Basics of Analytic SignalsNaoki SaitoProfessor of Department of
MathematicsUniversity of California, DavisFebruary 20,
[email protected] (UC Davis) Analytic Signals 02/20/2014 1
/ 10MotivationMany natural and man-made signals exhibit
time-varying frequencies(e.g., chirps, FM radio
waves).Characterization and analysis of such a signal,u(t ), based
oninstantaneous amplitudea(t ), instantaneous phase(t ),
andinstantaneous frequency(t ) :=
(t ), are very important:u(t ) =a(t ) cos(t
)[email protected] (UC Davis) Analytic Signals 02/20/2014 2 /
10MotivationMany natural and man-made signals exhibit time-varying
frequencies(e.g., chirps, FM radio waves).Characterization and
analysis of such a signal,u(t ), based oninstantaneous amplitudea(t
), instantaneous phase(t ), andinstantaneous frequency(t ) :=
(t ), are very important:u(t ) =a(t ) cos(t
)[email protected] (UC Davis) Analytic Signals 02/20/2014 2 /
10Analytic SignalIt is convenient to use a complexied version of
the signal whose realpart is a given real-valued signalu(t
).Givenu(t ), however, there are innitely many ways to dene
theinstantaneous amplitude and phase (IAP) pairs so thatu(t ) =a(t
) cos(t ).This is due to the arbitrariness of the complexied
version ofu, i.e.,f (t ) =u(t ) +iv(t )wherev(t ) is an arbitrary
real-valued signal; yet this yields the IAPrepresentation ofu(t )
viaa(t ) =
u2(t ) +v2(t ), (t ) =arctan v(t )u(t ).The instantaneous
frequency is dened as(t ) := ddt = u(t )v
(t ) u
(t )v(t )u2(t ) +v2(t )[email protected] (UC Davis)
Analytic Signals 02/20/2014 3 / 10Analytic SignalIt is convenient
to use a complexied version of the signal whose realpart is a given
real-valued signalu(t ).Givenu(t ), however, there are innitely
many ways to dene theinstantaneous amplitude and phase (IAP) pairs
so thatu(t ) =a(t ) cos(t ).This is due to the arbitrariness of the
complexied version ofu, i.e.,f (t ) =u(t ) +iv(t )wherev(t ) is an
arbitrary real-valued signal; yet this yields the IAPrepresentation
ofu(t ) viaa(t ) =
u2(t ) +v2(t ), (t ) =arctan v(t )u(t ).The instantaneous
frequency is dened as(t ) := ddt = u(t )v
(t ) u
(t )v(t )u2(t ) +v2(t )[email protected] (UC Davis)
Analytic Signals 02/20/2014 3 / 10Analytic SignalIt is convenient
to use a complexied version of the signal whose realpart is a given
real-valued signalu(t ).Givenu(t ), however, there are innitely
many ways to dene theinstantaneous amplitude and phase (IAP) pairs
so thatu(t ) =a(t ) cos(t ).This is due to the arbitrariness of the
complexied version ofu, i.e.,f (t ) =u(t ) +iv(t )wherev(t ) is an
arbitrary real-valued signal; yet this yields the IAPrepresentation
ofu(t ) viaa(t ) =
u2(t ) +v2(t ), (t ) =arctan v(t )u(t ).The instantaneous
frequency is dened as(t ) := ddt = u(t )v
(t ) u
(t )v(t )u2(t ) +v2(t )[email protected] (UC Davis)
Analytic Signals 02/20/2014 3 / 10Analytic SignalIt is convenient
to use a complexied version of the signal whose realpart is a given
real-valued signalu(t ).Givenu(t ), however, there are innitely
many ways to dene theinstantaneous amplitude and phase (IAP) pairs
so thatu(t ) =a(t ) cos(t ).This is due to the arbitrariness of the
complexied version ofu, i.e.,f (t ) =u(t ) +iv(t )wherev(t ) is an
arbitrary real-valued signal; yet this yields the IAPrepresentation
ofu(t ) viaa(t ) =
u2(t ) +v2(t ), (t ) =arctan v(t )u(t ).The instantaneous
frequency is dened as(t ) := ddt = u(t )v
(t ) u
(t )v(t )u2(t ) +v2(t )[email protected] (UC Davis)
Analytic Signals 02/20/2014 3 / 10Analytic SignalGabor (1946)
proposed to use the the Hilbert transform ofu(t ) asv(t ), and
called the complex-valuedf (t ) an analytic signal.Vakman (1972)
proved thatv(t ) must be of the Hilbert transform ofu(t ) if we
impose some a priori physical assumptions:1v(t ) must be derived
fromu(t ).2Amplitude continuity: a small change inu = a small
change ina(t ).3Phase independence of scale: ifcu(t ),c R arbitrary
scalar, then thephase does not change from that ofu(t ) and its
amplitude becomesctimes that ofu(t ).4Harmonic correspondence:
ifu(t ) =a0cos(0t +0), thena(t ) a0,(t ) 0t
[email protected] (UC Davis) Analytic Signals 02/20/2014 4
/ 10Analytic SignalGabor (1946) proposed to use the the Hilbert
transform ofu(t ) asv(t ), and called the complex-valuedf (t ) an
analytic signal.Vakman (1972) proved thatv(t ) must be of the
Hilbert transform ofu(t ) if we impose some a priori physical
assumptions:1v(t ) must be derived fromu(t ).2Amplitude continuity:
a small change inu = a small change ina(t ).3Phase independence of
scale: ifcu(t ),c R arbitrary scalar, then thephase does not change
from that ofu(t ) and its amplitude becomesctimes that ofu(t
).4Harmonic correspondence: ifu(t ) =a0cos(0t +0), thena(t ) a0,(t
) 0t [email protected] (UC Davis) Analytic Signals
02/20/2014 4 / 10Analytic SignalGabor (1946) proposed to use the
the Hilbert transform ofu(t ) asv(t ), and called the
complex-valuedf (t ) an analytic signal.Vakman (1972) proved
thatv(t ) must be of the Hilbert transform ofu(t ) if we impose
some a priori physical assumptions:1v(t ) must be derived fromu(t
).2Amplitude continuity: a small change inu = a small change ina(t
).3Phase independence of scale: ifcu(t ),c R arbitrary scalar, then
thephase does not change from that ofu(t ) and its amplitude
becomesctimes that ofu(t ).4Harmonic correspondence: ifu(t )
=a0cos(0t +0), thena(t ) a0,(t ) 0t [email protected] (UC
Davis) Analytic Signals 02/20/2014 4 / 10Analytic SignalGabor
(1946) proposed to use the the Hilbert transform ofu(t ) asv(t ),
and called the complex-valuedf (t ) an analytic signal.Vakman
(1972) proved thatv(t ) must be of the Hilbert transform ofu(t ) if
we impose some a priori physical assumptions:1v(t ) must be derived
fromu(t ).2Amplitude continuity: a small change inu = a small
change ina(t ).3Phase independence of scale: ifcu(t ),c R arbitrary
scalar, then thephase does not change from that ofu(t ) and its
amplitude becomesctimes that ofu(t ).4Harmonic correspondence:
ifu(t ) =a0cos(0t +0), thena(t ) a0,(t ) 0t
[email protected] (UC Davis) Analytic Signals 02/20/2014 4
/ 10Analytic SignalGabor (1946) proposed to use the the Hilbert
transform ofu(t ) asv(t ), and called the complex-valuedf (t ) an
analytic signal.Vakman (1972) proved thatv(t ) must be of the
Hilbert transform ofu(t ) if we impose some a priori physical
assumptions:1v(t ) must be derived fromu(t ).2Amplitude continuity:
a small change inu = a small change ina(t ).3Phase independence of
scale: ifcu(t ),c R arbitrary scalar, then thephase does not change
from that ofu(t ) and its amplitude becomesctimes that ofu(t
).4Harmonic correspondence: ifu(t ) =a0cos(0t +0), thena(t ) a0,(t
) 0t [email protected] (UC Davis) Analytic Signals
02/20/2014 4 / 10Analytic SignalGabor (1946) proposed to use the
the Hilbert transform ofu(t ) asv(t ), and called the
complex-valuedf (t ) an analytic signal.Vakman (1972) proved
thatv(t ) must be of the Hilbert transform ofu(t ) if we impose
some a priori physical assumptions:1v(t ) must be derived fromu(t
).2Amplitude continuity: a small change inu = a small change ina(t
).3Phase independence of scale: ifcu(t ),c R arbitrary scalar, then
thephase does not change from that ofu(t ) and its amplitude
becomesctimes that ofu(t ).4Harmonic correspondence: ifu(t )
=a0cos(0t +0), thena(t ) a0,(t ) 0t [email protected] (UC
Davis) Analytic Signals 02/20/2014 4 / 10Analytic Signal . . .For
simplicity, we assume that our signals are2-periodic in [, ).Hence,
we work on the unit circle and unit diskD inC=R2.Note that the
signals overR=(, ) can be treated similarly byconsidering the real
axis and the upper half plane ofC.The analytic signal of a given
signalu() R is often and simplyobtained via the Hilbert transform:f
() =u() +iHu(), Hu() :=12pv
u() cot2d.Note thatu() = a02 +
k1(ak cosk+bk sink) Hu() =
k1(ak sinkbk cosk).Furthermore,f () = a02 +
k1(akibk)[email protected] (UC Davis) Analytic Signals
02/20/2014 5 / 10Analytic Signal . . .For simplicity, we assume
that our signals are2-periodic in [, ).Hence, we work on the unit
circle and unit diskD inC=R2.Note that the signals overR=(, ) can
be treated similarly byconsidering the real axis and the upper half
plane ofC.The analytic signal of a given signalu() R is often and
simplyobtained via the Hilbert transform:f () =u() +iHu(), Hu()
:=12pv
u() cot2d.Note thatu() = a02 +
k1(ak cosk+bk sink) Hu() =
k1(ak sinkbk cosk).Furthermore,f () = a02 +
k1(akibk)[email protected] (UC Davis) Analytic Signals
02/20/2014 5 / 10Analytic Signal . . .For simplicity, we assume
that our signals are2-periodic in [, ).Hence, we work on the unit
circle and unit diskD inC=R2.Note that the signals overR=(, ) can
be treated similarly byconsidering the real axis and the upper half
plane ofC.The analytic signal of a given signalu() R is often and
simplyobtained via the Hilbert transform:f () =u() +iHu(), Hu()
:=12pv
u() cot2d.Note thatu() = a02 +
k1(ak cosk+bk sink) Hu() =
k1(ak sinkbk cosk).Furthermore,f () = a02 +
k1(akibk)[email protected] (UC Davis) Analytic Signals
02/20/2014 5 / 10Analytic Signal . . .For simplicity, we assume
that our signals are2-periodic in [, ).Hence, we work on the unit
circle and unit diskD inC=R2.Note that the signals overR=(, ) can
be treated similarly byconsidering the real axis and the upper half
plane ofC.The analytic signal of a given signalu() R is often and
simplyobtained via the Hilbert transform:f () =u() +iHu(), Hu()
:=12pv
u() cot2d.Note thatu() = a02 +
k1(ak cosk+bk sink) Hu() =
k1(ak sinkbk cosk).Furthermore,f () = a02 +
k1(akibk)[email protected] (UC Davis) Analytic Signals
02/20/2014 5 / 10Analytic Signal . . .For simplicity, we assume
that our signals are2-periodic in [, ).Hence, we work on the unit
circle and unit diskD inC=R2.Note that the signals overR=(, ) can
be treated similarly byconsidering the real axis and the upper half
plane ofC.The analytic signal of a given signalu() R is often and
simplyobtained via the Hilbert transform:f () =u() +iHu(), Hu()
:=12pv
u() cot2d.Note thatu() = a02 +
k1(ak cosk+bk sink) Hu() =
k1(ak sinkbk cosk).Furthermore,f () = a02 +
k1(akibk)[email protected] (UC Davis) Analytic Signals
02/20/2014 5 / 10Analytic Signal . . .For simplicity, we assume
that our signals are2-periodic in [, ).Hence, we work on the unit
circle and unit diskD inC=R2.Note that the signals overR=(, ) can
be treated similarly byconsidering the real axis and the upper half
plane ofC.The analytic signal of a given signalu() R is often and
simplyobtained via the Hilbert transform:f () =u() +iHu(), Hu()
:=12pv
u() cot2d.Note thatu() = a02 +
k1(ak cosk+bk sink) Hu() =
k1(ak sinkbk cosk).Furthermore,f () = a02 +
k1(akibk)[email protected] (UC Davis) Analytic Signals
02/20/2014 5 / 10Analytic Signal . . .We can gain a deeper insight
by viewing this as the boundary value of ananalytic functionF(z)
whereF(z) :=U(z) +i
U(z), z D,whereU(z) =U
r ei
=Pr u() =12
1r212r cos() +r2u() d,
U(z) = U
r ei
=Qr u() =12
2r sin()12r cos() +r2u() d.In other words, the original
signalu() =U ei
is the boundary value ofthe harmonic function UonD, which is
constructed by the Poissonintegral. Uand Qr() are referred to as
the conjugate harmonic functionand the conjugate Poisson kernel,
[email protected] (UC Davis) Analytic Signals
02/20/2014 6 / 10Analytic Signal . . . An Example:
u()[email protected] (UC Davis) Analytic Signals 02/20/2014 7
/ 10Analytic Signal . . . An Example: u() and
Hu()[email protected] (UC Davis) Analytic Signals 02/20/2014 8
/ 10Analytic Signal . . . An Example: U(z) and U(z)(a)U(z) (b)
U(z)[email protected] (UC Davis) Analytic Signals 02/20/2014 9
/ 10Analytic Signal . . .Even if we use the analytic signal, its
IAP representation is not unique asshown by Cohen, Loughlin, and
Vakman (1999):f () =a()ei(), wherea() =u() cos() +v() sin() may
benegative though() is continuous;f () =|a()|ei(()+()), where() is
an appropriate phase function,which may be
discontinuous.(a)Continuous phase(b)Nonnegative
[email protected] (UC Davis) Analytic Signals
02/20/2014 10 / 10Analytic Signal . . .Even if we use the analytic
signal, its IAP representation is not unique asshown by Cohen,
Loughlin, and Vakman (1999):f () =a()ei(), wherea() =u() cos() +v()
sin() may benegative though() is continuous;f () =|a()|ei(()+()),
where() is an appropriate phase function,which may be
discontinuous.(a)Continuous phase (b)Nonnegative
[email protected] (UC Davis) Analytic Signals
02/20/2014 10 / 10
