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A Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
Article in IEEE Access · March 2017
DOI: 10.1109/ACCESS.2017.2688467
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Received February 12, 2017, accepted March 22, 2017, date of publication March 31, 2017, date of current version May 17, 2017.
Digital Object Identifier 10.1109/ACCESS.2017.2688467
A Signal Theoretic Approach for Envelope
Analysis of Real-Valued Signals
YANLI YANG
Tianjin Key Laboratory of Optoelectronic Detection Technology and Systems, Tianjin Polytechnic University, Tianjin 300387, China
(yyl070805@163.com)
This work was supported in part by the National Natural Science Foundation of China under Grant 61401305 and in part by the Natural
Science Foundation of Tianjin, China, under Grant 15JCYBJC16500.
ABSTRACT Analytic envelope is the most prevalent definition of envelopes of real-valued signals. However,
analytic signals are not adopted by some envelope detectors in application. This paper investigates the
envelopes of real-valued signals from a signal processing perspective. We show that the upper and lower
envelopes of signals can be obtained by signal reconstruction after extrema sampling on signals. We prove
that extrema sampling is a sub-Nyquist sampling. We then conclude that the envelopes of real-valued
signals contain two parts, some low-frequency components of the original signal and some new components
generated by sub-Nyquist extrema samplings. Some examples are presented to compare with the analytic
envelope of signals.
INDEX TERMS Envelope of real-valued signals, analytic signal, extrema sampling, signal reconstruction,
digital signal processing.
I. INTRODUCTION
As an imaginary curve, the envelopes of a signal are the
boundary within which the signal is contained. Envelopes
contain some information of signals, though it is an
imaginary curve, for example, demodulating amplitude modulated (AM) signals by them. Envelope detection or demodulation method is a dominant approach in detecting faults in
rolling element bearings and gearboxes [1], [2].
There are various definitions about signal envelopes, such
as the analytic envelope [3]–[5], the pre-envelope [4], and
the nature envelope [5]. Among these definitions, the analytic
envelope based on the Hilbert transform is famous.
However, the concept of the envelope used in technical
problems is not as clear-cut as the geometrical concept of
the envelope of a family of curves [5]. Moreover, the Hilbert
transform is generally employed to obtain the analytic envelope of real-valued signals, which is not convenient in practice. Bedrosian [6] pointed out that the envelope of an actual
waveform has physical significance only for narrow-band
signals and cannot be measured precisely except for a pure
sinusoid.
In engineering, the detection of envelopes of real signals
sometimes does not rely upon analytic signals, though it is
uniquely defined based on analytic signals [7]. Most methods
for envelope demodulation depend on peak detection followed by a low-pass filter, typically a RC network, to remove
the carrier and provide smoothing [8]. The envelopes of real
signals are constructed by spline interpolation from extrema
VOLUME 5, 2017
sequences in the empirical mode decomposition (EMD) algorithm which is an adaptive non-stationary signal processing
method [9]. This paper aims to analyze envelopes of realvalued signals from the signal processing perspective.
II. ENVELOPES DEFINITION
In this section, the definition of analytic envelopes is reviewed
firstly. Then, a new definition of envelopes, called interpolative envelopes, is proposed.
A. THE ANALYTIC ENVELOPE
The definition of analytic envelope is associated with analytic
signals. The complex-valued signal m(t) is termed analytic if
and only if m̂(t) = −jm(t) where m̂(t) is the Hilbert transform
of m(t) [10]. It would be of more interest to mention that an
analytic signal contains no negative frequencies [6].
For a given real band-limited signal s(t), if s(t) ∈ L2 (R)
and its Fourier transform is S(f ), then its Hilbert transform is
defined by [11]
1
ŝ(t) = H [s(t)] = P
π
Z+∞
−∞
s(τ )
dτ
t −τ
(1)
where P is the Cauchy principal value. Hence, the complexvalued signal
m(t) = s(t) + jŝ(t)
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(2)
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Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
FIGURE 1. Block diagram of the process of forming envelopes.
is an analytic signal [6]. Then, (2) can be rewritten as [11]
m(t) = |m(t)| ejφ(t)
(3)
where
ŝ(t)
φ(t) = arctan
,
s(t)
h
i1/2
|m(t)| = s2 (t) + ŝ2 (t)
.
(4)
(5)
The analytic signal m(t) is referred to the pre-envelope
of s(t) [3], [4] or the complex analytic envelope of s(t) [5].
The modulus |m(t)| is termed the envelope of s(t) [3], [4] or
the analytic envelope of s(t) [5].
B. THE INTERPOLATIVE ENVELOPE
In the EMD algorithm, the upper and lower envelopes of
signals are obtained by using two steps: 1) identify all the
local extrema; 2) interpolate between extrema by a cubic
spline [9]. The interpolative envelope includes the upper and
lower envelopes. The first step is essentially a process of
extrema sampling, and the second step is used to recover the
signal from the extrema samples [12]–[16]. Then, the upper
and lower envelopes of real-valued signals can be given as
follows.
Def. 1: An upper (or lower) envelope of a real signal is a
function s : R → R; t → s(t) used for its reconstruction with
its local maxima (minima) sequence.
Based on Def. 1, an upper or lower envelope of a signal
can be obtained by two steps: extrema sampling and signal
reconstruction, as shown in Fig. 1. For the method of envelope
demodulation based on RC network, the peak detection is just
the process of extrema sampling, and the RC network is just
used to recover signals. It is clear that the upper and lower
envelopes can be formed by the same procedure. Without loss
of generality, only the upper envelope is analyzed in the rest
of this paper.
According to the Shannon’s sampling theorem for uniform
sampling and the folk theorem [17], or alternatively, the
Beurling-Landau theorem [18] for non-uniform sampling,
a component can be perfectly recovered when its frequency
is not greater than half of the maxima rate. As far as spline
interpolation is concerned, it is reported in [19] that the spline
interpolator of order n approaches an ideal sinc interpolator
as n goes to infinity. Therefore, the content of upper envelopes
is determined to a large extend by maxima samplings.
III. ANALYSIS ON ENVELOPES
A. EXTREMA SAMPLING
Extrema sampling includes the maxima and minima samplings. From the viewpoint of mathematical analysis, there
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should be a local minimum located between two adjacent
local maxima and a local maximum between two adjacent
local minima [20]. Hence, the minima sampling rate is equal
to the maxima sampling rate for a continuous signal. About
the extrema sampling, we have the following theorem.
Theorem 1: Consider x(t) ∈ L2 (R) a T -periodic signal that
satisfies the Dirichlet conditions. Then, the maxima (minima)
rate of x(t) is less than or equal to the highest frequency
of x(t).
Proof: Let fe denote the maxima (minima) rate of the
signal x(t). With no loss of generality, assume that f1 denotes
the highest frequency of x(t). The periodic signal x(t) can be
expressed by a finite Fourier series, in the form [21]:
M
X
x(t) =
cn ejnt
(6)
n=−M
where
2π
T
Z
1 T /2
cn =
x(t)e−jnt dt.
T −T /2
=
(7)
(8)
We have cn = 0 when n > T /T1 , where T1 = 1/f1 because of
the orthogonality of the basis functions of the Fourier series.
Then, for the sake of simplicity, let M = T /T1 .
If x(t) takes an extremum at t = te , then its first derivative is
x 0 (te ) =
M
X
jncn ejnte = 0.
(9)
n=−M
In fact, x 0 (t) can be regarded as a polynomial in ejt of
degree 2M . Using in (6) the mapping U = ejt , we derive
x 0 (U ) =
M
X
jncn U n .
(10)
n=−M
The number of zeros of x 0 (U ) is 2M and equals the trigonometric polynomial’s degree within a period T [22]. Therefore,
we have
T
N0 = 2M = 2
(11)
T1
where N0 denotes the number of zeros of x 0 (t) within a
period T . Let Ne represent the number of extrema of x(t)
within a period T . It is shown in [21] that not all the zeros
point of x 0 (t) are the extrema point of x(t). Then we have
Ne ≤ N0 .
(12)
For the continuous signal x(t), the numbers of maxima and
minima are equal in a period T . Then, we obtain
fe =
Ne /2
N0 /2
1
≤
=
= f1
T
T
T1
(13)
which brings the desired result.
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Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
B. COMPONENTS OF ENVELOPES
Theorem 1 shows that the maxima and minima sampling are
sub-Nyquist sampling. The highest frequency component of
signals can not be reconstructed perfectly by the upper and
lower envelopes. However, the components whose frequency
is no more than fe /2 can be recovered completely in the
envelopes. As a direct consequence, the upper and lower
envelopes will contain two parts, that are: components of the
original signal as well as news components generated by the
sub-Nyquist extrema sampling.
Considering a multicomponent signal in the form
n
X
xi (t),
(14)
x(t) = xh (t) + xl (t)
(15)
x(t) =
where ai , fi , and ϕi are, respectively, the amplitude, frequency
and phase of the component xi (t), 1 ≤ i ≤ 2. Without loss
of generality, assume f1 > f2 . Then, we have fe ≤ f1 which
implies that envelopes will contain some new components, as
shown in Fig. 2.
i=1
we can rewrite it as
where xh (t)and xl (t) denote the components whose frequency
is greater or lesser than fe /2, respectively. Let eu (t) and ed (t)
denote the upper and lower envelopes of a signal, respectively.
Based on the above analysis, the upper envelope of the multicomponent signal x(t) can be expressed as
eu (t) = xl (t) + nu (t)
(16)
where nd (t) represents the new components in the upper
envelope. Correspondingly, the lower envelope of x(t) can be
expressed as
ed (t) = xl (t) + nd (t)
(17)
where nd (t) is the new components in the lower envelope.
Especially, for the case of extrema spaced uniformly, the
upper and lower envelope of signals can be written as [17]
eu (t) =
ed (t) =
∞
X
k=−∞
∞
X
k=−∞
(18)
t − td
x(kTe + td )sinc(
− k)
Te
a2 f2 < a1 f1
(19)
IV. ILLUSTRATIVE EXAMPLES
In this section, the interpolative envelopes are analyzed
for two tones signals, double-sideband suppressed
carrier (DSB-SC) signals, conventional AM signals, and
non-stationary signals. The interpolative envelopes are also
compared with the analytic envelope in these signals.
A. ENVELOPES OF TWO TONES SIGNALS
A composite two tones signal model can be given by
x(t) = x1 (t) + x2 (t)
= a1 cos(2π f1 t + ϕ1 ) + a2 cos(2πf2 t + ϕ2 ) (20)
(21)
If f2 ≤ f1 /2, the signal x2 (t) can be reconstructed perfectly.
For the sake of simplifying the analysis, assume that extrema
are equally spaced. Using (18) and the property of the sinc
function, we then have
eu (t) =
∞
X
[x1 (kTe + tu ) + x2 (kTe + tu )] sin c(
t − tu
−k)
Te
= a1 a0 + a2 cos(2π f2 t + ϕ2 )
= a1 a0 + x2 (t)
where sinc(t) = sin(π t)/(π t), Te = 1/fe , tu (td ) denotes the
first maximum (minimum) point, respectively. Some examples and further discussion are shown in the next section.
VOLUME 5, 2017
For the upper envelope, we first consider the case that
fe = f1 which has a sufficient condition [23]
k=−∞
t − tu
− k)
Te
x(kTe + tu )sinc(
FIGURE 2. Illustration of new components in the upper envelopes. (a) The
two tones signal with f1 = 100 Hz, f2 = 90 Hz, a1 = 1, and a2 = 1. (b) The
spectrum of the upper envelope shown in (a). (c) The two tones signal
with f1 = 100 Hz, f2 = 60 Hz, a1 = 1, and a2 = 1. (d) The spectrum of the
upper envelope shown in (c).
(22)
where a0 = cos(2π f1 tu +ϕ1 ). In this case, the upper envelope
eu (t) only contains the low frequency signal x2 (t) and a
constant a0 . A simple illustration is shown in Fig. 3. However,
if f1 /2 < f2 < f1 , we have
eu (t) = a1 a0 + xa2 (t)
(23)
where xa2 (t) denotes the aliasing component of x2 (t) [17].
We then consider the case that fe < f1 . If f2 ≤ fe /2, the
signal x2 (t) can be reconstructed perfectly. Using (18), we
then have
eu (t) = x2 (t) + xa1 (t)
(24)
where xa1 (t) denotes the aliasing component of x1 (t). In this
case, the upper envelope eu (t) contains the low frequency
signal x2 (t) but with a new component xa1 (t). If f2 > fe /2,
the two components x1 (t) and x2 (t) cannot be reconstructed
perfectly. Then we have
eu (t) = xa1 (t) + xa2 (t)
(25)
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Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
It is quite clear from Fig. 4 that the analytic envelope contains
a new component rather than the two components of x(t).
For comparison’s sake, the upper envelope contains the low
frequency component x2 (t).
B. ENVELOPES OF DSB-SC SIGNALS
A DSB-SC signal is expressed mathematically as [24]
s(t) = Ac x(t) cos(2πfc t)
where Ac denotes the carrier amplitude and fc represents the
carrier frequency. The signal x(t) is usually known as the
message signal. Here, we consider x(t) = A cos(2π ft). Then
a simple DSB-SC signal model can be expressed as
FIGURE 3. Illustration of the upper envelopes without new components.
(a) The two tones signal with f1 = 100 Hz, f2 = 20 Hz, a1 = 1, and a2 = 1.
(b) The spectrum of the signal shown in (a).
s(t) = Ac A cos(2π ft) cos(2π fc t)
Ac A
cos[2π (fc − f )t]
=
2
Ac A
cos[2π (fc + f )t]
+
2
(27)
where fc >> f .
In fact, the signal shown in (27) is a two tones signal.
It is proved in [23] that fe = f1 if a1 f1 > a2 f2 for the two
tones signal expressed by (20). Hence, we have fe = fc + f
for the signal shown in (27). If assuming that extrema space
uniformly, by (18), the upper envelope of the DSB-SC signal
shown in (27) can be expressed as
eu (t) =
∞
Ac A X
{A0 + cos[4π f (kTe − t0 )]}
2
k=−∞
×sinc(
t − tu
− k)
Te
(28)
where A0 = cos[2π (f + fc )tu ] and t0 = (fc − f )tu /(2f ). Then,
we can obtain
eu (t) =
FIGURE 4. Example of envelopes of a two tones signal. (a) The original
signal and the analytic envelope. (b) The original signal and the upper
envelope. (c) The spectrum of the analytic envelope. (d) The spectrum of
the upper envelope.
Equation (25) shows that the upper envelope eu (t) only contains some new components.
The analytic envelope of x(t) shown in (20) is
h
i1/2
|m(t)| = x 2 (t) + x̂ 2 (t)
= {a21 + a22 + 2a1 a2 cos[2π (f1 − f2 )t]}1/2 (26)
where m(t) is the analytic signal. It is obvious from (26)
that the analytic envelope does not contain the signals
x1 (t) and x2 (t).
A simple example is shown in Fig. 4 to compare the interpolative envelope with the analytic envelope and to testify the
above theoretical analysis. The two tones of the signal used by
Fig. 4 are x1 (t) = cos(2π × 100t) and x2 (t) = cos(2π × 20t).
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Ac A
[A0 + cos(4π ft + ϕ)]
2
(29)
where ϕ is the phase of the upper envelope. The upper
envelope of s(t) contains the component whose frequency is
double that of the message signal x(t).
Nevertheless, the analytic envelope of s(t) is
h
i1/2
|m(t)| = s2 (t) + ŝ2 (t)
= Ac A |cos(2π ft)|
(30)
which also does not contain the message signal x(t). In addition, it has some discontinuous points at ωt = nπ + π/2
where ω = 2π f and n = 0, 1, 2, · · · . Hence, the envelope
detector cannot be used to demodulate DSB-SC signals.
An example is presented in Fig. 5 where fc = 120 and
f = 40 to compare the two types envelopes for DSB-SC
signals. It is clear from this Figure that the main component
of the two types of envelopes is the component of 80 Hz
instead of 40 Hz. However, the analytic envelope has some
high frequency components. Although they cannot be used
to detect the message signal x(t), the upper envelope is more
reasonable than the analytic envelope for the DSB-SC signal.
VOLUME 5, 2017
Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
FIGURE 5. Example of envelopes of a DSB-SC signal. The meaning of each
sub-panel is the same as in Fig. 4.
C. ENVELOPES OF CONVENTIONAL AM SIGNALS
Considering a conventional AM signal in which a carrier
signal of frequency fc is modulated by the message signal x(t),
the amplitude modulated signal is [24]
s(t) = Ac [1 + x(t)] cos(2π fc t)
where |x(t)| ≤ 1. Suppose that the bandwidth of x(t) is w1 ,
where we are using fc >> w1 and fe = fc for the signal s(t).
For simplicity, assume that extrema are equally spaced.
By using (18), the upper envelope of s(t) can be expressed as
eu (t) =
∞
X
Ac A0 [1 + x(kTe + tu )] sin c(
k=−∞
t − tu
− k)
Te
FIGURE 6. Illustration of the upper envelope of a conventional AM signal.
(a) The original signal and the upper envelope. (b) The spectrum of the
conventional AM signal. (c) The spectrum of the upper envelope.
where Ŝ(f ) denotes the spectrum of ŝ(t). By taking the inverse
Fourier transform of Ŝ(f ), we have
ŝ(t) = Ac [1 + x(t)] sin(2π fc t)
(34)
Hence, the analytic envelope of s(t) can be written as
h
i1/2
|m(t)| = s2 (t) + ŝ2 (t)
= Ac |1 + x(t)| .
(35)
As a consequence, the analytic envelope contains the message
signal x(t). Hence, it can be used to demodulate conventional
AM signals.
(31)
where A0 = cos(2π fc tu ) is a constant. Then, we can get
eu (t) = Ac A0 [1 + x(t)]
(32)
Equation (32) shows that the upper envelope of the conventional AM signal s(t) contains the messages signals x(t) and
a constant. Therefore, the envelope detector can be used to
demodulate.
A simple example is demonstrated in Fig. 6. The signal is
xc (t) = cos(2π × 100t) and x(t) = cos(2π × 20t). From this
figure, it is obvious that the main component of the upper
envelope is the component of 20 Hz, and we can be sure that
it is just the message signal.
The spectrum S(f ) is easy to obtain by taking the Fourier
transform of s(t). Then, we get
Ŝ(f ) = S(f ) · [−jsgn(f )]
Ac
= j [X (f + fc ) − X (f − fc )]
2
Ac
+ j [δ(f + fc ) − δ(f − fc )]
2
VOLUME 5, 2017
FIGURE 7. Example of envelopes of a AM signal. The meaning of each
sub-panel is the same as in Fig. 4.
(33)
The example to compare the interpolative envelope with
the analytic envelope for the conventional AM signal is shown
in Fig. 7. The signal is the same as used in Fig. 6. From this
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Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
figure, it is evident that the main component of the two types
of envelopes is the component of 20 Hz. Thus, we can be sure
that the two types of envelopes are almost the same, though
the upper envelope has a little high-frequency component.
D. ENVELOPES OF NON-STATIONARY SIGNALS
Non-stationary signals exist widely in practice, whose frequency contents vary with time. Typically, most vibration
signals in a fault rotate machine are non-stationary. Here, we
first consider a non-stationary signal which is written as
x(t) = 10 cos 2π (80t + 500t 2 ) + 10 cos 2π (460t + 310t 2 )
= x1 (t) + x2 (t)
where x1 (t) and x2 (t) are real chirp signals. The signal and
the short-time Fourier transform (STFT) are shown in Fig. 8.
contains the lower chirp signal component x1 (t). However, the
analytic envelop does not contain one of the two chirp signals.
Thus, the interpolative envelope is also fit for demodulating
non-stationary signals.
We then consider a simulation faulty bearing vibration
signal. According to [25] and [26], a vibration signal arising
from a rolling element bearing can be simulated as
q
−ξ ωn t
2
1 − ξ t + n(t) (36)
x(t) = a0 e
sin 2π fn
where ξ denotes the ratio of damping, fn represents the natural
frequency of bearings, and n(t) is the noise. In [1], the bearing
fault signals are added by some harmonic interferences which
is written as
h(t) =
3
X
Ai sin 2π fi t
i=1
For a0 = 10, ξ = 0.08, and fn = 3000, a series bearing
fault impulses repeated period of 0.008s with some harmonic
interferences is shown in Fig. 10. The fault characteristic frequency modulated by high frequency components is 125 Hz.
It is not easy from this Figure to find the fault characteristic
frequency component through the spectrum. The comparison
result on the two type envelopes is shown in Fig. 11, where
the values of some frequency points of envelopes are listed
in Table 1.
FIGURE 8. The non-stationary signal. (a) The signal. (b) The STFT.
FIGURE 9. Envelopes of the non-stationary signal shown in Fig. 8. (a) The
original signal and the analytic envelope. (b) The original signal and the
upper envelope. (c) The STFT of the analytic envelope. (d) The STFT of the
upper envelope.
The comparison result on the two type envelopes is shown
in Fig. 9. It is clear that the interpolative envelope only
5628
FIGURE 10. The simulated bearing fault signal. (a) A series of bearing
fault impulses. (b) The spectrum of (a). (c) The simulated fault signal
with some noise and harmonic interferences. (d) The spectrum
of (c).
In Fig. 11, we can find the fault characteristic frequency
and its harmonics from both the analytic and interpolative
envelopes. Nevertheless, the interpolative envelope contains
VOLUME 5, 2017
Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
FIGURE 11. Envelopes of the signal shown in Fig. 10. (a) The signal and
its analytic envelope. (b) The signal and its interpolative envelope. (c) The
spectrum of (a). (d) The spectrum of (b).
FIGURE 12. Envelopes of the signal shown in Fig. 10 filtered by a
high-pass filter. The meaning of each sub-panel is the same as in Fig. 11.
V. CONCLUSIONS
TABLE 1. The value of some points of Envelopes shown in Fig. 10.
the harmonic interferences, but the analytic envelop only contains part of the harmonic interferences whose frequencies are
100, 200, and 1000 Hz, respectively. Table 1 and Fig. 10 show
that the analytic envelop has some new components which are
marked out with box in this table.
In order to clearly extract the fault characteristic frequency embedded within the time waveform, a high-pass
filter is used to remove the low frequency component of
the signal. Then, the filtered signal is processed by the
analytic and interpolative envelopes. The result is shown
in Fig. 12.
Comparing this Figure with Fig. 11, it is obvious that the
fault characteristic frequency and its harmonics become very
visible for both the two type of envelopes. Furthermore, this
Figure shows that the interpolative envelope has the same
performance with the analytic envelope for this filtered signal. Hence, the interpolative envelope can be used to fault
diagnosis of bearing.
VOLUME 5, 2017
We investigated the envelopes of real-valued signals from the
viewpoint of signal processing. We have shown that the upper
and lower envelopes of a real-valued signal can be described
as the signals reconstructed from the maxima and minima
samples, respectively. We illustrated that the upper and lower
envelopes of signals can be obtained by two operations,
extrema sampling and signal reconstruction.
We have proven that the extrema sampling is a sub-Nyquist
sampling. Generally speaking, we concluded that envelopes
of real-valued signal contain some low-frequency components with some new components except certain special
cases. A typical case is that the envelope can be used to
demodulate conventional AM signals.
The proposed interpolative envelope, unlike the analytic
envelope, is not limited to narrow-band signals. The simulated results show that the interpolative envelope has a good
performance both for stationary and non-stationary signals.
Although the interpolative envelope has already many applications in engineering, further studies are needed to investigate envelope analysis through interpolation such as bearing
fault detection.
ACKNOWLEDGMENT
The authors would like to thank Prof. Ran Tao and
Jiahao Deng for helpful discussions. The authors also
acknowledge the anonymous reviewers for their valuable
comments, which helped us to improve the manuscript.
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Y. Yang: Signal Theoretic Approach for Envelope Analysis of Real-Valued Signals
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YANLI YANG received the master’s degree in
mechanical manufacture and automation from
Dalian Jiaotong University in 2006, and the Ph.D.
degree in mechatronic engineering from the Beijing Institute of Technology in 2010. He is currently an Associate Professor with the School of
Electronics and Information Engineering, Tianjin
Polytechnic University. His current research interests include signal processing and equipment state
monitoring. He was a recipient of the second prize
from the Chinese Machinery Industry Science and Technology in 2009.
VOLUME 5, 2017