Frequency and Instantaneous Frequency A Totally New View of Frequency

In search of frequency I found the trend and other information, e. g., quantification of nonlinearity Instantaneous Frequencies and Trends for Nonstationary Nonlinear Data IMA Hot Topic Conference 2011

Prevailing Views on Instantaneous Frequency The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer. J. Shekel, 1953 The uncertainty principle makes the concept of an Instantaneous Frequency impossible. K. Gröchennig, 2001

How to define frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena.

Definition of Frequency Given the period of a wave as T ; the frequency is defined as

Equivalence : The definition of frequency is equivalent to defining velocity as Velocity = Distance / Time But velocity should be V = dS / dt.

Traditional Definition of Frequency frequency = 1/period. Definition too crude Only work for simple sinusoidal waves Does not apply to nonstationary processes Does not work for nonlinear processes Does not satisfy the need for wave equations

Definitions of Frequency : 1 For any data from linear Processes

Jean-Baptiste-Joseph Fourier 1807 “On the Propagation of Heat in Solid Bodies” 1812 Grand Prize of Paris Institute “Théorie analytique de la chaleur” ‘... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigor. ’ 1817 Elected to Académie des Sciences 1822 Appointed as Secretary of Académie paper published Fourier’s work is a great mathematical poem. Lord Kelvin

Fourier Spectrum

Definition of Power Spectral Density Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a wide- sense stationary random process:

Problem with Fourier Frequency Limited to linear stationary cases: same spectrum for white noise and delta function. Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost. Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum.

The Importance of Phase

Random and Delta Functions

Fourier Components : Random Function

Fourier Components : Delta Function

Surrogate Signal I. Hello

The original data : Hello

The surrogate data : Hello

The Fourier Spectra : Hello

The IMF of Surrogate data : Hello

The Hilbert spectrum of Surrogate data : Hello

The IMF of original data : Hello

The Hilbert spectrum of original data : Hello

Surrogate Signal II. Duffing Wave

The original data : Duffing Pure Tone

Compare Duffing and Sine Duffing Sine

The surrogate data : Duffing

The Fourier Spectra : Duffing

The IMF of Surrogate data : Duffing

The Hilbert spectrum of Surrogate data : Duffing

The Hilbert spectrum of original data : Hello

Observations The sound qualities of the original and the surrogate is totally different, yet they have the same Fourier spectrum. The Hilbert spectra are totally different that reflect the different sound quality. Therefore, the ear should not perceive sound based on Fourier based analysis with the linear and stationary assumption.

Problems with Integral methods Frequency is not a function of time within the integral limit; therefore, the frequency variation could not be found in any differential equation, other than a constant. The integral transform pairs suffer the limitation imposed by the uncertainty principle.

Definitions of Frequency : 2 For Simple Dynamic System This is an system analysis but not a data analysis method.

Definitions of Frequency : 3 Instantaneous Frequency for IMF only

Teager Energy Operator : the Idea H. M. Teager, 1980: Some observations on oral air flow during phonation, IEEE Trans. Acoustics, Speech, Signal Processing, ASSP-28-5,

Generalized Zero-Crossing : By using intervals between all combinations of zero-crossings and extrema. T1T1 T2T2 T4T4

Generalized Zero-Crossing : Computing the weighted frequency.

Problems with TEO and GZC TEO has super time resolution but it is strictly for linear processes. GZC is robust but its resolution is still crude with resolution to ¼ wave length.

Definitions of Frequency : 4 Instantaneous Frequency for IMF only

Instantaneous Frequency

Instantaneous Frequency is indispensable for nonlinear Processes x

The Idea and the need of Instantaneous Frequency According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that Therefore, both wave number and frequency must have instantaneous values. But how to find φ(x, t)?

Ideal case for Instantaneous Frequency Obtain the analytic signal based on real valued function through Hilbert Transform. Compute the Instantaneous frequency by taking derivative of the phase function from AS. This is true only if the function is an IMF, and its imaginary part of the analytic signal is identical to the quadrature of the real part. Unfortunately, this is true only for very special and simple cases.

Hilbert Transform : Definition

Limitations for IF computed through Hilbert Transform Data must be expressed in terms of Intrinsic Mode Function. (Note : Traditional applications using band-pass filter distorts the wave form; therefore, it can only be used for linear processes.) IMF is only necessary but not sufficient. Bedrosian Theorem: Hilbert transform of a(t) cos θ(t) might not be exactly a(t) sin θ(t). Spectra of a(t) and cos θ(t) must be disjoint. Nuttall Theorem: Hilbert transform of cos θ(t) might not be sin θ(t) for an arbitrary function of θ(t). Quadrature and Hilbert Transform of arbitrary real functions are not necessarily identical. Therefore, a simple derivative of the phase of the analytic function for an arbitrary function may not work.

Data : Hello

Empirical Mode Decomposition Sifting to produce IMFs

Bedrosian Theorem Let f(x) and g(x) denotes generally complex functions in L 2 (-∞, ∞) of the real variable x. If (1) the Fourier transform F(ω) of f(x) vanished for │ω│> a and the Fourier transform G(ω) of g(x) vanishes for │ω│< a, where a is an arbitrary positive constant, or (2) f(x) and g(x) are analytic (i. e., their real and imaginary parts are Hilbert pairs), then the Hilbert transform of the product of f(x) and g(x) is given H { f(x) g(x) } = f(x) H { g(x) }. Bedrosian, E., 1963: A Product theorem for Hilbert Transform, Proceedings of the IEEE, 51,

Nuttall Theorem For any function x(t), having a quadrature xq(t), and a Hilbert transform xh(t); then, where Fq(ω) is the spectrum of xq(t). Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signal, Proc. IEEE, 54, 1458

Difficulties with the Existing Limitations Data are not necessarily IMFs. Even if we use EMD to decompose the data into IMFs. IMF is only necessary but not sufficient because of the following limitations: Bedrosian Theorem adds the requirement of not having strong amplitude modulations. Nuttall Theorem further points out the difference between analytic function and quadrature. The discrepancy, however, is given in term of the quadrature spectrum, which is an unknown quantity. Therefore, it cannot be evaluated. Nuttall Theorem provides a constant limit not a function of time; therefore, it is not very useful for non-stationary processes.

Analytic vs. Quadrature X(t) Y(t) Z(t) Analytic Hilbert Transform Q(t) Quadrature, not analytic No Known general method Analytic functions satisfy Cauchy-Reimann equation, but may be x 2 + y 2 ≠ 1. Then the arc-tangent would not recover the true phase function. Quadrature pairs are not analytic, but satisfy strict 90 o phase shift; therefore, x 2 + y 2 = 1, and the arc-tangent always gives the true phase function. For cosθ(t) with arbitrary function of θ(t) :

A Difficulty of Hilbert Transform Bedrosian Theorem

Data : Step-function with Carrier

Fourier Spectra for Step-function and Carrier

Hilbert Spectrum : Step-function with Carrier

Morlet Wavelet : Step-function with Carrier

Spectrogram : Step-function with Carrier

Data : Step-function with Carrier III

Hilbert Spectrum : Step-function with Carrier III

Problems with Hilbert Transform method If there is any amplitude change, the Fourier Spectrum for the envelope and carrier are not separable. Thus, we violated the limitations stated in the Bedrosian Theorem; drastic amplitude change produce drastic deteriorating results. Once we cannot separate the envelope and the carrier, the analytic signal through Hilbert Transform would not give the phase function of the carrier alone without the influence of the variation from the envelope. Therefore, the instantaneous frequency computed through the analytic signal ceases to have full physical meaning; it provides an approximation only.

Normalization To overcome the limitation imposed by Bedrosian Theorem

Why do we need Decomposition and Normalization : We need a method to reduce the data to Intrinsic Mode Functions; then we also need a method for AM FM decomposition to over come the difficulties stated in Bedrosian Theorem. An Example : Step-function with Carrier

Effects of Normalization Normalization method can give a true AM FM decomposition to over come the difficulties stated in Bedrosian Theorem, and also provide a sharper error index than Nuttall Theorem.

NHHT : Procedures Obtain IMF representation of the data from siftings. Find local maxima of the absolute value of IMF (to take advantage of using both upper and lower envelopes) and fix the end values as maxima to ameliorate the end effects. Construct a Spline Envelope (SE) through the maxima. When envelope goes under the data, straight line envelope will be used for that section of the SE. Normalize the data using SE : N-data = Data/SE. This steps can be repeated. Compute IF (FM) and Absolute Value (AV) from Hilbert Transform of N-data. Definition : Error Index = (AV-1) 2. Compute Instantaneous Frequency for SE (AM).

NHHT Procedures : 1. IMF from Data through siftings

NHHT Procedures : 2. Locate local maxima and fix the ends

NHHT Procedures : 3. Construct the Cubic Spline Envelope (CSE)

NHHT procedures:

NHHT Procedures : 4. Normalize the IMF through CSE

NHHT Procedures : 5. Compute IF through Hilbert Transform

NHHT Procedures : 6. Comparison of Hilbert Transforms of Data and Normalized data

NHHT Procedures : 7. Define the Error Index = (AV – 1) 2.

NHHT Procedures : 8. Define the IMF of Envelope

NHHT Procedures : 9. AM and FM of y=c3y(7001:8000,9)

NHHT : Procedures Obtain IMF representation of the data from siftings. Find local maxima of the absolute value of IMF (to take advantage of using both upper and lower envelopes) and fix the end values as maxima to ameliorate the end effects. Construct a Spline Envelope (SE) through the maxima. When envelope goes under the data, straight line envelope will be used for that section of the SE. Normalize the data using SE : N-data = Data/SE. This steps can be repeated. Compute IF (FM) and Absolute Value (AV) from Hilbert Transform of N-data. Definition : Error Index = (AV-1) 2. Compute Instantaneous Frequency for SE (AM).

Example : Exponentially Decaying Cubic Chirp Model function

Exponentially decaying cubic chirp : Equation

Exponentially decaying cubic chirp : Data

Exponentially decaying cubic chirp : Normalizing function

Exponentially decaying cubic chirp : Normalized carrier

Exponentially decaying cubic chirp : Phase Diagram

Exponentially decaying cubic chirp : Instantaneous Frequency

Exponentially decaying cubic chirp : Error Indices

Another difficulty of Hilbert Transform Nuttal Theorem

Nuttall Theorem For any function x(t), having a quadrature xq(t), and a Hilbert transform xh(t); then, where Fq(ω) is the spectrum of xq(t). Nuttall, A. H., 1966: On the quadrature approximation to the Hilbert Transform of modulated signal, Proc. IEEE, 54, 1458

Why do we need Quadrature : To over come the difficulties stated in Nuttall Theorem for complicate phase functions. An Example : Duffing Pendulum

Duffing : Model Equation

Duffing : Expansions of the Model Equation

Duffing : Data

Duffing : Data, Quadrature & Hilbert

Duffing : Amplitude

Duffing : Phase

Duffing : Frequency truth is given by quadrature

Quadrature To circumvent the limitation imposed by Nuttall Theorem

Quadrature : Procedures Normalize the IMFs as in the NHHT method. Compute IF (FM) from Quadrature of N-data as follows:

Validation of NHHT and Quadrature Methods Through examples using NHHT HHT GZC TEO Quadrature

Example : Duffing Equation Model function

Damped Chirp Duffing Model

Example : Speech Signal ‘ Hello ’ Real Data

Data : Hello

Data : Hello IMF

Hello : Data c3y(8)

Hello : Check Bedrosian Theorem

Hello : Instantaneous Frequency & data c3y(8)

Hello : Instantaneous Frequency & data Details c3y(8)

Some alternatives for Quadrature Different implement for IF

Hou’s Approach Let our data be x(t). Using Taylor’s expansion, we can write Therefore, we have

Hou’s Approach Thus, we should have the instantaneous frequency as the derivative of the phase function given by: This approach requires no normalization.. Hou, T. Y., M. P. Yan and Z. Wu, 2009: A variant of the EMD method for multi-scale data. Advances in Adaptive Data Analysis, 1,

Wu’s Approach In this method, we do not have to compute arc- cosine. After normalization of the IMF, we have therefore, we can also find

Wu’s Approach

Summary Instantaneous frequency could be computed accurately. Our implementation is basically according to Wu’s algorithm. In case of EEMD, the component might not be bona fide IMF

A Physical Example : Water Surface Waves Real Laboratory Data

The Idea and the need of Instantaneous Frequency According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that Therefore, both wave number and frequency must have instantaneous values.

The Idea and the need of Instantaneous Frequency According to the classic wave theory, there are other more important wave conservation laws for Energy and Action: Therefore, if frequency is a function of time, it has to satisfy certain condition for both laws to be valid.

Data

Governing Equations I:

Governing Equations II:

Governing Equations III:

Governing Equations IV: The 4 th order Nonlinear Schrodinger Equation

Dysthe, K. B., 1979: Note on a modification to the nonlinear Schrodinger equation for application to deep water waves. Proc. R. Soc. Lond., 369, Equation by perturbation up to 4 th order. But ω = constant.

Data and IF : Station #1

Data and IF : Station #3

Data and IF : Station #5

Phase Averaged Data and IF : Station #1

Phase Averaged Data and IF : Station #2

Phase Averaged Data and IF : Station #3

Phase Averaged Data and IF : Station #4

Summary Instantaneous frequency could be highly variable with high gradient. The assumption used in the classic wave theory might not be totally attainable. Coupled with the fusion of waves, we might need a new paradigm for water wave studies.

Comparisons of Different Methods TEO extremely local but for linear data only. GZC most stable but offers only smoothed frequency over ¼ wave period at most. HHT elegant and detailed, but suffers the limitations of Bedrosian and Nuttall Theorems. NHHT, with Normalized data, overcomes Bedrosian limitation, offers local, stable and detailed Instantaneous frequency and Error Index for nonlinear and nonstationary data. Quadrature is the best, but the sampling rate has to be sufficiently high.

Conclusions Instantaneous Frequency could be calculated routinely from the normalized IMFs through quadrature (for high data density) or Hilbert Transform (for low data density). For any signal, there might be more than one IF value at any given time. For data from nonlinear processes, there has to be intra-wave frequency modulations; therefore, the Instantaneous Frequency could be highly variable. This variations is essential for quantifying nonlinearity.