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f t
On
Analytic
Signals
with
Nonnegativ
e
Instan
taneous
F
requency
Xiang
Gen
Xia
and
Leon
Cohen
y
Abstract
In this paper we characterize all ana
lytic signals with band limited amplitudes and
polynomial phases We show that a signal
with band limited amplitude and polynomial
phase is analytic if and only if it has nonneg
ative constant instantaneous frequency i e
the derivative of the phase is a nonnegative
constant and the constant is greater than or
equal to the minimum bandwidth of the am
plitude
In
tro
duction
Both concepts of analytic signals and in
stantaneous frequencies play important roles
in many areas including communication sys
tems physics and joint time frequency analy
sis in signal processing They have been exten
sively studied see for example
A sig
nal is analytic if its Fourier spectrum vanishes
at negative frequencies This implies that
Department of Electrical and Computer En
gineering University of Delaware Newark DE
Phone Fax
Email
xxia ee udel edu His work was supported in part by
the Air Force O ce of Scienti c Research AFOSR
under Grant No F
and the
Of
ce of Naval Research ONR Young Investigator Pro
gram YIP under Grant N
y
Department of Physics and Astronomy Hunter
College and Graduate Center City University
of New York New York NY
Email
leon cohen hunter cuny edu His work was supported
in part by the O ce of Naval Research under Grant
N
a nonzero analytic signal must be complex
valued The instantaneous frequency IF of
a complex valued signal is commonly de ned
as the derivative of the phase of the signal
The IF is used as a generalization of the con
ventional frequency from the global sense to
the local sense With these two de nitions
it is not unusual to expect that the IF of an
analytic signal are nonnegative Trivial exam
ples of analytic signals with nonnegative IF
are single tone signals exp
j
0
t
for nonnega
tive constants
0
It is however not always
true for an analytic signal to have nonnegative
IFIn this paper we consider the class of sig
nals with band limited amplitudes and poly
nomial phases This class is quite broad in our
real applications such as in communication
systems We characterize all analytic signals
in the class as follows Let
f t
be a signal
with polynomial phase and band limited am
plitude of minimal bandwidth
B
It is shown
that the signal
f t
is analytic if and only if it
has a constant nonnegative IF
0
with
0
B
for all
t
except possibly isolated points of
t
Although this result looks simple it turns out
that the proof is not trivial
Main
Results
Before going to the main results let us re
call a characterization for a general analytic
signal In what follows the letter
z
always
stands for a complex value with its real part
x
and its imaginary part
y
i e
z x jy
Let be a region of the complex plane A function
f z
of the complex variable
z
is said
holomorphic
1
in if its rst derivative
f
0
z
exists for all the complex values
z
see for
example
Let
f t
be a signal or func
tion de ned for the real variable
t
Its com
plex extension
f z
is obtained from the signal
f t
by replacing the real variable
t
with the
complex variable
z
The function
f z
with
z x jy
belongs
to the Hardy class
H
+
2
if it is holomorphic in
the half plane
y
and satis es the following
inequality
M
sup
y
0
Z
1
1
j
f x jy
j
2
dx
The following result gives a characterization
of an analytic signal see for example
pg
Proposition
A nite energy signal
f t
is
analytic if and only if its complex extension
f z
belongs to the Hardy class
H
+
2
This Proposition is used in the proof of the
following main result
Theorem
Let a nite energy signal
f t A t
exp
jP t
where
A t
is real valued and band limited with
its minimal bandwidth
B
and
P t
is a poly
nomial of
t
with real coe cients Then signal
f t
is analytic if and only if
P t
0
0
t
where
0
and
0
are two real constants with
0
B
Proof
The su cient part is straightfor
ward We only need to prove the necessary
part
1
It is also called
analytic
in the mathematics lit
erature although the word analytic has a di erent
meaning in the signal processing literature
Since
A t
is band limited with minimal
bandwidth
B
and has nite energy we have
A t
Z
B
B
a e
jt
d
where
a
L
2
B B
We rst prove that
if
P t
bt
n
for
n
and a real nonzero
constant
b
then
f t
is not analytic To do
so we want to apply Prop
i e we want
to prove that such an
f t
is not in the Hardy
class
H
+
2
The complex extension of
f t
is
f x jy A x jy
exp
jb x jy
n
A x jy
exp
b
n
X
k
=0
n
k x
k
y
n k
j
n k
+1
We claim that
n
if
f t
is analytic
When
n
there is a term in the expo
nential
bx
n
3
y
3
j
4
bx
n
3
y
3
Since
A t
has the form
j
A x jy
j
2
has
the order at least
exp
B
q
x
2
y
2
In the meantime the exponential term has the
order at least
exp
bx
n
3
y
3
When
b
the order in
bx
n
3
y
3
is greater than the one in
B
p
x
2
y
2
when
y
is large enough where
x
is the variable
Thus the function
f x jy
of the variable
x
does not belong to
L
2
i e
f z
H
+
2
or
f t
is not analytic
When
b
and
n
is an odd number
we consider
Z
1
1
j
f x jy
j
2
dx
where the variable
x
runs from
to
Therefore when
x
runs negative values the
order in
bx
n
3
y
3
is greater than the one in
B
p
x
2
y
2
when
y
is large enough
Thus
f x jy L
2
and
f t
is not
analytic
When
b
and
n
is an even number
we have
n
In this case we consider the
term of the exponential
exp
bx
n
5
y
5
j
6
exp
bx
n
5
y
5
Since
b
clearly either
f x jy
L
2
when
n
or
M
sup
y
0
Z
1
1
j
f x jy
j
2
dx
sup
y
0
Z
1
1
j
A x jy
j
2
dx
exp
by
5
when
n
Thus
f t
is not analytic by
Prop
Thus we have proved the claim
When
n
exp
jb x jy
2
exp
jbx
2
bxy jy
2
In this case
j
f x jy
j
2
j
A t
j
2
e
2
bxy
No mater what the sign of
b
is similar to be
fore
f x jy
L
2
or
f t
is not
analytic
When
n
exp
jb x jy
3
exp
jbx
3
bx
2
y jbxy
2
by
3
When
b
similar to before
M
sup
y
0
Z
1
1
j
f x jy
j
2
dx
sup
y
0
Z
1
1
j
A x jy
j
2
exp
bx
2
y dx
exp
by
3
Thus
f t
is not analytic by Prop
When
b
similar to before exp
bx
2
y
is un
bounded in terms of
x
Thus
f x jy
L
2
and
f t
is not analytic
When the polynomial
P t
has mixed terms
of
t
n
we only need to consider the highest
order term It is because in the expansions of
x jy
n
either the order
x
or the order of
y
resulted from the highest order term is higher
than any one resulted from the lower order
terms in
P x jy
Therefore the highest
order term dominates the signal growth order
Using the above result to the highest order
term we have proved that
P t
0
0
t
Since
A t
has the minimal bandwidth
B
for
f t
to be analytic it is clear that
0
B
2
From Theorem we immediately have the
following corollary
Corollary
A nite energy signal with poly
nomial phase and band limited amplitude of
minimal bandwidth
B
is analytic if and only
if it has a nonnegative constant instantaneous
frequency with the constant greater than or
equal to
B
at all
t
but possibly isolated points
Proof
By Theorem
we only need to
prove that the IF of
f t
is
0
at all
t
but
possibly isolated points Since the amplitude
A t
is band limited it is an entire function
Therefore
A t
has only zeros at possibly
isolated points
t
n
for
n Z
with
j
t
n
t
m
j
s
for a positive constant
s
and
n m
In other
words
A t
has the same sign in each time
interval
t
n
t
n
+1
which proves Corollary
2
Consider a signal with the following form
f t
Z
b
a
F e
j t
d e
j
(
0
t
+
0
)
where
a b
0
is a constant
0
and
F
is complex conjugate symmetrical
around the center
c
F c
F c
for
a c
where
c a b
Clearly
f t
is analytic
Also
f t A t
exp
j
0
t
0 where
0
0
c
and
A t
Z
d
d
F
exp
j t d
where
d
b a
and
F
F
c
One can see that
A t
is real valued and band
limited Similar to the proof of Corollary
the IF of the above
f t
is
0
c
0
Thus the above family of signals are ana
lytic and have nonnegative IF Furthermore
by Corollary this family is the only family
of signals with band limited amplitudes and
polynomial phases that are analytic and have
nonnegative instantaneous frequencies
In the above we discussed the class of signals
with band limited amplitude and polynomial
phase One might want to ask what happens
when the phase is not a polynomial We have
the following result
Theorem
Let
A t
be a non zero real band
limited signal be a real constant and
f t A t e
je
t
Then
f t
is not analytic
The proof is similar to before by using Prop
Conclusions
In this paper we have characterized all sig
nals with band limited amplitudes and poly
nomial phases that are analytic It turns out
that such signals are analytic if and only if
they have nonnegative constant instantaneous
frequencies where the constants are greater
than or equal to the minimal bandwidths of
the amplitudes
References
J Shekel Instantaneous frequency Proc
IRE
vol
p
April
J
J
Hupert
Instantaneous frequency Proc IRE vol
p
Sept
W W Harman Instantaneous fre
quency Proc IRE vol p
April
H B Voelcker Toward a uni ed theory of
modulation Part I Phase envelope relation
ship Proc IEEE vol
pp
Mar
L Mandel Interpretation of Instanta
neous frequency Amer J Phys vol
pp B Boashash Estimating and interpret
ing the instantaneous frequency of a signal
Parts and
Pro