frequency and instantaneous frequency a totally new view of frequency
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Frequency and Instantaneous Frequency
A Totally New View of Frequency
Definition of Frequency
Given the period of a wave as T ; the frequency is defined as
T.
1
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 : 1For any data from linear Processes
j
Ti t
j 0
1. Fourier Analysis :
F( ) x( t ) e dt .
2. Wavelet Analysis
3. Wigner Ville Analysis
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
Random and Delta Functions
Fourier Components : Random Function
Fourier Components : Delta Function
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 : 2For Simple Dynamic System
TC
0
4. Dynamic System through Hamiltonian :
H( p,q,t ) and A( t ) pdq ;
H.
A
This is an system analysis but not a data analysis method.
Definitions of Frequency : 3Instantaneous Frequency for IMF
only
5. Teager Energy Operator
6. Period between zero cros sin gs and extrema
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, 599-601.
2c
2 2c
2 4c
c c
c c
[ x( t ) ] x ( t ) x( t ) x( t ) :
For x( t ) A sin t , then
[ x( t ) ] A .
[ x( t ) ] A .
[ x( t )] [ x( t )]; A
[ x( t )] [ x( t )]
Generalized Zero-Crossing :By using intervals between all combinations of zero-crossings and
extrema.
T1
T2
T4
Generalized Zero-Crossing :Computing the weighted frequency.
1 11
1 21 22 41 42
2i 2i2i
4 i 4 i
4
4 i
3 44
At any location, there are7 different int erval values :
11 T f ; we assign weight a factor 4
4T
12 T f ; we assigna weight factor 2
2T
14 T f ; we assigna weight facto
4 f 2( f f ) f f f ff
r
.2
1T
1
Problems with TEO and GZC
• TEO has super time resolution but it is strictly for linear processes.
• GZC is robust but its resolution is too crude.
Definitions of Frequency : 4Instantaneous Frequency for IMF only
ji ( t )
2 1
jj
j
8.Quadrature method
yNormalized x ; y
7. HHT Analysis :
dx( t ) a( t ) e .
dt
1 x ; ( t ) tanx
d ( t )( t ) .
dt
Instantaneous Frequency
distanceVelocity ; mean velocity
time
dxNewton v
dt
1Frequency ; mean frequency
period
dHH
So that both v and
T defines the p
can appear in differential equations.
hase functiondt
Instantaneous Frequency is indispensable for nonlinear Processes
xx t
dtx
d 22
2 c( os1 ) .
x
The Idea and the need of Instantaneous Frequency
k , ;
t
t
k0 .
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)?
Prevailing Views
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
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.
p
i ( t )
2 2 1 / 2 1
For any x( t ) L ,
1 x( )y( t ) d ,
t
then, x( t )and y( t )are complex conjugate :
z( t ) x( t ) i y( t ) a( t ) e ,
where
y( t )a( t ) x y and ( t ) tan .
x( t )
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 L2(-∞, ∞) 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, 868-869.
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
2
0
02
q
E xq( t ) xh( t ) dt
2 F ( ) d ,
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 x2+ y2 ≠ 1. Then the arc-tangent would not recover the true phase function.
Quadrature pairs are not analytic, but satisfy strict 90o phase shift; therefore, x2+ y2 = 1, and the arc-tangent always gives the true phase function.
For cosθ(t) with arbitrary function of θ(t) :
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
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 Spectra 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.
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:i
nn
n
n
F t f t is the empi
For an IMF x t withenvelope e t
x tf t
e t
x tf t e t x t
f te t e t e t e t
x tf t
e t e t e t
rical FM signal
A t e t e t e t is t
11
1 12
2 2 1 2
1 2
1 2
, ( ), ( ) :
( )( ) ;
( )
( )( ) ( ) ( )
( ) = ; ..
( ) ( ) ;
( ) ( )
.( ) ( ) ( ) ( )
( ) ...
( )( )
( ) ( ) ... (
( )
.)
he empirical AM signal .
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
3t t
x( t ) exp cos 2 ;128 512
t 0 : 1024 .
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
Quadrature
To circumvent the limitation imposed by Nuttall 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
2
0
02
q
E xq( t ) xh( t ) dt
2 F ( ) d ,
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 type wave mod el :
t 0 : 1024;
t tx( t ) cos 0.3 sin( ) .
128 64
Duffing : Expansions of the Model Equation
For 1, the Model equationcanbe expanded as
x( t ) cos t sin 2 t
cos t cos sin 2 t sin t sin sin 2 t
cos t sin 2 t sin t ..
1 cos t cos 3 t .
.
...2 2
.
Duffing : Data
Duffing : Data, Quadrature & Hilbert
Duffing : Amplitude
Duffing : Phase
Duffing : Frequency truth is given by quadrature
Quadrature : Procedures
• Normalize the IMFs as in the NHHT method.
• Compute IF (FM) from Quadrature of N-data as follows:
2 1
Quadrature method
yNormalized x ; y 1 x ; ( t ) tan
x
d ( t )( t ) .
dt
Validation of NHHT and Quadrature Methods
Through examples using• NHHT• HHT• GZC• TEO
• Quadrature
Example : Duffing Equation
Model function
Damped Chirp Duffing Model
2 2t t tx( t ) exp cos 32 0.3 sin 32 ,
256 64 512 32 512
with t 0 : 1024 .
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)
A Physical Example : Water Surface Waves
Real Laboratory Data
The Idea and the need of Instantaneous Frequency
k , ;
t
t
k0 .
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
g
g j
j
EC E 0 ; Energy
t x
C EE0 ; Action
t x
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:
2
22 2
2
, 0,
1, 0,
2
, .
z
z gt z t
zt z
Governing Equations II:
2
2
2
2
2 22 22
2
0, 0,
10, 0,
2
, , , 0 ,
1 1.
2 2
h h h h
h
z L L Qz t
zt z z
where L gt z x y
Q Lz t z
Governing Equations III:
2 22
22
..... . .
.... . .
.
kz i kz i
i i
Ae e A e e c c
Be B e c c
wherer k x t
Governing Equations IV: The 4th order Nonlinear Schrodinger
Equation
2 22
2 2
3 3
2 3
2
1 1 12
2 2 4
1 36
8 2
1.
2
A A A Ai A A
t x y x
A A A Ai iA A A
x y x x x
Ai A A i
x x z
Dysthe, K. B., 1979: Note on a modification to the nonlinear Schrodinger equation for
application to deep water waves. Proc. R. Soc. Lond., 369, 105-114.
Equation by perturbation up to 4th 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.