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首页瞬时频率分析:基础与应用
瞬时频率分析:基础与应用
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"这篇文献主要探讨了瞬时频率(IF)的概念,它在非稳态信号分析中的重要性,以及各种数学模型的建立和对应关系。" 瞬时频率是信号处理领域的一个关键概念,特别是在处理非正弦波信号或非稳态信号时。传统的频率定义适用于恒定频率的正弦波,但当信号的频率随时间变化时,这种定义就显得不足。在这种情况下,瞬时频率的概念应运而生,它能够描述信号频率随时间的变化情况。 在实际应用中,瞬时频率有着广泛的应用。例如,在地震学中,地震波的瞬时频率可以帮助分析地质结构;在雷达和声纳系统中,目标的距离和速度可以通过测量回波信号的瞬时频率来确定;在通信技术中,瞬时频率可以用于调制和解调,改善信号传输质量;在生物医学领域,如心电图或脑电图信号分析,瞬时频率可以揭示生理过程的动态变化。 文献深入讨论了瞬时频率的定义,指出其不仅仅是基频或者平均频率的概念,而是指信号在每个瞬间的实际频率值。它通常通过傅里叶变换、希尔伯特变换或其他频谱分析方法来估计。例如,希尔伯特变换可以提供一个信号的瞬时幅度和瞬时相位,从而推算出瞬时频率。 此外,文章还探讨了不同数学模型对瞬时频率的表述和它们之间的对应关系。这些模型可能包括微分方程、复指数函数的解析信号表示等。每种模型都有其适用范围和局限性,理解这些模型的关联和差异对于准确估计和解释瞬时频率至关重要。 瞬时频率是理解和分析非线性、非周期性信号的关键工具,其理论和计算方法对于科学研究和工程实践具有深远影响。通过深入学习和掌握瞬时频率的概念及其应用,我们可以更好地理解和处理那些复杂且变化无常的信号。
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1)
Interpretation
of
Instantaneous Frequency:
TO
provide
insight into the meaning of the IF, let us consider the
problem of positioning a signal,
s(t),
in the frequency
domain. We construct the analytic signal,
~(t)
=
a(t)
.
ej4(t),
as defined in (15). Its spectrum
Z(
f)
is given by
Z(
f)
=
/’“
z(t)e-jzXftdt
(23)
--03
The application of the stationary phase principle (see Ap-
pendix
B)
indicates that this integral will have its largest
value at the frequency,
fs,
for which the phase is stationary,
i.e., for
fs
such that
which leads to
671
UL.
This indicates that if
fs
is a function of
t,
fs(t)
provides
a measure of the frequency domain signal energy concen-
tration as a function of time. This measure equals the IF of
the signal; this property explains the importance of the IF
in signal recognition, tracking, estimation, and modeling.
However, the interpretation of the IF is often a subject
of controversy. Shekel [35], for example, argued that the
IF defined by
1
27~
dt
fi(t)
=
--
is not a unique function of time, since any amplitude
modulated (AM) wave, written in complex form, may be
expressed as either
m(t)ejZaft
or
moej4(t).
The latter
expression represents a wave with a constant amplitude
and a complicated IF law, while the former expression
represents a wave with complicated amplitude law and a
constant IF. The implication
is
that there are many ways
of constructing a complex signal starting from a real one.
A unique complex representation of a signal is obtained by
using the HT, as Gabor and Ville have noted [20], [42],
but whether or not it corresponds to any physical reality is
another question. This point is discussed further in Section
111.
Mandel [28] also challenged any physical interpretation
of the IF. He discussed its relationship with spectral fre-
quency in terms of Fourier decomposition. He argued that
there is no one-to-one relationship between these two kinds
of frequencies and provided examples of signals to prove
it. He also claimed that the only similarity between the IF
and Fourier decomposition frequency is that the average
frequency of the Fourier spectrum equals the time average
of the IF. This coincidence, however, does not extend
to higher order moments (as Ville [42] showed for the
second moment). Let us consider the signal Mandel chose
to illustrate his point:
.
(28)
z(t)
=
ale(jwo-Aw/2)t
+
a2ej(wo+Aw/2)t
If we express
~(t)
in terms of envelope and phase
a(t)
.
ej$(t)
we have
(29)
(-a1
+
u2)
sin
(Awl2)t
(a1
+
~2)
COS
(Aw/2)t
4(t)
=
tan-l
and now the instantaneous frequency given by Ville’s
definition (18) is
The Fourier spectrum of
~(t)
is obvious from (28)-it
consists of two components symmetrically placed with
respect to
fo.
The IF given by (30) varies with time, but
excursions of
fi(t)
about
fo
are not symmetrical; “they
are entirely upwards if
u2
>
a1
and entirely downwards if
a1
>
a2”
[28]. However, in this example, we argue that the
analytic signal in (28) corresponds to the bicomponent real
signal
s(t)
=
sl(t)
+s2(t).
Hence, the IF is not meaningful
for
s(t)
but only for the single component signals
sl(t)
and
s2(t)
taken separately. Therefore, the IF expressed by (30)
is outside the scope of the original definition (see Section
Mandel strongly promoted the idea that the IF and
Fourier frequencies are different quantities, and that
one source of their mutual confusion is the same
name-frequency-attached to both of them. Finally,
Mandel asks a question: Which of these two quantities is
most closely related to measurements? He also provides
the answer: It strongly depends “on the nature of the
experiment.”
Priestley indicated [32] that a nonstationary process in
general cannot be represented in a meaningful way by the
simple Fourier expansion as described by (5b). For exam-
ple, consider the nonstationary signal with time-varying
amplitude:
11-C).
-t2/2
y(t)
=
A.
e
cos(257fot
+
40).
(31)
The
FT
of
y(t)
consists of two Gaussian functions centered
at
fo
and
-
fo
and thus it contains Fourier components at
all
frequencies. It is possible to use an alternative form
for representing
y(t):
it consists of just two “frequency”
components (at
fo
and
-
fo),
with each component having
a time-varying amplitude
A
.
These two repre-
sentations of
y(t)
are equally valid. They correspond to
different “families” of basic orthogonal functions used for
representation. In the former case, the family consists of
sines and cosines with constant amplitudes, and in latter
case it consists of sines and cosines with time-varying
amplitudes.
According to the conventional definition, the term “fre-
quency” is associated specifically with the sine and cosine
functions. In order to apply the notion of frequency in
the analysis of nonstationary signals, it is necessary to
introduce a new basic family of functions which must be
nonstationary and still have an oscillatory form
so
that
the notion of “frequency” is applicable. Thus Priestley
BOASHASH:
INSTANTANEOUS FREQUENCY OF SIGNAL PART
1
523
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