Estimating and Interpreting
The
Instantaneous
Frequency
of
a
Signal-Part
1:
Fundamentals
BOUALEM BOASHASH,
SENIOR MEMBER, IEEE
The frequency of a sinusoidal signal is a well defined quantity.
However, often in practice, signals are not truly sinusoidal, or
even aggregates of sinusoidal components. Nonstationary signals
in particular do not lend themselves well to decomposition into
sinusoidal components. For such signals, the notion of frequency
loses its effectiveness, and one needs to use a parameter which
accounts for the time-varying nature of the process. This need has
given rise to the idea of instantaneous frequency.
The instantaneous frequency (IF) of a signal
is
a parame-
ter which is often of significant practical importance.
In
many
situations such as seismic, radar, sonar, communications, and
biomedical applications, the IF
is
a good descriptor of some
physical phenomenon.
This paper discusses the concept of instantaneous frequency, its
definitions, and the correspondence between the various mathe-
matical models formulated for representation of
IF.
The paper also
considers the extent to which the IF corresponds to our intuitive
expectation of reality.
A
historical review of the successive attempts to define the
IF is presented. Then the relationships between the IF and the
group-delay, analytic signal, and bandwidth-time
(BT)
product are
explored, as well as the relationship with time-frequency distribu-
tions. Finally, the notions of monocomponent and multicomponent
signals, and instantaneous bandwidth are discussed. It is shown
that all these notions are well described in the context of the theory
presented.
I.
INTRODUCTION
This paper presents a tutorial review of the theory nec-
essary to understand and utilize the notion of the instanta-
neous frequency (IF) of a signal. Like many other signal
processing concepts, the IF was originally defined in the
context
of
FM
modulation theory in communications.
The importance of the IF concept stems from the fact that
in many applications the signal analyst is confronted with
the task of processing signals whose spectral characteristics
(in particular the frequency of the spectral peaks) are
varying with time. These signals are often referred to as
“nonstationary,” a simple example being the chirp signal.
The latter may be conceptualized crudely as a sine wave
whose frequency sweeps with time. In seismic processing
Manuscript received January
25,
1991; revised November
6,
1991.
The author is with the Centre
for
Signal Processing Research, School
of
Electrical and Electronics Engineering, Queensland University
of
Tech-
nology, Brisbane, 4001, Australia.
IEEE Log Number 9107702.
these signals (referred to as “vibroseis” in geophysics)
are used as an alternative to explosive type signals, and
have the advantage that their spectral characteristics can
be accurately controlled in almost every regard including
duration, bandwidth and energy. These signals are also used
extensively for estimation of Doppler frequency shift in
radar returns, and for tracking the narrow-band components
of passive sonar. They also appear in various natural
situations, such as in the echo-location systems of bats.
For these signals, the
IF
is an important characteristic; it
is a time-varying parameter which defines the location of
the signal’s spectral peak as it varies with time. Conceptu-
ally it may be interpreted as the frequency of a sine wave
which locally fits the signal under analysis. Physically, it
has meaning only for monocomponent signals, where there
is only one frequency or a narrow range of frequencies
varying as a function of time. For multicomponent signals,
the notion of a single-valued IF becomes meaningless, and
a break-down into its components is needed.
A
theoretical treatment of instantaneous frequency is
presented in this paper, while in the sequel, the discrete
instantaneous frequency (DIF) is defined, the estimation
procedures for the
IF
are discussed, and a review of
applications is presented.
11.
THE
CONCEPT
OF
INSTANTANEOUS FREQUENCY
A.
The Concept
of
Frequency
In mechanics, the frequency of vibratory motion is de-
fined as the number of oscillations per unit time, where
vibratory motion is any to-and-fro motion and an oscillation
is a complete to-and-fro motion. In one oscillation, the
vibrating body moves from the equilibrium position to one
end of the path, then to the other end of path, and finally
back to the equilibrium position. Using this process as a
model, then, frequency may be defined for any arbitrary
vibratory motion.
A
special type of vibratory motion
is
a simple harmonic
motion in which the acceleration is proportional to the
displacement and always directed toward the equilibrium
position. When a body moves with uniform speed in a
circle, the projection of this motion on a diameter is simple
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0018-9219/92$03.00
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1992 IEEE
PROCEEDINGS
OF
THE
IEEE,
VOL.
80,
NO.
4,
APRIL
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