Introduction Difficulties Mathematical background A better definition Details Conclusions
An analytic approach to minimum phase signals
Michael P. Lamoureux and Gary F. Margrave
November 29, 2007
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I traditionally called minimum phase
I such signals can be recovered from the amplitude spectrum alone; the phase is uniquely determined.
I useful in seismic processing.
I Eg: in deconvolution, where the reflectivity and the wavelet are separated from the recorded seismic data.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Minimum phase introduction
I certain physical signals have energy concentrated at the front - impulsive seismic sources (hammers, dynamite, airgun blast) - signals traveling through lossy media
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I such signals can be recovered from the amplitude spectrum alone; the phase is uniquely determined.
I useful in seismic processing.
I Eg: in deconvolution, where the reflectivity and the wavelet are separated from the recorded seismic data.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Minimum phase introduction
I certain physical signals have energy concentrated at the front - impulsive seismic sources (hammers, dynamite, airgun blast) - signals traveling through lossy media
I traditionally called minimum phase
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I useful in seismic processing.
I Eg: in deconvolution, where the reflectivity and the wavelet are separated from the recorded seismic data.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Minimum phase introduction
I certain physical signals have energy concentrated at the front - impulsive seismic sources (hammers, dynamite, airgun blast) - signals traveling through lossy media
I traditionally called minimum phase
I such signals can be recovered from the amplitude spectrum alone; the phase is uniquely determined.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Eg: in deconvolution, where the reflectivity and the wavelet are separated from the recorded seismic data.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Minimum phase introduction
I certain physical signals have energy concentrated at the front - impulsive seismic sources (hammers, dynamite, airgun blast) - signals traveling through lossy media
I traditionally called minimum phase
I such signals can be recovered from the amplitude spectrum alone; the phase is uniquely determined.
I useful in seismic processing.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Minimum phase introduction
I certain physical signals have energy concentrated at the front - impulsive seismic sources (hammers, dynamite, airgun blast) - signals traveling through lossy media
I traditionally called minimum phase
I such signals can be recovered from the amplitude spectrum alone; the phase is uniquely determined.
I useful in seismic processing.
I Eg: in deconvolution, where the reflectivity and the wavelet are separated from the recorded seismic data.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
I impulse response has energy is maximally concentrated at the front.
I causal stable filter with causal stable inverse (def’n).
I all poles and zeros of the filter lie inside the unit circle (def’n).
I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
I or via Hilbert transform on log amplitude spectrum.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I impulse response has energy is maximally concentrated at the front.
I causal stable filter with causal stable inverse (def’n).
I all poles and zeros of the filter lie inside the unit circle (def’n).
I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
I or via Hilbert transform on log amplitude spectrum.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I causal stable filter with causal stable inverse (def’n).
I all poles and zeros of the filter lie inside the unit circle (def’n).
I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
I or via Hilbert transform on log amplitude spectrum.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
I impulse response has energy is maximally concentrated at the front.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I all poles and zeros of the filter lie inside the unit circle (def’n).
I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
I or via Hilbert transform on log amplitude spectrum.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
I impulse response has energy is maximally concentrated at the front.
I causal stable filter with causal stable inverse (def’n).
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
I or via Hilbert transform on log amplitude spectrum.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
I impulse response has energy is maximally concentrated at the front.
I causal stable filter with causal stable inverse (def’n).
I all poles and zeros of the filter lie inside the unit circle (def’n).
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I or via Hilbert transform on log amplitude spectrum.
Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
I impulse response has energy is maximally concentrated at the front.
I causal stable filter with causal stable inverse (def’n).
I all poles and zeros of the filter lie inside the unit circle (def’n).
I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Filter background
I terminology comes from filter theory.
I min phase filter has minimum group delay of all possible filters with a given amplitude spectrum (a misnomer).
I impulse response has energy is maximally concentrated at the front.
I causal stable filter with causal stable inverse (def’n).
I all poles and zeros of the filter lie inside the unit circle (def’n).
I FIR, IIR filter converted to min phase by reflecting poles, zeros across unit circle.
I or via Hilbert transform on log amplitude spectrum.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Introduction A better definition Details Conclusions Min phase vs zero phase
0.35 0.35 Min phase Zero phase
0.3 0.3
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
−0.05 −0.05
−0.1 −0.1
−50 0 50 −50 0 50
Figure: A minimum phase IIR filter response and zero phase equivalent.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Signals typically don’t have zeros and poles.
I Impossible to have a causal stable signal with causal stable inverse on R. I Hilbert transform not defined on arbitrary log spectrum.
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Fundamental difficulty: applying filter theory to signals
I Filter response is a poor model for general signals.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Impossible to have a causal stable signal with causal stable inverse on R. I Hilbert transform not defined on arbitrary log spectrum.
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Fundamental difficulty: applying filter theory to signals
I Filter response is a poor model for general signals.
I Signals typically don’t have zeros and poles.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Hilbert transform not defined on arbitrary log spectrum.
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Fundamental difficulty: applying filter theory to signals
I Filter response is a poor model for general signals.
I Signals typically don’t have zeros and poles.
I Impossible to have a causal stable signal with causal stable inverse on R.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Fundamental difficulty: applying filter theory to signals
I Filter response is a poor model for general signals.
I Signals typically don’t have zeros and poles.
I Impossible to have a causal stable signal with causal stable inverse on R. I Hilbert transform not defined on arbitrary log spectrum.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I But, f , g nice functions, implies f ∗ g also a function. NOT a distribution.
I For instance, locally integrable (“stable”) implies the convolution is locally integrable.
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why no causal stable signal with causal stable inverse?
I Signal f : R → R with inverse g means
f ∗ g = δ0, the Dirac delta function.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I For instance, locally integrable (“stable”) implies the convolution is locally integrable.
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why no causal stable signal with causal stable inverse?
I Signal f : R → R with inverse g means
f ∗ g = δ0, the Dirac delta function.
I But, f , g nice functions, implies f ∗ g also a function. NOT a distribution.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why no causal stable signal with causal stable inverse?
I Signal f : R → R with inverse g means
f ∗ g = δ0, the Dirac delta function.
I But, f , g nice functions, implies f ∗ g also a function. NOT a distribution.
I For instance, locally integrable (“stable”) implies the convolution is locally integrable.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I For those not in the know, it is some integral formula.
I Involves log(abs(Fourier transform)).
I Log blows up at the zeros of the Fourier transform.
I Try to fix by inserting a stability constant to remove zeros. log(abs(Fourier transform) + )
I Does that work?
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why Hilbert transform a problem?
I For those in the know, we use the Hilbert transform to compute min phase.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Involves log(abs(Fourier transform)).
I Log blows up at the zeros of the Fourier transform.
I Try to fix by inserting a stability constant to remove zeros. log(abs(Fourier transform) + )
I Does that work?
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why Hilbert transform a problem?
I For those in the know, we use the Hilbert transform to compute min phase.
I For those not in the know, it is some integral formula.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Log blows up at the zeros of the Fourier transform.
I Try to fix by inserting a stability constant to remove zeros. log(abs(Fourier transform) + )
I Does that work?
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why Hilbert transform a problem?
I For those in the know, we use the Hilbert transform to compute min phase.
I For those not in the know, it is some integral formula.
I Involves log(abs(Fourier transform)).
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Try to fix by inserting a stability constant to remove zeros. log(abs(Fourier transform) + )
I Does that work?
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why Hilbert transform a problem?
I For those in the know, we use the Hilbert transform to compute min phase.
I For those not in the know, it is some integral formula.
I Involves log(abs(Fourier transform)).
I Log blows up at the zeros of the Fourier transform.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Does that work?
Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why Hilbert transform a problem?
I For those in the know, we use the Hilbert transform to compute min phase.
I For those not in the know, it is some integral formula.
I Involves log(abs(Fourier transform)).
I Log blows up at the zeros of the Fourier transform.
I Try to fix by inserting a stability constant to remove zeros. log(abs(Fourier transform) + )
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Difficulties A better definition Details Conclusions Why Hilbert transform a problem?
I For those in the know, we use the Hilbert transform to compute min phase.
I For those not in the know, it is some integral formula.
I Involves log(abs(Fourier transform)).
I Log blows up at the zeros of the Fourier transform.
I Try to fix by inserting a stability constant to remove zeros. log(abs(Fourier transform) + )
I Does that work?
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals iω I F (e ) is just the usual Fourier transform. F (z) is an extension of the spectrum to the disk.
I F (z) is a power series, differentiable everywhere on the unit 1 disk. An analytic function. A function in Hardy space H (D).
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Analytic approach
I Given a causal, stable signal f = (f0, f1, f2,....), define
∞ X n F (z) = fnz , for complex numbers z with |z| < 1. n=0
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I F (z) is a power series, differentiable everywhere on the unit 1 disk. An analytic function. A function in Hardy space H (D).
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Analytic approach
I Given a causal, stable signal f = (f0, f1, f2,....), define
∞ X n F (z) = fnz , for complex numbers z with |z| < 1. n=0
iω I F (e ) is just the usual Fourier transform. F (z) is an extension of the spectrum to the disk.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Analytic approach
I Given a causal, stable signal f = (f0, f1, f2,....), define
∞ X n F (z) = fnz , for complex numbers z with |z| < 1. n=0
iω I F (e ) is just the usual Fourier transform. F (z) is an extension of the spectrum to the disk.
I F (z) is a power series, differentiable everywhere on the unit 1 disk. An analytic function. A function in Hardy space H (D).
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Similarly, a causal signal can’t have an interval of zeros in its spectrum.
I There are no band limited, causal signals.
I There are no band limited, minimum phase signals.
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Amazing facts in Hardy spaces
I An analytic function can’t be zero on an interval (or curve) in the disk, unless it is zero everywhere.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I There are no band limited, causal signals.
I There are no band limited, minimum phase signals.
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Amazing facts in Hardy spaces
I An analytic function can’t be zero on an interval (or curve) in the disk, unless it is zero everywhere.
I Similarly, a causal signal can’t have an interval of zeros in its spectrum.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I There are no band limited, minimum phase signals.
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Amazing facts in Hardy spaces
I An analytic function can’t be zero on an interval (or curve) in the disk, unless it is zero everywhere.
I Similarly, a causal signal can’t have an interval of zeros in its spectrum.
I There are no band limited, causal signals.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Amazing facts in Hardy spaces
I An analytic function can’t be zero on an interval (or curve) in the disk, unless it is zero everywhere.
I Similarly, a causal signal can’t have an interval of zeros in its spectrum.
I There are no band limited, causal signals.
I There are no band limited, minimum phase signals.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions No min phase, band limited spike
0.5
0
−0.5 0 10 20 30 40 50 60 70 80 90 100 0.5
0
−0.5 0 10 20 30 40 50 60 70 80 90 100 0.5
0
−0.5 0 10 20 30 40 50 60 70 80 90 100 0.5
0
−0.5 0 10 20 30 40 50 60 70 80 90 100
Figure: Trying to compute a min phase signal that does not exist. Stability factor goes to zero....
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions Existence conditions
Theorem: An amplitude spectrum |F (eiω)| is the spectrum of a causal signal if and only if R iω I |F (e )| dω < ∞, and R iω I log |F (e )| dω is finite.
Thus, the spectrum can not have an interval of zeros, since log 0 = −∞. Theorem: Any causal signal has a minimum phase equivalent.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Outer functions are like minimum phase filters.
I Inner functions are like all pass filters.
I The inner, outer definitions (complicated) apply to general signals, not just filters.
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions More facts about Hardy spaces.
I Each function F(z) can be factored as F(z) = G(z) H(z), where G is an outer function, H is an inner function.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Inner functions are like all pass filters.
I The inner, outer definitions (complicated) apply to general signals, not just filters.
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions More facts about Hardy spaces.
I Each function F(z) can be factored as F(z) = G(z) H(z), where G is an outer function, H is an inner function.
I Outer functions are like minimum phase filters.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I The inner, outer definitions (complicated) apply to general signals, not just filters.
Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions More facts about Hardy spaces.
I Each function F(z) can be factored as F(z) = G(z) H(z), where G is an outer function, H is an inner function.
I Outer functions are like minimum phase filters.
I Inner functions are like all pass filters.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Mathematical background A better definition Details Conclusions More facts about Hardy spaces.
I Each function F(z) can be factored as F(z) = G(z) H(z), where G is an outer function, H is an inner function.
I Outer functions are like minimum phase filters.
I Inner functions are like all pass filters.
I The inner, outer definitions (complicated) apply to general signals, not just filters.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Theorem: A discrete signal f = (f0, f1, f2,...) is front-loaded if and only if F (z) is an outer function.
Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions A better definition for “min phase” signals.
I Definition: A causal signal f = (f0, f1, f2,...) is front-loaded if its partial energies are maximized, relative to any other causal signal with the sample amplitude spectrum. That is,
N N X 2 X 2 |gn| ≤ |fn| , for each N = 0, 1, 2, . . .. n=0 n=0
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions A better definition for “min phase” signals.
I Definition: A causal signal f = (f0, f1, f2,...) is front-loaded if its partial energies are maximized, relative to any other causal signal with the sample amplitude spectrum. That is,
N N X 2 X 2 |gn| ≤ |fn| , for each N = 0, 1, 2, . . .. n=0 n=0
I Theorem: A discrete signal f = (f0, f1, f2,...) is front-loaded if and only if F (z) is an outer function.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I For causal stable signals: min phase implies front-loaded.
I Extra conditions: then front-loaded implies min phase.
I Signal f = (1, r, 0, 0,...) is min phase, front-loaded. (|r| < 1).
I Signal f = (1, 1, 0, 0,...) is not min phase, but is front-loaded.
I Signal f = (1, 1, 1, 1, 1, 1, 1, 0,...) is not min phase, but is front-loaded.
Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions Compare with old definition
Old definition works:
I For causal, stable filters: min phase implies front-loaded.
New definition more general:
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Extra conditions: then front-loaded implies min phase.
I Signal f = (1, r, 0, 0,...) is min phase, front-loaded. (|r| < 1).
I Signal f = (1, 1, 0, 0,...) is not min phase, but is front-loaded.
I Signal f = (1, 1, 1, 1, 1, 1, 1, 0,...) is not min phase, but is front-loaded.
Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions Compare with old definition
Old definition works:
I For causal, stable filters: min phase implies front-loaded.
I For causal stable signals: min phase implies front-loaded.
New definition more general:
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Signal f = (1, r, 0, 0,...) is min phase, front-loaded. (|r| < 1).
I Signal f = (1, 1, 0, 0,...) is not min phase, but is front-loaded.
I Signal f = (1, 1, 1, 1, 1, 1, 1, 0,...) is not min phase, but is front-loaded.
Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions Compare with old definition
Old definition works:
I For causal, stable filters: min phase implies front-loaded.
I For causal stable signals: min phase implies front-loaded.
I Extra conditions: then front-loaded implies min phase. New definition more general:
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Signal f = (1, 1, 0, 0,...) is not min phase, but is front-loaded.
I Signal f = (1, 1, 1, 1, 1, 1, 1, 0,...) is not min phase, but is front-loaded.
Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions Compare with old definition
Old definition works:
I For causal, stable filters: min phase implies front-loaded.
I For causal stable signals: min phase implies front-loaded.
I Extra conditions: then front-loaded implies min phase. New definition more general:
I Signal f = (1, r, 0, 0,...) is min phase, front-loaded. (|r| < 1).
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Signal f = (1, 1, 1, 1, 1, 1, 1, 0,...) is not min phase, but is front-loaded.
Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions Compare with old definition
Old definition works:
I For causal, stable filters: min phase implies front-loaded.
I For causal stable signals: min phase implies front-loaded.
I Extra conditions: then front-loaded implies min phase. New definition more general:
I Signal f = (1, r, 0, 0,...) is min phase, front-loaded. (|r| < 1).
I Signal f = (1, 1, 0, 0,...) is not min phase, but is front-loaded.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background A better definition A better definition Details Conclusions Compare with old definition
Old definition works:
I For causal, stable filters: min phase implies front-loaded.
I For causal stable signals: min phase implies front-loaded.
I Extra conditions: then front-loaded implies min phase. New definition more general:
I Signal f = (1, r, 0, 0,...) is min phase, front-loaded. (|r| < 1).
I Signal f = (1, 1, 0, 0,...) is not min phase, but is front-loaded.
I Signal f = (1, 1, 1, 1, 1, 1, 1, 0,...) is not min phase, but is front-loaded.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I A function F (z)is outer if
Z 1 2πiθ e + z 2πiθ F (z) = λ exp 2πiθ u(e )d θ 0 e − z where u is a real-valued integrable function on the unit circle, and λ is a complex number of modulus one.
I A function F (z) is inner if |F | ≡ 1 on the unit circle.
I Every function can be written as outer times inner.
Introduction Difficulties Mathematical background Details A better definition Details Conclusions Technical details
I We divide up signals into outer and inner parts. Outers have the energy concentration.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I A function F (z) is inner if |F | ≡ 1 on the unit circle.
I Every function can be written as outer times inner.
Introduction Difficulties Mathematical background Details A better definition Details Conclusions Technical details
I We divide up signals into outer and inner parts. Outers have the energy concentration.
I A function F (z)is outer if
Z 1 2πiθ e + z 2πiθ F (z) = λ exp 2πiθ u(e )d θ 0 e − z where u is a real-valued integrable function on the unit circle, and λ is a complex number of modulus one.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Every function can be written as outer times inner.
Introduction Difficulties Mathematical background Details A better definition Details Conclusions Technical details
I We divide up signals into outer and inner parts. Outers have the energy concentration.
I A function F (z)is outer if
Z 1 2πiθ e + z 2πiθ F (z) = λ exp 2πiθ u(e )d θ 0 e − z where u is a real-valued integrable function on the unit circle, and λ is a complex number of modulus one.
I A function F (z) is inner if |F | ≡ 1 on the unit circle.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Details A better definition Details Conclusions Technical details
I We divide up signals into outer and inner parts. Outers have the energy concentration.
I A function F (z)is outer if
Z 1 2πiθ e + z 2πiθ F (z) = λ exp 2πiθ u(e )d θ 0 e − z where u is a real-valued integrable function on the unit circle, and λ is a complex number of modulus one.
I A function F (z) is inner if |F | ≡ 1 on the unit circle.
I Every function can be written as outer times inner.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Details A better definition Details Conclusions Non singular formula for discrete, min phase signal.
From Hardy theory, we get alternate formulas for min phase signals. Signal coefficients from the outer function:
Z 1 1 2πiφ −2πinφ fn = n F (re )e dφ, any r < 1 . r 0 Signal coefficients from the amplitude spectrum:
Z 1 Z 1 2πiθ 2πiφ 1 e + re 2πiθ −2πinθ fn = n exp 2πiθ 2πiφ log |F (e )| dθ e dφ. r 0 0 e − re
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals + I Theorem: A causal signal f : R → R with spectrum F (z) an outer function, is front-loaded.
I Conjecture: This is if and only if.
Introduction Difficulties Mathematical background Details A better definition Details Conclusions Signals on the real line.
I Similarly, we have + Definition: A causal signal f : R → R is front-loaded if its partial energies are maximized, relative to any other causal signal with the sample amplitude spectrum. That is,
Z T Z T |g(t)|2 dt ≤ |f (t)|2 dt for each T > 0. 0 0
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Conjecture: This is if and only if.
Introduction Difficulties Mathematical background Details A better definition Details Conclusions Signals on the real line.
I Similarly, we have + Definition: A causal signal f : R → R is front-loaded if its partial energies are maximized, relative to any other causal signal with the sample amplitude spectrum. That is,
Z T Z T |g(t)|2 dt ≤ |f (t)|2 dt for each T > 0. 0 0
+ I Theorem: A causal signal f : R → R with spectrum F (z) an outer function, is front-loaded.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Details A better definition Details Conclusions Signals on the real line.
I Similarly, we have + Definition: A causal signal f : R → R is front-loaded if its partial energies are maximized, relative to any other causal signal with the sample amplitude spectrum. That is,
Z T Z T |g(t)|2 dt ≤ |f (t)|2 dt for each T > 0. 0 0
+ I Theorem: A causal signal f : R → R with spectrum F (z) an outer function, is front-loaded.
I Conjecture: This is if and only if.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Details A better definition Details Conclusions Computing approximately band-limited min phase signals.
3 1 Amplitude spectra Time series 2 0.5
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0 −0.5 0 0.2 0.4 0.6 0.8 1 0 10 20 30 3 1
2 0.5
1 0
0 −0.5 0 0.2 0.4 0.6 0.8 1 0 10 20 30 3 1
2 0.5
1 0
0 −0.5 0 0.2 0.4 0.6 0.8 1 0 10 20 30 3 1
2 0.5
1 0
0 −0.5 0 0.2 0.4 0.6 0.8 1 0 10 20 30
Figure: Step by step construction, using spectrum wrapping.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Front-loaded, energy concentration a useful alternative definition.
I Front-load signals equivalent to outer functions in Hardy space.
I No band-limited causal signals, no band-limited min phase signals.
I Hardy space theory gives useful formulations for computing min phase signals.
Introduction Difficulties Mathematical background Conclusion A better definition Details Conclusions Conclusions
I Mathematical difficulties in applying minimum phase definition to signals.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Front-load signals equivalent to outer functions in Hardy space.
I No band-limited causal signals, no band-limited min phase signals.
I Hardy space theory gives useful formulations for computing min phase signals.
Introduction Difficulties Mathematical background Conclusion A better definition Details Conclusions Conclusions
I Mathematical difficulties in applying minimum phase definition to signals.
I Front-loaded, energy concentration a useful alternative definition.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I No band-limited causal signals, no band-limited min phase signals.
I Hardy space theory gives useful formulations for computing min phase signals.
Introduction Difficulties Mathematical background Conclusion A better definition Details Conclusions Conclusions
I Mathematical difficulties in applying minimum phase definition to signals.
I Front-loaded, energy concentration a useful alternative definition.
I Front-load signals equivalent to outer functions in Hardy space.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals I Hardy space theory gives useful formulations for computing min phase signals.
Introduction Difficulties Mathematical background Conclusion A better definition Details Conclusions Conclusions
I Mathematical difficulties in applying minimum phase definition to signals.
I Front-loaded, energy concentration a useful alternative definition.
I Front-load signals equivalent to outer functions in Hardy space.
I No band-limited causal signals, no band-limited min phase signals.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background Conclusion A better definition Details Conclusions Conclusions
I Mathematical difficulties in applying minimum phase definition to signals.
I Front-loaded, energy concentration a useful alternative definition.
I Front-load signals equivalent to outer functions in Hardy space.
I No band-limited causal signals, no band-limited min phase signals.
I Hardy space theory gives useful formulations for computing min phase signals.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background End matters A better definition Details Conclusions References
I Helson, H., 1995, Harmonic Analysis.
I Hoffman, K., 1962, Banach Spaces of Analytic Functions.
I Karl, J., 1989, An Introduction to Digital Signal Processing.
I K¨orner,T., Fourier Analysis.
I Openheim, A. V. and Schafer, R. W., 1998, Discrete-time Signal Processing.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals Introduction Difficulties Mathematical background End matters A better definition Details Conclusions Acknowledgements
This research is supported by
I the industrial sponsors of CREWES and POTSI
I the funding agencies NSERC and MITACS.
Michael P. Lamoureux and Gary F. Margrave An analytic approach to minimum phase signals