMmx=Cqx{tilde over (x)}m. If n and m are chosen so that fn=fm, then, using Eqs. 36 and 15,

Cqx{tilde over (x)}m=1⁄2amCqxfn=1⁄2amfnHqx,n={tilde over (x)}mHqx,n  (39)

When Hqx,n for fn=fm is denoted by bQ,m (and the corresponding Hpx,n by bP,m), the model of Eq. 8 reduces to




  
    
      
        {
        
          
            
              
                
                  
                    Q
                    ~
                  
                  m
                
                =
                
                  
                    
                      
                        x
                        ~
                      
                      m
                    
                    ⁢
                    
                      b
                      
                        Q
                        ,
                        m
                      
                    
                  
                  +
                  
                    
                      Q
                      ~
                    
                    
                      e
                      ,
                      m
                    
                  
                
              
            
          
          
            
              
                
                  
                    P
                    ~
                  
                  m
                
                =
                
                  
                    
                      
                        x
                        ~
                      
                      m
                    
                    ⁢
                    
                      b
                      
                        P
                        ,
                        m
                      
                    
                  
                  +
                  
                    
                      P
                      ~
                    
                    
                      e
                      ,
                      m
                    
                  
                
              
            
          
        
      
    
    
      
        (
        40
        )
      
    
  




The impedance for fn=fm is

zm≡bP,m/bQ,m  (41)

A set of other oscillations can be added to the excitation source input of Eq. 35, provided that Δf of the used TFT is small enough so that these oscillations are well separated in the time-frequency domain.
Analysis of Variance in the Time-Frequency Domain.
The TFT permits an analysis of variance in the time-frequency domain (see the section below entitled “Analysis of variance in the time-frequency domain”). For instance, the squared norm of the noise in flow (see Eq. 8) can be partitioned into




  
    
      
        
          
            
            
              Q
              e
            
            
          
          2
        
        =
        
          
            1
            M
          
          ⁢
          
            
              ∑
              
                m
                =
                0
              
              
                M
                -
                1
              
            
            ⁢
            
              
                
                
                  
                    Q
                    ~
                  
                  
                    e
                    ,
                    m
                  
                
                
              
              2
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




The random variable ∥{tilde over (Q)}e,m∥2/N may be called the ‘TFT sample power spectrum’ of the noise Qe. Suppose that the transfer function of filter 1 from noise 1 to Qe,t (FIG. 2) is relatively flat in the passband of the TFT filter about fm. Then {tilde over (Q)}e,m is approximately distributed as if it were derived from white noise with zero mean and variance σQ,m2I. As a result, ∥{tilde over (Q)}e,m∥2/σQ,m2 follows an approximate chi-square distribution with equivalent degrees of freedom η. See section “Analysis of variance in the time-frequency domain” below for a derivation of η, where only those components of {tilde over (Q)}e,m are included that are independent of the circularity assumption (as well as the component at the complementary frequency fM-m).
Bivariate Least Squares in the Time-Frequency Domain.
It is now assumed that at every time t, a new temporarily stationary process starts that is described by the model of FIG. 2. The process encompasses N events and is assumed to be circular (that is, periodic in N). In the time-frequency domain, the model becomes (from Eq. 40)




  
    
      
        {
        
          
            
              
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                =
                
                  
                    
                      
                        x
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    ⁢
                    
                      b
                      
                        Q
                        ,
                        t
                        ,
                        m
                      
                    
                  
                  +
                  
                    
                      Q
                      ~
                    
                    
                      e
                      ,
                      t
                      ,
                      m
                    
                  
                
              
            
          
          
            
              
                
                  
                    P
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                =
                
                  
                    
                      
                        x
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    ⁢
                    
                      b
                      
                        P
                        ,
                        t
                        ,
                        m
                      
                    
                  
                  +
                  
                    
                      P
                      ~
                    
                    
                      e
                      ,
                      t
                      ,
                      m
                    
                  
                
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




where the vectors are ‘short-lasting’ N-vectors now, starting at time t. Note that the constants bQ,m and bP,m have acquired the index t as well, since they are assumed to be specific for the time at which the process starts. The constants, as well as the inputs to the model, may therefore be different at different times t (even if two processes overlap in time).

The best-fitting estimators of bQ,t,m and bP,t,m can be derived from {tilde over (x)}t,m, {tilde over (Q)}t,m and {tilde over (P)}t,m by minimizing the squared norm of the error vectors. This yields the least-squares estimators {circumflex over (B)}Q,t,m and {circumflex over (B)}P,t,m. The estimated relations are




  
    
      
        {
        
          
            
              
                
                  
                    
                      Q
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  =
                  
                    
                      
                        
                          x
                          ~
                        
                        
                          t
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          B
                          ^
                        
                        
                          Q
                          ,
                          t
                          ,
                          m
                        
                      
                    
                    +
                    
                      
                        Q
                        ^
                      
                      
                        e
                        ,
                        t
                        ,
                        m
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      P
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  =
                  
                    
                      
                        
                          x
                          ~
                        
                        
                          t
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          B
                          ~
                        
                        
                          P
                          ,
                          t
                          ,
                          m
                        
                      
                    
                    +
                    
                      
                        P
                        ^
                      
                      
                        e
                        ,
                        t
                        ,
                        m
                      
                    
                  
                
              
            
          
          ,
        
      
    
    
      
        (
        44
        )
      
    
  




where the estimated ‘predicted’ vector {circumflex over (Q)}p,t,m≡{tilde over (x)}t,m{circumflex over (B)}Q,t,m is orthogonal to the estimated ‘error’ {circumflex over (Q)}e,t,m (and, similarly, {circumflex over (P)}p,t,m is orthogonal to {circumflex over (P)}e,t,m). In FIG. 5, the vectors are depicted as arrows and all variables refer to time index t and frequency index m (corresponding to frequency fm=m/M). In particular, in FIG. 5, the excitation source input {tilde over (x)}t,m is a fixed variable, while the flow {tilde over (Q)}t,m and pressure {tilde over (P)}t,m are random variables. The vectors {circumflex over (Q)}p,t,m and {circumflex over (P)}p,t,m are the projections of {tilde over (Q)}t,m and {tilde over (P)}t,m onto the complex one-dimensional subspace spanned by {tilde over (x)}t,m (the line l). The impedance estimator follows from {circumflex over (P)}p,t,m={circumflex over (Q)}p,t,m{circumflex over (Z)}t,m. The error vectors {circumflex over (Q)}e,t,m and {circumflex over (P)}e,t,m are perpendicular to {tilde over (x)}t,m. Each vector is represented by an arrow whose length equals the norm. The angles between the vectors are drawn so that cos2φ and cos2θ equal the squared coherences between {tilde over (x)}t,m and, respectively, {tilde over (Q)}t,m and {tilde over (P)}t,m.

The degree of linear relation between {tilde over (x)}t,m and {tilde over (Q)}t,m comes to expression in the ‘TFT squared sample coherence’




  
    
      
        
          K
          
            Qx
            ,
            t
            ,
            m
          
          2
        
        ≡
        
          
            
              
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                ⁢
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
              
              
            
            2
          
          
            
              
                
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
              
              2
            
            ⁢
            
              
                
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
              
              2
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




The random variable {tilde over (x)}t,mH{tilde over (Q)}t,m/N may be called the ‘TFT sample cross spectrum’ from x to Q. Geometrically, KQx,t,m2 equals cos2φ, where φ is the angle from {tilde over (x)}t,m to {tilde over (Q)}t,m in FIG. 5. Similarly, KPx,t,m2 equals cos2θ. The least-squares estimators are derived in the standard way. This gives

{circumflex over (B)}Q,t,m={tilde over (x)}t,mH{tilde over (Q)}t,m/∥{tilde over (x)}t,m∥2  (46)

Accordingly, {circumflex over (B)}Q,t,m is the ratio of the sample cross spectrum from x to Q to the sample power spectrum of x. When {circumflex over (B)}P,t,m is derived in a similar way, the impedance is estimated by

{circumflex over (Z)}t,m={circumflex over (B)}P,t,m/{circumflex over (B)}Q,t,m  (47)

The ‘true’ impedance zt,m has a real part rt,m (‘resistance’) and imaginary part xt,m (‘reactance’). The corresponding estimators are the real and imaginary parts of {circumflex over (Z)}t,m,

{circumflex over (Z)}t,m={circumflex over (R)}t,m+i{circumflex over (X)}t,m  (48)

As is depicted in FIG. 6A, the 100(1−α)% confidence region for bQ,t,m is delimited by a circle in the complex plane, centered about {circumflex over (B)}Q,t,m. The confidence region is given by

|bQ,t,m−{circumflex over (B)}Q,t,m|2≦|{circumflex over (B)}Q,t,m|2AQ,t,m2,  (49)

where AQ,t,m2≡Fα·2(1−KQx,t,m2)/(η−2)KQx,t,m2 and Fα is the upper 100(1−α)% point of the F-distribution (see the section entitled “Derivation of confidence limit” below). The excitation source input xt is a dimensionless variable so {circumflex over (B)}Q,t,m has units of flow (since {circumflex over (Q)}p,t,m={tilde over (x)}t,m{circumflex over (B)}Q,t,m from FIG. 5). The values that depend on the circularity assumption are discarded here (the effective number of samples is NS=N−M+1). The relation will be rejected if the origin is within the circle (then {circumflex over (B)}Q,t,m is not significantly different from zero at the 100·α% confidence level). Hence, it follows from Eq. 49 that significance is obtained if AQ,t,m<1. The confidence region for {circumflex over (B)}P,t,m is delimited by a comparable circle and the relation between {tilde over (x)}t,m and {tilde over (P)}t,m is considered significant if AP,t,m<1.

A conservative estimate of the confidence region for {circumflex over (Z)}t,m is derived in the section entitled “Derivation of confidence limits” below, under the assumption that {tilde over (Q)}e,t,m and {tilde over (P)}e,t,m are uncorrelated. The region is described by




  
    
      
        
          
            
            
              
                z
                
                  t
                  ,
                  m
                
              
              -
              
                
                  
                    Z
                    ^
                  
                  
                    t
                    ,
                    m
                  
                
                ⁢
                
                  
                    1
                    +
                    
                      
                        A
                        
                          Q
                          ,
                          t
                          ,
                          m
                        
                      
                      ⁢
                      
                        A
                        
                          P
                          ,
                          t
                          ,
                          m
                        
                      
                    
                  
                  
                    1
                    -
                    
                      A
                      
                        Q
                        ,
                        t
                        ,
                        m
                      
                      2
                    
                  
                
              
            
            
          
          2
        
        ≤
        
          
            
            
              
                
                  Z
                  ^
                
                
                  t
                  ,
                  m
                
              
              ⁢
              
                
                  
                    A
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                  +
                  
                    A
                    
                      P
                      ,
                      t
                      ,
                      m
                    
                  
                
                
                  1
                  -
                  
                    A
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                    2
                  
                
              
            
            
          
          2
        
      
    
    
      
        (
        50
        )
      
    
  




Referring to FIG. 6B, the limiting circle is not concentric about the estimator {circumflex over (Z)}t,m. The real and imaginary parts of {circumflex over (Z)}t,m are represented by {circumflex over (R)}t,m and {circumflex over (X)}t,m. The vertical arrow indicates the upper confidence limit for the real part and the horizontal arrow indicates the lower confidence limit for the imaginary part of the impedance. The distribution of ‘true’ zt,m for a given realization of {circumflex over (Z)}t,m is apparently skew-symmetric. In FIG. 6B, the skewness is determined by the angle β. The skewness is entirely due to the noise in flow, as appears from the relation between β and the angle φ in FIG. 5,

tan2β=AQ,t,m2=tan2φ·Fα·2/(η−2)  (51)

If the noise in flow is zero, then φ=0, KQx,t,m2=1 and AQ,t,m=0, so that β=0 and the confidence region is symmetric about {circumflex over (Z)}t,m.
The section entitled “Derivation of confidence limits” below gives a further analysis of what happens if it is assumed that either the noise in flow or the noise in pressure is zero. The respective estimators would then be equal to

{circumflex over (Z)}t,m(Q)≡{tilde over (Q)}t,mH{tilde over (P)}t,m/∥{tilde over (Q)}t,m∥2 and {circumflex over (Z)}t,m(P)≡∥{tilde over (P)}t,m∥2/{tilde over (P)}t,mH{tilde over (Q)}t,m  (52)

This amounts to linear regression from flow to pressure (or conversely) in the time-frequency domain. The variables ∥{tilde over (Q)}t,m∥2/N, ∥{tilde over (P)}t,m∥2/N and {tilde over (Q)}t,mH{tilde over (P)}t,m/N are the respective sample power and cross spectra. The main difference between the estimators of Eq. 52 and {circumflex over (Z)}t,m lies in the magnitude of the estimated impedance. The magnitudes are related through

|{circumflex over (Z)}t,m(Q)/KQx,t,m2=|{circumflex over (Z)}t,m|=|{circumflex over (Z)}t,m(P)|KPx,t,m2  (53)

It follows that, if noise is present in both flow and pressure, the magnitude of impedance is underestimated if {circumflex over (Z)}t,m(Q) is used instead of {circumflex over (Z)}t,m (and overestimated if {circumflex over (Z)}t,m(P) is used).
Settings.
In use of the apparatus 2, flow and pressure were recorded at a sample rate of 800 Hz. A TFT was performed with Δt=50 (which amounts to 50/800=0.0625 s). This would require a width of the TFT filter of M=4πΔt2=31,416. However, the coefficients of the TFT filter were approximated by a Gaussian truncated at M′=10·Δt=500 (see the “Derivation of the uncertainty principle” section below). The associated uncertainty in frequency is Δf=1/(4πΔt)=0.0016 (or 0.0016×800=1.27 Hz). The TFT was derived for fm=0.01, 0.015, 0.02, 0.025, 0.03, corresponding to the imposed FOT frequencies of 8, 12, 16, 20, 24 Hz. Sample power and cross spectra were computed for NS=500 (that is, over a time interval of 500/800=0.625 s). Only data were used that are independent of the circularity assumption. This leads to an approximate chi-square distribution of the sample spectra with η=6.72. Confidence limits for impedance were derived for the 90% confidence level (Fα=4.05). Time-frequency spectra were aligned in time with the original time series.
EXAMPLES
FIG. 7 shows the real and imaginary parts of the estimated impedance {circumflex over (Z)}t,m as a function of time and frequency for a normal subject. The blue lines represent the real part {circumflex over (R)}t,m and the red lines represent the imaginary part {circumflex over (X)}t,m. The grey lines represent the confidence limits as depicted in FIG. 6B. The right hand axis of the 8 Hz frequency band indicates the scale of the components of impedance. At 8 Hz, the real part {circumflex over (R)}t,m is positive and fluctuates in the course of the respiratory cycle, while the imaginary part {circumflex over (X)}t,m is almost equal to zero and remains relatively constant. The confidence limits give an immediate impression of the statistical significance of the changes in {circumflex over (R)}t,m and {circumflex over (X)}t,m in the course of time. At higher frequencies, {circumflex over (R)}t,m decreases while {circumflex over (X)}t,m gradually becomes more positive. The signals were severely disturbed when the subject swallowed on request, which was accompanied by a large increase in the confidence region. The associated coherence spectra are shown in section “Derivation of confidence limits” below.
FIG. 8 shows the impedance for a patient with COPD with large negative swings in {circumflex over (X)}t,m during expiration (which are obviously significantly different from the inspiratory values in view of the confidence limits). Such negative swings have previously been described in COPD and are related to flow limitation due to submaximal collapse of the airways during expiration. The confidence region is maximal at the turning points from inspiration to expiration and vice versa, probably due to the high-frequency content of the patient's own breathing, which disturbs the forced oscillations. See section “Derivation of confidence limits” below for the coherence spectra.
Discussion
FIG. 9 is a flow chart that summarizes the steps in the method according to the invention. In this exemplary embodiment, pressure waves are generated using a loudspeaker connected to a mouthpiece as a patient interface device 4 that is in communication with an airway of a subject (step 101). The oscillations in flow and pressure of the gas passing through the mouthpiece 4 are measured to give a respective time series of measurements (step 103). In the steady state, the respiratory impedance can be inferred from forced pressure oscillations in the mouthpiece 4 of a quietly breathing subject. Since the mechanical properties of the respiratory system often change from inspiration to expiration, the method estimates the impedance under circumstances of temporary ‘stability’. The main problem is that impedance is a frequency-dependent quantity and that a high time resolution inevitably leads to a low frequency resolution, thereby distorting the estimation. Since the data consists of discrete time series (sequential measurements of flow and pressure in the mouthpiece 4), this asks for an optimal time-frequency analysis of short time series.
As described in the “Methods” section above, the present invention is based on the concept that a short time series can be stationary if time is assumed to be circular (see below). As described, a new version of the uncertainty principle has been derived for such time series, which poses a lower limit to the total uncertainty in the time and frequency domain. In step 105, this principle is used to partition the variance of each time series into components that are associated with a specific time and frequency (through transformation to the ‘discrete time-frequency domain’ or ‘lime-frequency domain’). Least squares analysis in the time-frequency domain then gives an unbiased estimator of impedance with optimal time-frequency resolution (step 107). Finally, confidence limits for impedance have been constructed as a function of time and frequency. The main steps are summarized and discussed below.
The Linear Model.
The estimation of respiratory impedance is based on a simple linear model (Eq. 3, FIG. 2). Oscillations in the mouthpiece 4 of the apparatus 2 are induced by a loudspeaker 6. The resulting changes in flow and pressure depend on the mechanical properties of the patient's respiratory system (the acoustics of that system). In the model, this is expressed by two linear time-invariant filters (2 and 3 in FIG. 2) that modify the loudspeaker input. The resulting fluctuations in flow and pressure are disturbed by two independent sources of noise in both flow and pressure. The respiratory impedance is determined by the two linear filters (2 and 3).
Possible sources of noise are: 1) components of the patient's breathing in the frequency range of the forced oscillations, which is probably the most important problem (see reference 2), 2) swallowing, coughing or other movements by the patient, 3) cardiogenic oscillations, in particular when the airway resistance is low, 4) time-dependent variations in respiratory impedance itself and 5) random measurement errors.
A linear model has been used previously to explain the frequency-dependent relation between flow and pressure during similar experiments (see reference 2), although there have been indications that nonlinear interactions in the respiratory system are not negligible. A linear description of the relation between flow and pressure in the FOT device 2 is however supported by the fact that the pressure differences are small compared to the mean absolute pressure. Using an impedance to describe the relation is a simplification of the interaction of standing waves that can be expected under these circumstances. The main restriction, however, seems to lie in the time during which the respiratory system can be considered as stable.
The Circularity Assumption.
To account for chance events, it has been assumed that flow and pressure are random variables (RVs) with a given probability distribution. A short sequence of N paired measurements of flow and pressure is thus considered as a sample from a finite bivariate stochastic process (a set of N chronologically ordered paired RVs). The frequency-dependent relation between flow and pressure in such a process can only be described by a stable impedance if the bivariate stochastic process is assumed to be (second-order) stationary. This means that the expected values of flow and pressure are constant, as well as the covariances between simultaneous and successive values. The covariances between successive values cannot be constant for a finite sequence, however, due to begin- and end-effects (unless these covariances are equal to zero).
However, this problem is solved by assuming that time is circular, in the sense that the last event precedes the first. In this way, the covariances between successive values may be constant, while the sequence still consists of a finite number of RVs. This is a direct consequence of the discrete and finite nature of the sequence. Think of a clock, at which it is only possible to look at the whole hours (a chronological sequence of N=12 events). If you look at one o'clock (time t1) and at two o'clock (time t2), it is possible that one hour has elapsed from t1 to t2, but it is also possible that 13 hours have passed or that the event at t2 took place 11 hours before t1. Without further knowledge, it is not possible to discern between a time difference of one hour and one hour plus or minus an integer multiple of 12. This indeterminacy is directly related to ‘aliasing’ (which happens if ones tries to reconstruct a continuous signal from such a time series). Still, accidental changes in the last part of the measured sequence may cause fluctuations in both variables that do not continue in the first part. This would cause a bias in the estimation of impedance, which is however ruled out in the applied time-frequency analysis (see below).
The Uncertainty Principle.
‘Uncertainty in time’ is a property of a time series that expresses how much the components of the series are spread in time. For a discrete, finite and circular time series, this gives a figure as shown in FIG. 3A. The components of the time series are represented by point masses placed at regular intervals around the time circle (each mass equals the squared absolute value of a component). The uncertainty Δt is defined with respect to a reference point (here the point where t=0, Eq. 19) and is closely related to the center of gravity (Eq. B3). If the time series has only one nonzero component (at time t=0), then the center of gravity is located at t=0 and Δt is zero. If all components have equal weight, then the center of gravity is located at the center and Δt is high (associated with maximal spread in the time domain).
Similarly, the ‘uncertainty in frequency’ Δf expresses the spread in the frequency domain (Eq. 22). Through the discrete Fourier transform, every time series with N components can be written as the sum of N harmonic oscillations with a given amplitude and frequency (representation in the discrete frequency domain). For a discrete time series, these frequencies are also circular. The spread in the frequency domain can be visualized by placing the squared amplitudes as N point masses around a circle comparable to that in FIG. 3A (each point corresponding to a specific frequency).
For continuous time and frequency, it has been shown that there is a trade-off between uncertainty in time and frequency, expressed by the uncertainty relation

ΔtΔf≧1/(4π)  (54)

This holds for continuous and infinite time series (under certain conditions). If Δt is small, Δf has to be large since the product should at least be equal to 1/(4π), and vice versa. This inequality is directly related to Heisenberg's famous uncertainty principle for position and momentum of a particle. For discrete time series, however, the lower limit of the product ΔtΔf is zero (see also “Derivation of the uncertainty principle” below). Still, there is a restriction on joint values of Δt and Δf. In the “Methods” section above (Eq. 25), it is shown that for discrete, finite and circular time series,

Δt2+N2Δf2≧σmin2  (55)

where σmin is a function of N. Thus, Δt and Δf cannot be both very small. The sum Δt2+N2Δf2 is minimal for a time series whose discrete Fourier transform is exactly equal to the original time series (FIG. 4A). As N gets large, this time series with minimal joint uncertainty in time and frequency approaches a Gaussian function of time. This time series is analogous to the minimal uncertainty state of two attracting particles in quantum mechanics (the so-called ‘quantum harmonic oscillator’). Forbes et al. (see reference 4) derived a similar time series from the wave equation for such an oscillator. In the present case, however, this minimal uncertainty state is simply derived from the definitions of Δt and Δf for discrete and finite time series. In “Derivation of the uncertainty principle” below, it is shown how the uncertainty principle of Eq. 55 relates to Gabor's uncertainty principle for large N (see FIG. 14).

The Discrete Time-Frequency Domain.
The presented time-frequency transform (TFT) partitions a discrete and finite time series into a set of short-lasting oscillations of M components, with M<N (a transformation to the time-frequency domain). Each oscillation is centered about a specific time and frequency, with minimal uncertainty in both domains according to the principle of Eq. 55. The TFT amounts to a windowed ‘short-term Fourier transform’ or ‘running Fourier transform’. The short-lasting oscillations maximally overlap each other in time so that the outcome is independent of the time at which the first oscillation starts. Only the oscillations that start before the beginning and end after the end of the time series are dependent on the circularity assumption (these are excluded in the analysis to avoid bias). In the “Derivation of the time-frequency transform” section below, it is shown that the transform is ‘orthonormal’ (Eq. 33), which implies that all information in the time series is preserved after transformation to the discrete time-frequency domain. Nothing is gained, nothing is lost. An inverse-transformation recovers the original time series.
To be precise, it is not possible to refer to ‘the’ discrete time-frequency domain, but only about the time-frequency domain for a chosen value of M. This integer determines the duration of each short-term oscillation, but it also equals the number of different frequencies about which these oscillations are centered (the TFT resonant frequencies). As shown in “Derivation of the uncertainty principle” below, the optimal Δt and Δf according to the uncertainty principle of Eq. 55 depends on M (substitute N by M in Eq. B8, see also FIG. 11). Not surprisingly, the larger M, the larger the corresponding optimal Δt and the smaller the corresponding Δf. In the present analysis, M is chosen so that Δf is small enough to be able to discern the different oscillations induced by the loudspeaker.
Input to the Loudspeaker.
In an exemplary embodiment, a set of five harmonic oscillations we used as an input to the loudspeaker 6 (at frequencies of 8, 12, 16, 20 and 24 Hz respectively). For the analysis, the value of M (actually an effective value, see Eq. B6) was chosen so that the uncertainties were, expressed in time units, Δt=0.0625 s and Δf=1.27 Hz. The TFT coefficients were only computed for these five frequencies, which amounts to band-pass filtering with five resonant frequencies. One prior art approach uses a single frequency for the loudspeaker input. This poses less requirements to the time-frequency analysis (they used a band-pass filter with rectangular window, with a larger Δf for a given Δt than in the present invention). For a single frequency, the frequency resolution is less critical, although it may still be useful to filter out the higher harmonics of the patient's breathing. On the other hand, the frequency dependence of impedance can provide additional information on respiratory mechanics. Alternatively, broad-band noise can be applied, which has the disadvantage that the power of the input signal is gradually distributed over all frequencies, with relatively less power at the studied frequencies (a lower signal-to-noise ratio at the same total power of the loudspeaker). In yet another approach, a ventilator waveform was applied that was built up of nonharmonic sinusoids to ventilated patients to prevent interaction between oscillations at harmonic frequencies. Such an approach is however not applicable in freely breathing subjects. Some interference of the respiratory system at harmonic frequencies may indeed occur in the present invention, although this has not been observed in the frequency response of the system after broad-band stimulation.
Analysis of Variance in the Time Frequency Domain.
The orthonormal property of the TFT makes it possible to partition the variance of a stochastic process into time-frequency dependent components, which gives the time-frequency power spectrum (see “Analysis of variance in the time-frequency domain” below). The TFT sample power expresses the contribution of a short-lasting oscillation about a given time and frequency to the total sample variance. Analysis of the TFT power spectrum thus amounts to an analysis of variance in the time-frequency domain (Eq. 42). In the model (Eq. 43), it is assumed that there are independent sources of noise in flow and pressure, together forming a bivariate circularly stationary stochastic process. When the values that depend on the circularity assumption are discarded, a number of NS=500 successive values in the time-frequency domain are considered as a sample from this process (an episode of 0.625 s at a sample rate of 800 Hz). It was assumed that at every moment (800 times per second) a new stochastic process of this type starts with a variance that is not necessarily equal to the previous one. Thus, the measurements were subdivided into maximally overlapping episodes of 500 values, each of which was considered as a sample from a different stationary stochastic process. The sources of noise were not assumed to be ‘white’, but it was assumed that the power was constant in each frequency band of the TFT filter. The mean power during each episode of 500 values then follows an approximate chi-square distribution for which the equivalent degrees of freedom were derived.
Bivariate Least Squares in the Time-Frequency Domain.
For each episode of NS=500 successive values, the coefficients of the linear relation between loudspeaker input and flow (bQ,t,m) and the relation between input and pressure (bP,t,m) were estimated by simple linear regression in the time-frequency domain (Eq. 44, FIG. 5). This results in unbiased estimators associated with time t and frequency fm (respectively, {circumflex over (B)}Q,t,m and {circumflex over (B)}P,t,m, see Eq. 46). Each estimator is based on time-averaged power and cross spectra over NS values in the time-frequency domain. Subsequently, the impedance was estimated by the ratio {circumflex over (Z)}t,m={circumflex over (B)}P,t,m/{circumflex over (B)}Q,t,m.
The choice of N depends on the time during which the system can be considered as stationary. For respiration, an episode of  ̃0.5 s seems plausible, assuming that the impedance is approximately stable in the midst of each inspiration and expiration. In the shown examples, this worked out well (FIGS. 7 and 8). In most prior art techniques, however, spectra are averaged over a longer episode, usually of more than 10 s. Some averaged spectral values derived from non-overlapping segments of  ̃0.65 s over a total episode of 16 s. The use of non-overlapping segments, however, gives considerably less degrees of freedom for an episode of the same length as compared to the maximally overlapping segments used here (i.e., the segments of M values used in the TFT). The major problem is that the use of an episode of 16 s averages out possible physiological differences between inspiration and expiration, which are usually more pronounced in diseases such as COPD. In that case, a low coherence between flow and pressure over 16 s is not only a reflection of noise, but also of within-breath variability of impedance (which is part of the disease). The duration of the supposed stationary should not be chosen too short, either. In the example of FIG. 8, the confidence intervals for are relatively wide at the turning points from inspiration to expiration and vice versa. This may be due to high-frequency components of the patient's own breathing. These changes are apparently shorter than 0.5 s, so that they appear as noise in the estimation. The duration of the supposed stationary should therefore be long enough to ‘interpret’ these changes as noise.
Since the impedance estimator {circumflex over (Z)}t,m is derived as the ratio of two normally distributed RVs ({circumflex over (B)}P,t,m and {circumflex over (B)}Q,t,m), it follows a Cauchy distribution which has no expected value. So, strictly speaking, {circumflex over (Z)}t,m has no bias in the sense of a difference between expected and true value. Still, since {circumflex over (B)}P,t,m and {circumflex over (B)}Q,t,m are unbiased normally distributed RVs, {circumflex over (Z)}t,m is by approximation normally distributed with expectation zt,m=bP,t,m/bQ,t,m. Daróczy and Hantos (reference 2) followed a similar approach, deriving the impedance estimator as the ratio of the regression coefficients from the loudspeaker input to respectively flow and pressure. Based on a simple model of the mechanical properties of patient and apparatus, they argued that a systematic error is introduced if it is assumed that noise only occurs in either flow or pressure. Experimental evidence for such bias has been found. Others, however, assumed that the loudspeaker generates a pressure wave and the main noise, the high-frequency component of the patient's own breathing, is a pure flow source (while the noise in pressure is zero). If the TFT would have been used, the estimator would be {circumflex over (Z)}t,m(P) in Eq. 52. On the other hand, other documents used an estimator analogous to {circumflex over (Z)}t,m(Q) in Eq. 52. Since {circumflex over (Z)}t,m(P) and {circumflex over (Z)}t,m(Q) are least-squares estimators under the assumption that the noise in either pressure or flow is zero, this leads to bias if this is not in agreement with reality, which was already pointed out by several authors (see reference 2). It has been argued that this type of error is minimal if the coherence between flow and pressure is high and recommended to discard the estimates for which this is not the case. This is in line with Eq. 53, from which it follows that {circumflex over (Z)}t,m(P) and {circumflex over (Z)}t,m(Q) both approach {circumflex over (Z)}t,m (and thus the true zt,m) if the coherence tends to unity. The discussion surrounding Eqs. 50 and 51 also shows that the noise in flow also affects the confidence region for impedance (in particular the skewness parameter tan β, see FIG. 6B). In one earlier document, band-pass filtered pressure was divided by flow to obtain impedance as a function of time (see above). The resulting impedance is numerically equal to a ‘total least squares’ estimate with equivalent contributions of the sample variances in flow and pressure. It can be expected that these values are often in the same range as those obtained with bivariate least squares (if the coherence from input to flow equals the coherence from input to pressure). This ‘estimator’ is however not derived from averaged values which makes it highly susceptible to random error.
Thus far, confidence limits on the estimated impedance have only been derived for relatively time long intervals. The time-frequency dependent confidence limits in embodiments of the present invention provide a continuous impression of the validity of the hypothesis that the short-lasting sequences are samples from a stationary stochastic process. They make it possible to test the significance of changes in impedance in the course of time and provide a basis to automatically reject unreliable estimates (like during swallowing in FIG. 7), which is of practical interest in real-time monitoring.
FIG. 10 is a flow chart illustrating the method performed by the apparatus 2 according to the invention (and particularly the processing steps performed by the computer 16) in more detail. As in FIG. 9, in step 121, pressure waves are generated using a loudspeaker 6 connected to a mouthpiece 4 that is being used by a subject, and the oscillations in flow and pressure of the air passing through the mouthpiece 4 are measured and digitized to give a respective time series of measurements (step 123). Thus flow and pressure are given as a function of discrete time t, qt and pt.
Then, in step 125, the time series are each transformed into the discrete time-frequency domain to give flow {tilde over (q)}t,m and pressure {tilde over (p)}t,m as a function of discrete time t and different frequencies fm=m/M, where in is the frequency index and M is the effective width of the time-frequency filter. The frequencies fm are chosen so that they match the frequencies of the imposed FOT frequencies.
Step 125 comprises evaluating:




  
    
      
        
          
            
              q
              ~
            
            
              t
              ,
              m
            
          
          =
          
            
              ∑
              
                l
                =
                
                  -
                  
                    K
                    1
                  
                
              
              
                K
                1
              
            
            ⁢
            
              
                ω
                l
              
              ⁢
              
                e
                
                  
                    -
                    2
                  
                  ⁢
                  
                    
                  
                  ⁢
                  πi
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ml
                    /
                    M
                  
                
              
              ⁢
              
                q
                
                  t
                  +
                  l
                
              
            
          
        
        ,
      
    
    
      
        (
        56
        )
      
    
  




for the flow time series qt where i2=−1 and the weighting factor wl is approximated by a Gaussian function of time, wl=e−(l/Δt)2 and Δt is the chosen uncertainty in time. The filter is truncated at K1=5·Δt. While the weighting factor wl is described here as being approximated by a Gaussian function, it is to be understood that other windows or weighting factors are contemplated by the present invention, such as a triangle window, a piece-wise linear approximation, or a polynomial function.

The pressure {tilde over (p)}t,m as a function of time and frequency is derived analogously from




  
    
      
        
          
            p
            ~
          
          
            t
            ,
            m
          
        
        =
        
          
            ∑
            
              l
              =
              
                -
                
                  K
                  1
                
              
            
            
              K
              1
            
          
          ⁢
          
            
              w
              l
            
            ⁢
            
              e
              
                
                  -
                  2
                
                ⁢
                
                  
                
                ⁢
                πi
                ⁢
                
                  
                
                ⁢
                
                  ml
                  ⁢
                  
                    
                  
                  /
                  M
                
              
            
            ⁢
            
              p
              
                t
                +
                l
              
            
          
        
      
    
    
      
        (
        
          56
          ⁢
          a
        
        )
      
    
  




Then, in step 127, the power and cross spectra are derived as a function of time and frequency. The power of the flow is given by:




  
    
      
        
          
            P
            
              q
              ,
              t
              ,
              m
            
          
          =
          
            
              1
              
                N
                S
              
            
            ⁢
            
              
                ∑
                
                  l
                  =
                  
                    -
                    
                      K
                      2
                    
                  
                
                
                  K
                  2
                
              
              ⁢
              
                
                  
                  
                    
                      q
                      ~
                    
                    
                      
                        t
                        +
                        l
                      
                      ,
                      m
                    
                  
                  
                
                2
              
            
          
        
        ,
      
    
    
      
        (
        57
        )
      
    
  




where NS is the number of samples in the time-frequency domain, K2=1⁄2(NS−1) if NS is odd and |•| stands for the absolute value.

The power of the pressure, represented by Pp,t,m is derived analogously from




  
    
      
        
          P
          
            p
            ,
            t
            ,
            m
          
        
        =
        
          
            1
            
              N
              S
            
          
          ⁢
          
            
              ∑
              
                l
                =
                
                  -
                  
                    K
                    2
                  
                
              
              
                K
                2
              
            
            ⁢
            
              
                
                
                  
                    p
                    ~
                  
                  
                    
                      t
                      +
                      l
                    
                    ,
                    m
                  
                
                
              
              2
            
          
        
      
    
    
      
        (
        
          57
          ⁢
          a
        
        )
      
    
  




The cross spectrum from the loudspeaker input {tilde over (x)}t,m to the flow {tilde over (q)}t,m in the time-frequency domain is given by:




  
    
      
        
          
            C
            
              xq
              ,
              t
              ,
              m
            
          
          =
          
            
              1
              
                N
                S
              
            
            ⁢
            
              
                ∑
                
                  l
                  =
                  
                    -
                    
                      K
                      2
                    
                  
                
                
                  K
                  2
                
              
              ⁢
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  *
                
                ⁢
                
                  
                    q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        58
        )
      
    
  




where the asterisk denotes the complex conjugate.

The cross spectrum from loudspeaker input to pressure, Cxp,t,m, is derived from {tilde over (x)}t,m and {tilde over (p)}t,m in a similar way:




  
    
      
        
          C
          
            xp
            ,
            t
            ,
            m
          
        
        =
        
          
            1
            
              N
              S
            
          
          ⁢
          
            
              ∑
              
                l
                =
                
                  -
                  
                    K
                    2
                  
                
              
              
                K
                2
              
            
            ⁢
            
              
                
                  x
                  ~
                
                
                  t
                  ,
                  m
                
                *
              
              ⁢
              
                
                  p
                  ~
                
                
                  t
                  ,
                  m
                
              
            
          
        
      
    
    
      
        (
        
          58
          ⁢
          a
        
        )
      
    
  




In step 129, the power and cross spectra are used to determine the transfer functions from the loudspeaker input to flow and pressure respectively.
In particular, the transfer function from loudspeaker input to flow is given by the ratio of the cross spectra to the power:




  
    
      
        
          
            B
            
              xq
              ,
              t
              ,
              m
            
          
          =
          
            
              C
              
                xq
                ,
                t
                ,
                m
              
            
            
              P
              
                x
                ,
                t
                ,
                m
              
            
          
        
        ,
      
    
    
      
        (
        59
        )
      
    
  




and the transfer function from loudspeaker input to pressure, Bxp,t,m, is derived analogously:





  
    
      
        
          B
          
            xp
            ,
            t
            ,
            m
          
        
        =
        
          
            C
            
              xp
              ,
              t
              ,
              m
            
          
          
            P
            
              x
              ,
              t
              ,
              m
            
          
        
      
    
    
      
        (
        
          59
          ⁢
          a
        
        )
      
    
  




Thus, in step 131, the impedance of the respiratory system can be determined from the ratio of the transfer functions:




  
    
      
        
          
            Zrs
            
              t
              ,
              m
            
          
          =
          
            
              B
              
                xp
                ,
                t
                ,
                m
              
            
            
              B
              
                xq
                ,
                t
                ,
                m
              
            
          
        
        ,
      
    
    
      
        (
        60
        )
      
    
  




with real component Rrst,m and imaginary component Xrst,m.

Furthermore, confidence limits on the respiratory impedance can be derived (step 133) as follows.
The equivalent number of degrees of freedom η of the power and cross spectra is given by:




  
    
      
        
          η
          =
          
            
              
                (
                
                  tr
                  ⁢
                  
                    {
                    
                      
                        P
                        H
                      
                      ⁢
                      P
                    
                    }
                  
                
                )
              
              2
            
            
              tr
              ⁢
              
                {
                
                  
                    (
                    
                      
                        P
                        H
                      
                      ⁢
                      P
                    
                    )
                  
                  2
                
                }
              
            
          
        
        ,
      
    
    
      
        (
        61
        )
      
    
  




where P is an N×N matrix, H stands for the Hermitian transpose and tr{•} for the trace of the matrix. The matrix P is defined as

P=S(Mm+MM-m)/NS.  (62)

where S is a diagonal matrix S that has NS ones on the main diagonal (and the rest zeros), i.e.

S=diag{0, . . . ,0,1, . . . ,1,0, . . . ,0}.  (63)

The (shifted) time-frequency transform matrix Mm is a circulant whose entries on the first row are wle2πi ml/M, for l=−K3, . . . , K3 and K3=1⁄2(N−1) where N is odd. Fα is labeled as the upper 100(1−α) % point of the F-distribution on 2,η−2 degrees of freedom.
The squared coherence between {tilde over (x)}t,m and {tilde over (q)}t,m is given by:




  
    
      
        
          
            K
            
              xq
              ,
              t
              ,
              m
            
            2
          
          =
          
            
              
                
                
                  C
                  
                    xq
                    ,
                    t
                    ,
                    m
                  
                
                
              
              2
            
            
              
                P
                
                  x
                  ,
                  t
                  ,
                  m
                
              
              ·
              
                P
                
                  q
                  ,
                  t
                  ,
                  m
                
              
            
          
        
        ,
      
    
    
      
        (
        64
        )
      
    
  




and the squared coherence between {tilde over (x)}t,m and {tilde over (p)}t,m, Kxp,t,m2, is derived analogously from





  
    
      
        
          K
          
            xp
            ,
            t
            ,
            m
          
          2
        
        =
        
          
            
              
              
                C
                
                  xp
                  ,
                  t
                  ,
                  m
                
              
              
            
            2
          
          
            
              P
              
                x
                ,
                t
                ,
                m
              
            
            ·
            
              P
              
                p
                ,
                t
                ,
                m
              
            
          
        
      
    
    
      
        (
        
          64
          ⁢
          a
        
        )
      
    
  




The flow-related variable Aq,t,m2 is defined as




  
    
      
        
          
            A
            
              q
              ,
              t
              ,
              m
            
            2
          
          =
          
            
              
                F
                α
              
              ⁡
              
                (
                
                  2
                  
                    η
                    -
                    2
                  
                
                )
              
            
            ⁢
            
              (
              
                
                  1
                  -
                  
                    K
                    
                      xq
                      ,
                      t
                      ,
                      m
                    
                    2
                  
                
                
                  K
                  
                    xq
                    ,
                    t
                    ,
                    m
                  
                  2
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        65
        )
      
    
  




and the analogous pressure-related variable, denoted Ap,t,m2, is derived in a similar way from





  
    
      
        
          A
          
            p
            ,
            t
            ,
            m
          
          2
        
        =
        
          
            
              F
              α
            
            ⁡
            
              (
              
                2
                
                  η
                  -
                  2
                
              
              )
            
          
          ⁢
          
            (
            
              
                1
                -
                
                  K
                  
                    xp
                    ,
                    t
                    ,
                    m
                  
                  2
                
              
              
                K
                
                  xp
                  ,
                  t
                  ,
                  m
                
                2
              
            
            )
          
        
      
    
    
      
        (
        
          65
          ⁢
          a
        
        )
      
    
  




Therefore, the 100(1−α) % confidence limits for the real and imaginary parts of the respiratory impedance are given by, respectively,

Rrs·c1±c2 and Xrs·c1±c2,

with c1=1+Aq,t,m·c3,c2=|Zrst,m|·c3, and





  
    c
    3
  
  =
  
    
      
        
          A
          
            q
            ,
            t
            ,
            m
          
        
        +
        
          A
          
            p
            ,
            t
            ,
            m
          
        
      
      
        1
        -
        
          A
          
            q
            ,
            t
            ,
            m
          
          2
        
      
    
    .
  




The estimates of respiratory impedance for a given time and frequency are rejected as not significant if either Aq,t,m≧a first threshold, or Ap,t,m≧a second threshold. In an exemplary embodiment, the first threshold and the second threshold are set to 1 (one), so that the estimates of respiratory impedance for a given time and frequency are rejected as not significant if either Aq,t,m≧1 or Ap,t,m≧1.
To deal with begin- and end-effects, the algorithm contains several circular data buffers.
The Measurement Apparatus
Measurements.
The airflow can be measured with a pneumotach head 8 and built-in pressure transducer 10 (for example a Jaeger Masterscreen pneumotach type BF/IEC 601-1, Hoechberg, Germany). The pressure at the mouthpiece 4 can be measured with reference to ambient air with a differential pressure transducer 12 (for example a Hans Rudolph Pneumotach amplifier 1 series 1110, Shawnee, Kans.). The pressure oscillations can be generated in the FOT device 2 by a loudspeaker 6 (for example a Jaeger Masterscreen IOS, Hoechberg, Germany) that is controlled by an analog output signal from a personal computer 16 (for example a Hewlett Packard Compac dc 7600, Palo Alto, Calif.) through an analog-digital conversion card 14 (for example a National Instruments PCI-6221, Dallas, Tex.), which is amplified by an amplifier 18 (for example a Harman Kardon HK 970 amplifier, Washington D.C.)). Subjects can breathe in and out through the mesh-wired resistance 9 connected to the pneumotach head 8. Analog input signals can be converted to digital sequences through the same analog-digital conversion card 14 at a sample rate of 800 Hz and stored in the personal computer 16. The generation of the output signal and the analysis of the data can be performed by a computer or processor 16 executing appropriate computer software.
Measurements can be made using the apparatus 2 during a period covering a number of breathing cycles, for example 90 seconds of quiet breathing.
Derivation of the Uncertainty Principle
Geometrical Interpretation of Uncertainty in Time and Frequency.
According to the definition of Eq. 20, the uncertainty Δt is directly related to the ‘center of gravity’ of the time series x≡{xt}. The events in the time series are represented as point masses with value |xt|2, placed at regular intervals around the perimeter of a circle with radius rt in the complex plane, centered about the origin. Each event occurs at a point rtωt in the complex plane, with rt≡N/(2π) and ω≡exp(2πi/N). The weighted mean ωt can be defined as




  
    
      
        
          
            ω
            _
          
          t
        
        ≡
        
          
            ∑
            
              t
              =
              0
            
            
              N
              -
              1
            
          
          ⁢
          
            
              ω
              t
            
            ⁢
            
              
                
                  
                  
                    x
                    t
                  
                  
                
                2
              
              /
              
                
                  ∑
                  
                    t
                    =
                    0
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                    
                    
                      x
                      t
                    
                    
                  
                  2
                
              
            
          
        
      
    
    
      
        (
        B1
        )
      
    
  




The center of gravity is located at rtωt. In matrix formulation,

ωt≡xHΩx/∥x∥2,  (B2)

where Ω≡diag{1, ω, . . . , ωN−1}. Using the fact that Ω is a unitary matrix, the squared uncertainty Δt2 can then be reduced to





  
    
      
        
          
            
              
                Δ
                t
                2
              
              ≡
              
              ⁢
              
                
                  r
                  t
                  2
                
                ⁢
                
                  
                    
                      x
                      H
                    
                    ⁡
                    
                      (
                      
                        Ω
                        -
                        I
                      
                      )
                    
                  
                  H
                
                ⁢
                
                  (
                  
                    Ω
                    -
                    I
                  
                  )
                
                ⁢
                
                  x
                  /
                  
                    
                      
                      x
                      
                    
                    2
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  t
                  2
                
                ⁢
                
                  
                    x
                    H
                  
                  ⁡
                  
                    (
                    
                      
                        
                          Ω
                          H
                        
                        ⁢
                        Ω
                      
                      +
                      I
                      -
                      
                        Ω
                        H
                      
                      -
                      Ω
                    
                    )
                  
                
                ⁢
                
                  x
                  /
                  
                    
                      
                      x
                      
                    
                    2
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  t
                  2
                
                ⁢
                
                  
                    x
                    H
                  
                  ⁡
                  
                    [
                    
                      
                        2
                        ⁢
                        I
                      
                      -
                      
                        
                          2
                          ·
                          
                            1
                            2
                          
                        
                        ⁢
                        
                          (
                          
                            
                              Ω
                              H
                            
                            +
                            Ω
                          
                          )
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  x
                  /
                  
                    
                      
                      x
                      
                    
                    2
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                2
                ⁢
                
                  
                    
                      r
                      t
                      2
                    
                    ⁡
                    
                      (
                      
                        1
                        -
                        
                          Re
                          ⁢
                          
                            {
                            
                              
                                ω
                                _
                              
                              t
                            
                            }
                          
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        B3
        )
      
    
  




This is in line with the ‘parallel axes theorem’ for the second moment of inertia about an axis (perpendicular to the plane of the circle) through the reference point R in FIG. 12. FIG. 12 illustrates the relation between uncertainty in time Δt and the centers of gravity of the eigenvectors of CHC for N=16. The eigenvectors are mapped along the time circle in the complex plane with radius rt, centered about the origin. The centers of gravity of each eigenvector are depicted by black dots. The corresponding singular value of C is minimal for the center of gravity at the right (near to the reference point R) and gradually increases to the left of the Figure. The uncertainty Δt for each eigenvector is the oblique distance from R to one of the intersection points of the vertical line through the center of gravity and the circle.
In the representation of FIG. 13, it follows from Eq. 11 that Δt2=2rt·AR, where AR is the distance from A to R in FIG. 12. Using the Pythagorean theorem, it follows that Δt=CR, since

CR2=AC2+AR2=rt2−(rt−AR)2+AR2=2rt·AR

In the frequency domain, the uncertainty Δf can be geometrically interpreted in a similar manner.
Singular Values of the Matrix C.
The uncertainty principle of Eq. 25 is determined by σmin2 and σmax2, the squared minimal and maximal singular values of C. The singular values σ can be derived from

det(CHC−σ2I)=0  (B4)

where det(•) stands for the determinant of the matrix. For N=2, it readily follows that σmin2=2(2−√{square root over (2)})/π2 and σmax2=2(2+√{square root over (2)})/π2. For higher N, various strategies have been developed to derive the singular values of C (the eigenvalues of CHC). FIG. 11 shows σmin and σmax as a function of N. It turns out that for larger N (say, N>15), σmin approaches √{square root over (N/2π)} and σmax approaches N√{square root over (2)}/π. In terms of the radius of the time circle, this means that σmin≈√{square root over (rt)} and σmax≈2√{square root over (2)}rt.

Let v be a unit eigenvector of CHC that corresponds to singular value σ. The normal equations (CHC−σ2I)v=0 can be rewritten as




  
    
      
        
          
            
              
                
                  
                    
                      r
                      t
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              (
                              
                                Ω
                                -
                                I
                              
                              )
                            
                            H
                          
                          ⁢
                          
                            (
                            
                              Ω
                              -
                              I
                            
                            )
                          
                        
                        +
                        
                          
                            
                              (
                              
                                
                                  T
                                  
                                    -
                                    1
                                  
                                
                                -
                                I
                              
                              )
                            
                            H
                          
                          ⁢
                          
                            (
                            
                              
                                T
                                
                                  -
                                  1
                                
                              
                              -
                              I
                            
                            )
                          
                        
                      
                      ]
                    
                  
                   
                
                ⁢
                
                   
                  
                    v
                    -
                    
                       
                       
                    
                  
                   
                
                ⁢
                
                  σ
                  2
                
                ⁢
                v
              
              ⁢
              
                
              
               
            
            ⁢
            
              
            
            =
          
        
        
          
            (
            B5
            )
          
        
      
    
    ⁢
    
      
    
     
  
  ⁢
  
    
  
  ⁢
  
     
     
  
  ⁢
  
     
    
      
        r
        t
        2
      
      ⁢
      
         
        
          
            
              
                [
                
                  
                    
                      Ω
                      H
                    
                    ⁢
                    Ω
                  
                  +
                  I
                  -
                  
                    Ω
                    H
                  
                  -
                  Ω
                  +
                  
                    TT
                    
                      -
                      1
                    
                  
                  +
                  I
                  -
                  T
                  -
                  
                    T
                    
                      -
                      1
                    
                  
                
                ]
              
              ⁢
              v
            
            -
            
              
                σ
                2
              
              ⁢
              v
            
          
          =
          
            
              
                
                  [
                  
                    
                      
                        r
                        t
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            2
                            ⁢
                            I
                          
                          -
                          
                            Ω
                            H
                          
                          -
                          Ω
                        
                        )
                      
                    
                    -
                    
                      σ
                      2
                    
                  
                  ]
                
                ⁢
                v
              
              -
              
                
                  
                    r
                    t
                    2
                  
                  ⁡
                  
                    (
                    
                      T
                      -
                      
                        2
                        ⁢
                        I
                      
                      +
                      
                        T
                        
                          -
                          1
                        
                      
                    
                    )
                  
                
                ⁢
                v
              
            
            =
            0.
          
        
      
    
  




The first term depends on the departure of the components of v from the reference point R along the real axis (cf Eq. B3). The second term can be seen as the second derivative of the components of v (for discrete and circular time). Equation B5 is comparable to the Schrödinger equation for a quantum harmonic oscillator. The solutions can be regarded as discrete orthogonal Mathieu functions. The eigenvector vmin that corresponds to σmin is a unimodal function of time (as in FIG. 4A). For large N, vmin can be approximated by a vector vmin′, whose components νmin,t′ are a Gaussian function of time,

νmin,t′=νmin,0′exp[−(t/2Δt)2]  (B6)

where N has to be subtracted from t if t≧1⁄2N. Since the components νmin,t′ are close to zero if t>5·Δt, the Gaussian can be truncated by using only the 10·Δt largest values about the maximal value and setting the rest to zero. Calling the resulting approximated eigenvector v′, and the difference vector d≡v−v′, the norm ∥d∥ can be taken as a measure for the error made by the approximation. Using appropriate mathematical software, it follows that the relative error ∥d∥/∥v∥ is less than 0.005 if N>75.

Since CHC commutes with F, these matrices share the same eigenvectors. Since F4=I, the eigenvalues of F are 1, −1, i and −i. As a result, for every unit eigenvector v, the uncertainty in time is directly related to the uncertainty in frequency,

Δt=rt∥(Ω−I)v∥=N/2π∥(Ω−I)Fv∥=NΔf  (B7)

This means that for every eigenvector v, the total uncertainty is evenly distributed over time and frequency,

σ2=vHCHCv=Δt2+N2Δf2=2Δt2=2N2Δf2  (B8)

So, for every eigenvector v, Δt=NΔf=σ/√{square root over (2)}. Note that the Gaussian approximation of vmin according to Eq. B6 can be rewritten as νmin,t′=νmin,0′exp[−1⁄2(t/σmin)2]. The total uncertainty σmin2 is therefore equal to the variance of this Gaussian function. Equation B8 also implies that, for N>15, the values of Δt that correspond to minimal and maximal total uncertainty are, respectively, Δt,min=σmin/√{square root over (2)}≈√{square root over (rt/2)} and Δt,max≈σmax/√{square root over (2)}≈2rt. The latter is intuitively reasonable since 2rt is the largest possible Δt within the time circle (see FIG. 13). FIG. 13 illustrates the relation between uncertainty in time Δt and uncertainty in frequency Δf for N=16. The accessible region for all possible N-vectors x is bounded by the two circles with radius σmin and σmax (the minimal and maximal singular value of C). In FIG. 13, rt, is the radius of the time circle, the black dots belong to the eigenvectors of CHC, vmin and vmax are eigenvectors corresponding to σmin and σmax e0 and e8 are canonical vectors corresponding to t=0 and t=8, f0 and f8 are harmonic oscillations with frequencies fn=0 and fn=8/16=1⁄2.

As described above, FIG. 12 shows the centers of gravity for all eigenvectors v in the time circle for N=16. Since CHC is a normal matrix, it has N orthogonal eigenvectors according to the spectral theorem. As appears from Eq. B3, the corresponding Δt depends on the horizontal distance from R to the center of gravity in the complex plane. It can be read from FIG. 12 as the length of the chord from R to the intersection of the circle with the vertical line through the center of gravity. The center of gravity closest to R corresponds to σmin. Centers of gravity that lie further away correspond to increasing values of σ(and Δt=σ/√{square root over (2)}). For relatively large N, the distance between the center of gravity of vmin and R approaches 1⁄4 which follows from Eq. B3 and the finding that Δt,min approaches √{square root over (rt/2)}. There is a multiplicity of eigenvalues of CHC in the sense that there are two orthogonal eigenvectors with the same eigenvalue σ2=N2/π2. Such multiplicity of eigenvalues probably only occurs if N is divisible by four. The centers of gravity of these two orthogonal eigenvectors are both located on the imaginary axis. They have the same Δt, equal to σ/√{square root over (2)}=N/(√{square root over (2)}π)=√{square root over (2)}rt. The corresponding Δf equals Δt/N=1/(√{square root over (2)}π)=√{square root over (2)}rf. These two eigenvectors have the largest possible spread in both time and frequency domain (identical to the values that would be obtained if the center of gravity would be located at the center of both the time and frequency circle). Eigenvectors corresponding to higher values of σ are associated with larger ‘uncertainty’ Δt (relative to R), but with a smaller spread in time. The weight of the vectors actually becomes more and more concentrated at the opposite side of both time and frequency circle, until the maximal total uncertainty σmax2 is reached for eigenvector vmax and Δt is almost equal to 2rt. Note the apparent symmetry in FIG. 12 (the centers of gravity are reflected in the imaginary axis).
FIG. 13 shows NΔf as a function of Δt. Due to the uncertainty principle of Eq. 25, the accessible region is bounded by two circles centered about the origin, with radius σmin and σmax. This does not mean that all values between these circles are possible, but that all values outside the circles are impossible. The coordinates that belong to the eigenvectors of CHC are all located on the identity line (due to Eq. B7). Some extreme cases are also shown. One is the canonical vector e0≡{1, 0, . . . , 0}. Its ‘weight’∥x∥2=|x0|2+ . . . +|xN−1|2 is entirely concentrated at t=0, so Δt is zero. Transformation to the frequency domain gives




  
    
      
        
          Fe
          0
        
        =
        
          
            
              
                1
                
                  N
                
              
              ⁡
              
                [
                
                  
                    
                      
                        f
                        0
                        *
                      
                    
                    
                      ...
                    
                    
                      
                        f
                        
                          N
                          -
                          1
                        
                        *
                      
                    
                  
                
                ]
              
            
            ⁢
            
              e
              0
            
          
          =
          
            
              
                1
                
                  N
                
              
              ⁢
              
                f
                0
                *
              
            
            =
            
              
                1
                
                  N
                
              
              ⁢
              1
            
          
        
      
    
    
      
        (
        B9
        )
      
    
  




where fn is defined as in Eq. 9, the ‘*’ sign stands for the complex conjugate and 1 is the N-vector that contains only ones. Thus, the weight of e0 is evenly distributed over all frequencies in the frequency domain and Δf=√{square root over (2)}rf or NΔf=√{square root over (2)}rt. Another extreme is e8. (Let the canonical vector et be defined as {0, . . . , 0, 1, 0, . . . , 0}, where the one stands on the tth place, starting from zero.) The weight of e8 is entirely concentrated at the point (−rt,0) in the complex plane and so Δt is maximal, equal to 2rt. In the frequency domain,





  
    
      
        
          Fe
          8
        
        =
        
          
            
              
                1
                
                  N
                
              
              ⁡
              
                [
                
                  
                    
                      
                        f
                        0
                        *
                      
                    
                    
                      ...
                    
                    
                      
                        f
                        
                          N
                          -
                          1
                        
                        *
                      
                    
                  
                
                ]
              
            
            ⁢
            
              e
              8
            
          
          =
          
            
              1
              
                N
              
            
            ⁢
            
              f
              8
              *
            
          
        
      
    
    
      
        (
        B10
        )
      
    
  




The column vector f8* describes a harmonic oscillation with frequency fn=8/16=1⁄2. Its squared components are also evenly distributed over all frequencies, so NΔf=√{square root over (2)}rt. Starting from the time domain, other extremes are the oscillations f0, whose weight is evenly distributed in the time domain and sharply concentrated in the frequency domain at fn=0 (Δt=√{square root over (2)}rt and Δf=0), and f8, whose weight is also evenly distributed in the time domain but sharply concentrated in the frequency domain at fn=1⁄2 (so that Δt=√{square root over (2)}rt and NΔf=2rt). Since possible values for Δt and NΔf are confined to the range [0,2rt], it can be expected that all possible combinations (Δt,NΔf) are confined to a region in FIG. 13 that is enclosed by the coordinates for vmin, e0, f8, vmax, e8 and f0. These extreme vectors are also illustrations of the general rule: taking the Fourier transform of a time series vector x (through premultiplication by F) exchanges the (Δt, NΔf) coordinates in the plot of FIG. 13. Differently put: the coordinates for x and F x are reflected in the identity line. This is readily verified using the definitions of Eqs. 21 and 23.
Comparison with Gabor's Uncertainty Principle.
The uncertainty principle that was described by Gabor in 1946 (see reference 5 below) determines a lower limit for the product ΔtΔf for continuous time and frequency,




  
    
      
        
          
            Δ
            t
          
          ⁢
          
            Δ
            f
          
        
        ≥
        
          1
          
            4
            ⁢
            π
          
        
      
    
    
      
        (
        B11
        )
      
    
  




How does this inequality relate to the uncertainty principle for discrete time and frequency according to Eq. 25? In Eq. B11, Δt2 is also defined as the second moment of inertia about an axis perpendicular to the point where t=0, although time is mapped on an infinite line now (or a circle with infinite radius). In the derivation of Eq. B11, it is assumed that the mean time is zero (the center of gravity is located at t=0). The same holds for the spread in the frequency domain (regardless of the fact that frequency is not discrete and periodic but continuous and infinite).
The main difference between the inequalities of Eq. B11 and Eq. 25 lies in the fact that according to Gabon's uncertainty principle, the lower limit of ΔtΔf is attained for Gaussian functions of t (of the same form as in Eq. B6) with any nonzero value of Δt (see reference 5). This is a set of linearly independent (infinite) vectors. The lower limit of the total uncertainty Δt2+NΔf2 according to Eq. 25, however, is attained for only one eigenspace of CHC (the complex one-dimensional subspace spanned by vmin), which corresponds to one single value of Δt (equal to σmin/√{square root over (2)}). On the other hand, the vectors in this eigenspace do attain the lower limit of Gabor's uncertainty principle (in the limit situation as N→∞). For every eigenvector of CHC, it follows from Eq. B8 that

ΔtΔf=Δt2/N=σ2/(2N)  (B12)

For vmin (or any other vector in the corresponding eigenspace), a approaches √{square root over (N/(2π))} as N gets large (FIG. 11). Numerical approximation then shows that, as N→∞,




  
    
      
        
          
            Δ
            
              t
              ,
              
                
              
              ⁢
              
                m
                ⁢
                
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                n
              
            
          
          ⁢
          
            
              
                Δ
                
                  f
                  ,
                  
                    
                  
                  ⁢
                  
                    m
                    ⁢
                    
                      
                    
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
              
              ↑
              
                
              
              ⁢
              
                N
                
                  2
                  ⁢
                  π
                
              
            
            ·
            
              1
              
                2
                ⁢
                N
              
            
          
        
        =
        
          1
          
            4
            ⁢
            π
          
        
      
    
    
      
        (
        B13
        )
      
    
  




For finite N however, 1/(4π) is not the absolute lower limit for ΔtΔf. The product is zero for e0 and f0 and addition of small random numbers shows that it is close to zero for slightly different vectors.
The relation between the two uncertainty principles becomes more obvious if we consider the N-vectors that are derived from vmin vectors that were obtained for shorter time series (consisting of M events, with 1<M≦N). See the example of FIG. 14, where vmin was derived from the 8×8 matrix CHC. The components of F vmin are shown as a function of frequency fn (black dots). These frequencies are multiples of 1⁄8. Insert a block of 8 zeros between the two smallest components of the 8-vector vmin (so that the unimodal structure of the vector remains intact on a circular time base), which results in the 16-vector

{νmin,0,νmin,1, . . . ,νmin,4,0, . . . ,0,νmin,5, . . . ,νmin,8}  (B14)

This gives a slight increase in Δt (from 0.7583 to 0.7900) and an almost insignificant increase in Δf (from 0.0948 to 0.0949). The open circles in FIG. 14 represent interpolated values in the frequency domain when the block of eight zeros is inserted between the two smallest values of the eigenvector in the time domain, the continuous line represents interpolated values as the number of added zeros in the time domain tends to infinity. The resulting 16-vector leads to interpolation in the frequency domain, at frequencies that are multiples of 1/16 (open circles). This is a well-known form of interpolation in the frequency domain (through ‘zero-padding’ in the time domain).
In this way, for a given N, a number of N−1 vectors can be derived by adding zeros to smaller M-dimensional vmin vectors that are obtained from M×M matrices CHC. A trivial example is e0 that is derived by adding zeros to the ‘1-vector’ vmin=1 for M=1. FIG. 15 shows the corresponding uncertainties in the plot of NΔf vs. Δt for N=16. These values appear in zone I, delimited by the lines Δt=0, Δt=√{square root over (rt/2)}, NΔf=√{square root over (rt/2)} and NΔf=√{square root over (2)}·rt. The value for M=1 coincides with e0, the value for M=2 is indicated by an arrow and values for higher M gradually approach the value for M=N=16, located at the identity line (which corresponds to vmin for N=16). For relatively large M, Δt and Δf are hardly changed by the addition of zeros, while the associated σ approaches √{square root over (M/(2π))} so that




  
    
      
        
          
            
              
                Δ
                t
              
              ·
              N
            
            ⁢
            
              
            
            ⁢
            
              Δ
              f
            
          
          ≈
          
            
              
                
                  M
                  
                    4
                    ⁢
                    π
                  
                
              
              ·
              N
              ·
              
                1
                M
              
            
            ⁢
            
              
                M
                
                  4
                  ⁢
                  π
                
              
            
          
        
        =
        
          N
          
            4
            ⁢
            π
          
        
      
    
    
      
        (
        B15
        )
      
    
  




As a result, the values for relatively large M are plotted slightly below the hyperbola x·y=N/(4π) in FIG. 15. As time proceeds and N gets larger, a trail of values grows along this hyperbola, with new values appearing at the identity line. If Δf is plotted against Δt, the related hyperbola is given by x·y=1/(4π) and contains all possible values corresponding to different Gaussians at the lower limit of Gabor's uncertainty principle.
Without formal mathematical proof, one may imagine that the N-vectors derived from vmin for M<N pose a further limit to the accessible region in the (Δt,NΔf)-plane of FIG. 15. Since these vectors have the minimally possible total uncertainty for a sequence of M nonzero events, it can be expected that values outside the region marked by the black dots in FIG. 15 will not occur. This also holds for the values in zone II, which were derived by premultiplying the vectors in zone I by Ω8, which circularly shifts the vectors in the frequency domain (to the opposite part of the frequency circle), without a change of the weights in the time domain. Thus, Δf of these vectors is maximized without a change in Δt. Conversely, the points in zone VI are obtained from the vectors in zone I, premultiplied by T8, which circularly shifts the vectors in the time domain (thereby maximizing Δt without a change in Δf). The vectors in zone V follow from those in zone II through premultiplication by T8. The vectors in zone VIII follow from those in zone I by taking the Fourier transform (premultiplication by F means reflection in the identity line). Premultiplying these vectors by Ω8 gives the vectors in zone III, premultiplying by T8 gives the vectors in zone VII, and premultiplying the latter again by Ω8 gives those in zone IV. (It will be clear that there are more ways to go from one time-frequency zone to another, e.g., taking the Fourier transform of the vectors in zone VI moves them to zone III). In conclusion, it can be expected that possible combinations of Δt, NΔf are confined to the spade-like region outlined by the black dots (and open circles) in FIG. 15.
The Uncertainty Principle for Nonzero Mean Time and Frequency.
If the reference time tR in Eq. 19 is allowed to be nonzero, then

Δt2≡rt2xH(Ω−ωtRI)H(Ω−ωtRI)x/∥x∥2, or

Δt≡rt∥(Ω−ωtRI)x∥/∥x∥  (B16)

Similarly, for nonzero reference frequency fR≡nR/N,




  
    
      
        
          
            
              Δ
              f
            
            ≡
            
              
                r
                f
              
              ⁢
              
                
                  
                  
                    
                      (
                      
                        Ω
                        -
                        
                          
                            ω
                            
                              n
                              R
                            
                          
                          ⁢
                          I
                        
                      
                      )
                    
                    ⁢
                    Fx
                  
                  
                
                /
                
                  
                  x
                  
                
              
            
          
          =
          
            
              r
              f
            
            ⁢
            
              
                
                
                  
                    (
                    
                      
                        T
                        
                          -
                          1
                        
                      
                      -
                      
                        
                          ω
                          
                            n
                            R
                          
                        
                        ⁢
                        I
                      
                    
                    )
                  
                  ⁢
                  x
                
                
              
              /
              
                
                x
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              Let
              ⁢
              
                
              
              ⁢
              
                A
                R
              
            
            ≡
            
              Ω
              -
              
                
                  ω
                  
                    t
                    R
                  
                
                ⁢
                I
              
            
          
          ,
          
            
              B
              R
            
            ≡
            
              
                T
                
                  -
                  1
                
              
              -
              
                
                  ω
                  
                    n
                    R
                  
                
                ⁢
                I
                ⁢
                
                  
                
                ⁢
                and
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            C
            R
          
          ≡
          
            
              r
              t
            
            ⁡
            
              [
              
                
                  
                    
                      A
                      R
                    
                  
                
                
                  
                    
                      B
                      R
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        B17
        )
      
    
  




When C is still defined with respect to tR=0 and fR=0, then CRHCR and CHC turn out to be similar,

(CRHCR)TtRΩnR=TtRΩnR(CHC)  (B18)

This is readily shown using the relation TkΩn=ω−nkΩnTk, which is valid for any integer k, n. So, if (σ2,v) is an eigenvalue-eigenvector pair of CHC, then

(CRHCR)TtRΩnRv=σ2TtRΩnRv  (B19)

meaning that TtRΩnRv is an eigenvector of CRHCR with eigenvalue σ2. It follows that the uncertainty of TtRΩnRv relative to tR in the time domain and fR in the frequency domain is equal to the uncertainty of v relative to zero in both domains. This is no surprise. The squared absolute values (the ‘weights’) of the components of TtRΩnRv are identical to those of v, albeit after a circular shift in the frequency domain (by ΩnR) and in the time domain (by TtR).

Derivation of the Time-Frequency Transform

The columns of the time-frequency transform matrix M are orthonormal,

MHM=I  (C1)

This can be proved as follows. The product can be written as a sum of circulant matrices,




  
    
      
        
          
            
              
                
                  M
                  H
                
                ⁢
                M
              
              =
              
              ⁢
              
                
                  
                    1
                    M
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            M
                            0
                            H
                          
                        
                        
                          ...
                        
                        
                          
                            M
                            
                              M
                              -
                              1
                            
                            H
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          M
                          0
                        
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        
                          M
                          
                            M
                            -
                            1
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  1
                  M
                
                ⁢
                
                  
                    ∑
                    
                      m
                      =
                      0
                    
                    
                      M
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      M
                      m
                      H
                    
                    ⁢
                    
                      
                        M
                        m
                      
                      .
                    
                  
                
              
            
          
        
      
    
    
      
        (
        C2
        )
      
    
  




Since postmultiplication of hmH by T−1 means that its components are circularly shifted to the right, the circulant Mm can also be written




  
    
      
        
          M
          m
        
        =
        
          
            [
            
              
                
                  
                    h
                    m
                    H
                  
                
              
              
                
                  
                    
                      h
                      m
                      H
                    
                    ⁢
                    
                      T
                      
                        -
                        1
                      
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    
                      h
                      m
                      H
                    
                    ⁢
                    
                      T
                      
                        
                          -
                          N
                        
                        +
                        1
                      
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        C3
        )
      
    
  




Since circulants are normal matrices, they commute with their Hermitian transpose,




  
    
      
        
          
            
              
                
                  M
                  m
                  H
                
                ⁢
                
                  M
                  m
                
              
              =
              
              ⁢
              
                
                  M
                  m
                
                ⁢
                
                  M
                  m
                  H
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          h
                          m
                          H
                        
                      
                    
                    
                      
                        
                          
                            h
                            m
                            H
                          
                          ⁢
                          
                            T
                            
                              -
                              1
                            
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                    
                    
                      
                        
                          
                            h
                            m
                            H
                          
                          ⁢
                          
                            T
                            
                              
                                
                                  -
                                  1
                                
                                ⁢
                                N
                              
                              +
                              1
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          h
                          m
                        
                      
                      
                        
                          Th
                          m
                        
                      
                      
                        ...
                      
                      
                        
                          
                            T
                            
                              N
                              -
                              1
                            
                          
                          ⁢
                          
                            h
                            m
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          h
                          m
                        
                      
                    
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          Th
                          m
                        
                      
                    
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          T
                          2
                        
                        ⁢
                        
                          h
                          m
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          T
                          
                            N
                            -
                            1
                          
                        
                        ⁢
                        
                          h
                          m
                        
                      
                    
                  
                  
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          T
                          
                            N
                            -
                            1
                          
                        
                        ⁢
                        
                          h
                          m
                        
                      
                    
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          h
                          m
                        
                      
                    
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          Th
                          m
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          h
                          m
                          H
                        
                        ⁢
                        
                          T
                          
                            N
                            -
                            2
                          
                        
                        ⁢
                        
                          h
                          m
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  circ
                  ⁢
                  
                    {
                    
                      g
                      m
                      H
                    
                    }
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        C4
        )
      
    
  




where gm≡{hmHhm hmHT−1hm . . . hmHT−N+1hm} is the ‘complex autocorrelation’ of hm.

Its components are




  
    
      
        
          g
          
            m
            ,
            u
          
        
        ≡
        
          {
          
            
              
                
                  
                    
                      h
                      m
                      H
                    
                    ⁢
                    
                      T
                      
                        -
                        u
                      
                    
                    ⁢
                    
                      h
                      m
                    
                  
                  ,
                
              
              
                
                  
                    
                      
                        
                          
                            0
                            ≤
                            u
                            ≤
                            
                              M
                              -
                              1
                            
                          
                          ,
                          or
                        
                        ⁢
                        
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            N
                            -
                            M
                            +
                            2
                          
                          ≤
                          u
                          ≤
                          
                            N
                            -
                            1
                          
                        
                        ;
                      
                    
                  
                
              
            
            
              
                
                  0
                  ,
                
              
              
                
                  otherwise
                  .
                
              
            
          
        
      
    
    
      
        (
        C5
        )
      
    
  




According to the definition of the N-vector hm (Eqs. 26 and 27), it can also be written as ΩmN/Mh0. Using TkΩn=ω−nkΩnTk, it follows that the nonzero components are equal to




  
    
      
        
          
            
              h
              m
              H
            
            ⁢
            
              T
              
                -
                u
              
            
            ⁢
            
              h
              m
            
          
          =
          
            
              
                h
                0
                H
              
              ⁢
              
                Ω
                
                  
                    -
                    mN
                  
                  /
                  M
                
              
              ⁢
              
                T
                
                  -
                  u
                
              
              ⁢
              
                Ω
                
                  mN
                  /
                  M
                
              
              ⁢
              
                h
                0
              
            
            =
            
              
                
                  ω
                  
                    muN
                    /
                    M
                  
                
                ⁢
                
                  h
                  0
                  H
                
                ⁢
                
                  T
                  
                    -
                    u
                  
                
                ⁢
                
                  h
                  0
                
              
              =
              
                
                  e
                  
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      mu
                      /
                      M
                    
                  
                
                ⁢
                
                  a
                  u
                
              
            
          
        
        ,
      
    
    
      
        (
        C6
        )
      
    
  




where au≡h0HT−uh0. When these components are summed over m, we have





  
    
      
        
          
            ∑
            
              m
              =
              0
            
            
              M
              -
              1
            
          
          ⁢
          
            g
            
              m
              ,
              u
            
          
        
        =
        
          
            
              ∑
              
                m
                =
                0
              
              
                M
                -
                1
              
            
            ⁢
            
              
                e
                
                  2
                  ⁢
                  πi
                  ⁢
                  
                    
                  
                  ⁢
                  
                    mu
                    /
                    M
                  
                
              
              ⁢
              
                a
                u
              
            
          
          =
          
            {
            
              
                
                  
                    
                      
                        a
                        0
                      
                      ⁢
                      M
                    
                    ,
                  
                
                
                  
                    
                      u
                      =
                      0
                    
                    ;
                  
                
              
              
                
                  
                    0
                    ,
                  
                
                
                  
                    otherwise
                    ,
                  
                
              
            
          
        
      
    
    
      
        (
        C7
        )
      
    
  




which follows from the fact that {e2πimu/M} is a geometric progression for m=0, . . . , M−1. Note that a0=h0HT0h0=∥h0∥2=1 since hm is taken to be a unit vector. As a result, the sum in Eq. C2 can be rewritten as





  
    
      
        
          
            
              1
              M
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  0
                
                
                  M
                  -
                  1
                
              
              ⁢
              
                
                  M
                  m
                  H
                
                ⁢
                
                  M
                  m
                
              
            
          
          =
          
            
              
                1
                M
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    0
                  
                  
                    M
                    -
                    1
                  
                
                ⁢
                
                  circ
                  ⁢
                  
                    {
                    
                      g
                      m
                      H
                    
                    }
                  
                
              
            
            =
            
              
                
                  1
                  M
                
                ⁢
                circ
                ⁢
                
                  {
                  
                    
                      ∑
                      
                        m
                        =
                        0
                      
                      
                        M
                        -
                        1
                      
                    
                    ⁢
                    
                      g
                      m
                      H
                    
                  
                  }
                
              
              =
              
                
                  
                    1
                    M
                  
                  ⁢
                  circ
                  ⁢
                  
                    {
                    
                      M
                      ,
                      0
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      0
                    
                    }
                  
                
                =
                I
              
            
          
        
        ,
      
    
    
      
        (
        C8
        )
      
    
  




which completes the proof of Eq. C1.

The orthonormality property (Eq. C1) still holds if h0 is approximated by a truncated normalized Gaussian with uncertainty Δt and M′=10·Δt nonzero components (using Eq. B4), since it does not depend on the choice of h0 (provided ∥h0∥=1).
Time-Frequency Synthesis.
A direct corollary of the orthonormality property is that a given time series vector x can be written as the weighted sum of short-lasting oscillations (Eq. 34). Using the definition of Eq. 28,




  
    
      
        x
        =
        
          Ix
          =
          
            
              
                M
                H
              
              ⁢
              M
              ⁢
              
                
              
              ⁢
              x
            
            =
            
              
                
                  1
                  M
                
                ⁢
                
                  
                    ∑
                    
                      m
                      =
                      0
                    
                    
                      M
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      M
                      m
                      H
                    
                    ⁢
                    
                      M
                      m
                    
                  
                
              
              =
              
                
                  
                    1
                    M
                  
                  ⁢
                  
                    
                      ∑
                      
                        m
                        =
                        0
                      
                      
                        M
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        [
                        
                          
                            
                              
                                h
                                
                                  m
                                  ,
                                  0
                                
                              
                            
                            
                              ...
                            
                            
                              
                                h
                                
                                  m
                                  ,
                                  
                                    N
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                h
                                
                                  m
                                  ,
                                  0
                                
                                H
                              
                            
                          
                          
                            
                              ⋮
                            
                          
                          
                            
                              
                                h
                                
                                  m
                                  ,
                                  
                                    N
                                    -
                                    1
                                  
                                
                                H
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                =
                
                  
                    
                      1
                      M
                    
                    ⁢
                    
                      
                        ∑
                        
                          m
                          =
                          0
                        
                        
                          M
                          -
                          1
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            t
                            =
                            0
                          
                          
                            N
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            h
                            
                              m
                              ,
                              t
                            
                          
                          ⁢
                          
                            h
                            
                              m
                              ,
                              t
                            
                            H
                          
                          ⁢
                          x
                        
                      
                    
                  
                  =
                  
                    
                      1
                      M
                    
                    ⁢
                    
                      
                        ∑
                        
                          m
                          =
                          0
                        
                        
                          M
                          -
                          1
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            t
                            =
                            0
                          
                          
                            N
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            h
                            
                              m
                              ,
                              t
                            
                          
                          ⁢
                          
                            
                              
                                x
                                ~
                              
                              
                                m
                                ,
                                t
                              
                            
                            .
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        C9
        )
      
    
  




Analysis of Variance in the Time-Frequency Domain

Another direct corollary of the orthonormality property of the TFT is that the ‘energy’ (the squared norm) of the N-vector x is preserved after the transformation,

∥x∥2=xHIx=xHMHMx=∥Mx∥2=∥{tilde over (x)}∥2  (D1)

This permits an ‘analysis of variance’ (ANOVA) in the time-frequency domain. The above also means that the energy of x can be partitioned into M frequency-dependent components,




  
    
      
        
          
            
            x
            
          
          2
        
        =
        
          
            
              1
              M
            
            ⁢
            
              
                x
                H
              
              ⁡
              
                (
                
                  
                    ∑
                    
                      m
                      =
                      0
                    
                    
                      M
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      M
                      m
                      H
                    
                    ⁢
                    
                      M
                      m
                    
                  
                
                )
              
            
            ⁢
            x
          
          =
          
            
              
                1
                M
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    0
                  
                  
                    M
                    -
                    1
                  
                
                ⁢
                
                  
                    x
                    H
                  
                  ⁢
                  
                    M
                    m
                    H
                  
                  ⁢
                  
                    M
                    m
                  
                  ⁢
                  x
                
              
            
            =
            
              
                
                  1
                  M
                
                ⁢
                
                  
                    ∑
                    
                      m
                      =
                      0
                    
                    
                      M
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                      
                        
                          M
                          m
                        
                        ⁢
                        x
                      
                      
                    
                    2
                  
                
              
              =
              
                
                  1
                  M
                
                ⁢
                
                  
                    ∑
                    
                      m
                      =
                      0
                    
                    
                      M
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        
                        
                          
                            x
                            ~
                          
                          m
                        
                        
                      
                      2
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        D2
        )
      
    
  




Suppose the random N-vector X has a multivariate normal distribution with expectation E{X}=0 and variance var{X}=σ2I. Then the quadratic form ∥X∥2/σ2 follows a chi-square distribution on N degrees of freedom. According to Eq. D2, ∥X∥2 can be partitioned into M frequency-dependent components ∥{tilde over (X)}m∥2. The quadratic form ∥{tilde over (X)}m∥2/N may be called the ‘TFT sample power spectrum’ of X for frequency fm. For white noise, the expectation of this RV is



  
    
      E
      ⁢
      
        {
        
          
            
              
              
                
                  X
                  ~
                
                m
              
              
            
            2
          
          /
          N
        
        }
      
    
    =
    
      
        E
        ⁢
        
          
            {
            
              
                
                
                  
                    M
                    m
                  
                  ⁢
                  X
                
                
              
              2
            
            }
          
          /
          N
        
      
      =
      
        E
        ⁢
        
          
            {
            
              
                X
                H
              
              ⁢
              
                M
                m
                H
              
              ⁢
              
                M
                m
              
              ⁢
              X
            
            }
          
          /
          N
        
      
    
  




  
    
      
        σ
        2
      
      ⁢
      tr
      ⁢
      
        
          {
          
            
              M
              m
              H
            
            ⁢
            
              M
              m
            
          
          }
        
        /
        N
      
    
    ,
  



where tr{•} stands for the trace of the matrix. Using Eq. C4, it follows that tr{MmHMm}=NhmHhm=N, so that

E{∥{tilde over (X)}m∥2/N}=σ2  (D3)

The sample power spectrum according to the definition above is however sensitive to bias due to the circularity assumption. Note that the components of {tilde over (X)}m≡MmX are derived from X by circular convolution with a linear time-invariant filter with M nonzero elements (Eqs. 26-31). The structure of this filter is such that the first M−K−1 and the last K components of {tilde over (X)}m are dependent on the circularity assumption (provided these integers are larger than zero), where K is the largest integer less than or equal to M/2. A power estimator free of bias due to the circularity assumption is thus obtained with the ‘selection matrix’ S, defined as




  
    
      
        
          S
          ≡
          
            diag
            ⁢
            
              
                {
                
                  0
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  0
                
              
              
                }
                
                  M
                  -
                  K
                  -
                  1
                
              
            
          
        
        ,
        
          
            1
            ,
            ...
            ⁢
            
              
            
            ,
            1
          
          
            }
            
              N
              -
              M
              +
              1
            
          
        
        ,
        
          
            
              0
              ,
              ...
              ⁢
              
                
              
              ,
              0
            
            }
          
          
            }
            K
          
        
      
    
    
      
        (
        D4
        )
      
    
  




Premultiplication of {tilde over (X)}m by S selects the components of {tilde over (X)}m that are independent of the circularity assumption, which gives the unbiased quadratic form ∥S Mm X∥2. When the complementary frequency fM-m′ is also included, the ‘total’ unbiased sample power may be defined as




  
    
      
        
          P
          
            X
            ,
            m
          
        
        ≡
        
          {
          
            
              
                
                  
                    
                      
                        
                        
                          
                            SM
                            m
                          
                          ⁢
                          X
                        
                        
                      
                      2
                    
                    /
                    
                      N
                      S
                    
                  
                  ,
                
              
              
                
                  
                    m
                    =
                    
                      
                        0
                        ⁢
                        
                          
                        
                        ⁢
                        or
                        ⁢
                        
                          
                        
                        ⁢
                        m
                      
                      =
                      
                        1
                        2
                      
                    
                  
                  ;
                
              
            
            
              
                
                  
                    
                      
                        
                        
                          
                            S
                            ⁡
                            
                              (
                              
                                
                                  M
                                  m
                                
                                +
                                
                                  M
                                  
                                    M
                                    -
                                    m
                                  
                                
                              
                              )
                            
                          
                          ⁢
                          X
                        
                        
                      
                      2
                    
                    /
                    
                      (
                      
                        2
                        ⁢
                        
                          N
                          S
                        
                      
                      )
                    
                  
                  ,
                
              
              
                
                  otherwise
                  .
                
              
            
          
        
      
    
    
      
        (
        D5
        )
      
    
  




The number of ‘samples’ in the t-f domain is NS=N−M+1.
The sample power (divided by σ2) follows an approximate chi-square distribution with equivalent degrees of freedom (EDOFs)

η=2·(E{PX,m})2/var{PX,m}  (D6)

Letting P≡S(Mm+MM-m)/(2NS) for m≠0 and m≠1⁄2, it follows that for zero mean white noise X,




  
    
      
        
          
            E
            ⁢
            
              {
              
                P
                
                  X
                  ,
                  m
                
              
              }
            
          
          =
          
            
              σ
              2
            
            ⁢
            tr
            ⁢
            
              {
              
                
                  P
                  H
                
                ⁢
                P
              
              }
            
          
        
        ,
        
          

        
        ⁢
        
          
            var
            ⁢
            
              {
              
                P
                
                  X
                  ,
                  m
                
              
              }
            
          
          =
          
            2
            ⁢
            
              
            
            ⁢
            
              σ
              4
            
            ⁢
            tr
            ⁢
            
              {
              
                
                  (
                  
                    
                      P
                      H
                    
                    ⁢
                    P
                  
                  )
                
                2
              
              }
            
          
        
        ,
      
    
    
      
        (
        D7
        )
      
    
  




which gives

η=(tr{PHP})2/tr{(PHP)2}  (D8)

In the present paper, η was numerically computed according to Eq. D8 using mathematical computer software.
Derivation of Confidence Limits
Mean and variance of {circumflex over (B)}Q,t,m and {circumflex over (B)}P,t,m. Since the noise in flow {tilde over (Q)}e,t,m is assumed to be zero mean white noise, the expectation of {tilde over (Q)}t,m is, using Eq. 43,




  
    
      
        
          E
          ⁢
          
            {
            
              
                Q
                ~
              
              
                t
                ,
                m
              
            
            }
          
        
        =
        
          
            E
            ⁢
            
              {
              
                
                  
                    
                      x
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  ⁢
                  
                    b
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                
                +
                
                  
                    Q
                    ~
                  
                  
                    e
                    ,
                    t
                    ,
                    m
                  
                
              
              }
            
          
          =
          
            
              
                E
                ⁢
                
                  {
                  
                    
                      
                        x
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    ⁢
                    
                      b
                      
                        Q
                        ,
                        t
                        ,
                        m
                      
                    
                  
                  }
                
              
              +
              
                E
                ⁢
                
                  {
                  
                    
                      Q
                      ~
                    
                    
                      e
                      ,
                      t
                      ,
                      m
                    
                  
                  }
                
              
            
            =
            
              
                
                  
                    
                      x
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  ⁢
                  
                    b
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                
                +
                0
              
              =
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                ⁢
                
                  
                    b
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        E1
        )
      
    
  




The variance of {tilde over (Q)}e,t,m is σQ,m2I. Since the product {tilde over (x)}t,mbQ,t,m is deterministic,

var{{tilde over (Q)}t,m}=var{{tilde over (x)}t,mbQ,t,m+{tilde over (Q)}e,t,m}=var{{tilde over (Q)}e,t,m}=σQ,m2I  (E2)

The mean and variance of {circumflex over (B)}Q,t,m now follow from Eq. 46,




  
    
      
        
          
            E
            ⁢
            
              {
              
                
                  B
                  ^
                
                
                  Q
                  ,
                  t
                  ,
                  m
                
              
              }
            
          
          =
          
            
              E
              ⁢
              
                {
                
                  
                    
                      x
                      ~
                    
                    
                      t
                      ,
                      m
                    
                    H
                  
                  ⁢
                  
                    
                      
                        Q
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    /
                    
                      
                        
                        
                          
                            x
                            ~
                          
                          
                            t
                            ,
                            m
                          
                        
                        
                      
                      2
                    
                  
                
                }
              
            
            =
            
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                ⁢
                E
                ⁢
                
                  
                    {
                    
                      
                        Q
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    }
                  
                  /
                  
                    
                      
                      
                        
                          x
                          ~
                        
                        
                          t
                          ,
                          m
                        
                      
                      
                    
                    2
                  
                
              
              =
              
                
                  
                    
                      x
                      ~
                    
                    
                      t
                      ,
                      m
                    
                    H
                  
                  ⁢
                  
                    
                      x
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  ⁢
                  
                    
                      b
                      
                        Q
                        ,
                        t
                        ,
                        m
                      
                    
                    /
                    
                      
                        
                        
                          
                            x
                            ~
                          
                          
                            t
                            ,
                            m
                          
                        
                        
                      
                      2
                    
                  
                
                =
                
                  b
                  
                    Q
                    ,
                    t
                    ,
                    m
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        E3
        )
      
    
  
  
    
      
        
          var
          ⁢
          
            {
            
              
                B
                ^
              
              
                Q
                ,
                t
                ,
                m
              
            
            }
          
          ⁢
          
            
              
                x
                ~
              
              
                t
                ,
                m
              
              H
            
            
              
                
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
              
              2
            
          
          ⁢
          var
          ⁢
          
            {
            
              
                Q
                ~
              
              
                t
                ,
                m
              
            
            }
          
          ⁢
          
            
              
                x
                ~
              
              
                t
                ,
                m
              
            
            
              
                
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
              
              2
            
          
        
        =
        
          
            
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                
                  
                    
                    
                      
                        x
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    
                  
                  2
                
              
              ·
              
                σ
                
                  Q
                  ,
                  m
                
                2
              
            
            ⁢
            
              I
              ·
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
                  
                    
                    
                      
                        x
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    
                  
                  2
                
              
            
          
          =
          
            
              σ
              
                Q
                ,
                m
              
              2
            
            
              
                
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
              
              2
            
          
        
      
    
    
      
        (
        E4
        )
      
    
  




As a result, {circumflex over (B)}Q,t,m is an unbiased estimator of bQ,t,m. Mean and variance for {circumflex over (B)}P,t,m follow in an analogous manner,

E{{circumflex over (B)}P,t,m}=bP,t,m and var{{circumflex over (B)}P,t,m}=σP,m2/∥{tilde over (x)}t,m∥2  (E5)

Confidence Limits.
The confidence limits for {circumflex over (B)}Q,t,m and {circumflex over (B)}P,t,m follow from the standard approach in least squares analysis. Combination of Eqs. 43 and 44 gives

{tilde over (Q)}e,t,m={tilde over (Q)}t,m−{tilde over (x)}t,mbQ,t,m={tilde over (x)}t,m({circumflex over (B)}Q,t,m−bQ,t,m)+{circumflex over (Q)}e,t,m  (E6)

Due to the orthogonality between {tilde over (x)}t,m and {circumflex over (Q)}e,t,m (see FIG. 5),

∥{tilde over (Q)}e,t,m∥2=∥{tilde over (x)}t,m∥2·|{circumflex over (B)}Q,t,m−bQ,t,m|2+∥{circumflex over (Q)}e,t,m∥2  (E7)

The random variable ∥{tilde over (x)}t,m∥2|{circumflex over (B)}Q,t,m−bQ,t,m|2/σQ,m2 has the form

({circumflex over (B)}Q,t,m−E{{circumflex over (B)}Q,t,m})H(var{{circumflex over (B)}Q,t,m})−1({circumflex over (B)}Q,t,m−E{{circumflex over (B)}Q,t,m})

and is thus (exactly) distributed as a chi-square variable (on one degree of freedom). When {tilde over (x)}t,M-m at the complementary frequency fM-m=1−m/M is also included in the estimation, the resulting chi-square variable has two degrees of freedom. When the noise is estimated by the total unbiased power spectrum (Eq. D5), Equation E7 decomposes the total power of the noise into orthogonal components with EDOFs η=2+(η−2). Then the ratio of the two orthogonal components at the right-hand side of Eq. E7 follows an F-distribution on (2,η−2) degrees of freedom. Let the upper 100(1−α) % point of this distribution be denoted by Fα. Then





  
    
      
        
          
            
              
                
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
                
              
              2
            
            ⁢
            
              
                
                
                  
                    
                      B
                      ^
                    
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                  -
                  
                    b
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                
                
              
              2
            
          
          
            
              
              
                
                  Q
                  ^
                
                
                  e
                  ,
                  t
                  ,
                  m
                
              
              
            
            2
          
        
        ≤
        
          
            F
            α
          
          ⁢
          
            2
            
              η
              -
              2
            
          
        
      
    
    
      
        (
        E8
        )
      
    
  




Using the definition of the TFT sample coherence (Eq. 45) and assuming that KQ,t,m2>0, this can be rewritten as




  
    
      
        
          
            
              
              
                
                  
                    B
                    ^
                  
                  
                    Q
                    ,
                    t
                    ,
                    m
                  
                
                -
                
                  b
                  
                    Q
                    ,
                    t
                    ,
                    m
                  
                
              
              
            
            2
          
          
            
              
              
                b
                
                  Q
                  ,
                  t
                  ,
                  m
                
              
              
            
            2
          
        
        ≤
        
          
            F
            α
          
          ⁢
          
            2
            
              η
              -
              2
            
          
          ⁢
          
            
              1
              -
              
                K
                
                  Qx
                  ,
                  t
                  ,
                  m
                
                2
              
            
            
              K
              
                Qx
                ,
                t
                ,
                m
              
              2
            
          
        
      
    
    
      
        (
        E9
        )
      
    
  




Denoting the right-hand side of this inequality by AQ,t,m2,

|{circumflex over (B)}Q,t,m−bQ,t,m|2≦|{circumflex over (B)}Q,t,m|2AQ,t,m2  (E10)

The 100(1−α) % confidence region for {circumflex over (B)}Q,t,m is thus described by a circle in the complex plane with center {circumflex over (B)}Q,t,m and radius |{circumflex over (B)}Q,t,m|AQ,t,m. The confidence region for {circumflex over (B)}P,t,m is derived in a similar manner,

|{circumflex over (B)}P,t,m−bP,t,m|2≦|{circumflex over (B)}P,t,m|2AP,t,m2  (E11)

To arrive at a confidence region for {circumflex over (Z)}t,m, multiply both sides of Eq. 10 by |{circumflex over (Z)}t,m/bQ,t,m|2 and substitute U≡{circumflex over (Z)}t,m{circumflex over (B)}Q,t,m/bQ,t,m, which gives

|U−{circumflex over (Z)}t,m|2≦|U|2AQ,t,m2  (E12)

Using the real and imaginary parts of the complex variables, this can be rearranged into




  
    
      
        
          
            
            
              U
              -
              
                
                  
                    Z
                    ^
                  
                  
                    t
                    ,
                    m
                  
                
                
                  1
                  -
                  
                    A
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                    2
                  
                
              
            
            
          
          2
        
        ≤
        
          
            
              
              
                
                  
                    Z
                    ^
                  
                  
                    t
                    ,
                    m
                  
                
                
                  1
                  -
                  
                    A
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                    2
                  
                
              
              
            
            2
          
          ⁢
          
            A
            
              Q
              ,
              t
              ,
              m
            
            2
          
        
      
    
    
      
        (
        E13
        )
      
    
  




Next divide both sides of Eq. E11 by |bQ,t,m|2. Replacing {circumflex over (B)}P,t,m by {circumflex over (Z)}t,m {circumflex over (B)}Q,t,m (Eq. 47) and {circumflex over (Z)}t,m{circumflex over (B)}Q,t,m/bQ,t,m by U,

|U−zt,m|2≦|U|2AP,t,m2  (E14)

The confidence region for {circumflex over (Z)}t,m can now be derived in a two-step approach. According to Eq. E13, U lies within a circle with center Cu≡{circumflex over (Z)}t,m/(1−AQ,t,m2) and radius ru≡|{circumflex over (Z)}t,m|AQ,t,m/(1−AQ,t,m2), on 100(1−α)% of occasions (FIG. 16A). Suppose that {tilde over (Q)}e,t,m and {tilde over (P)}e,t,m are uncorrelated. Then, for a given value of U, zt,m lies within a second circle, with center U and radius |U|AP,t,m, on 100(1−α) % of occasions (Eq. E14). Consider the two extreme values for |U|, which are the magnitudes of Umin and Umax in FIG. 16A. The corresponding extreme values for |zt,m| are the magnitudes of zmin and zmax in FIG. 16B. From Eqs. E13 and E14,




  
    
      
        
          
            z
            min
          
          =
          
            
              
                Z
                ^
              
              
                t
                ,
                m
              
            
            ⁢
            
              
                1
                -
                
                  A
                  
                    P
                    ,
                    t
                    ,
                    m
                  
                
              
              
                1
                +
                
                  A
                  
                    Q
                    ,
                    t
                    ,
                    m
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            z
            max
          
          =
          
            
              
                Z
                ^
              
              
                t
                ,
                m
              
            
            ⁢
            
              
                1
                +
                
                  A
                  
                    P
                    ,
                    t
                    ,
                    m
                  
                
              
              
                1
                -
                
                  A
                  
                    Q
                    ,
                    t
                    ,
                    m
                  
                
              
            
          
        
      
    
    
      
        (
        E15
        )
      
    
  




A conservative limit of the 100(1−α) % confidence region for {circumflex over (Z)}t,m is now given by the circle that contains zmin and zmax, and whose center lies on the line through {circumflex over (Z)}t,m and the origin. As a result, the corresponding confidence region is described by




  
    
      
        
          
            
            
              
                z
                
                  t
                  ,
                  m
                
              
              -
              
                
                  
                    Z
                    ^
                  
                  
                    t
                    ,
                    m
                  
                
                ⁢
                
                  
                    1
                    +
                    
                      
                        A
                        
                          Q
                          ,
                          t
                          ,
                          m
                        
                      
                      ⁢
                      
                        A
                        
                          P
                          ,
                          t
                          ,
                          m
                        
                      
                    
                  
                  
                    1
                    -
                    
                      A
                      
                        Q
                        ,
                        t
                        ,
                        m
                      
                      2
                    
                  
                
              
            
            
          
          2
        
        ≤
        
          
            
            
              
                
                  Z
                  ^
                
                
                  t
                  ,
                  m
                
              
              ⁢
              
                
                  
                    A
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                  
                  +
                  
                    A
                    
                      P
                      ,
                      t
                      ,
                      m
                    
                  
                
                
                  1
                  -
                  
                    A
                    
                      Q
                      ,
                      t
                      ,
                      m
                    
                    2
                  
                
              
            
            
          
          2
        
      
    
    
      
        (
        E16
        )
      
    
  




This region is delimited by the circle in FIGS. 16B and 16C with center C2≡{circumflex over (Z)}t,m(1+AQ,t,mAP,t,m)/(1−AQ,t,m2) and radius r2≡|{circumflex over (Z)}t,m|(AQ,t,m+AP,t,m)/(1−AQ,t,m2). This circle is not concentric about {circumflex over (Z)}t,m. The distribution of possible values for zt,m for a given {circumflex over (Z)}t,m is skew-symmetric about {circumflex over (Z)}t,m (FIG. 16D). The skewness is expressed by tan β in FIG. 16C. From Eq. E16, it follows that

tan β=|C2−{circumflex over (Z)}t,m|/r2=AQ,t,m  (E17)

Special Case 1: No Noise Inflow.
If the noise in flow is zero, {tilde over (Q)}e,t,m=0, then {tilde over (Q)}t,m={tilde over (x)}t,mbQ,t,m and {circumflex over (B)}Q,t,m=bQ,t,m (Eqs. 43 and 46). Let the corresponding estimator of zt,m be denoted by {circumflex over (Z)}t,m(Q). Assuming that bQ,t,m≠0,




  
    
      Z
      ^
    
    
      t
      ,
      m
    
    
      (
      Q
      )
    
  
  =
  
    
      
        
          B
          ^
        
        
          P
          ,
          t
          ,
          m
        
      
      
        
          B
          ^
        
        
          Q
          ,
          t
          ,
          m
        
      
    
    =
    
      
        
          
            x
            ~
          
          
            t
            ,
            m
          
          H
        
        ⁢
        
          
            P
            ~
          
          
            t
            ,
            m
          
        
      
      
        
          
            
            
              
                x
                ~
              
              
                t
                ,
                m
              
            
            
          
          2
        
        ⁢
        
          b
          
            Q
            ,
            t
            ,
            m
          
        
      
    
  




Since in this case {tilde over (x)}t,m={tilde over (Q)}t,m/bQ,t,m,




  
    
      
        
          
            Z
            ^
          
          
            t
            ,
            m
          
          
            (
            Q
            )
          
        
        =
        
          
            
              
                Q
                ~
              
              
                t
                ,
                m
              
              H
            
            ⁢
            
              
                P
                ~
              
              
                t
                ,
                m
              
            
          
          
            
              
              
                
                  Q
                  ~
                
                
                  t
                  ,
                  m
                
              
              
            
            2
          
        
      
    
    
      
        (
        E18
        )
      
    
  




This is the ‘ordinary least squares estimator’. Then KQx,t,m2=1 and AQ,t,m=0. The confidence limits are still described by Eq. E16, although with AQ,t,m=0. The confidence circle is centered about {circumflex over (Z)}t,m(Q) now, with radius |{circumflex over (Z)}t,m(Q)|AP,t,m. The skewness parameter tan β is equal to zero.
Special Case 2: No Noise in Pressure.
If the noise in pressure is zero, {tilde over (P)}e,t,m=0, then {tilde over (P)}t,m={tilde over (x)}t,mbP,t,m and {circumflex over (B)}P,t,m=bP,t,m. When the corresponding estimator of zt,m is denoted by {circumflex over (Z)}t,m(P),




  
    
      
        
          
            Z
            ^
          
          
            t
            ,
            m
          
          
            (
            P
            )
          
        
        =
        
          
            
              
                B
                ^
              
              
                P
                ,
                t
                ,
                m
              
            
            
              
                B
                ^
              
              
                Q
                ,
                t
                ,
                m
              
            
          
          =
          
            
              
                
                  b
                  
                    P
                    ,
                    t
                    ,
                    m
                  
                
                ⁢
                
                  
                    
                    
                      
                        x
                        ~
                      
                      
                        t
                        ,
                        m
                      
                    
                    
                  
                  2
                
              
              
                
                  
                    x
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                ⁢
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
              
            
            =
            
              
                
                  
                  
                    
                      P
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  
                
                2
              
              
                
                  
                    P
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                ⁢
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
              
            
          
        
      
    
    
      
        (
        E19
        )
      
    
  




This is the ‘data least squares estimator’. Now KPx,t,m2=1 and AP,t,m=0. From Eq. E16, the confidence circle has center {circumflex over (Z)}t,m(P)/(1−AQ,t,m2) and radius |{circumflex over (Z)}t,m(P)|/(1−AQ,t,m2). The skewness parameter tan β equals AQ,t,m. The two estimators {circumflex over (Z)}t,m(Q) and {circumflex over (Z)}t,m(P) are related through the squared coherence between flow and pressure,




  
    
      
        
          
            K
            
              PQ
              ,
              t
              ,
              m
            
            2
          
          ≡
          
            
              
                
                
                  
                    
                      Q
                      ~
                    
                    
                      t
                      ,
                      m
                    
                    H
                  
                  ⁢
                  
                    
                      P
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                
                
              
              2
            
            
              
                
                  
                  
                    
                      Q
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  
                
                2
              
              ⁢
              
                
                  
                  
                    
                      P
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  
                
                2
              
            
          
        
        =
        
          
            
              
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                ⁢
                
                  
                    P
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
              
              
                
                  
                  
                    
                      Q
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  
                
                2
              
            
            ·
            
              
                
                  
                    P
                    ~
                  
                  
                    t
                    ,
                    m
                  
                  H
                
                ⁢
                
                  
                    Q
                    ~
                  
                  
                    t
                    ,
                    m
                  
                
              
              
                
                  
                  
                    
                      P
                      ~
                    
                    
                      t
                      ,
                      m
                    
                  
                  
                
                2
              
            
          
          =
          
            
              
                Z
                ^
              
              
                t
                ,
                m
              
              
                (
                Q
                )
              
            
            
              
                Z
                ^
              
              
                t
                ,
                m
              
              
                (
                P
                )
              
            
          
        
      
    
    
      
        (
        E20
        )
      
    
  




Examples
Squared Coherences as a Function of Time and Frequency
FIG. 17 shows the squared coherences for the same recording as in FIG. 7. In this example, the coherence between loudspeaker input and flow was generally lower than the coherence between input and pressure. As is apparent from the figure, the coherences change as a function of both time and frequency. The smallest coherences are obtained at the turning points from inspiration to expiration and vice versa, and at the lowest frequencies. Swallowing has a deep impact on coherence, especially between input and flow. FIG. 18 shows the squared coherences as a function of time and frequency for the estimated impedance shown in FIG. 8 for a patient with COPD. The drops in coherence at the turning points between the respiratory phases are much more pronounced than in the normal example of FIG. 17. In both FIGS. 17 and 18, the bold lines represent the squared coherence KQx,t,m2 between loudspeaker input and flow as a function of time for different resonant frequencies (8 to 24 Hz) and the thin lines represent the squared coherence KPx,t,m2 between loudspeaker input and pressure. “Flow” represents the low-frequency component of airflow, “insp” represents inspiration and “exp” represents expiration.
In conclusion, a method is presented to estimate the transfer function of the respiratory system of a subject with optimal time-frequency resolution.
As described above, the estimate of respiratory impedance may be made by a diagnostic tool that uses the estimate to assess obstruction of the airways or to estimate the severity of disease. The diagnostic tool may therefore also be used to assess the effectiveness of treatments (whether pharmacological or otherwise) that should affect the respiratory impedance.
For example, the estimates of respiratory impedance can be used to detect (i) expiratory flow limitation in chronic obstructive lung disease (COPD); (ii) the severity of airway obstruction in COPD or asthma, which itself be used to evaluate the effect of inhaled airway dilators (e.g. sympathicomimetic or parasympathicolytic drugs) over the course of time (this can be applicable to a research setting and also to a clinical setting where the patient is unable to perform the standard forced respiratory maneuvers; (iii) airway obstruction in asthma or COPD during sleep; or (iv) upper airway obstruction during sleep in patients suspected of sleep apnea-hypopnea syndrome.
The estimates of respiratory impedance may also or alternatively be used to adapt the settings of machine used in the treatment of a medical condition. For example, the estimates of respiratory impedance can be used in the adjustment of settings of noninvasive ventilation in COPD. The respiratory impedance can be used as an input to the ventilator to provide information on the severity of the airway impedance on a continuous basis (with unreliable values, as indicated by the confidence values, being discarded). This information can also be used to adjust the level of bilevel positive airway pressure with which the expiratory flow limitation is just overcome. As an alternative example, the estimates of respiratory impedance can be used as an aid to guide the level of continuous positive airway pressure in patients with obstructive sleep apnea syndrome.
While the invention has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the invention is not limited to the disclosed embodiments.
Variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single processor or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. A computer program may be stored/distributed on a suitable medium, such as an optical storage medium or a solid-state medium supplied together with or as part of other hardware, but may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems. Any reference signs in the claims should not be construed as limiting the scope.
REFERENCES


    1. Dellacà et al. Expiratory flow limitation detected by forced oscillation and negative expiratory pressure, European Respiratory Journal Vol. 29 No. 2, pages 363-374
    2. Daróczy B, Hantos Z. An improved forced oscillation estimation of respiratory impedance. Int J Biomed Comput 13: 221-235, 1982.
    3. Forbes G W, Alonso M A. Consistent analogs of the Fourier uncertainty relation. Am J Physics 69: 340-347, 2001.
    4. Forbes G W, Alonso M A, Siegman A E. Uncertainty relations and minimal uncertainty states for the discrete Fourier transform and the Fourier series. J Phys A: Math Gen 36: 7027-7047, 2003.
    5. Gabor D. Theory of communications. J Inst Electr Eng 93: 429-457, 1946.

Table 1.

Symbols and Abbreviations.

Time and Frequency



fm, fn, discrete frequency m/M (or n/N)
fR reference frequency for Δf 
fn N-vector that describes an harmonic oscillation with frequency fn 
m frequency index for the TFT
M (M′) width of the TFT filter (or effective width using a truncated Gaussian)
n frequency index for the ODFT
N number of events in a time series
NS number of samples in the t-f domain from a stochastic process
rt, rf radius of the time circle (or frequency circle)
t discrete time index
tR reference time for Δt 
A, B, C matrices related to uncertainty in time and frequency
F ODFT matrix
M (Mm) TFT matrix (or partial TFT matrix for frequency fm)
T time-shift operator; circ{0, . . . , 0, 1}
Δt, Δf uncertainty in time or frequency
ω complex exponential; exp(2πi/N)
σmin, σmax minimal and maximal singular value of C
Ω diagonal matrix with ωt on the main diagonal; diag{ω0, . . . , ωN−1}
Names of Variables
p pressure
q flow
x loudspeaker input (or an unknown variable)
z respiratory impedance













Types of variables
Examples








deterministic variable as a function of discrete time
qt



random variable as a function of discrete time
Qt



deterministic N-vector (lowercase)
q



random N-vector (uppercase)
Q



vector in the t-f domain (centered about frequency fm)
{tilde over (Q)}m



vector in the t-f domain (centered about frequency fm
{tilde over (Q)}t,.m



and starting at time t)




deterministic vector ‘predicted’ by linear 
{tilde over (q)}p,t,m



regression in t-f domain




random ‘error’ for linear regression in t-f domain
{tilde over (Q)}e,t,m



estimator of the error in t-f domain
{circumflex over (Q)}e,t,m









Linear Filters

hqx vector with the periodized impulse response of filter from x to q
Hqx,n transfer function from x to q for frequency fn 
Cqx circulant matrix that describes the filter between loudspeaker input and flow
Λxq diagonal matrix with Hqx,n on the main diagonal
Statistics
AQ,t,m, variable in t-f domain that determines the contribution of noise in flow to the confidence region for {circumflex over (Z)}t,m 
Fα upper 100(1−α)% confidence limit for the F-test on (2,η−2) degrees of freedom
KQx,t,m2 squared sample coherence between loudspeaker input and flow in t-f domain
{circumflex over (Z)}t,m bivariate least squares estimator of impedance zt,m as a function of time and frequency
{circumflex over (Z)}t,m(Q) estimator of zt,m under the assumption that the noise in flow is zero
α type II error
β angle that determines the deviation of {circumflex over (Z)}t,m from the center of the confidence region
η equivalent degrees of freedom
μQ mean flow
σQ2 variance of the noise in flow
General Mathematical Symbols
i imaginary number (i2=−1)
I N×N identity matrix
≡ is by definition equal to
|xt| I magnitude of complex-valued xt 
∥x∥ 2-norm of N-vector x
xH Hermitian transpose of the vector x
x* complex conjugate of x
{•} sequence (usually expressed as a column vector)
diag{•} diagonal matrix with the sequence on the main diagonal
circ{•} circulant matrix with the sequence in the first row
Abbreviations
FOT forced oscillation technique
t-f domain time-frequency domain
TFT time-frequency transform
ODFT orthonormal discrete Fourier transform
RV random variable

## Claims

1. A method of estimating respiratory impedance with a system comprising a patient interface device, an excitation source operatively coupled to the patient interface, a ventilator, and one or more hardware processors, the method comprising:
coupling the patient interface device to an airway of a subject who is using the ventilator to create a pneumatic system that includes the patient interface device and an airway of such a subject;
generating, with the excitation source, pressure oscillations, flow oscillations, or volume of gas oscillations in the airway of the subject;
determining, with the one or more hardware processors, the flow and pressure of the gas in the pneumatic system to produce a respective time series representing the flow and the pressure;
transforming, with the one or more hardware processors, the respective time series to the time-frequency domain to create a transformed time series, wherein the transforming determines flow and pressure as a function of discrete time and different frequencies, wherein the different frequencies are a function of a frequency index and an effective width of a time-frequency filter, and wherein the transforming is based on the frequency index, the time frequency filter, and a weighting factor that is approximated by a windowing function of time, the weighting factor being a function of a time uncertainty, wherein the time series of measurements for flow and pressure are denoted qt and pt respectively, and the step of transforming the respective time series to the time-frequency domain comprises evaluating





  
    
      q
      ~
    
    
      t
      ,
      m
    
  
  =
  
    
      ∑
      
        l
        =
        
          -
          
            K
            1
          
        
      
      
        K
        1
      
    
    ⁢
    
      
        w
        l
      
      ⁢
      
        e
        
          
            -
            2
          
          ⁢
          π
          ⁢
          
            
          
          ⁢
          i
          ⁢
          
            
          
          ⁢
          m
          ⁢
          
            
          
          ⁢
          
            l
            /
            M
          
        
      
      ⁢
      
        q
        
          t
          +
          l
        
      
    
  




for the flow time series and




  
    
      p
      ~
    
    
      t
      ,
      m
    
  
  =
  
    
      ∑
      
        l
        =
        
          -
          
            K
            1
          
        
      
      
        K
        1
      
    
    ⁢
    
      
        w
        l
      
      ⁢
      
        e
        
          
            -
            2
          
          ⁢
          πi
          ⁢
          
            
          
          ⁢
          m
          ⁢
          
            
          
          ⁢
          
            l
            /
            M
          
        
      
      ⁢
      
        p
        
          t
          +
          l
        
      
    
  




for the pressure time series, wherein the different frequencies are give by fm=m/M, where m is the frequency index and M is the effective width of the time-frequency filter, i2=−1, and wherein the weighting factor wl is given by wl=e−(l/Δt)2, and Δt is the uncertainty in time;
estimating, with the one or more hardware processors, power of the flow and the pressure from the transformed time series;
estimating, with the one or more hardware processors, respective cross spectra of the flow and the pressure based the transformed time series;
estimating, with the one or more hardware processors, the respiratory impedance of the subject from the estimated power and the estimated cross spectra; and
providing, with the one or more hardware processors, input to the ventilator based on the estimated respiratory impedance, the provided input causing an adjustment of one or more settings of the ventilator.

2. A method as claimed in claim 1, wherein the step of estimating the respiratory impedance of the subject from the estimated power and the estimated cross spectra comprises:
determining transfer functions from the pressure oscillations to the flow and pressure respectively from the respective estimated power and estimated cross spectra; and
estimating the respiratory impedance from the flow and pressure transfer functions.

3. A method as claimed in claim 1, wherein the method further comprises a step of determining confidence limits on the estimated impedance.

4. A method as claimed in claim 1, wherein the windowing function is a Gaussian function, a triangle function, a piece-wise linear approximation function, or a polynomial function.

5. A method as claimed in claim 1, wherein the step of estimating the power of the flow and pressure comprises evaluating




  
    P
    
      q
      ,
      t
      ,
      m
    
  
  =
  
    
      1
      
        N
        S
      
    
    ⁢
    
      
        ∑
        
          l
          =
          
            -
            
              K
              2
            
          
        
        
          K
          2
        
      
      ⁢
      
        
          
          
            
              q
              ~
            
            
              
                t
                +
                l
              
              ,
              m
            
          
          
        
        2
      
    
  




for the flow and




  
    P
    
      p
      ,
      t
      ,
      m
    
  
  =
  
    
      1
      
        N
        S
      
    
    ⁢
    
      
        ∑
        
          l
          =
          
            -
            
              K
              2
            
          
        
        
          K
          2
        
      
      ⁢
      
        
          
          
            
              p
              ~
            
            
              
                t
                +
                l
              
              ,
              m
            
          
          
        
        2
      
    
  




for the pressure, where NS is the number of samples in the time-frequency domain, K2=1⁄2(NS−1) where NS is odd and |•| stands for the absolute value.

6. A method as claimed in claim 5, wherein the step of estimating the cross spectra of the flow and pressure with the generated pressure waves comprises evaluating




  
    C
    
      xq
      ,
      t
      ,
      m
    
  
  =
  
    
      1
      
        N
        S
      
    
    ⁢
    
      
        ∑
        
          l
          =
          
            -
            
              K
              2
            
          
        
        
          K
          2
        
      
      ⁢
      
        
          
            x
            ~
          
          
            t
            ,
            m
          
          *
        
        ⁢
        
          
        
        ⁢
        
          
            q
            ~
          
          
            t
            ,
            m
          
        
      
    
  




for the flow and




  
    C
    
      xp
      ,
      t
      ,
      m
    
  
  =
  
    
      1
      
        N
        S
      
    
    ⁢
    
      
        ∑
        
          l
          =
          
            -
            
              K
              2
            
          
        
        
          K
          2
        
      
      ⁢
      
        
          
            x
            ~
          
          
            t
            ,
            m
          
          *
        
        ⁢
        
          
        
        ⁢
        
          
            p
            ~
          
          
            t
            ,
            m
          
        
      
    
  




for the pressure, where the asterisk denotes the complex conjugate and {tilde over (x)}t,m represents the pressure waves.

7. A method as claimed in claim 6, wherein the step of estimating the respiratory impedance of the subject from the estimated power and cross spectra comprises determining a transfer function from the pressure waves to the flow by evaluating




  
    B
    
      xq
      ,
      t
      ,
      m
    
  
  =
  
    
      C
      
        xq
        ,
        t
        ,
        m
      
    
    
      P
      
        x
        ,
        t
        ,
        m
      
    
  




and a transfer function from the pressure waves to the pressure by evaluating




  
    
      B
      
        xp
        ,
        t
        ,
        m
      
    
    =
    
      
        C
        
          xp
          ,
          t
          ,
          m
        
      
      
        P
        
          x
          ,
          t
          ,
          m
        
      
    
  
  ,




and wherein the respiratory impedance is estimated by evaluating




  
    Zrs
    
      t
      ,
      m
    
  
  =
  
    
      B
      
        xp
        ,
        t
        ,
        m
      
    
    
      B
      
        xq
        ,
        t
        ,
        m
      
    
  




with real component Rrst,m and imaginary component Xrst,m.

8. A method as claimed in claim 7, further comprising a step of determining confidence limits on the estimated impedance which includes:
determining the squared coherence between the pressure waves and the flow using





  
    
      K
      
        xq
        ,
        t
        ,
        m
      
      2
    
    =
    
      
        
          
          
            C
            
              xq
              ,
              t
              ,
              m
            
          
          
        
        2
      
      
        
          P
          
            x
            ,
            t
            ,
            m
          
        
        ·
        
          P
          
            
              q
              ,
              t
              ,
              m
            
            ⁢
            
              
            
          
        
      
    
  
  ;



determining the squared coherence between the pressure waves and the pressure using





  
    
      K
      
        xp
        ,
        t
        ,
        m
      
      2
    
    =
    
      
        
          
          
            C
            
              xp
              ,
              t
              ,
              m
            
          
          
        
        2
      
      
        
          P
          
            x
            ,
            t
            ,
            m
          
        
        ·
        
          P
          
            p
            ,
            t
            ,
            m
          
        
      
    
  
  ;



determining the number of degrees of freedom, η, of the power and cross spectra;
determining a flow-related variable Ap,t,m2 from





  
    
      A
      
        q
        ,
        t
        ,
        m
      
      2
    
    =
    
      
        
          F
          α
        
        ⁡
        
          (
          
            2
            
              η
              -
              2
            
          
          )
        
      
      ⁢
      
        (
        
          
            1
            -
            
              K
              
                xq
                ,
                t
                ,
                m
              
              2
            
          
          
            K
            
              xq
              ,
              t
              ,
              m
            
            2
          
        
        )
      
    
  
  ;



determining a pressure-related variable Ap,t,m2 from





  
    
      A
      
        p
        ,
        t
        ,
        m
      
      2
    
    =
    
      
        
          F
          α
        
        ⁡
        
          (
          
            2
            
              η
              -
              2
            
          
          )
        
      
      ⁢
      
        (
        
          
            1
            -
            
              K
              
                xp
                ,
                t
                ,
                m
              
              2
            
          
          
            K
            
              xp
              ,
              t
              ,
              m
            
            2
          
        
        )
      
    
  
  ;



determining 100(1−α)% confidence limits for the real and imaginary parts of the estimated impedance from, respectively, Rrs·c1±c2 and Xrs·c1±c2, where c1=1+Aq,t,m·c3,





  
    c
    2
  
  =
  
    
      
        
          
          
            Zrs
            
              t
              ,
              m
            
          
          
        
        ·
        
          c
          3
        
      
      ⁢
      
        
      
      ⁢
      and
      ⁢
      
        
      
      ⁢
      
        c
        3
      
    
    =
    
      
        
          
            A
            
              q
              ,
              t
              ,
              m
            
          
          +
          
            A
            
              p
              ,
              t
              ,
              m
            
          
        
        
          1
          -
          
            A
            
              q
              ,
              t
              ,
              m
            
            2
          
        
      
      .

9. A method as claimed in claim 7, further comprising the step of rejecting the estimates of respiratory impedance for a given time and frequency if either Aq,t,m≧a first threshold or Ap,t,m≧a second threshold.

10. A method as claimed in claim 9, wherein the first threshold is 1 and wherein the second threshold is 1.

11. A method as claimed in claim 1, wherein estimating the respective cross spectra of the flow and the pressure based the transformed time series is done based on a least squares criterion using a model with the oscillating pressure, flow, or volume of gas as an input and the flow and the pressure as outputs.

12. A method as claimed in claim 1, wherein determining the flow of the gas in the pneumatic system is accomplished by measuring the flow using a flow sensor operatively coupled to the pneumatic system, and wherein determining the pressure of the gas in the pneumatic system is accomplished using a pressure sensor operatively coupled to the pneumatic system.

13. An apparatus for estimating respiratory impedance of a subject, the apparatus comprising:
a patient interface device;
a ventilator;
an excitation source for generating pressure oscillations, flow oscillations, or volume of gas oscillations in the airway of the subject;
means for determining the flow and the pressure of gas in a pneumatic circuit defined by the patient interface device and an airway of such a subject and for outputting respective time series of values representing the flow or the pressure;
a processor configured to:
transform the respective time series to a time-frequency domain, wherein the transforming determines flow and pressure as a function of discrete time and different frequencies, wherein the different frequencies are a function of a frequency index and an effective width of a time-frequency filter, and wherein the transforming is based on the frequency index, the time frequency filter, and a weighting factor that is approximated by a windowing function of time the weighting factor being a function of a time uncertainty, wherein the time series of measurements for flow and pressure are denoted qt and pt respectively, and the step of transforming the respective time series to the time-frequency domain comprises evaluating






  
    
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          -
          
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        K
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        l
      
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          ⁢
          
            l
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            M
          
        
      
      ⁢
      
        q
        
          t
          +
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 for the flow time series and






  
    
      p
      ~
    
    
      t
      ,
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      ∑
      
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    ⁢
    
      
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 for the pressure time series, wherein the different frequencies are given by fm=m/M, where m is the frequency index and M is the effective width of the time-frequency filter, i2=−1, and wherein the weighting factor wl is given by wl=e−(l/Δt)2, and Δt is the uncertainty in time;
estimate power of the flow and the pressure based on the respective transformed time series;
estimate respective cross spectra of the flow and pressure based the respective transformed time series;
estimate the respiratory impedance of the subject from the estimated power and cross spectra; and
provide input to the ventilator based on the estimated respiratory impedance, the provided input causing an adjustment of one or more settings of the ventilator.

14. An apparatus as claimed in claim 13, wherein the means for determining the flow and the pressure of gas in the pneumatic circuit comprises:
a flow sensor operatively coupled to the pneumatic system; and
a pressure sensor operatively coupled to the pneumatic system.

15. An apparatus as claimed in claim 13, wherein estimating the respective cross spectra of the flow and the pressure based the transformed time series is done based on a least squares criterion using a model with the oscillating pressure, flow, or volume of gas as an input and the flow and the pressure as outputs.

16. An apparatus as claimed in claim 13, wherein estimating the respiratory impedance of the subject from the estimated power and the estimated cross spectra comprises:
determining transfer functions from the pressure oscillations to the flow and pressure respectively from the respective estimated power and estimated cross spectra; and
estimating the respiratory impedance from the flow and pressure transfer functions.

17. An apparatus as claimed in claim 13, wherein the processor is further configured to determine confidence limits for the estimated respiratory impedance.

18. A method as claimed in claim 1, wherein providing the input to the ventilator comprises providing the estimated respiratory impedance as input to the ventilator to cause the adjustment of one or more settings of the ventilator based on the estimated respiratory impedance.

19. A method as claimed in claim 1, wherein providing the input to the ventilator comprising providing input to the ventilator on a continuous basis based on the estimated respiratory impedance to cause the ventilator to adjust positive airway pressure effectuated by the ventilator for the subject.